RSS feed for Scattering and tunnelling
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection0
This RSS feed contains all the sections in Scattering and tunnelling
Moodle
Copyright © 2016 The Open University
https://www.open.edu/openlearn/ocw/theme/image.php/_s/openlearnng/core/1555574462/i/rsssitelogo
moodle
https://www.open.edu/openlearn/ocw
140
35
engbThu, 18 Apr 2019 15:00:41 +0100Thu, 18 Apr 2019 15:00:41 +010020190418T15:00:41+01:00The Open UniversityengbCopyright © 2016 The Open UniversityCopyright © 2016 The Open University
Introduction
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection0
Wed, 13 Apr 2016 23:00:00 GMT
<p>In this course we shall consider two physical phenomena of fundamental importance: <i>scattering</i> and <i>tunnelling</i>. Each will be treated using both a stationarystate approach <i>and</i> a wavepacket approach.</p><p>We can consider two approaches to describing the state of a system in wave mechanics. In cases where the probability distributions are independent of time, a <i>stationarystate</i> approach can be used. In other cases, where probabilities are timedependent and motion is really taking place, a <i>wavepacket</i> approach can be used. The two approaches are related but different. In many situations the choice of approach is obvious and straightforward, but that is not always the case, as you will soon see.</p><p>You will need to be familiar with some mathematical topics to gain the most from this course. The most important are differential equations, in particular the solution of partial differential equations using the technique of separation of variables, and complex numbers. This material is available in the Mathematics and Statistics topic of OpenLearn, in the courses MST209_10 <i>Modelling with Fourier series</i> and M337_1 <i>Introduction to complex numbers</i>.</p><p>You may also find it useful to refer to the original glossary and Physics Toolkit as you work through this course. PDFs of these documents have been attached in the <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection6">Summary</a>.</p><p>This OpenLearn course is an adapted extract from the Open University course : <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/sm358.htm?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">SM358 <i>The quantum world</i></a></span>.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection0
IntroductionSM358_1<p>In this course we shall consider two physical phenomena of fundamental importance: <i>scattering</i> and <i>tunnelling</i>. Each will be treated using both a stationarystate approach <i>and</i> a wavepacket approach.</p><p>We can consider two approaches to describing the state of a system in wave mechanics. In cases where the probability distributions are independent of time, a <i>stationarystate</i> approach can be used. In other cases, where probabilities are timedependent and motion is really taking place, a <i>wavepacket</i> approach can be used. The two approaches are related but different. In many situations the choice of approach is obvious and straightforward, but that is not always the case, as you will soon see.</p><p>You will need to be familiar with some mathematical topics to gain the most from this course. The most important are differential equations, in particular the solution of partial differential equations using the technique of separation of variables, and complex numbers. This material is available in the Mathematics and Statistics topic of OpenLearn, in the courses MST209_10 <i>Modelling with Fourier series</i> and M337_1 <i>Introduction to complex numbers</i>.</p><p>You may also find it useful to refer to the original glossary and Physics Toolkit as you work through this course. PDFs of these documents have been attached in the <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection6">Summary</a>.</p><p>This OpenLearn course is an adapted extract from the Open University course : <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www3.open.ac.uk/study/undergraduate/course/sm358.htm?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">SM358 <i>The quantum world</i></a></span>.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

Learning outcomes
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsectionlearningoutcomes
Wed, 13 Apr 2016 23:00:00 GMT
<p>After studying this course, you should be able to:</p><ul><li><p>explain the meanings of key terms and use them appropriately</p></li><li><p>describe the behaviour of wave packets when they encounter potential energy steps, barriers and wells</p></li><li><p>describe how stationarystate solutions of the Schrödinger equation can be used to analyse scattering and tunnelling</p></li><li><p> for a range of simple potential energy functions obtain the solution of the timeindependent Schrödinger equation and use continuity boundary conditions to find reflection and transmission coefficients</p></li><li><p>present information about solutions of the timeindependent Schrödinger equation in graphical terms.</p></li></ul>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsectionlearningoutcomes
Learning outcomesSM358_1<p>After studying this course, you should be able to:</p><ul><li><p>explain the meanings of key terms and use them appropriately</p></li><li><p>describe the behaviour of wave packets when they encounter potential energy steps, barriers and wells</p></li><li><p>describe how stationarystate solutions of the Schrödinger equation can be used to analyse scattering and tunnelling</p></li><li><p> for a range of simple potential energy functions obtain the solution of the timeindependent Schrödinger equation and use continuity boundary conditions to find reflection and transmission coefficients</p></li><li><p>present information about solutions of the timeindependent Schrödinger equation in graphical terms.</p></li></ul>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

1 What are scattering and tunnelling?
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1
Wed, 13 Apr 2016 23:00:00 GMT
<p>The phenomenon of <b>scattering</b> was an important topic in physics long before the development of wave mechanics. In its most general sense, scattering is a process in which incident particles (or waves) are affected by interaction with some kind of target, quite possibly another particle (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007001">Figure 1</a>). The interaction can affect an incident particle in a number of ways: it may change its speed, direction of motion or state of internal excitation. Particles can even be created, destroyed or absorbed.</p><div class="oucontentfigure oucontentmediamini" id="fig007_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0a6bd6c2/sm358_1_001i.jpg" alt="Figure 1 The phenomenon of scattering" width="275" height="430" style="maxwidth:275px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7217952"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 1 The phenomenon of scattering</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7217952&clicked=1">Long description</a></div><a id="back_longdesc_idp7217952"></a></div><p>It can be argued that almost everything we know about the world is learnt as a result of scattering. When we look at a nonluminous object such as a book or a building we see it because of the light that is scattered from its surface. The sky is blue because the particles in the Earth's atmosphere are more effective at scattering blue light (relatively short wavelengths) than yellow or red light (longer wavelengths). This is also the reason why sunsets are red (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007002">Figure 2</a>). As the Sun approaches the horizon, its light has to traverse a lengthening path through the Earth's atmosphere; as a consequence, shorter wavelengths are increasingly scattered out of the beam until all that remains is red.</p><div class="oucontentfigure oucontentmediamini" id="fig007_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2dffd8c3/sm358_1_002i.jpg" alt="Figure 2 Red sunsets are a direct consequence of the scattering of sunlight" width="190" height="176" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7225680"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Jon Arnold Images/Alamy
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Jon Arnold Images/Alamy</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 2 Red sunsets are a direct consequence of the scattering of sunlight</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7225680&clicked=1">Long description</a></div><a id="back_longdesc_idp7225680"></a></div><p>Much of what we know about the structure of matter has been derived from scattering experiments. For example, the scattering of alpha particles from a gold foil, observed by Geiger and Marsden in 1909, led Rutherford to propose the first nuclear model of an atom. More recent scattering experiments, involving giant particle accelerators, have provided insight into the fundamental constituents of matter such as the quarks and gluons found inside protons and neutrons. Even our knowledge of cosmology – the study of the Universe as a whole – is deeply dependent on scattering. One of the main sources of precise cosmological information is the study of the <i>surface of last scattering</i> observed all around us at microwave wavelengths (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007003">Figure 3</a>).</p><div class="oucontentfigure" style="width:461px;" id="fig007_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/fb745293/sm358_1_003i.jpg" alt="Figure 3 Microwave image of the surface of last scattering" width="461" height="242" style="maxwidth:461px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp7235024"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, <i>Astrophysical Journal</i> (submitted) © 2003 The American Astronomical Society
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, <i>Astrophysical Journal</i> (submitted) © 2003 The American Astronomical Society</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 3 Microwave image of the surface of last scattering</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7235024&clicked=1">Long description</a></div><a id="back_longdesc_idp7235024"></a></div><p>In our detailed discussions of scattering we shall not consider cases where the scattering changes the number or nature of the scattered particles, since that requires the use of <i>quantum field theory</i>, a part of quantum physics beyond the scope of this course. Rather, we shall mainly restrict ourselves to onedimensional problems in which an incident beam or particle is either <i>transmitted</i> (allowed to pass) or <i>reflected</i> (sent back the way it came) as a result of scattering from a target. Moreover, that target will generally be represented by a fixed potential energy function, typically a finite well or a finite barrier of the kind indicated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007004">Figure 4</a>. Despite these restrictions, our discussion of quantummechanical scattering will contain many surprises. For instance, you will see that a finite potential energy barrier of height <i>V</i><sub>0</sub> can reflect a particle of energy <i>E</i><sub>0</sub>, even when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>. Perhaps even more amazingly, you will see that unbound particles can be reflected when they encounter a finite well.</p><div class="oucontentfigure" style="width:511px;" id="fig007_004"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=thumbnailfigure_idp7241888" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8cd82813/sm358_1_004i.small.jpg" alt="Figure 4 (a) Particles with energy E0 > V0, encountering a finite barrier of height V0, have some probability of being reflected. (b) Similarly, unbound particles with energy E0 > 0 can be reflected by a finite well" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp7250672"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=thumbnailfigure_idp7241888">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 4 (a) Particles with energy <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>, encountering a finite barrier of height <i>V</i><sub>0</sub>, have some probability of being reflected. (b) Similarly, unbound particles with energy <i>E</i><sub>0</sub> > 0 can be reflected by a finite well</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7250672&clicked=1">Long description</a></div><a id="back_longdesc_idp7250672"></a><a id="back_thumbnailfigure_idp7241888"></a></div><p>The phenomenon of <b>tunnelling</b> is entirely quantummechanical with no analogue in classical physics. It is an extension of the phenomenon of <i>barrier penetration</i>, which may be familiar in the context of particles bound in potential energy wells. Barrier penetration involves the appearance of particles in <i>classically forbidden regions</i>. In cases of tunnelling, such as that shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007005">Figure 5</a>, a particle with energy <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> can penetrate a potential energy barrier of height <i>V</i><sub>0</sub>, pass through the classically forbidden region within the barrier, and have some finite probability of emerging into the classically allowed region on the far side.</p><div class="oucontentfigure oucontentmediamini" id="fig007_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/937551e6/sm358_1_005i.jpg" alt="Figure 5 Particles with energy E0 < V0, encountering a finite barrier of height V0, have some probability of being transmitted by tunnelling through the barrier. Such a process is forbidden in classical physics" width="272" height="219" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7265824"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 5 Particles with energy <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>, encountering a finite barrier of height <i>V</i><sub>0</sub>, have some probability of being transmitted by tunnelling through the barrier. Such a process is forbidden in classical physics</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7265824&clicked=1">Long description</a></div><a id="back_longdesc_idp7265824"></a></div><p>Tunnelling phenomena are common in many areas of physics. In this course you will see how tunnelling provides an explanation of the <i>alpha decay</i> of radioactive nuclei, and is also an essential part of the <i>nuclear fusion</i> processes by which stars produce light. Finally you will see how quantum tunnelling has allowed the development of instruments called <i>scanning tunnelling microscopes</i> (STMs) that permit the positions of individual atoms on a surface to be mapped in stunning detail.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1
1 What are scattering and tunnelling?SM358_1<p>The phenomenon of <b>scattering</b> was an important topic in physics long before the development of wave mechanics. In its most general sense, scattering is a process in which incident particles (or waves) are affected by interaction with some kind of target, quite possibly another particle (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007001">Figure 1</a>). The interaction can affect an incident particle in a number of ways: it may change its speed, direction of motion or state of internal excitation. Particles can even be created, destroyed or absorbed.</p><div class="oucontentfigure oucontentmediamini" id="fig007_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0a6bd6c2/sm358_1_001i.jpg" alt="Figure 1 The phenomenon of scattering" width="275" height="430" style="maxwidth:275px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7217952"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 1 The phenomenon of scattering</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7217952&clicked=1">Long description</a></div><a id="back_longdesc_idp7217952"></a></div><p>It can be argued that almost everything we know about the world is learnt as a result of scattering. When we look at a nonluminous object such as a book or a building we see it because of the light that is scattered from its surface. The sky is blue because the particles in the Earth's atmosphere are more effective at scattering blue light (relatively short wavelengths) than yellow or red light (longer wavelengths). This is also the reason why sunsets are red (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007002">Figure 2</a>). As the Sun approaches the horizon, its light has to traverse a lengthening path through the Earth's atmosphere; as a consequence, shorter wavelengths are increasingly scattered out of the beam until all that remains is red.</p><div class="oucontentfigure oucontentmediamini" id="fig007_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2dffd8c3/sm358_1_002i.jpg" alt="Figure 2 Red sunsets are a direct consequence of the scattering of sunlight" width="190" height="176" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7225680"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Jon Arnold Images/Alamy
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Jon Arnold Images/Alamy</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 2 Red sunsets are a direct consequence of the scattering of sunlight</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7225680&clicked=1">Long description</a></div><a id="back_longdesc_idp7225680"></a></div><p>Much of what we know about the structure of matter has been derived from scattering experiments. For example, the scattering of alpha particles from a gold foil, observed by Geiger and Marsden in 1909, led Rutherford to propose the first nuclear model of an atom. More recent scattering experiments, involving giant particle accelerators, have provided insight into the fundamental constituents of matter such as the quarks and gluons found inside protons and neutrons. Even our knowledge of cosmology – the study of the Universe as a whole – is deeply dependent on scattering. One of the main sources of precise cosmological information is the study of the <i>surface of last scattering</i> observed all around us at microwave wavelengths (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007003">Figure 3</a>).</p><div class="oucontentfigure" style="width:461px;" id="fig007_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/fb745293/sm358_1_003i.jpg" alt="Figure 3 Microwave image of the surface of last scattering" width="461" height="242" style="maxwidth:461px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp7235024"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, <i>Astrophysical Journal</i> (submitted) © 2003 The American Astronomical Society
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, <i>Astrophysical Journal</i> (submitted) © 2003 The American Astronomical Society</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 3 Microwave image of the surface of last scattering</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7235024&clicked=1">Long description</a></div><a id="back_longdesc_idp7235024"></a></div><p>In our detailed discussions of scattering we shall not consider cases where the scattering changes the number or nature of the scattered particles, since that requires the use of <i>quantum field theory</i>, a part of quantum physics beyond the scope of this course. Rather, we shall mainly restrict ourselves to onedimensional problems in which an incident beam or particle is either <i>transmitted</i> (allowed to pass) or <i>reflected</i> (sent back the way it came) as a result of scattering from a target. Moreover, that target will generally be represented by a fixed potential energy function, typically a finite well or a finite barrier of the kind indicated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007004">Figure 4</a>. Despite these restrictions, our discussion of quantummechanical scattering will contain many surprises. For instance, you will see that a finite potential energy barrier of height <i>V</i><sub>0</sub> can reflect a particle of energy <i>E</i><sub>0</sub>, even when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>. Perhaps even more amazingly, you will see that unbound particles can be reflected when they encounter a finite well.</p><div class="oucontentfigure" style="width:511px;" id="fig007_004"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=thumbnailfigure_idp7241888" title="View larger image"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8cd82813/sm358_1_004i.small.jpg" alt="Figure 4 (a) Particles with energy E0 > V0, encountering a finite barrier of height V0, have some probability of being reflected. (b) Similarly, unbound particles with energy E0 > 0 can be reflected by a finite well" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp7250672"/></a><div class="oucontentfiguretext"><div class="oucontentthumbnaillink"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=thumbnailfigure_idp7241888">View larger image</a></div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 4 (a) Particles with energy <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>, encountering a finite barrier of height <i>V</i><sub>0</sub>, have some probability of being reflected. (b) Similarly, unbound particles with energy <i>E</i><sub>0</sub> > 0 can be reflected by a finite well</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7250672&clicked=1">Long description</a></div><a id="back_longdesc_idp7250672"></a><a id="back_thumbnailfigure_idp7241888"></a></div><p>The phenomenon of <b>tunnelling</b> is entirely quantummechanical with no analogue in classical physics. It is an extension of the phenomenon of <i>barrier penetration</i>, which may be familiar in the context of particles bound in potential energy wells. Barrier penetration involves the appearance of particles in <i>classically forbidden regions</i>. In cases of tunnelling, such as that shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007005">Figure 5</a>, a particle with energy <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> can penetrate a potential energy barrier of height <i>V</i><sub>0</sub>, pass through the classically forbidden region within the barrier, and have some finite probability of emerging into the classically allowed region on the far side.</p><div class="oucontentfigure oucontentmediamini" id="fig007_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/937551e6/sm358_1_005i.jpg" alt="Figure 5 Particles with energy E0 < V0, encountering a finite barrier of height V0, have some probability of being transmitted by tunnelling through the barrier. Such a process is forbidden in classical physics" width="272" height="219" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7265824"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 5 Particles with energy <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>, encountering a finite barrier of height <i>V</i><sub>0</sub>, have some probability of being transmitted by tunnelling through the barrier. Such a process is forbidden in classical physics</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7265824&clicked=1">Long description</a></div><a id="back_longdesc_idp7265824"></a></div><p>Tunnelling phenomena are common in many areas of physics. In this course you will see how tunnelling provides an explanation of the <i>alpha decay</i> of radioactive nuclei, and is also an essential part of the <i>nuclear fusion</i> processes by which stars produce light. Finally you will see how quantum tunnelling has allowed the development of instruments called <i>scanning tunnelling microscopes</i> (STMs) that permit the positions of individual atoms on a surface to be mapped in stunning detail.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

2.1 Overview
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>Session 2 discusses the scattering of a particle using wave packets. We shall restrict attention to one dimension and suppose that the incident particle is initially free, described by a wave packet of the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ae720a5a/sm358_1_e001i.gif" alt=""/></div><p>This is a superposition of de Broglie waves, with the function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8cfd79ec/sm358_1_ie031i.gif" alt="" width="75" height="20" style="maxwidth:75px;" class="oucontentinlinefigureimage"/></span>corresponding to momentum <i>p<sub>x</sub></i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i>k</i> and energy <i>E<sub>k</sub></i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i><sup>2</sup>k<sup>2</sup></i>/2<i>m</i>, where <i>k</i> = 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/λ is the wave number. The momentum amplitude function <i>A</i>(<i>k</i>) determines the blend of de Broglie waves in the initial free wave packet, but when the wave packet encounters a change in the potential energy function, the blend of de Broglie waves changes, and scattering takes place.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.1
2.1 OverviewSM358_1<p>Session 2 discusses the scattering of a particle using wave packets. We shall restrict attention to one dimension and suppose that the incident particle is initially free, described by a wave packet of the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ae720a5a/sm358_1_e001i.gif" alt=""/></div><p>This is a superposition of de Broglie waves, with the function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8cfd79ec/sm358_1_ie031i.gif" alt="" width="75" height="20" style="maxwidth:75px;" class="oucontentinlinefigureimage"/></span>corresponding to momentum <i>p<sub>x</sub></i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i>k</i> and energy <i>E<sub>k</sub></i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i><sup>2</sup>k<sup>2</sup></i>/2<i>m</i>, where <i>k</i> = 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/λ is the wave number. The momentum amplitude function <i>A</i>(<i>k</i>) determines the blend of de Broglie waves in the initial free wave packet, but when the wave packet encounters a change in the potential energy function, the blend of de Broglie waves changes, and scattering takes place.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

2.2 Wave packets and scattering in one dimension
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> shows the scattering of a wave packet, incident from the left, on a target represented by a potential energy function of the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/1b084a28/sm358_1_e002i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/74a3aa7d/sm358_1_e003i.gif" alt=""/></div><p>Potential energy functions of this type are called <b>finite square barriers</b>. They are simple idealisations of the more general kind of finite barrier shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007004">Figure 4</a>a. The de Broglie waves that make up the wave packet extend over a range of energy and momentum values. In the case illustrated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>, the expectation value of the energy, 〈<i>E</i>〉, has a value <i>E</i><sub>0</sub> that is greater than the height of the potential energy barrier <i>V</i><sub>0</sub>. The classical analogue of the process illustrated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> would be a particle of energy <i>E</i><sub>0</sub> scattering from a repulsive target. The unrealistically steep sides of the potential energy function imply that the encounter is sudden and impulsive – not like the encounter between two negatively charged particles, for instance; there is no gradual slope for the incident particle to ‘climb’, nor for it to descend after the interaction. Still, in the classical case, the fact that <i>E</i><sub>0</sub> is greater than <i>V</i><sub>0</sub>, implies that the incident particle has enough energy to overcome the resistance offered by the target, and is certain to emerge on the far side of it.</p><p>The quantum analysis tells a different story. Based on numerical solutions of Schrödinger's equation, the computergenerated results in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> show successive snapshots of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup>, each of which represents the probability density of the particle at a particular instant. Examining the sequence of pictures it is easy to visualise the probability as a sort of fluid that flows from one part of the <i>x</i>axis to another. Initially, as the wave packet approaches the barrier, the probability is concentrated in a single ‘blob’, flowing from left to right. Then, as the wave packet encounters the barrier, something odd starts to happen; the probability distribution develops closely spaced peaks. These peaks are a consequence of reflection – part of the quantum wave packet passes through the barrier, but another part is reflected; the reflected part interferes with the part still advancing from the left and results in the spiky graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup>. Eventually, however, the interference subsides and what remains are two distinct ‘blobs’ of probability density: one returning to the left, the other progressing to the right.</p><div class="oucontentfigure oucontentmediamini" id="fig007_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e75c3813/sm358_1_006i.jpg" alt="Figure 6 The scattering of a wave packet with 〈E〉 = E0 by a finite square barrier of height V0 when E0 > V0. The probability density Ψ2 is shown in a sequence of snapshots with time increasing from top to bottom. The barrier has been made narrow in this example but a greater width could have been chosen" width="190" height="663" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7327968"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 6 The scattering of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> by a finite square barrier of height <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>. The probability density Ψ<sup>2</sup> is shown in a sequence of snapshots with time increasing from top to bottom. The barrier has been made narrow in this example but a greater width could have been chosen</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7327968&clicked=1">Long description</a></div><a id="back_longdesc_idp7327968"></a></div><p>It's important to realise that the splitting of the wave packet illustrated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> does <i>not</i> represent a splitting of the particle described by the wave packet. The normalisation of the wave packet is preserved throughout the scattering process (the area under each of the graphs in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> is equal to 1); there is only ever one particle being scattered. The splitting of the wave packet simply indicates that, following the scattering, there are two distinct regions in which the particle might be found. In contrast to the certainty of transmission in the classical case, the quantum calculation predicts some probability of transmission, but also some probability of reflection. Indeterminacy is, of course, a characteristic feature of quantum mechanics.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 1</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Simply judging by eye, what are the respective probabilities of reflection and transmission as the final outcome of the scattering process shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The two bumps in the probability density have roughly equal areas. This indicates that the probability of transmission is roughly equal to the probability of reflection. Since the sum of these two probabilities must be equal to 1, the probability of each outcome is about 0.5.</p></div></div></div></div><p>The probability that a given incident particle is reflected is called the <b>reflection coefficient</b>, <i>R</i>, while the probability that it is transmitted is called the <b>transmission coefficient</b>, <i>T</i>. Since one or other of these outcomes must occur, we always have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ebb51b6d/sm358_1_e004i.gif" alt=""/></div><p>The values of <i>R</i> and <i>T</i> that apply in any particular case of onedimensional scattering can be worked out by examining the solution of the relevant Schrödinger equation that applies long after the incident particle has encountered the target. For properly normalised wave packets, the values of <i>R</i> and <i>T</i> are found by measuring the areas under the graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> that are located to the left and to the right of the target at that time.</p><p>Reflection and transmission coefficients can also describe scattering from an attractive target. The idealised case of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> > 0 encountering a finite square well of depth <i>V</i><sub>0</sub> and width <i>L</i> is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007007">Figure 7</a>. Again, the wave packet is incident from the left, and the wave packet is partly reflected, giving rise to interference effects. Eventually, however, the interference abates, leaving two distinct parts of the wave packet. The areas of these two parts determine the reflection and transmission coefficients.</p><div class="oucontentfigure oucontentmediamini" id="fig007_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/73bdeb48/sm358_1_007i.jpg" alt="Figure 7 The scattering of a wave packet with 〈E〉 = E0 by a finite square well of depth V0 when E0 > 0. The probability density Ψ2 is shown in a sequence of snapshots with time increasing from top to bottom. The well has been made narrow in this example but a greater width could have been chosen" width="190" height="669" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7361648"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 7 The scattering of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> by a finite square well of depth <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> > 0. The probability density Ψ<sup>2</sup> is shown in a sequence of snapshots with time increasing from top to bottom. The well has been made narrow in this example but a greater width could have been chosen</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7361648&clicked=1">Long description</a></div><a id="back_longdesc_idp7361648"></a></div><p>Of course, wave mechanics is probabilistic, so although reflection and transmission coefficients can be calculated in any given situation, it is never possible to predict what will happen to any individual particle (unless <i>R</i> or <i>T</i> happen to be equal to zero in the situation considered). Indeed, the detection of the scattered particle, after the scattering has taken place, is exactly the kind of measurement that brings about the <i>collapse of the wave function</i> – a sudden, abrupt and unpredictable change that is not described by Schrödinger's equation. After such a collapse has occurred, the particle is localised on one side of the barrier or the other.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2
2.2 Wave packets and scattering in one dimensionSM358_1<p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> shows the scattering of a wave packet, incident from the left, on a target represented by a potential energy function of the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/1b084a28/sm358_1_e002i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/74a3aa7d/sm358_1_e003i.gif" alt=""/></div><p>Potential energy functions of this type are called <b>finite square barriers</b>. They are simple idealisations of the more general kind of finite barrier shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1#fig007004">Figure 4</a>a. The de Broglie waves that make up the wave packet extend over a range of energy and momentum values. In the case illustrated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>, the expectation value of the energy, 〈<i>E</i>〉, has a value <i>E</i><sub>0</sub> that is greater than the height of the potential energy barrier <i>V</i><sub>0</sub>. The classical analogue of the process illustrated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> would be a particle of energy <i>E</i><sub>0</sub> scattering from a repulsive target. The unrealistically steep sides of the potential energy function imply that the encounter is sudden and impulsive – not like the encounter between two negatively charged particles, for instance; there is no gradual slope for the incident particle to ‘climb’, nor for it to descend after the interaction. Still, in the classical case, the fact that <i>E</i><sub>0</sub> is greater than <i>V</i><sub>0</sub>, implies that the incident particle has enough energy to overcome the resistance offered by the target, and is certain to emerge on the far side of it.</p><p>The quantum analysis tells a different story. Based on numerical solutions of Schrödinger's equation, the computergenerated results in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> show successive snapshots of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup>, each of which represents the probability density of the particle at a particular instant. Examining the sequence of pictures it is easy to visualise the probability as a sort of fluid that flows from one part of the <i>x</i>axis to another. Initially, as the wave packet approaches the barrier, the probability is concentrated in a single ‘blob’, flowing from left to right. Then, as the wave packet encounters the barrier, something odd starts to happen; the probability distribution develops closely spaced peaks. These peaks are a consequence of reflection – part of the quantum wave packet passes through the barrier, but another part is reflected; the reflected part interferes with the part still advancing from the left and results in the spiky graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup>. Eventually, however, the interference subsides and what remains are two distinct ‘blobs’ of probability density: one returning to the left, the other progressing to the right.</p><div class="oucontentfigure oucontentmediamini" id="fig007_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e75c3813/sm358_1_006i.jpg" alt="Figure 6 The scattering of a wave packet with 〈E〉 = E0 by a finite square barrier of height V0 when E0 > V0. The probability density Ψ2 is shown in a sequence of snapshots with time increasing from top to bottom. The barrier has been made narrow in this example but a greater width could have been chosen" width="190" height="663" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7327968"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 6 The scattering of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> by a finite square barrier of height <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>. The probability density Ψ<sup>2</sup> is shown in a sequence of snapshots with time increasing from top to bottom. The barrier has been made narrow in this example but a greater width could have been chosen</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7327968&clicked=1">Long description</a></div><a id="back_longdesc_idp7327968"></a></div><p>It's important to realise that the splitting of the wave packet illustrated in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> does <i>not</i> represent a splitting of the particle described by the wave packet. The normalisation of the wave packet is preserved throughout the scattering process (the area under each of the graphs in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a> is equal to 1); there is only ever one particle being scattered. The splitting of the wave packet simply indicates that, following the scattering, there are two distinct regions in which the particle might be found. In contrast to the certainty of transmission in the classical case, the quantum calculation predicts some probability of transmission, but also some probability of reflection. Indeterminacy is, of course, a characteristic feature of quantum mechanics.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_001"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 1</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Simply judging by eye, what are the respective probabilities of reflection and transmission as the final outcome of the scattering process shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The two bumps in the probability density have roughly equal areas. This indicates that the probability of transmission is roughly equal to the probability of reflection. Since the sum of these two probabilities must be equal to 1, the probability of each outcome is about 0.5.</p></div></div></div></div><p>The probability that a given incident particle is reflected is called the <b>reflection coefficient</b>, <i>R</i>, while the probability that it is transmitted is called the <b>transmission coefficient</b>, <i>T</i>. Since one or other of these outcomes must occur, we always have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ebb51b6d/sm358_1_e004i.gif" alt=""/></div><p>The values of <i>R</i> and <i>T</i> that apply in any particular case of onedimensional scattering can be worked out by examining the solution of the relevant Schrödinger equation that applies long after the incident particle has encountered the target. For properly normalised wave packets, the values of <i>R</i> and <i>T</i> are found by measuring the areas under the graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> that are located to the left and to the right of the target at that time.</p><p>Reflection and transmission coefficients can also describe scattering from an attractive target. The idealised case of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> > 0 encountering a finite square well of depth <i>V</i><sub>0</sub> and width <i>L</i> is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007007">Figure 7</a>. Again, the wave packet is incident from the left, and the wave packet is partly reflected, giving rise to interference effects. Eventually, however, the interference abates, leaving two distinct parts of the wave packet. The areas of these two parts determine the reflection and transmission coefficients.</p><div class="oucontentfigure oucontentmediamini" id="fig007_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/73bdeb48/sm358_1_007i.jpg" alt="Figure 7 The scattering of a wave packet with 〈E〉 = E0 by a finite square well of depth V0 when E0 > 0. The probability density Ψ2 is shown in a sequence of snapshots with time increasing from top to bottom. The well has been made narrow in this example but a greater width could have been chosen" width="190" height="669" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7361648"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 7 The scattering of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> by a finite square well of depth <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> > 0. The probability density Ψ<sup>2</sup> is shown in a sequence of snapshots with time increasing from top to bottom. The well has been made narrow in this example but a greater width could have been chosen</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7361648&clicked=1">Long description</a></div><a id="back_longdesc_idp7361648"></a></div><p>Of course, wave mechanics is probabilistic, so although reflection and transmission coefficients can be calculated in any given situation, it is never possible to predict what will happen to any individual particle (unless <i>R</i> or <i>T</i> happen to be equal to zero in the situation considered). Indeed, the detection of the scattered particle, after the scattering has taken place, is exactly the kind of measurement that brings about the <i>collapse of the wave function</i> – a sudden, abrupt and unpredictable change that is not described by Schrödinger's equation. After such a collapse has occurred, the particle is localised on one side of the barrier or the other.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

3.1 Overview
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>Scattering calculations using wave packets are so laborious that they are generally done numerically, using a computer. However, in many cases, scattering phenomena can be adequately treated using a procedure based on stationary states. This approach can give valuable insight into the scattering process without the need for computer simulations.</p><p>Session 3 introduces the stationarystate approach to scattering. The discussion is mainly confined to one dimension, so a stationarystate solution to the Schrödinger equation can be written in the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn001_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f9266fe3/sm358_1_ie001i.gif" alt=""/></div><p>where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) satisfies the appropriate timeindependent Schrödinger equation at energy <i>E</i>. The first challenge is to find a way of interpreting stationarystate solutions that makes them relevant to an inherently timedependent phenomenon like scattering.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.1
3.1 OverviewSM358_1<p>Scattering calculations using wave packets are so laborious that they are generally done numerically, using a computer. However, in many cases, scattering phenomena can be adequately treated using a procedure based on stationary states. This approach can give valuable insight into the scattering process without the need for computer simulations.</p><p>Session 3 introduces the stationarystate approach to scattering. The discussion is mainly confined to one dimension, so a stationarystate solution to the Schrödinger equation can be written in the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn001_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f9266fe3/sm358_1_ie001i.gif" alt=""/></div><p>where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) satisfies the appropriate timeindependent Schrödinger equation at energy <i>E</i>. The first challenge is to find a way of interpreting stationarystate solutions that makes them relevant to an inherently timedependent phenomenon like scattering.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

3.2 Stationary states and scattering in one dimension
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>The key idea of the stationarystate approach is to avoid treating individual particles, and to consider instead the scattering of a steady intense beam of particles, each particle having the same energy <i>E</i><sub>0</sub>. It is not possible to predict the exact behaviour of any individual particle but, if the incident beam is sufficiently intense, the result of the scattering will be reflected and transmitted beams with steady intensities that are determined by the reflection and transmission coefficients we are aiming to evaluate. Provided we consider the beams as a whole, nothing in this arrangement depends on time. A snapshot of the setup taken at one time would be identical to a similar snapshot taken at another time. In contrast to the wavepacket approach, there are no moving ‘blobs of probability density’, so the whole process can be described in terms of stationary states.</p><p>For a onedimensional beam, we define the <b>intensity</b> <i>j</i> to be the number of beam particles that pass a given point per unit time. We also define the <b>linear number density</b> <i>n</i> of the beam to be the number of beam particles per unit length. Then, thinking in classical terms for a moment, if all the particles in a beam have the same speed <i>v</i>, the beam intensity is given by <i>j</i> = <i>vn</i>. Specialising this relationship to the incident, reflected and transmitted beams, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2b81651c/sm358_1_ue001i.gif" alt=""/></div><p>In the stationarystate approach, the reflection and transmission coefficients can be expressed in terms of beam intensity ratios, as follows:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d8754684/sm358_1_e005i.gif" alt=""/></div><p>If all the incident particles are scattered, and no particles are created or destroyed, it must be the case that <i>j</i><sub>inc</sub> = <i>j</i><sub>ref</sub> + <i>j</i><sub>trans</sub>. Dividing both sides by <i>j</i><sub>inc</sub> and rearranging gives <i>R</i> + <i>T</i> = 1, as expected from our earlier discussions.</p><p>We now need to relate these steady beam intensities to stationarystate solutions of the relevant Schrödinger equation. This requires some care, since Schrödinger's equation is normally used to describe individual particles, rather than beams of particles. To make the steps in the analysis as clear as possible, we shall begin by considering a particularly simple kind of onedimensional scattering target.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.2
3.2 Stationary states and scattering in one dimensionSM358_1<p>The key idea of the stationarystate approach is to avoid treating individual particles, and to consider instead the scattering of a steady intense beam of particles, each particle having the same energy <i>E</i><sub>0</sub>. It is not possible to predict the exact behaviour of any individual particle but, if the incident beam is sufficiently intense, the result of the scattering will be reflected and transmitted beams with steady intensities that are determined by the reflection and transmission coefficients we are aiming to evaluate. Provided we consider the beams as a whole, nothing in this arrangement depends on time. A snapshot of the setup taken at one time would be identical to a similar snapshot taken at another time. In contrast to the wavepacket approach, there are no moving ‘blobs of probability density’, so the whole process can be described in terms of stationary states.</p><p>For a onedimensional beam, we define the <b>intensity</b> <i>j</i> to be the number of beam particles that pass a given point per unit time. We also define the <b>linear number density</b> <i>n</i> of the beam to be the number of beam particles per unit length. Then, thinking in classical terms for a moment, if all the particles in a beam have the same speed <i>v</i>, the beam intensity is given by <i>j</i> = <i>vn</i>. Specialising this relationship to the incident, reflected and transmitted beams, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2b81651c/sm358_1_ue001i.gif" alt=""/></div><p>In the stationarystate approach, the reflection and transmission coefficients can be expressed in terms of beam intensity ratios, as follows:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d8754684/sm358_1_e005i.gif" alt=""/></div><p>If all the incident particles are scattered, and no particles are created or destroyed, it must be the case that <i>j</i><sub>inc</sub> = <i>j</i><sub>ref</sub> + <i>j</i><sub>trans</sub>. Dividing both sides by <i>j</i><sub>inc</sub> and rearranging gives <i>R</i> + <i>T</i> = 1, as expected from our earlier discussions.</p><p>We now need to relate these steady beam intensities to stationarystate solutions of the relevant Schrödinger equation. This requires some care, since Schrödinger's equation is normally used to describe individual particles, rather than beams of particles. To make the steps in the analysis as clear as possible, we shall begin by considering a particularly simple kind of onedimensional scattering target.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

3.3 Scattering from a finite square step
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3
Wed, 13 Apr 2016 23:00:00 GMT
<p>The kind of onedimensional scattering target we shall be concerned with in this section is called a <b>finite square step</b>. It can be represented by the potential energy function</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e84db789/sm358_1_e006i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6c92e379/sm358_1_e007i.gif" alt=""/></div><p>The finite square step (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#fig007008">Figure 8</a>) provides a simplified model of the potential energy function that confronts an electron as it crosses the interface between two homogeneous media. The discontinuous change in the potential energy at <i>x</i> = 0 is, of course, unrealistic, but this is the feature that makes the finite square step simple to treat mathematically. The fact that we are dealing with a square step means that we shall only have to consider <i>two</i> regions of the <i>x</i>axis: Region 1 where <i>x</i> ≤ 0, and Region 2 where <i>x</i> > 0.</p><div class="oucontentfigure oucontentmediamini" id="fig007_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b35ad807/sm358_1_008i.jpg" alt="Figure 8 A finite square step of height V0 < E0" width="273" height="155" style="maxwidth:273px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7413856"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 8 A finite square step of height <i>V</i><sub>0</sub> < <i>E</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7413856&clicked=1">Long description</a></div><a id="back_longdesc_idp7413856"></a></div><p>Classically, when a finite square step of height <i>V</i><sub>0</sub> scatters a rightward moving beam in which each particle has energy <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>, each of the particles will continue moving to the right but will be suddenly slowed as it passes the point <i>x</i> = 0. The transmitted particles are slowed because, in the region <i>x</i> > 0, each particle has an increased potential energy, and hence a reduced kinetic energy. The intensity of each beam is the product of the linear number density and the speed of the particles in that beam. To avoid any accumulation of particles at the step, the incident and transmitted beams must have equal intensities; the slowing of the transmitted beam therefore implies that it has a greater linear number density than the incident beam.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>In general terms, how would you expect the outcome of the quantum scattering process to differ from the classical outcome?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>In view of the quantum behaviour of individual particles (as represented by wave packets) when they meet a finite square barrier, it is reasonable to expect that there is some chance that the particles encountering a finite square step will be reflected. In the case of quantum scattering we should therefore expect the outcome to include a reflected beam as well as a transmitted beam, even though <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>.</p></div></div></div></div><p>We start our analysis by writing down the relevant Schrödinger equation:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/488c4794/sm358_1_e008i.gif" alt=""/></div><p>where <i>V</i>(<i>x</i>) is the finite square step potential energy function given in Equations <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007006">7.6</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007007">7.7</a>. We seek stationarystate solutions of the form <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bc9b6898/sm358_1_ie002i.gif" alt="" width="134" height="22" style="maxwidth:134px;" class="oucontentinlinefigureimage"/></span>, where <i>E</i><sub>0</sub> is the fixed energy of each beam particle. The task of solving <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007008">Equation 7.8</a> then reduces to that of solving the timeindependent Schrödinger equations</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6d0ee4ef/sm358_1_e009i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/45e9af4d/sm358_1_e010i.gif" alt=""/></div><p>A simple rearrangement gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/09613da3/sm358_1_ue002i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/09a5cbcb/sm358_1_ue003i.gif" alt=""/></div><p>and it is easy to see that these equations have the general solutions</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/82e3c4b4/sm358_1_e011i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4753be07/sm358_1_e012i.gif" alt=""/></div><p>where <i>A, B, C</i> and <i>D</i> are arbitrary complex constants, and the wave numbers in Region 1 and Region 2 are respectively</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4dd18b03/sm358_1_e013i.gif" alt=""/></div><p>You may wonder why we have expressed these solutions in terms of complex exponentials rather than sines and cosines (recall the identity <i>e</i><sup>i<i>x</i></sup> = cos <i>x</i> + i sin <i>x</i>). The reason is that the individual terms in Equations <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007011">7.11</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007012">7.12</a> have simple interpretations in terms of the incident, reflected and transmitted beams. To see how this works, it is helpful to note that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4e3f895d/sm358_1_ue004i.gif" alt=""/></div><p>where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/481f5f0c/sm358_1_ie100i.gif" alt="" width="21" height="31" style="maxwidth:21px;" class="oucontentinlinefigureimage"/></span>is the momentum operator in the <i>x</i> direction.</p><p>It therefore follows that terms proportional to e<sup>i<i>kx</i></sup> are associated with particles moving rightward at speed <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cba06d56/sm358_1_ie032i.gif" alt="" width="35" height="16" style="maxwidth:35px;" class="oucontentinlinefigureimage"/></span>, while terms proportional to e<sup>−i<i>kx</i></sup> are associated with particles moving leftward at speed <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cba06d56/sm358_1_ie032i.gif" alt="" width="35" height="16" style="maxwidth:35px;" class="oucontentinlinefigureimage"/></span>.</p><p>These directions of motion can be confirmed by writing down the corresponding stationarystate solutions, which take the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/345a2d76/sm358_1_e014i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/05cf2e96/sm358_1_e015i.gif" alt=""/></div><p>where <i>ω</i> = <i>E</i><sub>0</sub>/<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span>. We can then identify terms of the form e<sup>i(<i>kx</i>−<i>ωt</i>)</sup> as plane waves travelling in the positive <i>x</i>direction, while terms of the form e<sup>−i(<i>kx</i>+<i>ωt</i>)</sup> are plane waves travelling in the negative <i>x</i>direction. None of these waves can be normalised, so they cannot describe individual particles, but you will see that they can describe steady beams of particles.</p><p>In most applications of wave mechanics, the wave function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>) describes the state of a single particle, and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> represents the probability density for that particle. In the steadystate approach to scattering, however, it is assumed that the wave function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>) describes steady beams of particles, with <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> interpreted as the number of particles per unit length – that is, the linear number density of particles. We know that the wave function is not normalisable, and this corresponds to the fact that the steady beams extend indefinitely to the left and right of the step and therefore contain an infinite number of particles. This will not concern us, however, because we only need to know the linear number density of particles, and this is given by the square of the modulus of the wave function.</p><p>Looking at <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007014">Equation 7.14</a>, and recalling that the first term <i>A</i>e<sup>i(<i>k</i><sub>1</sub><i>x</i>−<i>ωt</i>)</sup> represents a wave travelling in the positive <i>x</i>direction for <i>x</i> ≤ 0, we identify this term as representing the incident wave in Region 1 (<i>x</i> ≤ 0). We can say that each particle in the beam travels to the right with speed <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/aac549d8/sm358_1_ie033i.gif" alt="" width="75" height="17" style="maxwidth:75px;" class="oucontentinlinefigureimage"/></span>, and that the linear number density of particles in the beam is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/26755042/sm358_1_ue005i.gif" alt=""/></div><p>(You will find further justification of this interpretation in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4">Section 3.4</a>.)</p><p>Similarly, the second term on the right of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007014">Equation 7.14</a> can be interpreted as representing the reflected beam in Region 1 (<i>x</i> ≤ 0). This beam travels to the left with speed <i>v</i><sub>ref</sub> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i>k</i><sub>1</sub>/<i>m</i> and has linear number density <i>n</i><sub>ref</sub> = <i>B</i><sup>2</sup>.</p><p>The first term on the right of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007015">Equation 7.15</a> represents the transmitted beam in Region 2 (<i>x</i> > 0). This beam travels to the right with speed <i>v</i><sub>trans</sub> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i>k</i><sub>2</sub>/<i>m</i> and has linear number density <i>n</i><sub>trans</sub> = <i>C</i><sup>2</sup>. The second term on the right of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007015">Equation 7.15</a> would represent a leftward moving beam in the region <i>x</i> > 0. On physical grounds, we do not expect there to be any such beam, so we ensure its absence by setting <i>D</i> = 0 in our equations.</p><p>Using these interpretations, we see that the beam intensities are:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5dee3215/sm358_1_e016i.gif" alt=""/></div><p>Expressions for the reflection and transmission coefficients then follow from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.2#eqn007005">Equation 7.5</a>:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2889bb7f/sm358_1_e017i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/264babea/sm358_1_e018i.gif" alt=""/></div><p>It is worth noting that the expression for the transmission coefficient includes the wave numbers <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub>, which are proportional to the speeds of the beams in Regions 1 and 2. The wave numbers cancel in the expression for the reflection coefficient because the incident and reflected beams both travel in the same region.</p><p>To calculate <i>R</i> and <i>T</i>, we need to find the ratios <i>B/A</i> and <i>C/A</i>. To achieve this, we must eliminate unwanted arbitrary constants from our solutions to the timeindependent Schrödinger equation. This can be done by requiring that the solutions satisfy continuity boundary conditions:</p><ul class="oucontentbulleted"><li>
<p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><i>(x)</i> is continuous everywhere.</p>
</li><li>
<p>
d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><i>(x)</i>/d<i>x</i> is continuous where the potential energy function is finite.</p>
</li></ul><p>The first of these conditions tells us that our two expressions for <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><i>(x)</i> must match at their common boundary <i>x</i> = 0. From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007011">Equations 7.11</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007012">7.12</a>, we therefore obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b58a2379/sm358_1_e019i.gif" alt=""/></div><p>Taking the derivatives of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007011">Equations 7.11</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007012">7.12</a>,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a7d8db2b/sm358_1_ue006i.gif" alt=""/></div><p>so requiring the continuity of d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = 0 implies that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/44c11d3a/sm358_1_e020i.gif" alt=""/></div><p>After some manipulation, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007019">Equations 7.19</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007020">7.20</a> allow us to express <i>B</i> and <i>C</i> in terms of <i>A</i>
</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c418fba3/sm358_1_ue008i.gif" alt=""/></div><p>Combining these expressions with <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007017">Equations 7.17</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007018">7.18</a>, we finally obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_021"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2fba61ac/sm358_1_e021i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bf9c3675/sm358_1_e022i.gif" alt=""/></div><p>Since <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b4d696bc/sm358_1_ie003i.gif" alt="" width="96" height="21" style="maxwidth:96px;" class="oucontentinlinefigureimage"/></span> and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ed06ab7b/sm358_1_ie004i.gif" alt="" width="140" height="23" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span>, where <i>E</i><sub>0</sub> is the incident particle energy and <i>V</i><sub>0</sub> is the height of the step, we have now managed to express <i>R</i> and <i>T</i> entirely in terms of given quantities. The transmission coefficient, <i>T</i>, is plotted against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub> in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#fig007009">Figure 9</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig007_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/870f1a47/sm358_1_009i.jpg" alt="Figure 9 A graph of the transmission coefficient T against E0/V0 for a finite square step of height V0" width="272" height="205" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7620896"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 9 A graph of the transmission coefficient <i>T</i> against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub> for a finite square step of height <i>V</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7620896&clicked=1">Long description</a></div><a id="back_longdesc_idp7620896"></a></div><p>The above results have been derived by considering a rightward moving beam incident on an upward step of the kind shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#fig007008">Figure 8</a>. However, almost identical calculations can be carried out for leftward moving beams or downward steps. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equations 7.21</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007022">7.22</a> continue to apply in all these cases, provided we take <i>k</i><sub>1</sub> to be the wave number of the incident beam and <i>k</i><sub>2</sub> to be the wave number of the transmitted beam.</p><p>The formulae for <i>R</i> and <i>T</i> are symmetrical with respect to an interchange of <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub>, so a beam of given energy, incident on a step of given magnitude, is reflected to the same extent <i>no matter whether the step is upwards or downwards</i>. This may seem strange, but you should note that the reflection is a purely quantum effect, and has nothing to do with any classical forces provided by the step.</p><p>Another surprising feature of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equation 7.21</a> is that <i>R</i> is independent of <i>m</i> and so does not vanish as the particle mass <i>m</i> becomes very large. However, we know from experience that macroscopic objects are <i>not</i> reflected by small changes in their potential energy function – you can climb upstairs without serious risk of being reflected! How can such everyday experiences be reconciled with wave mechanics?</p><p>This puzzle can be resolved by noting that our calculation assumes an <i>abrupt</i> step. Detailed quantummechanical calculations show that <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equation 7.21</a> provides a good approximation to reflections from a diffuse step <i>provided that</i> the wavelength of the incident particles is much longer than the distance over which the potential energy function varies. For example, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equation 7.21</a> accurately describes the reflection of an electron with a wavelength of 1 nm from a finite step that varies over a distance of order 0.1 nm. However, macroscopic particles have wavelengths that are much shorter than the width of any realistic step, so the above calculation does not apply to them. Detailed calculations show that macroscopic particles are not reflected to any appreciable extent so, in this macroscopic limit, quatum mechanics agrees with both classical physics and everyday experience.</p><p>Although we have been discussing the behaviour of beams of particles in this section, it is important to realise that these beams are really no more than a convenient fiction. The beams were simply used to provide a physical interpretation of de Broglie waves that could not be normalised. The crucial point is that we have arrived at explicit expressions for <i>R</i> and <i>T</i>, and we have done so using relatively simple stationarystate methods based on the timeindependent Schrödinger equation rather than computationally complicated wave packets. Moreover, as you will see, the method we have used in this section can be generalised to other onedimensional scattering problems.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><ul class="oucontentunnumbered"><li><p>(a) Use <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equations 7.21</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007022">7.22</a> to show that <i>R</i> + <i>T</i> = 1.</p></li><li><p>(b) Evaluate <i>R</i> and <i>T</i> in the case that <i>E</i><sub>0</sub> = 2<i>V</i><sub>0</sub>, and confirm that their sum is equal to 1 in this case.</p></li></ul></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><ul class="oucontentunnumbered"><li><p>(a) From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equations 7.21</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007022">7.
22</a>,</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_033"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d3a93c20/sm358_1_ue033i.gif" alt=""/></div></li><li><p>(b) From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007013">Equation 7.13</a>, we have</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_034"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d346df81/sm358_1_ue034i.gif" alt=""/></div></li><li><p>So, in this case, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/044bce15/sm358_1_ie018i.gif" alt="" width="61" height="19" style="maxwidth:61px;" class="oucontentinlinefigureimage"/></span>. Therefore</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_035"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3a354e12/sm358_1_ue035i.gif" alt=""/></div></li><li><p>So we can check that <i>R</i> + <i>T</i> = 1 in this case.</p></li></ul></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_004"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 4</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Consider the case where <i>k</i><sub>2</sub> = <i>k</i><sub>1</sub>/2.</p><ul class="oucontentunnumbered"><li><p>(a) Express <i>B</i> and <i>C</i> in terms of <i>A</i>.</p></li><li><p>(b) Show that in the region <i>x</i> > 0, we have <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> = 16A<sup>2</sup>/9 = constant, while in the region <i>x</i> ≤ 0, we have <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/70180910/sm358_1_ie005i.gif" alt="" width="215" height="23" style="maxwidth:215px;" class="oucontentinlinefigureimage"/></span>.</p></li><li><p>(c) What general physical phenomenon is responsible for the spatial variation of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> to the left of the step?</p></li><li><p>(d) If the linear number density in the incident beam is 1.00 × 10<sup>24</sup> m<sup>−1</sup>, what are the linear number densities in the reflected and transmitted beams?</p></li></ul></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><ul class="oucontentunnumbered"><li><p>(a) When <i>k</i><sub>2</sub> = <i>k</i><sub>1</sub>/2, we have</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_036"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/78a93488/sm358_1_ue036i.gif" alt=""/></div></li><li><p>(b) From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007015">Equation 7.15</a> with <i>D</i> = 0,</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_037"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b01a0e23/sm358_1_ue037i.gif" alt=""/></div></li><li><p>Similarly, from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007014">Equation 7.14</a>,</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_038"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/378af7d3/sm358_1_ue038i.gif" alt=""/></div></li><li><p>Since <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a137bdda/sm358_1_ie019i.gif" alt="" width="57" height="26" style="maxwidth:57px;" class="oucontentinlinefigureimage"/></span>, we have</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_039"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3ff95327/sm358_1_ue039i.gif" alt=""/></div></li><li><p>Multiplying out the brackets, we find</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_040"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/65c4ce56/sm358_1_ue040i.gif" alt=""/></div></li><li><p>(c) The variation indicated by the cosinedependence to the left of the step is a result of <i>interference</i> between the incident and reflected beams. The presence of interference effects was noted earlier when we were discussing the scattering of wave packets but there the effect was transitory. In the stationarystate approach interference is a permanent feature.</p></li><li><p>(d) The linear number densities in the incident, reflected and transmitted beams are given by <i>A</i><sup>2</sup>, <i>B</i><sup>2</sup> and <i>C</i><sup>2</sup>. The question tells us that <i>A</i><sup>2</sup> = 1.00 × 10<sup>24</sup> m<sup>−1</sup>, so the linear number density in the reflected and transmitted beams are</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_041"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9b78ae22/sm358_1_ue041i.gif" alt=""/></div></li><li><p>Note that the transmitted beam is denser than the incident beam: <i>C</i><sup>2</sup> > <i>A</i><sup>2</sup>. However, since <i>k</i><sub>2</sub> = <i>k</i><sub>1</sub>/2, we have <i>j</i><sub>trans</sub> < <i>j</i><sub>inc</sub>. The transmitted beam is less intense than the incident beam because it travels much more slowly.</p></li></ul></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_005"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 5</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Based on the solution to <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#exe007004">Exercise 4</a>, sketch a graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> that indicates its behaviour both to the left and to the right of a finite square step.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>A suitable graph is shown in Figure 10.</p><div class="oucontentfigure oucontentmediamini" id="fig0s7_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4b2608b8/sm358_1_024i.jpg" alt="Figure 10 Ψ2 plotted against x for a finite square step at x = 0 when E0 > V0" width="315" height="198" style="maxwidth:315px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7743808"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 10 Ψ<sup>2</sup> plotted against <i>x</i> for a finite square step at <i>x</i> = 0 when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7743808&clicked=1">Long description</a></div><a id="back_longdesc_idp7743808"></a></div></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3
3.3 Scattering from a finite square stepSM358_1<p>The kind of onedimensional scattering target we shall be concerned with in this section is called a <b>finite square step</b>. It can be represented by the potential energy function</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e84db789/sm358_1_e006i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_007"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6c92e379/sm358_1_e007i.gif" alt=""/></div><p>The finite square step (<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#fig007008">Figure 8</a>) provides a simplified model of the potential energy function that confronts an electron as it crosses the interface between two homogeneous media. The discontinuous change in the potential energy at <i>x</i> = 0 is, of course, unrealistic, but this is the feature that makes the finite square step simple to treat mathematically. The fact that we are dealing with a square step means that we shall only have to consider <i>two</i> regions of the <i>x</i>axis: Region 1 where <i>x</i> ≤ 0, and Region 2 where <i>x</i> > 0.</p><div class="oucontentfigure oucontentmediamini" id="fig007_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b35ad807/sm358_1_008i.jpg" alt="Figure 8 A finite square step of height V0 < E0" width="273" height="155" style="maxwidth:273px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7413856"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 8 A finite square step of height <i>V</i><sub>0</sub> < <i>E</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7413856&clicked=1">Long description</a></div><a id="back_longdesc_idp7413856"></a></div><p>Classically, when a finite square step of height <i>V</i><sub>0</sub> scatters a rightward moving beam in which each particle has energy <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>, each of the particles will continue moving to the right but will be suddenly slowed as it passes the point <i>x</i> = 0. The transmitted particles are slowed because, in the region <i>x</i> > 0, each particle has an increased potential energy, and hence a reduced kinetic energy. The intensity of each beam is the product of the linear number density and the speed of the particles in that beam. To avoid any accumulation of particles at the step, the incident and transmitted beams must have equal intensities; the slowing of the transmitted beam therefore implies that it has a greater linear number density than the incident beam.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_002"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 2</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>In general terms, how would you expect the outcome of the quantum scattering process to differ from the classical outcome?</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>In view of the quantum behaviour of individual particles (as represented by wave packets) when they meet a finite square barrier, it is reasonable to expect that there is some chance that the particles encountering a finite square step will be reflected. In the case of quantum scattering we should therefore expect the outcome to include a reflected beam as well as a transmitted beam, even though <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>.</p></div></div></div></div><p>We start our analysis by writing down the relevant Schrödinger equation:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/488c4794/sm358_1_e008i.gif" alt=""/></div><p>where <i>V</i>(<i>x</i>) is the finite square step potential energy function given in Equations <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007006">7.6</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007007">7.7</a>. We seek stationarystate solutions of the form <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bc9b6898/sm358_1_ie002i.gif" alt="" width="134" height="22" style="maxwidth:134px;" class="oucontentinlinefigureimage"/></span>, where <i>E</i><sub>0</sub> is the fixed energy of each beam particle. The task of solving <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007008">Equation 7.8</a> then reduces to that of solving the timeindependent Schrödinger equations</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6d0ee4ef/sm358_1_e009i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/45e9af4d/sm358_1_e010i.gif" alt=""/></div><p>A simple rearrangement gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/09613da3/sm358_1_ue002i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/09a5cbcb/sm358_1_ue003i.gif" alt=""/></div><p>and it is easy to see that these equations have the general solutions</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/82e3c4b4/sm358_1_e011i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4753be07/sm358_1_e012i.gif" alt=""/></div><p>where <i>A, B, C</i> and <i>D</i> are arbitrary complex constants, and the wave numbers in Region 1 and Region 2 are respectively</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4dd18b03/sm358_1_e013i.gif" alt=""/></div><p>You may wonder why we have expressed these solutions in terms of complex exponentials rather than sines and cosines (recall the identity <i>e</i><sup>i<i>x</i></sup> = cos <i>x</i> + i sin <i>x</i>). The reason is that the individual terms in Equations <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007011">7.11</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007012">7.12</a> have simple interpretations in terms of the incident, reflected and transmitted beams. To see how this works, it is helpful to note that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_004"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4e3f895d/sm358_1_ue004i.gif" alt=""/></div><p>where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/481f5f0c/sm358_1_ie100i.gif" alt="" width="21" height="31" style="maxwidth:21px;" class="oucontentinlinefigureimage"/></span>is the momentum operator in the <i>x</i> direction.</p><p>It therefore follows that terms proportional to e<sup>i<i>kx</i></sup> are associated with particles moving rightward at speed <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cba06d56/sm358_1_ie032i.gif" alt="" width="35" height="16" style="maxwidth:35px;" class="oucontentinlinefigureimage"/></span>, while terms proportional to e<sup>−i<i>kx</i></sup> are associated with particles moving leftward at speed <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cba06d56/sm358_1_ie032i.gif" alt="" width="35" height="16" style="maxwidth:35px;" class="oucontentinlinefigureimage"/></span>.</p><p>These directions of motion can be confirmed by writing down the corresponding stationarystate solutions, which take the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/345a2d76/sm358_1_e014i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/05cf2e96/sm358_1_e015i.gif" alt=""/></div><p>where <i>ω</i> = <i>E</i><sub>0</sub>/<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span>. We can then identify terms of the form e<sup>i(<i>kx</i>−<i>ωt</i>)</sup> as plane waves travelling in the positive <i>x</i>direction, while terms of the form e<sup>−i(<i>kx</i>+<i>ωt</i>)</sup> are plane waves travelling in the negative <i>x</i>direction. None of these waves can be normalised, so they cannot describe individual particles, but you will see that they can describe steady beams of particles.</p><p>In most applications of wave mechanics, the wave function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>) describes the state of a single particle, and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> represents the probability density for that particle. In the steadystate approach to scattering, however, it is assumed that the wave function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>) describes steady beams of particles, with <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> interpreted as the number of particles per unit length – that is, the linear number density of particles. We know that the wave function is not normalisable, and this corresponds to the fact that the steady beams extend indefinitely to the left and right of the step and therefore contain an infinite number of particles. This will not concern us, however, because we only need to know the linear number density of particles, and this is given by the square of the modulus of the wave function.</p><p>Looking at <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007014">Equation 7.14</a>, and recalling that the first term <i>A</i>e<sup>i(<i>k</i><sub>1</sub><i>x</i>−<i>ωt</i>)</sup> represents a wave travelling in the positive <i>x</i>direction for <i>x</i> ≤ 0, we identify this term as representing the incident wave in Region 1 (<i>x</i> ≤ 0). We can say that each particle in the beam travels to the right with speed <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/aac549d8/sm358_1_ie033i.gif" alt="" width="75" height="17" style="maxwidth:75px;" class="oucontentinlinefigureimage"/></span>, and that the linear number density of particles in the beam is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_005"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/26755042/sm358_1_ue005i.gif" alt=""/></div><p>(You will find further justification of this interpretation in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4">Section 3.4</a>.)</p><p>Similarly, the second term on the right of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007014">Equation 7.14</a> can be interpreted as representing the reflected beam in Region 1 (<i>x</i> ≤ 0). This beam travels to the left with speed <i>v</i><sub>ref</sub> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i>k</i><sub>1</sub>/<i>m</i> and has linear number density <i>n</i><sub>ref</sub> = <i>B</i><sup>2</sup>.</p><p>The first term on the right of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007015">Equation 7.15</a> represents the transmitted beam in Region 2 (<i>x</i> > 0). This beam travels to the right with speed <i>v</i><sub>trans</sub> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span><i>k</i><sub>2</sub>/<i>m</i> and has linear number density <i>n</i><sub>trans</sub> = <i>C</i><sup>2</sup>. The second term on the right of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007015">Equation 7.15</a> would represent a leftward moving beam in the region <i>x</i> > 0. On physical grounds, we do not expect there to be any such beam, so we ensure its absence by setting <i>D</i> = 0 in our equations.</p><p>Using these interpretations, we see that the beam intensities are:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5dee3215/sm358_1_e016i.gif" alt=""/></div><p>Expressions for the reflection and transmission coefficients then follow from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.2#eqn007005">Equation 7.5</a>:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2889bb7f/sm358_1_e017i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/264babea/sm358_1_e018i.gif" alt=""/></div><p>It is worth noting that the expression for the transmission coefficient includes the wave numbers <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub>, which are proportional to the speeds of the beams in Regions 1 and 2. The wave numbers cancel in the expression for the reflection coefficient because the incident and reflected beams both travel in the same region.</p><p>To calculate <i>R</i> and <i>T</i>, we need to find the ratios <i>B/A</i> and <i>C/A</i>. To achieve this, we must eliminate unwanted arbitrary constants from our solutions to the timeindependent Schrödinger equation. This can be done by requiring that the solutions satisfy continuity boundary conditions:</p><ul class="oucontentbulleted"><li>
<p>
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><i>(x)</i> is continuous everywhere.</p>
</li><li>
<p>
d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><i>(x)</i>/d<i>x</i> is continuous where the potential energy function is finite.</p>
</li></ul><p>The first of these conditions tells us that our two expressions for <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><i>(x)</i> must match at their common boundary <i>x</i> = 0. From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007011">Equations 7.11</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007012">7.12</a>, we therefore obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b58a2379/sm358_1_e019i.gif" alt=""/></div><p>Taking the derivatives of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007011">Equations 7.11</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007012">7.12</a>,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_006"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a7d8db2b/sm358_1_ue006i.gif" alt=""/></div><p>so requiring the continuity of d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = 0 implies that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/44c11d3a/sm358_1_e020i.gif" alt=""/></div><p>After some manipulation, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007019">Equations 7.19</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007020">7.20</a> allow us to express <i>B</i> and <i>C</i> in terms of <i>A</i>
</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_008"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c418fba3/sm358_1_ue008i.gif" alt=""/></div><p>Combining these expressions with <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007017">Equations 7.17</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007018">7.18</a>, we finally obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_021"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2fba61ac/sm358_1_e021i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bf9c3675/sm358_1_e022i.gif" alt=""/></div><p>Since <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b4d696bc/sm358_1_ie003i.gif" alt="" width="96" height="21" style="maxwidth:96px;" class="oucontentinlinefigureimage"/></span> and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ed06ab7b/sm358_1_ie004i.gif" alt="" width="140" height="23" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span>, where <i>E</i><sub>0</sub> is the incident particle energy and <i>V</i><sub>0</sub> is the height of the step, we have now managed to express <i>R</i> and <i>T</i> entirely in terms of given quantities. The transmission coefficient, <i>T</i>, is plotted against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub> in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#fig007009">Figure 9</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig007_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/870f1a47/sm358_1_009i.jpg" alt="Figure 9 A graph of the transmission coefficient T against E0/V0 for a finite square step of height V0" width="272" height="205" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7620896"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 9 A graph of the transmission coefficient <i>T</i> against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub> for a finite square step of height <i>V</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7620896&clicked=1">Long description</a></div><a id="back_longdesc_idp7620896"></a></div><p>The above results have been derived by considering a rightward moving beam incident on an upward step of the kind shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#fig007008">Figure 8</a>. However, almost identical calculations can be carried out for leftward moving beams or downward steps. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equations 7.21</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007022">7.22</a> continue to apply in all these cases, provided we take <i>k</i><sub>1</sub> to be the wave number of the incident beam and <i>k</i><sub>2</sub> to be the wave number of the transmitted beam.</p><p>The formulae for <i>R</i> and <i>T</i> are symmetrical with respect to an interchange of <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub>, so a beam of given energy, incident on a step of given magnitude, is reflected to the same extent <i>no matter whether the step is upwards or downwards</i>. This may seem strange, but you should note that the reflection is a purely quantum effect, and has nothing to do with any classical forces provided by the step.</p><p>Another surprising feature of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equation 7.21</a> is that <i>R</i> is independent of <i>m</i> and so does not vanish as the particle mass <i>m</i> becomes very large. However, we know from experience that macroscopic objects are <i>not</i> reflected by small changes in their potential energy function – you can climb upstairs without serious risk of being reflected! How can such everyday experiences be reconciled with wave mechanics?</p><p>This puzzle can be resolved by noting that our calculation assumes an <i>abrupt</i> step. Detailed quantummechanical calculations show that <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equation 7.21</a> provides a good approximation to reflections from a diffuse step <i>provided that</i> the wavelength of the incident particles is much longer than the distance over which the potential energy function varies. For example, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equation 7.21</a> accurately describes the reflection of an electron with a wavelength of 1 nm from a finite step that varies over a distance of order 0.1 nm. However, macroscopic particles have wavelengths that are much shorter than the width of any realistic step, so the above calculation does not apply to them. Detailed calculations show that macroscopic particles are not reflected to any appreciable extent so, in this macroscopic limit, quatum mechanics agrees with both classical physics and everyday experience.</p><p>Although we have been discussing the behaviour of beams of particles in this section, it is important to realise that these beams are really no more than a convenient fiction. The beams were simply used to provide a physical interpretation of de Broglie waves that could not be normalised. The crucial point is that we have arrived at explicit expressions for <i>R</i> and <i>T</i>, and we have done so using relatively simple stationarystate methods based on the timeindependent Schrödinger equation rather than computationally complicated wave packets. Moreover, as you will see, the method we have used in this section can be generalised to other onedimensional scattering problems.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_003"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 3</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><ul class="oucontentunnumbered"><li><p>(a) Use <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equations 7.21</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007022">7.22</a> to show that <i>R</i> + <i>T</i> = 1.</p></li><li><p>(b) Evaluate <i>R</i> and <i>T</i> in the case that <i>E</i><sub>0</sub> = 2<i>V</i><sub>0</sub>, and confirm that their sum is equal to 1 in this case.</p></li></ul></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><ul class="oucontentunnumbered"><li><p>(a) From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007021">Equations 7.21</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007022">7.
22</a>,</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_033"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d3a93c20/sm358_1_ue033i.gif" alt=""/></div></li><li><p>(b) From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007013">Equation 7.13</a>, we have</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_034"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d346df81/sm358_1_ue034i.gif" alt=""/></div></li><li><p>So, in this case, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/044bce15/sm358_1_ie018i.gif" alt="" width="61" height="19" style="maxwidth:61px;" class="oucontentinlinefigureimage"/></span>. Therefore</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_035"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3a354e12/sm358_1_ue035i.gif" alt=""/></div></li><li><p>So we can check that <i>R</i> + <i>T</i> = 1 in this case.</p></li></ul></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_004"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 4</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Consider the case where <i>k</i><sub>2</sub> = <i>k</i><sub>1</sub>/2.</p><ul class="oucontentunnumbered"><li><p>(a) Express <i>B</i> and <i>C</i> in terms of <i>A</i>.</p></li><li><p>(b) Show that in the region <i>x</i> > 0, we have <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> = 16A<sup>2</sup>/9 = constant, while in the region <i>x</i> ≤ 0, we have <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/70180910/sm358_1_ie005i.gif" alt="" width="215" height="23" style="maxwidth:215px;" class="oucontentinlinefigureimage"/></span>.</p></li><li><p>(c) What general physical phenomenon is responsible for the spatial variation of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> to the left of the step?</p></li><li><p>(d) If the linear number density in the incident beam is 1.00 × 10<sup>24</sup> m<sup>−1</sup>, what are the linear number densities in the reflected and transmitted beams?</p></li></ul></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><ul class="oucontentunnumbered"><li><p>(a) When <i>k</i><sub>2</sub> = <i>k</i><sub>1</sub>/2, we have</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_036"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/78a93488/sm358_1_ue036i.gif" alt=""/></div></li><li><p>(b) From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007015">Equation 7.15</a> with <i>D</i> = 0,</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_037"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b01a0e23/sm358_1_ue037i.gif" alt=""/></div></li><li><p>Similarly, from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007014">Equation 7.14</a>,</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_038"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/378af7d3/sm358_1_ue038i.gif" alt=""/></div></li><li><p>Since <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a137bdda/sm358_1_ie019i.gif" alt="" width="57" height="26" style="maxwidth:57px;" class="oucontentinlinefigureimage"/></span>, we have</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_039"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3ff95327/sm358_1_ue039i.gif" alt=""/></div></li><li><p>Multiplying out the brackets, we find</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_040"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/65c4ce56/sm358_1_ue040i.gif" alt=""/></div></li><li><p>(c) The variation indicated by the cosinedependence to the left of the step is a result of <i>interference</i> between the incident and reflected beams. The presence of interference effects was noted earlier when we were discussing the scattering of wave packets but there the effect was transitory. In the stationarystate approach interference is a permanent feature.</p></li><li><p>(d) The linear number densities in the incident, reflected and transmitted beams are given by <i>A</i><sup>2</sup>, <i>B</i><sup>2</sup> and <i>C</i><sup>2</sup>. The question tells us that <i>A</i><sup>2</sup> = 1.00 × 10<sup>24</sup> m<sup>−1</sup>, so the linear number density in the reflected and transmitted beams are</p></li><li><div class="oucontentlistitemspacer"> </div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_041"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9b78ae22/sm358_1_ue041i.gif" alt=""/></div></li><li><p>Note that the transmitted beam is denser than the incident beam: <i>C</i><sup>2</sup> > <i>A</i><sup>2</sup>. However, since <i>k</i><sub>2</sub> = <i>k</i><sub>1</sub>/2, we have <i>j</i><sub>trans</sub> < <i>j</i><sub>inc</sub>. The transmitted beam is less intense than the incident beam because it travels much more slowly.</p></li></ul></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_005"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 5</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Based on the solution to <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#exe007004">Exercise 4</a>, sketch a graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> that indicates its behaviour both to the left and to the right of a finite square step.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>A suitable graph is shown in Figure 10.</p><div class="oucontentfigure oucontentmediamini" id="fig0s7_001"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4b2608b8/sm358_1_024i.jpg" alt="Figure 10 Ψ2 plotted against x for a finite square step at x = 0 when E0 > V0" width="315" height="198" style="maxwidth:315px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7743808"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 10 Ψ<sup>2</sup> plotted against <i>x</i> for a finite square step at <i>x</i> = 0 when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7743808&clicked=1">Long description</a></div><a id="back_longdesc_idp7743808"></a></div></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

3.4 Probability currents
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4
Wed, 13 Apr 2016 23:00:00 GMT
<p>The expressions we have derived for reflection and transmission coefficients were based on the assumption that the intensity of a beam is the product of the speed of its particles and their linear number density. This assumption seems very natural from the viewpoint of classical physics, but we should always be wary about carrying over classical ideas into quantum physics. In this section we shall establish a general quantummechanical formula for the beam intensity. The formula will be consistent with the assumptions made so far, but is also more general, applying in regions where a classical beam would not exist and for localised wave packets as well as steady beams.</p><p>At the heart of our analysis lies the idea that matter is conserved. Neglecting relativistic processes in which particles can be created or destroyed, the total number of particles remains fixed. This is built deep into the formalism of quantum mechanics: if the wave function describing a particle is normalised now, it will remain normalised forever because particles do not simply disappear. The conservation of particles applies locally as well as globally, so if the number of particles in a small region changes, this must be due to particles entering or leaving the region by crossing its boundaries. We shall now express this idea in mathematical terms.</p><p>Let us first consider the onedimensional flow of a fluid along the <i>x</i>axis. At each point, we define a fluid current <i>j<sub>x</sub></i>(<i>x, t</i>) that represents the rate of flow of fluid particles along the <i>x</i>axis. If the fluid is compressible, like air, this fluid current may vary in space and time.</p><div class="oucontentfigure oucontentmediamini" id="fig007_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/12976930/sm358_1_010i.jpg" alt="Figure 11 Any change in the number of particles in a small region is due to fluid currents that carry particles into or out of the region" width="234" height="80" style="maxwidth:234px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp7756448"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 11 Any change in the number of particles in a small region is due to fluid currents that carry particles into or out of the region</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7756448&clicked=1">Long description</a></div><a id="back_longdesc_idp7756448"></a></div><p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#fig007010">Figure 11</a>, above, shows a small onedimensional region between <i>x</i> and <i>x</i> + <i>δ</i><i>x</i>. The number of particles in this region can be written as <i>n</i>(<i>x, t</i>) <i>δ</i><i>x</i>, where <i>n</i>(<i>x, t</i>) is the linear number density of particles. The <i>change</i> in the number of particles in the region during a small time interval <i>δ</i><i>t</i> is then</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/88734c56/sm358_1_e023i.gif" alt=""/></div><p>where, for flow in the positive <i>x</i>direction, the first term on the righthand side represents the number of particles <i>entering</i> the region from the left and the second term represents the number of particles <i>leaving</i> the region to the right. Rearranging <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007023">Equation 7.23</a> gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3fb559a6/sm358_1_ue009i.gif" alt=""/></div><p>and, on taking the limit as <i>δ</i><i>x</i> and <i>δ</i><i>t</i> tend to zero we see that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_024"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c2f01efa/sm358_1_e024i.gif" alt=""/></div><p>This result is called the <b>equation of continuity</b> in one dimension. With a little care, it can be extended to quantum mechanics.</p><p>For a singleparticle wave packet in quantum mechanics, the flowing quantity is <i>probability density</i>. This is evident from images of wave packets as ‘blobs’ of moving probability density (e.g. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>). Now we know that probability density is represented in quantum mechanics by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, so we should be able to construct the appropriate equation of continuity by examining the time derivative of this quantity.</p><p>Obviously, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_025"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6b2f6ffb/sm358_1_e025i.gif" alt=""/></div><p>where Schrödinger's equation dictates that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/413d4d75/sm358_1_ue010i.gif" alt=""/></div><p>Dividing through by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e2843f0d/sm358_1_ie034i.gif" alt="" width="13" height="13" style="maxwidth:13px;" class="oucontentinlinefigureimage"/></span>, the rate of change of the wave function is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_026"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a52becbc/sm358_1_e026i.gif" alt=""/></div><p>Substituting this equation, and its complex conjugate, into <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007025">Equation 7.25</a>, we then obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/1403075d/sm358_1_ue011i.gif" alt=""/></div><p>(since the potential energy function <i>V</i>(<i>x</i>) is real and cancels out), and a further manipulation gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5d224b1f/sm358_1_ue012i.gif" alt=""/></div><p>This equation can be written in the form of an equation of continuity:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_027"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a2904df0/sm358_1_e027i.gif" alt=""/></div><p>provided that we interpret <i>n</i>(<i>x, t</i>) as the probability density <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, with a corresponding current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_028"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8bd66253/sm358_1_e028i.gif" alt=""/></div><p>In the onedimensional situations we are considering, <i>j<sub>x</sub></i>(<i>x, t</i>) is called the <b>probability current</b>. In one dimension, the probability density <i>n</i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span> is a probability per unit length, and therefore has the dimensions of [L]<sup>−1</sup>. It follows from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007027">Equation 7.27</a> that the probability current has dimensions of</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a90cef45/sm358_1_ue013i.gif" alt=""/></div><p>and therefore has SI unit ‘per second’, as expected for a current of particles.</p><p>These ideas can be readily extended to steady beams of particles. Snapshots of a steady beam would not reveal any changes from one moment to the next but the beam nevertheless carries a steady flow of particles, just as a steadily flowing river carries a current of water. For a particle beam, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span> represents the linear number density of particles and the probability current is the rate of flow of particles in the positive <i>x</i>direction. For a steady beam, described by a stationarystate wave function,
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9ca801be/sm358_1_ie006i.gif" alt="" width="127" height="21" style="maxwidth:127px;" class="oucontentinlinefigureimage"/></span>, the timedependent phase factors cancel out in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">Equation 7.28</a>, and the probability current can be written more simply as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_029"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4a82a455/sm358_1_e029i.gif" alt=""/></div><p>which is independent of time. It is important to realise that <i>j</i><sub>x</sub> is a <i>signed</i> quantity; it is positive for a beam travelling in the positive <i>x</i>direction, and negative for a beam travelling in the negative <i>x</i>direction. This is unlike the beam intensity <i>j</i> introduced earlier, which is always positive. It is natural to define the beam intensity of a steady beam to be the <i>magnitude</i> of the probability current: <i>j</i> = <i>j<sub>x</sub></i>. It is this definition that gives us a way of calculating beam intensities without making unwarranted classical assumptions.</p><p>In fact, each beam intensity calculated using <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">Equation 7.29</a> turns out to be precisely what we have always assumed – the product of a particle speed and a linear number density – as you can check by tackling <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#exe007007">Exercise 7</a> below. So our analysis adds rigour, but contains no surprises. However, the really significant feature of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">Equation 7.28</a> is its generality; it applies to singleparticle wave packets as well as to steady beams, and (as you will see later) it will also apply in cases of tunnelling, where a classical beam does not exist.</p><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_006"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 6</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Show that <i>j<sub>x</sub></i>(<i>x, t</i>) as defined in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">Equation 7.28</a> is a real quantity.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>One way of showing that a quantity is real is to show that it is equal to its own complex conjugate. Taking the complex conjugate of each factor in <i>j<sub>x</sub></i>, we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_042"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8f8786ad/sm358_1_ue042i.gif" alt=""/></div><p>Since <i>j*<sub>x</sub></i> = <i>j<sub>x</sub></i>, we conclude that <i>j<sub>x</sub></i> is real.</p></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_007"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 7</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Using the solutions to the Schrödinger equation that were obtained in the stationarystate approach to scattering from a finite square step, evaluate the probability current in the regions <i>x</i> > 0 and <i>x</i> ≤ 0. Interpret your results in terms of the beam intensities in these two regions.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>In the region <i>x</i> > 0,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_043"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b9e39ede/sm358_1_ue043i.gif" alt=""/></div><p>so <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">Equation 7.29</a> gives the probability current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_044"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/14530d3d/sm358_1_ue044i.gif" alt=""/></div><p>In the region <i>x</i> ≤ 0,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_045"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d361e8a3/sm358_1_ue045i.gif" alt=""/></div><p>and</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_046"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8f3fe8ed/sm358_1_ue046i.gif" alt=""/></div><p>so <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">Equation 7.29</a> gives the probability current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_047"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ad3439f3/sm358_1_ue047i.gif" alt=""/></div><p>Simplifying this expression, we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_048"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/36f4a37f/sm358_1_ue048i.gif" alt=""/></div><p>This can be interpreted as the sum of an incident probability current, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a3970171/sm358_1_ie020i.gif" alt="" width="71" height="23" style="maxwidth:71px;" class="oucontentinlinefigureimage"/></span>, and a reflected probability current, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0df92ca7/sm358_1_ie021i.gif" alt="" width="76" height="23" style="maxwidth:76px;" class="oucontentinlinefigureimage"/></span>. These two contributions have opposite signs because they flow in opposite directions. Note that, in each region, the probability currents are consistent with the incident, reflected and transmitted beam intensities assumed earlier (and now justified).</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4
3.4 Probability currentsSM358_1<p>The expressions we have derived for reflection and transmission coefficients were based on the assumption that the intensity of a beam is the product of the speed of its particles and their linear number density. This assumption seems very natural from the viewpoint of classical physics, but we should always be wary about carrying over classical ideas into quantum physics. In this section we shall establish a general quantummechanical formula for the beam intensity. The formula will be consistent with the assumptions made so far, but is also more general, applying in regions where a classical beam would not exist and for localised wave packets as well as steady beams.</p><p>At the heart of our analysis lies the idea that matter is conserved. Neglecting relativistic processes in which particles can be created or destroyed, the total number of particles remains fixed. This is built deep into the formalism of quantum mechanics: if the wave function describing a particle is normalised now, it will remain normalised forever because particles do not simply disappear. The conservation of particles applies locally as well as globally, so if the number of particles in a small region changes, this must be due to particles entering or leaving the region by crossing its boundaries. We shall now express this idea in mathematical terms.</p><p>Let us first consider the onedimensional flow of a fluid along the <i>x</i>axis. At each point, we define a fluid current <i>j<sub>x</sub></i>(<i>x, t</i>) that represents the rate of flow of fluid particles along the <i>x</i>axis. If the fluid is compressible, like air, this fluid current may vary in space and time.</p><div class="oucontentfigure oucontentmediamini" id="fig007_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/12976930/sm358_1_010i.jpg" alt="Figure 11 Any change in the number of particles in a small region is due to fluid currents that carry particles into or out of the region" width="234" height="80" style="maxwidth:234px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp7756448"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 11 Any change in the number of particles in a small region is due to fluid currents that carry particles into or out of the region</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7756448&clicked=1">Long description</a></div><a id="back_longdesc_idp7756448"></a></div><p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#fig007010">Figure 11</a>, above, shows a small onedimensional region between <i>x</i> and <i>x</i> + <i>δ</i><i>x</i>. The number of particles in this region can be written as <i>n</i>(<i>x, t</i>) <i>δ</i><i>x</i>, where <i>n</i>(<i>x, t</i>) is the linear number density of particles. The <i>change</i> in the number of particles in the region during a small time interval <i>δ</i><i>t</i> is then</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/88734c56/sm358_1_e023i.gif" alt=""/></div><p>where, for flow in the positive <i>x</i>direction, the first term on the righthand side represents the number of particles <i>entering</i> the region from the left and the second term represents the number of particles <i>leaving</i> the region to the right. Rearranging <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007023">Equation 7.23</a> gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_009"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3fb559a6/sm358_1_ue009i.gif" alt=""/></div><p>and, on taking the limit as <i>δ</i><i>x</i> and <i>δ</i><i>t</i> tend to zero we see that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_024"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c2f01efa/sm358_1_e024i.gif" alt=""/></div><p>This result is called the <b>equation of continuity</b> in one dimension. With a little care, it can be extended to quantum mechanics.</p><p>For a singleparticle wave packet in quantum mechanics, the flowing quantity is <i>probability density</i>. This is evident from images of wave packets as ‘blobs’ of moving probability density (e.g. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>). Now we know that probability density is represented in quantum mechanics by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, so we should be able to construct the appropriate equation of continuity by examining the time derivative of this quantity.</p><p>Obviously, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_025"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6b2f6ffb/sm358_1_e025i.gif" alt=""/></div><p>where Schrödinger's equation dictates that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_010"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/413d4d75/sm358_1_ue010i.gif" alt=""/></div><p>Dividing through by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e2843f0d/sm358_1_ie034i.gif" alt="" width="13" height="13" style="maxwidth:13px;" class="oucontentinlinefigureimage"/></span>, the rate of change of the wave function is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_026"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a52becbc/sm358_1_e026i.gif" alt=""/></div><p>Substituting this equation, and its complex conjugate, into <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007025">Equation 7.25</a>, we then obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/1403075d/sm358_1_ue011i.gif" alt=""/></div><p>(since the potential energy function <i>V</i>(<i>x</i>) is real and cancels out), and a further manipulation gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5d224b1f/sm358_1_ue012i.gif" alt=""/></div><p>This equation can be written in the form of an equation of continuity:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_027"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a2904df0/sm358_1_e027i.gif" alt=""/></div><p>provided that we interpret <i>n</i>(<i>x, t</i>) as the probability density <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, with a corresponding current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_028"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8bd66253/sm358_1_e028i.gif" alt=""/></div><p>In the onedimensional situations we are considering, <i>j<sub>x</sub></i>(<i>x, t</i>) is called the <b>probability current</b>. In one dimension, the probability density <i>n</i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span> is a probability per unit length, and therefore has the dimensions of [L]<sup>−1</sup>. It follows from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007027">Equation 7.27</a> that the probability current has dimensions of</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a90cef45/sm358_1_ue013i.gif" alt=""/></div><p>and therefore has SI unit ‘per second’, as expected for a current of particles.</p><p>These ideas can be readily extended to steady beams of particles. Snapshots of a steady beam would not reveal any changes from one moment to the next but the beam nevertheless carries a steady flow of particles, just as a steadily flowing river carries a current of water. For a particle beam, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>* <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span> represents the linear number density of particles and the probability current is the rate of flow of particles in the positive <i>x</i>direction. For a steady beam, described by a stationarystate wave function,
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9ca801be/sm358_1_ie006i.gif" alt="" width="127" height="21" style="maxwidth:127px;" class="oucontentinlinefigureimage"/></span>, the timedependent phase factors cancel out in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">Equation 7.28</a>, and the probability current can be written more simply as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_029"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4a82a455/sm358_1_e029i.gif" alt=""/></div><p>which is independent of time. It is important to realise that <i>j</i><sub>x</sub> is a <i>signed</i> quantity; it is positive for a beam travelling in the positive <i>x</i>direction, and negative for a beam travelling in the negative <i>x</i>direction. This is unlike the beam intensity <i>j</i> introduced earlier, which is always positive. It is natural to define the beam intensity of a steady beam to be the <i>magnitude</i> of the probability current: <i>j</i> = <i>j<sub>x</sub></i>. It is this definition that gives us a way of calculating beam intensities without making unwarranted classical assumptions.</p><p>In fact, each beam intensity calculated using <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">Equation 7.29</a> turns out to be precisely what we have always assumed – the product of a particle speed and a linear number density – as you can check by tackling <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#exe007007">Exercise 7</a> below. So our analysis adds rigour, but contains no surprises. However, the really significant feature of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">Equation 7.28</a> is its generality; it applies to singleparticle wave packets as well as to steady beams, and (as you will see later) it will also apply in cases of tunnelling, where a classical beam does not exist.</p><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_006"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 6</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Show that <i>j<sub>x</sub></i>(<i>x, t</i>) as defined in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">Equation 7.28</a> is a real quantity.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>One way of showing that a quantity is real is to show that it is equal to its own complex conjugate. Taking the complex conjugate of each factor in <i>j<sub>x</sub></i>, we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_042"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8f8786ad/sm358_1_ue042i.gif" alt=""/></div><p>Since <i>j*<sub>x</sub></i> = <i>j<sub>x</sub></i>, we conclude that <i>j<sub>x</sub></i> is real.</p></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_007"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 7</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Using the solutions to the Schrödinger equation that were obtained in the stationarystate approach to scattering from a finite square step, evaluate the probability current in the regions <i>x</i> > 0 and <i>x</i> ≤ 0. Interpret your results in terms of the beam intensities in these two regions.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>In the region <i>x</i> > 0,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_043"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b9e39ede/sm358_1_ue043i.gif" alt=""/></div><p>so <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">Equation 7.29</a> gives the probability current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_044"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/14530d3d/sm358_1_ue044i.gif" alt=""/></div><p>In the region <i>x</i> ≤ 0,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_045"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d361e8a3/sm358_1_ue045i.gif" alt=""/></div><p>and</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_046"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8f3fe8ed/sm358_1_ue046i.gif" alt=""/></div><p>so <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">Equation 7.29</a> gives the probability current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_047"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ad3439f3/sm358_1_ue047i.gif" alt=""/></div><p>Simplifying this expression, we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_048"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/36f4a37f/sm358_1_ue048i.gif" alt=""/></div><p>This can be interpreted as the sum of an incident probability current, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a3970171/sm358_1_ie020i.gif" alt="" width="71" height="23" style="maxwidth:71px;" class="oucontentinlinefigureimage"/></span>, and a reflected probability current, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0df92ca7/sm358_1_ie021i.gif" alt="" width="76" height="23" style="maxwidth:76px;" class="oucontentinlinefigureimage"/></span>. These two contributions have opposite signs because they flow in opposite directions. Note that, in each region, the probability currents are consistent with the incident, reflected and transmitted beam intensities assumed earlier (and now justified).</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

3.5 Scattering from finite square wells and barriers
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5
Wed, 13 Apr 2016 23:00:00 GMT
<p>The procedure used to analyse scattering from a finite square step can also be applied to scattering from finite square wells or barriers, or indeed to any combination of finite square steps, wells and barriers. The general procedure is as follows:</p><ul class="oucontentbulleted"><li>
<p>Divide the <i>x</i>axis into the minimum possible number of regions of constant potential energy.</p>
</li><li>
<p>Write down the general solution of the relevant timeindependent Schrödinger equation in each of these regions, remembering to use the appropriate value of the wave number <i>k</i> in each region and introducing arbitrary constants as necessary.</p>
</li><li>
<p>Use continuity boundary conditions to determine all but one of the arbitrary constants. The one remaining constant is associated with the incident beam, which may enter from the right or the left.</p>
</li><li>
<p>Obtain expressions for all the beam intensities relative to the intensity of the incident beam.</p>
</li><li>
<p>Determine the reflection and transmission coefficients from ratios of beam intensities.</p>
</li></ul><p>The best way to become familiar with this procedure is by means of examples and exercises. Below is a worked example involving a finite square well of the kind shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007011">Figure 12</a>.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box007_006"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Worked Example 1</h2><div class="oucontentinnerbox"><p>A particle of mass <i>m</i> with positive energy <i>E</i><sub>0</sub> is scattered by a onedimensional finite square well of depth <i>V</i><sub>0</sub> and width <i>L</i> (shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007011">Figure 12</a>). Derive an expression for the probability that the particle will be transmitted across the well.</p><div class="oucontentfigure oucontentmediamini" id="fig007_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8c83acf3/sm358_1_011i.jpg" alt="Figure 12 A finite square well of depth V0 and width L. A beam of particles, each of energy E0, is scattered by the well" width="283" height="102" style="maxwidth:283px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7922000"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">Figure 12 A finite square well of depth <i>V</i><sub>0</sub> and width <i>L</i>. A beam of particles, each of energy <i>E</i><sub>0</sub>, is scattered by the well</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7922000&clicked=1">Long description</a></div><a id="back_longdesc_idp7922000"></a></div><h3 class="oucontenth4 oucontentbasic">Solution</h3><p>Suppose the particle concerned to be part of an intense steady beam of identical particles, each having energy <i>E</i><sub>0</sub> and each incident from the left on a well located between <i>x</i> = 0 and <i>x</i> = <i>L</i>.</p><p>Divide the xaxis into three regions: Region 1, (<i>x</i> < 0), where <i>V</i> = 0; Region 2, (0 ≤ <i>x</i> ≤ <i>L</i>), where <i>V</i> = −<i>V</i><sub>0</sub>; and Region 3, (<i>x</i> > <i>L</i>), where <i>V</i> = 0. In each region the timeindependent Schrödinger equation takes the general form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_030"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cb0050d4/sm358_1_e030i.gif" alt=""/></div><p>so the solution in each region is:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_031"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/291d21a9/sm358_1_e031i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_032"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/fb832488/sm358_1_e032i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_033"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7774ba50/sm358_1_e033i.gif" alt=""/></div><p>where <i>A, B, C, D, F</i> and <i>G</i> are arbitrary constants, and the wave numbers in Region 1 and Region 2 are</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_034"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9687e453/sm358_1_e034i.gif" alt=""/></div><p>Note that the wave number in Region 3 is the same as that in Region 1. This is because the potential energy function is the same in these two regions.</p><p>There is no leftward moving beam in Region 3, so we set <i>G</i> = 0. Since the potential energy function is finite everywhere, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> must be continuous everywhere. Continuity of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = 0 implies that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_035"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/dcd429c0/sm358_1_e035i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_036"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f1147f8e/sm358_1_e036i.gif" alt=""/></div><p>while continuity of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = <i>L</i> gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_037"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/96f5112b/sm358_1_e037i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_038"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ac7a68f0/sm358_1_e038i.gif" alt=""/></div><p>Since the wave numbers in Regions 1 and 3 are both equal to <i>k</i><sub>1</sub>, the intensities of the incident and transmitted beams are</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_039"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7c6e3b26/sm358_1_e039i.gif" alt=""/></div><p>and the required transmission coefficient is given by <i>T</i> = <i>F</i><sup>2</sup>/<i>A</i><sup>2</sup>.</p><p>The mathematical task is now to eliminate <i>B, C</i> and <i>D</i> from Equations 7.35 to 7.38 in order to find the ratio <i>F/A</i>. To achieve this, we note that the constant <i>B</i> only appears in the first two equations, so we take the opportunity of eliminating it immediately. Multiplying <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007035">Equation 7.35</a> by i<i>k</i><sub>1</sub> and adding the result to Equation 7.36 we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_040"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8b64bca0/sm358_1_e040i.gif" alt=""/></div><p>Now we must eliminate <i>C</i> and <i>D</i> from the remaining equations. To eliminate <i>D</i>, we multiply <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007037">Equation 7.37</a> by i<i>k</i><sub>2</sub> and add the result to <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007038">Equation 7.38</a>. This gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_041"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ecbb1bfc/sm358_1_e041i.gif" alt=""/></div><p>Similarly, multiplying <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007037">Equation 7.37</a> by i<i>k</i><sub>2</sub> and subtracting the result from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007038">Equation 7.38</a> we see that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_042"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b9378312/sm358_1_e042i.gif" alt=""/></div><p>Finally, substituting <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007041">Equations 7.41</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007042">7.42</a> into <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007040">Equation 7.40</a> and rearranging slightly we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a5640118/sm358_1_ue014i.gif" alt=""/></div><p>so the transmission coefficient is given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_043"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5ee6e36a/sm358_1_e043i.gif" alt=""/></div><p>This is the expression we have been seeking; it only involves <i>k</i><sub>1</sub>, <i>k</i><sub>2</sub> and <i>L</i>, and both <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub> can be written in terms of <i>m</i>, <i>E</i><sub>0</sub> and <i>V</i><sub>0</sub>.</p></div></div></div><p>Although <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007043">Equation 7.43</a> provides a complete answer to the worked example, it is expressed in a rather opaque form. After several pages of substitutions and manipulations (which are not a good investment of your time) it is possible to recast and simplify this formula. We just quote the final result:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_044"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a1ac585c/sm358_1_e044i.gif" alt=""/></div><p>Treating <i>E</i><sub>0</sub> as an independent variable and <i>V</i><sub>0</sub> as a given constant, this function can be displayed as a graph of <i>T</i> against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub>, as shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007012">Figure 13</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig007_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/579cf123/sm358_1_012i.jpg" alt="Figure 13 The transmission coefficient T for a particle of energy E0 scattering from a finite square well of depth V0, plotted against E0/V0. To plot this graph, we have taken the particle to have the mass of an electron and taken the well to have a depth of 8.6 eV and a width of 1.0 nm" width="271" height="201" style="maxwidth:271px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8046592"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 13 The transmission coefficient <i>T</i> for a particle of energy <i>E</i><sub>0</sub> scattering from a finite square well of depth <i>V</i><sub>0</sub>, plotted against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub>. To plot this graph, we have taken the particle to have the mass of an electron and taken the well to have a depth of 8.6 eV and a width of 1.0 nm</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8046592&clicked=1">Long description</a></div><a id="back_longdesc_idp8046592"></a></div><p>As the incident energy is increased significantly above 0 (the top of the well), transmission becomes relatively more likely, but there is generally some chance of reflection (<i>R</i> =1 − <i>T</i> ≠ 0), and for some energies the reflection probability may be quite high. However, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007012">Figure 13</a> also shows that there are some special incident particle energies at which <i>T</i> = 1, so that transmission is certain to occur. Although it is a rather poor use of terminology, these peaks of high transmission are usually called <b>transmission resonances</b>. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007044">Equation 7.44</a> shows that transmission resonances occur when sin(<i>k</i><sub>2</sub><i>L</i>) = 0, that is when</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_045"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5725cb4d/sm358_1_e045i.gif" alt=""/></div><p>Recalling the relationship <i>k</i> = 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/<i>λ</i> between wave number and wavelength, we can also express this condition as <i>Nλ</i><sub>2</sub> = 2<i>L</i>; in other words:</p><div class="oucontentquote oucontentsbox" id="quo001"><blockquote><p>A transmission resonance occurs when a whole number of wavelengths occupies the full path length 2<i>L</i> of a wave that crosses the width of the well and is reflected back again.</p></blockquote></div><p>This condition can be interpreted in terms of interference between waves reflected at <i>x</i> = 0 and <i>x</i> = L. The interference turns out to be destructive because reflection at the <i>x</i> = 0 interface is accompanied by a phase change of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>. Of course, suppression of the reflected beam is accompanied by an enhancement of the transmitted beam. The effect is similar to one found in optics, where destructive interference between waves reflected from the front and back surfaces of a thin transparent film accounts for the success of the antireflective coatings on lenses and mirrors.</p><p>Before leaving the subject of scattering from a finite well there is one other point that deserves attention. This concerns the precise form of the functions <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) in the three regions we identified earlier. The value of <i>A</i><sup>2</sup> is equal to the linear number density of particles in the incident beam. If we regard <i>A</i> as being known, the values of the constants <i>B, C, D</i> and <i>F</i> can be evaluated using the continuity boundary conditions and graphs of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) can be drawn.</p><div class="oucontentfigure oucontentmediamini" id="fig007_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/31c6f7ed/sm358_1_013i.jpg" alt="Figure 14 A typical example of the real part Re(ψ) of the function ψ(x) for the kind of square well shown in Figure 12" width="270" height="157" style="maxwidth:270px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8087312"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 14 A typical example of the real part Re(<i>ψ</i>) of the function <i>ψ</i>(<i>x</i>) for the kind of square well shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007011">Figure 12</a></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8087312&clicked=1">Long description</a></div><a id="back_longdesc_idp8087312"></a></div><p>Since <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is generally complex there are several possible graphs that might be of interest, including the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, the imaginary part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup>. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007013">Figure 14</a>, above, shows a typical plot of the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span> for chosen values of <i>m</i>, <i>E</i><sub>0</sub>, <i>V</i><sub>0</sub> and <i>L</i>; the following points should be noted:</p><ol class="oucontentnumbered"><li>
<p>In each region, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is a periodic function of <i>x</i>.</p>
</li><li>
<p>The wavelength is smaller inside the well than outside. The shorter wavelength corresponds to a higher wave number, and higher momentum, and echoes the classical increase in speed that a particle experiences as it enters the well.</p>
</li><li>
<p>The amplitude of the wave is smaller inside the well than outside. This is because the beam moves more rapidly inside the well, and has a lower linear number density there.</p>
</li></ol><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_008"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 8</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>With one modification, the stationarystate method that was applied to scattering from a finite square well can also be applied to scattering from a finite square barrier of height <i>V</i><sub>0</sub> and width <i>L</i>, when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>. Describe the modification required, draw a figure analogous to <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007013">Figure 14</a>, but for the case of a square barrier, and comment on any differences between the two graphs.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The main difference is that in the case of a finite square barrier <i>V</i><sub>0</sub> must be replaced by −<i>V</i><sub>0</sub> throughout the analysis. The wave number in Region 2 is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_049"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0e5f4982/sm358_1_ue049i.gif" alt=""/></div><p>Adapting <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007044">Equation 7.44</a>, the transmission coefficient is then given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_050"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bd2d960d/sm358_1_ue050i.gif" alt=""/></div><p>A typical graph of the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) for scattering by a finite square barrier is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.6#fig007014">Figure 15</a>, below. For a barrier, the wave number in Region 2 is decreased (corresponding to a reduced momentum eigenvalue) and the amplitude of the wave is generally greater in Region 2 than outside it (unless <i>T</i> = 1).</p><div class="oucontentfigure oucontentmediamini" id="fig0s7_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7dc74651/sm358_1_025i.jpg" alt="Figure 15 A typical graph of the real part Re(ψ) of ψ(x) for a finite square barrier" width="308" height="238" style="maxwidth:308px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8141104"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 15 A typical graph of the real part Re(<i>ψ</i>) of <i>ψ</i>(<i>x</i>) for a finite square barrier</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8141104&clicked=1">Long description</a></div><a id="back_longdesc_idp8141104"></a></div></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_009"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 9</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Draw two sketches to illustrate the form of the function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> in the case of the stationarystate approach to scattering from (a) a finite square well, and (b) a finite square barrier. Comment on the timedependence of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x,t</i>)<sup>2</sup> in each case.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The graphs of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> plotted against <i>x</i> for each case are shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 16</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig00s7_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8f00661b/sm358_1_026i.jpg" alt="Figure 16 Ψ(x, t)2 plotted against x for (a) a finite square well and (b) a finite square barrier" width="310" height="489" style="maxwidth:310px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8163664"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 16 Ψ(<i>x, t</i>)<sup>2</sup> plotted against <i>x</i> for (a) a finite square well and (b) a finite square barrier</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8163664&clicked=1">Long description</a></div><a id="back_longdesc_idp8163664"></a></div><p>In Regions 1 and 2 interference effects lead to periodic spatial variations with a period related to the relevant wave number. In Region 3 there is only one beam, so there are no interference effects. Note that there is no simple relationship between the plots of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> and the corresponding plots of the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>). This is because <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> is partly determined by the imaginary part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x,t</i>). In both cases <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> is independent of time. This must be the case, despite the timedependence of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x,t</i>), because we are dealing with <i>stationary states</i>.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5
3.5 Scattering from finite square wells and barriersSM358_1<p>The procedure used to analyse scattering from a finite square step can also be applied to scattering from finite square wells or barriers, or indeed to any combination of finite square steps, wells and barriers. The general procedure is as follows:</p><ul class="oucontentbulleted"><li>
<p>Divide the <i>x</i>axis into the minimum possible number of regions of constant potential energy.</p>
</li><li>
<p>Write down the general solution of the relevant timeindependent Schrödinger equation in each of these regions, remembering to use the appropriate value of the wave number <i>k</i> in each region and introducing arbitrary constants as necessary.</p>
</li><li>
<p>Use continuity boundary conditions to determine all but one of the arbitrary constants. The one remaining constant is associated with the incident beam, which may enter from the right or the left.</p>
</li><li>
<p>Obtain expressions for all the beam intensities relative to the intensity of the incident beam.</p>
</li><li>
<p>Determine the reflection and transmission coefficients from ratios of beam intensities.</p>
</li></ul><p>The best way to become familiar with this procedure is by means of examples and exercises. Below is a worked example involving a finite square well of the kind shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007011">Figure 12</a>.</p><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box007_006"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Worked Example 1</h2><div class="oucontentinnerbox"><p>A particle of mass <i>m</i> with positive energy <i>E</i><sub>0</sub> is scattered by a onedimensional finite square well of depth <i>V</i><sub>0</sub> and width <i>L</i> (shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007011">Figure 12</a>). Derive an expression for the probability that the particle will be transmitted across the well.</p><div class="oucontentfigure oucontentmediamini" id="fig007_011"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8c83acf3/sm358_1_011i.jpg" alt="Figure 12 A finite square well of depth V0 and width L. A beam of particles, each of energy E0, is scattered by the well" width="283" height="102" style="maxwidth:283px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp7922000"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">Figure 12 A finite square well of depth <i>V</i><sub>0</sub> and width <i>L</i>. A beam of particles, each of energy <i>E</i><sub>0</sub>, is scattered by the well</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp7922000&clicked=1">Long description</a></div><a id="back_longdesc_idp7922000"></a></div><h3 class="oucontenth4 oucontentbasic">Solution</h3><p>Suppose the particle concerned to be part of an intense steady beam of identical particles, each having energy <i>E</i><sub>0</sub> and each incident from the left on a well located between <i>x</i> = 0 and <i>x</i> = <i>L</i>.</p><p>Divide the xaxis into three regions: Region 1, (<i>x</i> < 0), where <i>V</i> = 0; Region 2, (0 ≤ <i>x</i> ≤ <i>L</i>), where <i>V</i> = −<i>V</i><sub>0</sub>; and Region 3, (<i>x</i> > <i>L</i>), where <i>V</i> = 0. In each region the timeindependent Schrödinger equation takes the general form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_030"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cb0050d4/sm358_1_e030i.gif" alt=""/></div><p>so the solution in each region is:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_031"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/291d21a9/sm358_1_e031i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_032"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/fb832488/sm358_1_e032i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_033"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7774ba50/sm358_1_e033i.gif" alt=""/></div><p>where <i>A, B, C, D, F</i> and <i>G</i> are arbitrary constants, and the wave numbers in Region 1 and Region 2 are</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_034"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9687e453/sm358_1_e034i.gif" alt=""/></div><p>Note that the wave number in Region 3 is the same as that in Region 1. This is because the potential energy function is the same in these two regions.</p><p>There is no leftward moving beam in Region 3, so we set <i>G</i> = 0. Since the potential energy function is finite everywhere, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> must be continuous everywhere. Continuity of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = 0 implies that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_035"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/dcd429c0/sm358_1_e035i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_036"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f1147f8e/sm358_1_e036i.gif" alt=""/></div><p>while continuity of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = <i>L</i> gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_037"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/96f5112b/sm358_1_e037i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_038"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ac7a68f0/sm358_1_e038i.gif" alt=""/></div><p>Since the wave numbers in Regions 1 and 3 are both equal to <i>k</i><sub>1</sub>, the intensities of the incident and transmitted beams are</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_039"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7c6e3b26/sm358_1_e039i.gif" alt=""/></div><p>and the required transmission coefficient is given by <i>T</i> = <i>F</i><sup>2</sup>/<i>A</i><sup>2</sup>.</p><p>The mathematical task is now to eliminate <i>B, C</i> and <i>D</i> from Equations 7.35 to 7.38 in order to find the ratio <i>F/A</i>. To achieve this, we note that the constant <i>B</i> only appears in the first two equations, so we take the opportunity of eliminating it immediately. Multiplying <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007035">Equation 7.35</a> by i<i>k</i><sub>1</sub> and adding the result to Equation 7.36 we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_040"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8b64bca0/sm358_1_e040i.gif" alt=""/></div><p>Now we must eliminate <i>C</i> and <i>D</i> from the remaining equations. To eliminate <i>D</i>, we multiply <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007037">Equation 7.37</a> by i<i>k</i><sub>2</sub> and add the result to <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007038">Equation 7.38</a>. This gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_041"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ecbb1bfc/sm358_1_e041i.gif" alt=""/></div><p>Similarly, multiplying <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007037">Equation 7.37</a> by i<i>k</i><sub>2</sub> and subtracting the result from <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007038">Equation 7.38</a> we see that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_042"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b9378312/sm358_1_e042i.gif" alt=""/></div><p>Finally, substituting <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007041">Equations 7.41</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007042">7.42</a> into <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007040">Equation 7.40</a> and rearranging slightly we obtain</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a5640118/sm358_1_ue014i.gif" alt=""/></div><p>so the transmission coefficient is given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_043"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5ee6e36a/sm358_1_e043i.gif" alt=""/></div><p>This is the expression we have been seeking; it only involves <i>k</i><sub>1</sub>, <i>k</i><sub>2</sub> and <i>L</i>, and both <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub> can be written in terms of <i>m</i>, <i>E</i><sub>0</sub> and <i>V</i><sub>0</sub>.</p></div></div></div><p>Although <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007043">Equation 7.43</a> provides a complete answer to the worked example, it is expressed in a rather opaque form. After several pages of substitutions and manipulations (which are not a good investment of your time) it is possible to recast and simplify this formula. We just quote the final result:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_044"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a1ac585c/sm358_1_e044i.gif" alt=""/></div><p>Treating <i>E</i><sub>0</sub> as an independent variable and <i>V</i><sub>0</sub> as a given constant, this function can be displayed as a graph of <i>T</i> against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub>, as shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007012">Figure 13</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig007_012"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/579cf123/sm358_1_012i.jpg" alt="Figure 13 The transmission coefficient T for a particle of energy E0 scattering from a finite square well of depth V0, plotted against E0/V0. To plot this graph, we have taken the particle to have the mass of an electron and taken the well to have a depth of 8.6 eV and a width of 1.0 nm" width="271" height="201" style="maxwidth:271px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8046592"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 13 The transmission coefficient <i>T</i> for a particle of energy <i>E</i><sub>0</sub> scattering from a finite square well of depth <i>V</i><sub>0</sub>, plotted against <i>E</i><sub>0</sub>/<i>V</i><sub>0</sub>. To plot this graph, we have taken the particle to have the mass of an electron and taken the well to have a depth of 8.6 eV and a width of 1.0 nm</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8046592&clicked=1">Long description</a></div><a id="back_longdesc_idp8046592"></a></div><p>As the incident energy is increased significantly above 0 (the top of the well), transmission becomes relatively more likely, but there is generally some chance of reflection (<i>R</i> =1 − <i>T</i> ≠ 0), and for some energies the reflection probability may be quite high. However, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007012">Figure 13</a> also shows that there are some special incident particle energies at which <i>T</i> = 1, so that transmission is certain to occur. Although it is a rather poor use of terminology, these peaks of high transmission are usually called <b>transmission resonances</b>. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007044">Equation 7.44</a> shows that transmission resonances occur when sin(<i>k</i><sub>2</sub><i>L</i>) = 0, that is when</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_045"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5725cb4d/sm358_1_e045i.gif" alt=""/></div><p>Recalling the relationship <i>k</i> = 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/<i>λ</i> between wave number and wavelength, we can also express this condition as <i>Nλ</i><sub>2</sub> = 2<i>L</i>; in other words:</p><div class="oucontentquote oucontentsbox" id="quo001"><blockquote><p>A transmission resonance occurs when a whole number of wavelengths occupies the full path length 2<i>L</i> of a wave that crosses the width of the well and is reflected back again.</p></blockquote></div><p>This condition can be interpreted in terms of interference between waves reflected at <i>x</i> = 0 and <i>x</i> = L. The interference turns out to be destructive because reflection at the <i>x</i> = 0 interface is accompanied by a phase change of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>. Of course, suppression of the reflected beam is accompanied by an enhancement of the transmitted beam. The effect is similar to one found in optics, where destructive interference between waves reflected from the front and back surfaces of a thin transparent film accounts for the success of the antireflective coatings on lenses and mirrors.</p><p>Before leaving the subject of scattering from a finite well there is one other point that deserves attention. This concerns the precise form of the functions <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) in the three regions we identified earlier. The value of <i>A</i><sup>2</sup> is equal to the linear number density of particles in the incident beam. If we regard <i>A</i> as being known, the values of the constants <i>B, C, D</i> and <i>F</i> can be evaluated using the continuity boundary conditions and graphs of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) can be drawn.</p><div class="oucontentfigure oucontentmediamini" id="fig007_013"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/31c6f7ed/sm358_1_013i.jpg" alt="Figure 14 A typical example of the real part Re(ψ) of the function ψ(x) for the kind of square well shown in Figure 12" width="270" height="157" style="maxwidth:270px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8087312"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 14 A typical example of the real part Re(<i>ψ</i>) of the function <i>ψ</i>(<i>x</i>) for the kind of square well shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007011">Figure 12</a></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8087312&clicked=1">Long description</a></div><a id="back_longdesc_idp8087312"></a></div><p>Since <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is generally complex there are several possible graphs that might be of interest, including the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, the imaginary part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>, and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup>. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007013">Figure 14</a>, above, shows a typical plot of the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span> for chosen values of <i>m</i>, <i>E</i><sub>0</sub>, <i>V</i><sub>0</sub> and <i>L</i>; the following points should be noted:</p><ol class="oucontentnumbered"><li>
<p>In each region, <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is a periodic function of <i>x</i>.</p>
</li><li>
<p>The wavelength is smaller inside the well than outside. The shorter wavelength corresponds to a higher wave number, and higher momentum, and echoes the classical increase in speed that a particle experiences as it enters the well.</p>
</li><li>
<p>The amplitude of the wave is smaller inside the well than outside. This is because the beam moves more rapidly inside the well, and has a lower linear number density there.</p>
</li></ol><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_008"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 8</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>With one modification, the stationarystate method that was applied to scattering from a finite square well can also be applied to scattering from a finite square barrier of height <i>V</i><sub>0</sub> and width <i>L</i>, when <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>. Describe the modification required, draw a figure analogous to <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#fig007013">Figure 14</a>, but for the case of a square barrier, and comment on any differences between the two graphs.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The main difference is that in the case of a finite square barrier <i>V</i><sub>0</sub> must be replaced by −<i>V</i><sub>0</sub> throughout the analysis. The wave number in Region 2 is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_049"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0e5f4982/sm358_1_ue049i.gif" alt=""/></div><p>Adapting <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#eqn007044">Equation 7.44</a>, the transmission coefficient is then given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_050"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bd2d960d/sm358_1_ue050i.gif" alt=""/></div><p>A typical graph of the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) for scattering by a finite square barrier is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.6#fig007014">Figure 15</a>, below. For a barrier, the wave number in Region 2 is decreased (corresponding to a reduced momentum eigenvalue) and the amplitude of the wave is generally greater in Region 2 than outside it (unless <i>T</i> = 1).</p><div class="oucontentfigure oucontentmediamini" id="fig0s7_002"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7dc74651/sm358_1_025i.jpg" alt="Figure 15 A typical graph of the real part Re(ψ) of ψ(x) for a finite square barrier" width="308" height="238" style="maxwidth:308px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8141104"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 15 A typical graph of the real part Re(<i>ψ</i>) of <i>ψ</i>(<i>x</i>) for a finite square barrier</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8141104&clicked=1">Long description</a></div><a id="back_longdesc_idp8141104"></a></div></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_009"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 9</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Draw two sketches to illustrate the form of the function <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> in the case of the stationarystate approach to scattering from (a) a finite square well, and (b) a finite square barrier. Comment on the timedependence of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x,t</i>)<sup>2</sup> in each case.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>The graphs of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> plotted against <i>x</i> for each case are shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 16</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig00s7_003"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8f00661b/sm358_1_026i.jpg" alt="Figure 16 Ψ(x, t)2 plotted against x for (a) a finite square well and (b) a finite square barrier" width="310" height="489" style="maxwidth:310px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8163664"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 16 Ψ(<i>x, t</i>)<sup>2</sup> plotted against <i>x</i> for (a) a finite square well and (b) a finite square barrier</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8163664&clicked=1">Long description</a></div><a id="back_longdesc_idp8163664"></a></div><p>In Regions 1 and 2 interference effects lead to periodic spatial variations with a period related to the relevant wave number. In Region 3 there is only one beam, so there are no interference effects. Note that there is no simple relationship between the plots of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> and the corresponding plots of the real part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>). This is because <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> is partly determined by the imaginary part of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x,t</i>). In both cases <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x, t</i>)<sup>2</sup> is independent of time. This must be the case, despite the timedependence of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8593d351/psicap.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x,t</i>), because we are dealing with <i>stationary states</i>.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

3.6 Scattering in three dimensions
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.6
Wed, 13 Apr 2016 23:00:00 GMT
<p>Sophisticated methods have been developed to analyse scattering in threedimensions. The complexity of these methods makes them unsuitable for inclusion in this course but it is appropriate to say something about the basic quantities involved.</p><p>In three dimensions, we are obliged to think in terms of scattering at a given angle, rather than in terms of onedimensional reflection or transmission. We distinguish between the incident particles (some of which may be unaffected by the target) and the scattered particles which are affected by the target in some way (changing their direction of motion, energy or state of internal excitation). The detectors for the scattered particles are placed far from the target, well outside the range of interaction of the incident beam and the target, so the scattering process is complete by the time the particles are detected. The incident beam is assumed to be uniform and broad enough to cover all the regions in which the beam particles interact with the target. The incident beam is characterised by its <b>flux</b>; this is the rate of flow of particles <i>per unit time per unit area perpendicular to the beam</i>.</p><p>If we consider a particular scattering experiment (electronproton scattering, for example), one of the main quantities of interest is the <b>total crosssection</b>, <i>σ</i>. This is the total rate at which scattered particles emerge from the target, <i>per unit time per unit incident flux</i>. The total crosssection has the dimensions of area. Very loosely, you can think of it as representing the ‘effective’ area that the target presents to an incident projectile, but you should not give too much weight to this classical interpretation, as most total crosssections vary markedly with the energy of the incident particles. An acceptable SI unit for the measurement of total crosssections would be m<sup>2</sup>, but measured crosssections are generally so small that physicists prefer to use a much smaller unit called the <b>barn</b>, defined by the relation 1 barn = 1 × 10<sup>−28</sup> m<sup>2</sup>. The name is intended as a joke, 1 barn being such a large crosssection in particle and nuclear physics that it can be considered to be ‘as big as a barn door’. Many crosssections are measured in millibarn (mb), microbarn (μb) or even nanobarn (nb).</p><p>Scattering processes that conserve the total kinetic energy of the colliding particles are said to be examples of <b>elastic scattering</b>. They may be contrasted with cases of <b>inelastic scattering</b> where the particles may change their internal state of excitation or be absorbed; particles may even be created or destroyed, especially at very high energies. In reality, total crosssections often contain both elastic and inelastic contributions.</p><p>Scattering experiments are often analysed in great detail. The total crosssection arises as a sum of contributions from particles scattered in different directions. For each direction, we can define a quantity called the <b>differential crosssection</b>, which tells us the rate of scattering in a small cone of angles around the given direction. The integral of the differential crosssection, taken over all directions, is equal to the total crosssection. We can also vary the energy of the incident beam. Both the total crosssection and the differential crosssection depend on the energies of the incident particles. There is therefore a wealth of experimental information to collect, interpret and explain.</p><p>In exploring the microscopic world of atoms, nuclei and elementary particles, physicists have few options, other than to carry out a scattering experiment. This process has been compared with that of trying to find out how a finely crafted watch works by the expedient of hurling stones at it and seeing what bits come flying out. It is not a delicate business, but by collecting all the data that a scattering experiment provides, and by comparing these data with the predictions of quantum physics, physicists have learnt an amazing amount about matter on the scale of atoms and below. One early discovery in the scattering of electrons from noble gas atoms (such as xenon) was a sharp dip in the measured crosssection at an energy of about 1 eV. The experimental discovery of this <b>RamsauerTownsend effect</b> in the early 1920s was an early indication from elastic scattering that some new theoretical insight was needed that would take physics beyond the classical domain. The effect is now recognised as a threedimensional analogue of the transmission resonance we met earlier.</p><p>At the much higher collision energies made available by modern particle accelerators, such as those at the CERN laboratory in Geneva, total crosssections become dominated by inelastic effects, as new particles are produced. As an example, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.6#fig007014">Figure 17</a> shows some data concerning the scattering of K<sup>−</sup> mesons by protons. The upper curve shows the variation of the total crosssection over a very wide range of energies, up to several GeV (1 GeV = 10<sup>9</sup> eV). The lower curve shows the contribution from elastic scattering alone. As the collision energy increases the contribution from elastic scattering becomes less and less important as inelastic processes become more common.</p><div class="oucontentfigure" style="width:511px;" id="fig007_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5bdd4b2b/sm358_1_014i.jpg" alt="Figure 17 The total crosssection (upper curve) and the elastic contribution alone (lower curve), plotted against collision energy, for the scattering of K− mesons by protons" width="511" height="284" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp8209200"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 17 The total crosssection (upper curve) and the elastic contribution alone (lower curve), plotted against collision energy, for the scattering of K<sup>−</sup> mesons by protons</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8209200&clicked=1">Long description</a></div><a id="back_longdesc_idp8209200"></a></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.6
3.6 Scattering in three dimensionsSM358_1<p>Sophisticated methods have been developed to analyse scattering in threedimensions. The complexity of these methods makes them unsuitable for inclusion in this course but it is appropriate to say something about the basic quantities involved.</p><p>In three dimensions, we are obliged to think in terms of scattering at a given angle, rather than in terms of onedimensional reflection or transmission. We distinguish between the incident particles (some of which may be unaffected by the target) and the scattered particles which are affected by the target in some way (changing their direction of motion, energy or state of internal excitation). The detectors for the scattered particles are placed far from the target, well outside the range of interaction of the incident beam and the target, so the scattering process is complete by the time the particles are detected. The incident beam is assumed to be uniform and broad enough to cover all the regions in which the beam particles interact with the target. The incident beam is characterised by its <b>flux</b>; this is the rate of flow of particles <i>per unit time per unit area perpendicular to the beam</i>.</p><p>If we consider a particular scattering experiment (electronproton scattering, for example), one of the main quantities of interest is the <b>total crosssection</b>, <i>σ</i>. This is the total rate at which scattered particles emerge from the target, <i>per unit time per unit incident flux</i>. The total crosssection has the dimensions of area. Very loosely, you can think of it as representing the ‘effective’ area that the target presents to an incident projectile, but you should not give too much weight to this classical interpretation, as most total crosssections vary markedly with the energy of the incident particles. An acceptable SI unit for the measurement of total crosssections would be m<sup>2</sup>, but measured crosssections are generally so small that physicists prefer to use a much smaller unit called the <b>barn</b>, defined by the relation 1 barn = 1 × 10<sup>−28</sup> m<sup>2</sup>. The name is intended as a joke, 1 barn being such a large crosssection in particle and nuclear physics that it can be considered to be ‘as big as a barn door’. Many crosssections are measured in millibarn (mb), microbarn (μb) or even nanobarn (nb).</p><p>Scattering processes that conserve the total kinetic energy of the colliding particles are said to be examples of <b>elastic scattering</b>. They may be contrasted with cases of <b>inelastic scattering</b> where the particles may change their internal state of excitation or be absorbed; particles may even be created or destroyed, especially at very high energies. In reality, total crosssections often contain both elastic and inelastic contributions.</p><p>Scattering experiments are often analysed in great detail. The total crosssection arises as a sum of contributions from particles scattered in different directions. For each direction, we can define a quantity called the <b>differential crosssection</b>, which tells us the rate of scattering in a small cone of angles around the given direction. The integral of the differential crosssection, taken over all directions, is equal to the total crosssection. We can also vary the energy of the incident beam. Both the total crosssection and the differential crosssection depend on the energies of the incident particles. There is therefore a wealth of experimental information to collect, interpret and explain.</p><p>In exploring the microscopic world of atoms, nuclei and elementary particles, physicists have few options, other than to carry out a scattering experiment. This process has been compared with that of trying to find out how a finely crafted watch works by the expedient of hurling stones at it and seeing what bits come flying out. It is not a delicate business, but by collecting all the data that a scattering experiment provides, and by comparing these data with the predictions of quantum physics, physicists have learnt an amazing amount about matter on the scale of atoms and below. One early discovery in the scattering of electrons from noble gas atoms (such as xenon) was a sharp dip in the measured crosssection at an energy of about 1 eV. The experimental discovery of this <b>RamsauerTownsend effect</b> in the early 1920s was an early indication from elastic scattering that some new theoretical insight was needed that would take physics beyond the classical domain. The effect is now recognised as a threedimensional analogue of the transmission resonance we met earlier.</p><p>At the much higher collision energies made available by modern particle accelerators, such as those at the CERN laboratory in Geneva, total crosssections become dominated by inelastic effects, as new particles are produced. As an example, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.6#fig007014">Figure 17</a> shows some data concerning the scattering of K<sup>−</sup> mesons by protons. The upper curve shows the variation of the total crosssection over a very wide range of energies, up to several GeV (1 GeV = 10<sup>9</sup> eV). The lower curve shows the contribution from elastic scattering alone. As the collision energy increases the contribution from elastic scattering becomes less and less important as inelastic processes become more common.</p><div class="oucontentfigure" style="width:511px;" id="fig007_014"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/5bdd4b2b/sm358_1_014i.jpg" alt="Figure 17 The total crosssection (upper curve) and the elastic contribution alone (lower curve), plotted against collision energy, for the scattering of K− mesons by protons" width="511" height="284" style="maxwidth:511px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp8209200"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 17 The total crosssection (upper curve) and the elastic contribution alone (lower curve), plotted against collision energy, for the scattering of K<sup>−</sup> mesons by protons</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8209200&clicked=1">Long description</a></div><a id="back_longdesc_idp8209200"></a></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

4.1 Overview
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>One of the most surprising aspects of quantum physics is the ability of particles to pass through regions that they are classically forbidden from entering. This is the phenomenon of quantummechanical tunnelling that was mentioned in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1">Session 1</a>.</p><p>In Session 4 we first demonstrate the phenomenon of tunnelling with the aid of wave packets. We then go on to examine some of its quantitative features using stationarystate methods, similar to those used in our earlier discussion of scattering from wells and barriers.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.1
4.1 OverviewSM358_1<p>One of the most surprising aspects of quantum physics is the ability of particles to pass through regions that they are classically forbidden from entering. This is the phenomenon of quantummechanical tunnelling that was mentioned in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1">Session 1</a>.</p><p>In Session 4 we first demonstrate the phenomenon of tunnelling with the aid of wave packets. We then go on to examine some of its quantitative features using stationarystate methods, similar to those used in our earlier discussion of scattering from wells and barriers.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

4.2 Wave packets and tunnelling in one dimension
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 18</a> shows a sequence of images captured from a wave packet simulation program. The sequence involves a Gaussian wave packet, with energy expectation value 〈<i>E</i>〉 = <i>E</i><sub>0</sub>, incident from the left on a finite square barrier of height <i>V</i><sub>0</sub>. The sequence is broadly similar to that shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>, which involved a similar wave packet and a similar barrier, but with one important difference; in the earlier process <i>E</i><sub>0</sub> was greater than <i>V</i><sub>0</sub>, so transmission was classically allowed, but in the case of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 18</a> <i>E</i><sub>0</sub> is less than <i>V</i><sub>0</sub> and transmission is classically forbidden. The bottom image shows that transmission can occur in quantum mechanics.</p><div class="oucontentfigure oucontentmediamini" id="fig007_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7ee94b99/sm358_1_015i.jpg" alt="Figure 18 The passage of a wave packet with 〈E〉 = E0 through a finite square barrier of height V0 when E0 < V0. The probability density Ψ2 is shown in a sequence of snapshots with time increasing from top to bottom" width="190" height="499" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8230864"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 18 The passage of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> through a finite square barrier of height <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>. The probability density Ψ<sup>2</sup> is shown in a sequence of snapshots with time increasing from top to bottom</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8230864&clicked=1">Long description</a></div><a id="back_longdesc_idp8230864"></a></div><p>In the case shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 18</a>, part of the reason for transmission is that the wave packet has a spread of energies, some of which lie above the top of the barrier. However, there is a second reason, which applies even for wave packets with energies wholly below the top of the barrier; there is the possibility that a particle, with insufficient energy to surmount the barrier, may nevertheless <i>tunnel through</i> it. For a given wave packet, the probability of tunnelling decreases with the height of the barrier and it also decreases very markedly with its thickness. We shall now use stationarystate methods to investigate this phenomenon.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2
4.2 Wave packets and tunnelling in one dimensionSM358_1<p>
<a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 18</a> shows a sequence of images captured from a wave packet simulation program. The sequence involves a Gaussian wave packet, with energy expectation value 〈<i>E</i>〉 = <i>E</i><sub>0</sub>, incident from the left on a finite square barrier of height <i>V</i><sub>0</sub>. The sequence is broadly similar to that shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.2#fig007006">Figure 6</a>, which involved a similar wave packet and a similar barrier, but with one important difference; in the earlier process <i>E</i><sub>0</sub> was greater than <i>V</i><sub>0</sub>, so transmission was classically allowed, but in the case of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 18</a> <i>E</i><sub>0</sub> is less than <i>V</i><sub>0</sub> and transmission is classically forbidden. The bottom image shows that transmission can occur in quantum mechanics.</p><div class="oucontentfigure oucontentmediamini" id="fig007_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7ee94b99/sm358_1_015i.jpg" alt="Figure 18 The passage of a wave packet with 〈E〉 = E0 through a finite square barrier of height V0 when E0 < V0. The probability density Ψ2 is shown in a sequence of snapshots with time increasing from top to bottom" width="190" height="499" style="maxwidth:190px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8230864"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 18 The passage of a wave packet with 〈<i>E</i>〉 = <i>E</i><sub>0</sub> through a finite square barrier of height <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>. The probability density Ψ<sup>2</sup> is shown in a sequence of snapshots with time increasing from top to bottom</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8230864&clicked=1">Long description</a></div><a id="back_longdesc_idp8230864"></a></div><p>In the case shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.2#fig007015">Figure 18</a>, part of the reason for transmission is that the wave packet has a spread of energies, some of which lie above the top of the barrier. However, there is a second reason, which applies even for wave packets with energies wholly below the top of the barrier; there is the possibility that a particle, with insufficient energy to surmount the barrier, may nevertheless <i>tunnel through</i> it. For a given wave packet, the probability of tunnelling decreases with the height of the barrier and it also decreases very markedly with its thickness. We shall now use stationarystate methods to investigate this phenomenon.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

4.3 Stationary states and barrier penetration
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3
Wed, 13 Apr 2016 23:00:00 GMT
<p>The example of tunnelling we have just been examining can be regarded as a special case of scattering; it just happens to have <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>. As long as we keep this energy range in mind, we can apply the same stationarystate methods to the study of tunnelling that we used earlier when studying scattering.</p><p>As before, we shall start by considering the finite square step, whose potential energy function was defined in Equations <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007006">7.6</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007007">7.7</a>. This is shown for the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007016">Figure 19</a>. The potential energy function divides the <i>x</i>axis into two regions: Region 1 (<i>x</i> ≤ 0) which contains the incident and reflected beams, and Region 2 (<i>x</i> > 0) which contains what is effectively an infinitely wide barrier. There is no possibility of tunnelling <i>through</i> the barrier in this case since there is no classically allowed region on the far side, but the finite square step nonetheless constitutes a valuable limiting case, as you will see.</p><div class="oucontentfigure oucontentmediamini" id="fig007_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/75e638e5/sm358_1_016i.jpg" alt="Figure 19 A finite square step potential energy function. Also shown is the energy E0 of an incident particle for the case E0 < V0, where V0 is the height of the step" width="271" height="123" style="maxwidth:271px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8254080"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 19 A finite square step potential energy function. Also shown is the energy <i>E</i><sub>0</sub> of an incident particle for the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>, where <i>V</i><sub>0</sub> is the height of the step</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8254080&clicked=1">Long description</a></div><a id="back_longdesc_idp8254080"></a></div><p>Proceeding as before, we seek stationarystate solutions of the Schrödinger equation of the form <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3920d2ae/sm358_1_ie007i.gif" alt="" width="139" height="22" style="maxwidth:139px;" class="oucontentinlinefigureimage"/></span>, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is a solution of the corresponding timeindependent Schrödinger equation. In this case we might try to use exactly the same solution as in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3">Section 3.3</a> but doing so would complicate the analysis since we would find that in the region <i>x</i> > 0 the wave number
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ed06ab7b/sm358_1_ie008i.gif" alt="" width="140" height="23" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span> would be imaginary. In view of this, it is better to recognise that <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> generally implies a combination of exponentially growing and exponentially decaying terms in the region <i>x</i> > 0, and write the solutions as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_046"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/edaf93b0/sm358_1_e046i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_047"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/173d4397/sm358_1_e047i.gif" alt=""/></div><p>where, <i>A, B, C</i> and <i>D</i> are arbitrary complex constants, while <i>k</i><sub>1</sub> and α are <i>real</i> quantities given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_048"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f8a55c9a/sm358_1_e048i.gif" alt=""/></div><p>We require <i>D</i> to be zero on physical grounds, to avoid having any part of the solution that grows exponentially as <i>x</i> approaches infinity. To determine the values of <i>B</i> and <i>C</i> relative to that of <i>A</i> we impose the usual requirement (for a finite potential energy function) that both <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>), and its derivative d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i>, must be continuous everywhere. Applying these conditions at <i>x</i> = 0 we find:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9a1815bc/sm358_1_ue015i.gif" alt=""/></div><p>from which it follows that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/26096f99/sm358_1_ue016i.gif" alt=""/></div><p>The reflection coefficient is given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/06bf0f6c/sm358_1_ue017i.gif" alt=""/></div><p>(<i>Note:</i> For any complex number, <i>z</i>, )</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn001_110"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8cffa1f0/sm358_1_ie010i.gif" alt=""/></div><p>So, if particles of energy <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> encounter a finite square step of height <i>V</i><sub>0</sub>, reflection is certain. There is no transmission and no possibility of particles becoming lodged inside the step; everything must eventually be reflected. Note however that <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is not zero inside the step (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007017">Figure 20</a>). Rather, it decreases exponentially over a length scale determined by the quantity
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/80cb5fa2/sm358_1_ie009i.gif" alt="" width="133" height="23" style="maxwidth:133px;" class="oucontentinlinefigureimage"/></span>, which is usually called the <b>attenuation coefficient</b>. This is an example of the phenomenon of <i>barrier penetration</i>. It is not the same as tunnelling since there is no transmitted beam, but it is what makes tunnelling possible, and the occurrence of exponentially decaying solutions in a classically forbidden region suggests why tunnelling probabilities decline rapidly as barrier width increases.</p><div class="oucontentfigure oucontentmediamini" id="fig007_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c5b84ec7/sm358_1_017i.jpg" alt="Figure 20 The quantity ψ(x)2 plotted against x for a finite square step in the case E0 < V0. There is a finite probability that the particle will penetrate the step, even though there is no possibility of tunnelling through it" width="272" height="231" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8328208"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 20 The quantity <i>ψ</i>(<i>x</i>)<sup>2</sup> plotted against <i>x</i> for a finite square step in the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>. There is a finite probability that the particle will penetrate the step, even though there is no possibility of tunnelling through it</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8328208&clicked=1">Long description</a></div><a id="back_longdesc_idp8328208"></a></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_0010"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 10</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Show that the stationarystate probability density <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>)<sup>2</sup> in Region 1 of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007017">Figure 20</a> is a periodic function of <i>x</i> with minima separated by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/<i>k</i><sub>1</sub>.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#eqn007046">Equation 7.46</a>
</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_051"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e3343279/sm358_1_ue051i.gif" alt=""/></div><p>and</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_052"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c0d71b07/sm358_1_ue052i.gif" alt=""/></div><p>so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_053"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e8c36844/sm358_1_ue053i.gif" alt=""/></div><p>This is a periodic function that runs through the same range of values each time <i>x</i> increases by 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/2<i>k</i><sub>1</sub>, so its minima are separated by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/<i>k</i><sub>1</sub>.</p></div></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_0011"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 11</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Show that the probability current in Region 2 of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007017">Figure 20</a> is zero.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>Using <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#eqn007047">Equation 7.47</a>, with <i>D</i> = 0 and <i>α</i> real, and recalling the definition of probability current given in Equation <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">7.28</a>, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_054"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3a094410/sm358_1_ue054i.gif" alt=""/></div></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3
4.3 Stationary states and barrier penetrationSM358_1<p>The example of tunnelling we have just been examining can be regarded as a special case of scattering; it just happens to have <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>. As long as we keep this energy range in mind, we can apply the same stationarystate methods to the study of tunnelling that we used earlier when studying scattering.</p><p>As before, we shall start by considering the finite square step, whose potential energy function was defined in Equations <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007006">7.6</a> and <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3#eqn007007">7.7</a>. This is shown for the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007016">Figure 19</a>. The potential energy function divides the <i>x</i>axis into two regions: Region 1 (<i>x</i> ≤ 0) which contains the incident and reflected beams, and Region 2 (<i>x</i> > 0) which contains what is effectively an infinitely wide barrier. There is no possibility of tunnelling <i>through</i> the barrier in this case since there is no classically allowed region on the far side, but the finite square step nonetheless constitutes a valuable limiting case, as you will see.</p><div class="oucontentfigure oucontentmediamini" id="fig007_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/75e638e5/sm358_1_016i.jpg" alt="Figure 19 A finite square step potential energy function. Also shown is the energy E0 of an incident particle for the case E0 < V0, where V0 is the height of the step" width="271" height="123" style="maxwidth:271px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8254080"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 19 A finite square step potential energy function. Also shown is the energy <i>E</i><sub>0</sub> of an incident particle for the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>, where <i>V</i><sub>0</sub> is the height of the step</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8254080&clicked=1">Long description</a></div><a id="back_longdesc_idp8254080"></a></div><p>Proceeding as before, we seek stationarystate solutions of the Schrödinger equation of the form <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3920d2ae/sm358_1_ie007i.gif" alt="" width="139" height="22" style="maxwidth:139px;" class="oucontentinlinefigureimage"/></span>, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is a solution of the corresponding timeindependent Schrödinger equation. In this case we might try to use exactly the same solution as in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.3">Section 3.3</a> but doing so would complicate the analysis since we would find that in the region <i>x</i> > 0 the wave number
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ed06ab7b/sm358_1_ie008i.gif" alt="" width="140" height="23" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span> would be imaginary. In view of this, it is better to recognise that <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> generally implies a combination of exponentially growing and exponentially decaying terms in the region <i>x</i> > 0, and write the solutions as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_046"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/edaf93b0/sm358_1_e046i.gif" alt=""/></div><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_047"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/173d4397/sm358_1_e047i.gif" alt=""/></div><p>where, <i>A, B, C</i> and <i>D</i> are arbitrary complex constants, while <i>k</i><sub>1</sub> and α are <i>real</i> quantities given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_048"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f8a55c9a/sm358_1_e048i.gif" alt=""/></div><p>We require <i>D</i> to be zero on physical grounds, to avoid having any part of the solution that grows exponentially as <i>x</i> approaches infinity. To determine the values of <i>B</i> and <i>C</i> relative to that of <i>A</i> we impose the usual requirement (for a finite potential energy function) that both <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>), and its derivative d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i>, must be continuous everywhere. Applying these conditions at <i>x</i> = 0 we find:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_015"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/9a1815bc/sm358_1_ue015i.gif" alt=""/></div><p>from which it follows that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_016"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/26096f99/sm358_1_ue016i.gif" alt=""/></div><p>The reflection coefficient is given by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/06bf0f6c/sm358_1_ue017i.gif" alt=""/></div><p>(<i>Note:</i> For any complex number, <i>z</i>, )</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn001_110"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8cffa1f0/sm358_1_ie010i.gif" alt=""/></div><p>So, if particles of energy <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> encounter a finite square step of height <i>V</i><sub>0</sub>, reflection is certain. There is no transmission and no possibility of particles becoming lodged inside the step; everything must eventually be reflected. Note however that <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) is not zero inside the step (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007017">Figure 20</a>). Rather, it decreases exponentially over a length scale determined by the quantity
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/80cb5fa2/sm358_1_ie009i.gif" alt="" width="133" height="23" style="maxwidth:133px;" class="oucontentinlinefigureimage"/></span>, which is usually called the <b>attenuation coefficient</b>. This is an example of the phenomenon of <i>barrier penetration</i>. It is not the same as tunnelling since there is no transmitted beam, but it is what makes tunnelling possible, and the occurrence of exponentially decaying solutions in a classically forbidden region suggests why tunnelling probabilities decline rapidly as barrier width increases.</p><div class="oucontentfigure oucontentmediamini" id="fig007_017"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c5b84ec7/sm358_1_017i.jpg" alt="Figure 20 The quantity ψ(x)2 plotted against x for a finite square step in the case E0 < V0. There is a finite probability that the particle will penetrate the step, even though there is no possibility of tunnelling through it" width="272" height="231" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8328208"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 20 The quantity <i>ψ</i>(<i>x</i>)<sup>2</sup> plotted against <i>x</i> for a finite square step in the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub>. There is a finite probability that the particle will penetrate the step, even though there is no possibility of tunnelling through it</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8328208&clicked=1">Long description</a></div><a id="back_longdesc_idp8328208"></a></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_0010"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 10</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Show that the stationarystate probability density <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>)<sup>2</sup> in Region 1 of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007017">Figure 20</a> is a periodic function of <i>x</i> with minima separated by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/<i>k</i><sub>1</sub>.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#eqn007046">Equation 7.46</a>
</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_051"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e3343279/sm358_1_ue051i.gif" alt=""/></div><p>and</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_052"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c0d71b07/sm358_1_ue052i.gif" alt=""/></div><p>so</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_053"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e8c36844/sm358_1_ue053i.gif" alt=""/></div><p>This is a periodic function that runs through the same range of values each time <i>x</i> increases by 2<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/2<i>k</i><sub>1</sub>, so its minima are separated by <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span>/<i>k</i><sub>1</sub>.</p></div></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_0011"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 11</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Show that the probability current in Region 2 of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#fig007017">Figure 20</a> is zero.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>Using <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.3#eqn007047">Equation 7.47</a>, with <i>D</i> = 0 and <i>α</i> real, and recalling the definition of probability current given in Equation <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007028">7.28</a>, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_054"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3a094410/sm358_1_ue054i.gif" alt=""/></div></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

4.4 Stationary states and tunnelling in one dimension
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4
Wed, 13 Apr 2016 23:00:00 GMT
<p>We will now use the stationarystate approach to analyse the tunnelling of particles of energy <i>E</i><sub>0</sub> through a finite square barrier of width <i>L</i> and height <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#fig007018">Figure 21</a>).</p><div class="oucontentfigure oucontentmediamini" id="fig007_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ca418c50/sm358_1_018i.jpg" alt="Figure 21 A finite square barrier of width L and height V0, together with the energy E0 of each tunnelling particle" width="272" height="118" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8382000"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 21 A finite square barrier of width <i>L</i> and height <i>V</i><sub>0</sub>, together with the energy <i>E</i><sub>0</sub> of each tunnelling particle</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8382000&clicked=1">Long description</a></div><a id="back_longdesc_idp8382000"></a></div><p>Our main aim will be to find an expression for the transmission coefficient. By now, you should be familiar with the general technique for dealing with problems of this kind, including the existence of exponentially growing and decaying solutions in the classically forbidden region, so we shall immediately write down the solution of the relevant timeindependent Schrödinger equation:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_049"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ce29fa7/sm358_1_e049i.gif" alt=""/></div><p>where <i>A, B, C, D</i> and <i>F</i> are arbitrary constants and</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ed354ea/sm358_1_ue018i.gif" alt=""/></div><p>Note that the term that might describe a leftward moving beam in Region 3 has already been omitted, and the wave number in Region 3 has been set equal to that in Region 1.</p><p>Requiring the continuity of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = 0 and at <i>x</i> = <i>L</i> leads to the following four relations:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4021de68/sm358_1_ue019i.gif" alt=""/></div><p>After some lengthy algebra, similar to that in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#box007006">Worked Example 1</a>, these four equations can be reduced to a relationship between <i>F</i> and <i>A</i>, from which it is possible to obtain the following expression for the transmission coefficient <i>T</i> = <i>F</i>/<i>A</i><sup>2</sup>.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_052"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ac92ff67/sm358_1_e052i.gif" alt=""/></div><p>When <i>αL</i> ≫ 1, the denominator of this expression can be approximated by
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/175836a0/sm358_1_ie011i.gif" alt="" width="55" height="30" style="maxwidth:55px;" class="oucontentinlinefigureimage"/></span>, and the transmission coefficient through the barrier is welldescribed by the useful relationship</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_053"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/dc2b8fc5/sm358_1_e053i.gif" alt=""/></div><p>(Remember: sinh <i>x</i> = (<i>e<sup>x</sup></i> − <i>e<sup>−x</sup>)</i>/2.)</p><p>This shows the exponential behaviour that might have been expected on the basis of our earlier results for barrier penetration into a square step, but in this case it is a true tunnelling result. It tells us that tunnelling will occur, but indicates that the tunnelling probability will generally be rather small when <i>αL</i> ≫ 1, and will decrease rapidly as the barrier width <i>L</i> increases.</p><p>A graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> plotted against <i>x</i> for a finite square barrier in the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> will look something like <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#fig007019">Figure 22</a>. Note that because the incident and reflected beams have different intensities, the minimum value of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> in Region 1 is always greater than zero. Also note that for a square barrier of finite width the declining curve in Region 2 is not described by a simple exponential function; there are both exponentially decreasing and exponentially increasing contributions to <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) in that region.</p><div class="oucontentfigure oucontentmediamini" id="fig007_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0918f4d0/sm358_1_019i.jpg" alt="Figure 22 The quantity ψ2 plotted against x for a finite square barrier in the case E0 < V0" width="272" height="217" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8444112"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 22 The quantity <i>ψ</i><sup>2</sup> plotted against <i>x</i> for a finite square barrier in the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8444112&clicked=1">Long description</a></div><a id="back_longdesc_idp8444112"></a></div><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box007_011"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Worked Example 2 </h2><div class="oucontentinnerbox"><p>Electrons with a kinetic energy of 5 eV are incident upon a finite square barrier with a height of 10 eV and a width of 0.5 nm. Estimate the value of <i>T</i> and hence the probability that any particular electron will tunnel through the barrier.</p><h3 class="oucontenth4 oucontentbasic">Solution</h3><p>In this case, <i>V</i><sub>0</sub> − <i>E</i><sub>0</sub> = (10 − 5) eV = 5 × 1.6 × 10<sup>−19</sup>J. It follows that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/270e8c82/sm358_1_ue020i.gif" alt=""/></div><p>With <i>L</i> = 5 × 10<sup>−10</sup> m, it follows that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_021"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0a003667/sm358_1_ue021i.gif" alt=""/></div><p>Since this is much larger than 1, we can use <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#eqn007053">Equation 7.53</a> to estimate</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8db4efaa/sm358_1_ue022i.gif" alt=""/></div><p>This is the probability that any particular electron will tunnel through the barrier. </p></div></div></div><div class=" oucontentactivity oucontentsheavybox1 oucontentsbox " id="exe007_012"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 12</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Determine the probability current in each of the three regions in the case of tunnelling through a finite square barrier. Comment on the significance of your result for Region 2.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Equation <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">7.29</a>,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_055"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6f3156e5/sm358_1_ue055i.gif" alt=""/></div><p>In Region 1, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e3343279/sm358_1_ie022i.gif" alt="" width="146" height="21" style="maxwidth:146px;" class="oucontentinlinefigureimage"/></span> and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2ac7fcb9/sm358_1_ie023i.gif" alt="" width="186" height="21" style="maxwidth:186px;" class="oucontentinlinefigureimage"/></span>, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_056"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/83813d96/sm358_1_ue056i.gif" alt=""/></div><p>Simplifying gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_057"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/567f5135/sm358_1_ue057i.gif" alt=""/></div><p>which can be interpreted as the sum of the probability density currents associated with the incident and reflected beams.</p><p>Similarly, in Region 2, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ee1a383/sm358_1_ie024i.gif" alt="" width="135" height="19" style="maxwidth:135px;" class="oucontentinlinefigureimage"/></span> and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/52a74047/sm358_1_ie025i.gif" alt="" width="177" height="19" style="maxwidth:177px;" class="oucontentinlinefigureimage"/></span>, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_058"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c6a226ae/sm358_1_ue058i.gif" alt=""/></div><p>which simplifies to</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_059"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c8567167/sm358_1_ue059i.gif" alt=""/></div><p>Finally, in Region 3, where
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d952e73c/sm358_1_ie026i.gif" alt="" width="90" height="21" style="maxwidth:90px;" class="oucontentinlinefigureimage"/></span>
</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_060"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0a853d69/sm358_1_ue060i.gif" alt=""/></div><p>The result for Region 2 is a real nonzero quantity, so there is a probability current inside the barrier. This is not really surprising since particles must pass through the barrier to produce a transmitted beam in Region 3. In fact, conservation of probability <i>requires</i> that the probability current should be the same in all three regions.</p></div></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4
4.4 Stationary states and tunnelling in one dimensionSM358_1<p>We will now use the stationarystate approach to analyse the tunnelling of particles of energy <i>E</i><sub>0</sub> through a finite square barrier of width <i>L</i> and height <i>V</i><sub>0</sub> when <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#fig007018">Figure 21</a>).</p><div class="oucontentfigure oucontentmediamini" id="fig007_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ca418c50/sm358_1_018i.jpg" alt="Figure 21 A finite square barrier of width L and height V0, together with the energy E0 of each tunnelling particle" width="272" height="118" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8382000"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 21 A finite square barrier of width <i>L</i> and height <i>V</i><sub>0</sub>, together with the energy <i>E</i><sub>0</sub> of each tunnelling particle</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8382000&clicked=1">Long description</a></div><a id="back_longdesc_idp8382000"></a></div><p>Our main aim will be to find an expression for the transmission coefficient. By now, you should be familiar with the general technique for dealing with problems of this kind, including the existence of exponentially growing and decaying solutions in the classically forbidden region, so we shall immediately write down the solution of the relevant timeindependent Schrödinger equation:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_049"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ce29fa7/sm358_1_e049i.gif" alt=""/></div><p>where <i>A, B, C, D</i> and <i>F</i> are arbitrary constants and</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_018"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ed354ea/sm358_1_ue018i.gif" alt=""/></div><p>Note that the term that might describe a leftward moving beam in Region 3 has already been omitted, and the wave number in Region 3 has been set equal to that in Region 1.</p><p>Requiring the continuity of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) and d<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>/d<i>x</i> at <i>x</i> = 0 and at <i>x</i> = <i>L</i> leads to the following four relations:</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4021de68/sm358_1_ue019i.gif" alt=""/></div><p>After some lengthy algebra, similar to that in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.5#box007006">Worked Example 1</a>, these four equations can be reduced to a relationship between <i>F</i> and <i>A</i>, from which it is possible to obtain the following expression for the transmission coefficient <i>T</i> = <i>F</i>/<i>A</i><sup>2</sup>.</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_052"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ac92ff67/sm358_1_e052i.gif" alt=""/></div><p>When <i>αL</i> ≫ 1, the denominator of this expression can be approximated by
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/175836a0/sm358_1_ie011i.gif" alt="" width="55" height="30" style="maxwidth:55px;" class="oucontentinlinefigureimage"/></span>, and the transmission coefficient through the barrier is welldescribed by the useful relationship</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_053"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/dc2b8fc5/sm358_1_e053i.gif" alt=""/></div><p>(Remember: sinh <i>x</i> = (<i>e<sup>x</sup></i> − <i>e<sup>−x</sup>)</i>/2.)</p><p>This shows the exponential behaviour that might have been expected on the basis of our earlier results for barrier penetration into a square step, but in this case it is a true tunnelling result. It tells us that tunnelling will occur, but indicates that the tunnelling probability will generally be rather small when <i>αL</i> ≫ 1, and will decrease rapidly as the barrier width <i>L</i> increases.</p><p>A graph of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> plotted against <i>x</i> for a finite square barrier in the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub> will look something like <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#fig007019">Figure 22</a>. Note that because the incident and reflected beams have different intensities, the minimum value of <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span><sup>2</sup> in Region 1 is always greater than zero. Also note that for a square barrier of finite width the declining curve in Region 2 is not described by a simple exponential function; there are both exponentially decreasing and exponentially increasing contributions to <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/db34bacd/psilower_ital.gif" alt="" width="9" height="10" style="maxwidth:9px;" class="oucontentinlinefigureimage"/></span>(<i>x</i>) in that region.</p><div class="oucontentfigure oucontentmediamini" id="fig007_019"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0918f4d0/sm358_1_019i.jpg" alt="Figure 22 The quantity ψ2 plotted against x for a finite square barrier in the case E0 < V0" width="272" height="217" style="maxwidth:272px;" class="oucontentfigureimage" longdesc="view.php&extra=longdesc_idp8444112"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 22 The quantity <i>ψ</i><sup>2</sup> plotted against <i>x</i> for a finite square barrier in the case <i>E</i><sub>0</sub> < <i>V</i><sub>0</sub></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8444112&clicked=1">Long description</a></div><a id="back_longdesc_idp8444112"></a></div><div class="oucontentbox oucontentsheavybox1 oucontentsbox " id="box007_011"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Worked Example 2 </h2><div class="oucontentinnerbox"><p>Electrons with a kinetic energy of 5 eV are incident upon a finite square barrier with a height of 10 eV and a width of 0.5 nm. Estimate the value of <i>T</i> and hence the probability that any particular electron will tunnel through the barrier.</p><h3 class="oucontenth4 oucontentbasic">Solution</h3><p>In this case, <i>V</i><sub>0</sub> − <i>E</i><sub>0</sub> = (10 − 5) eV = 5 × 1.6 × 10<sup>−19</sup>J. It follows that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/270e8c82/sm358_1_ue020i.gif" alt=""/></div><p>With <i>L</i> = 5 × 10<sup>−10</sup> m, it follows that</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_021"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0a003667/sm358_1_ue021i.gif" alt=""/></div><p>Since this is much larger than 1, we can use <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#eqn007053">Equation 7.53</a> to estimate</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8db4efaa/sm358_1_ue022i.gif" alt=""/></div><p>This is the probability that any particular electron will tunnel through the barrier. </p></div></div></div><div class="
oucontentactivity
oucontentsheavybox1 oucontentsbox " id="exe007_012"><div class="oucontentouterbox"><h2 class="oucontenth3 oucontentheading oucontentnonumber">Exercise 12</h2><div class="oucontentinnerbox"><div class="oucontentsaqquestion"><p>Determine the probability current in each of the three regions in the case of tunnelling through a finite square barrier. Comment on the significance of your result for Region 2.</p></div>
<div class="oucontentsaqanswer"><h3 class="oucontenth4">Answer</h3><p>From Equation <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.4#eqn007029">7.29</a>,</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_055"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/6f3156e5/sm358_1_ue055i.gif" alt=""/></div><p>In Region 1, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/e3343279/sm358_1_ie022i.gif" alt="" width="146" height="21" style="maxwidth:146px;" class="oucontentinlinefigureimage"/></span> and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2ac7fcb9/sm358_1_ie023i.gif" alt="" width="186" height="21" style="maxwidth:186px;" class="oucontentinlinefigureimage"/></span>, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_056"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/83813d96/sm358_1_ue056i.gif" alt=""/></div><p>Simplifying gives</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_057"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/567f5135/sm358_1_ue057i.gif" alt=""/></div><p>which can be interpreted as the sum of the probability density currents associated with the incident and reflected beams.</p><p>Similarly, in Region 2, where <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ee1a383/sm358_1_ie024i.gif" alt="" width="135" height="19" style="maxwidth:135px;" class="oucontentinlinefigureimage"/></span> and <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/52a74047/sm358_1_ie025i.gif" alt="" width="177" height="19" style="maxwidth:177px;" class="oucontentinlinefigureimage"/></span>, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_058"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c6a226ae/sm358_1_ue058i.gif" alt=""/></div><p>which simplifies to</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_059"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/c8567167/sm358_1_ue059i.gif" alt=""/></div><p>Finally, in Region 3, where
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d952e73c/sm358_1_ie026i.gif" alt="" width="90" height="21" style="maxwidth:90px;" class="oucontentinlinefigureimage"/></span>
</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_060"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/0a853d69/sm358_1_ue060i.gif" alt=""/></div><p>The result for Region 2 is a real nonzero quantity, so there is a probability current inside the barrier. This is not really surprising since particles must pass through the barrier to produce a transmitted beam in Region 3. In fact, conservation of probability <i>requires</i> that the probability current should be the same in all three regions.</p></div></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

5.1 Overview
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.1
Wed, 13 Apr 2016 23:00:00 GMT
<p>The discovery that quantum mechanics permits the tunnelling of particles was of great significance. It has deep implications for our understanding of the physical world and many practical applications, particularly in electronics and the developing field of nanotechnology. This section introduces some of these implications and applications. Applications naturally involve the three dimensions of the real world, and realistic potential energy functions are never perfectly square. Despite these added complexities, the principles developed in the last section provide a good basis for the discussion that follows.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.1
5.1 OverviewSM358_1<p>The discovery that quantum mechanics permits the tunnelling of particles was of great significance. It has deep implications for our understanding of the physical world and many practical applications, particularly in electronics and the developing field of nanotechnology. This section introduces some of these implications and applications. Applications naturally involve the three dimensions of the real world, and realistic potential energy functions are never perfectly square. Despite these added complexities, the principles developed in the last section provide a good basis for the discussion that follows.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

5.2 Alpha decay
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2
Wed, 13 Apr 2016 23:00:00 GMT
<p>You have probably met the law of radioactive decay, which says that, given a sample of <i>N</i><sub>0</sub> similar nuclei at time <i>t</i> = 0, the number remaining at time <i>t</i> is <i>N</i>(<i>t</i>) = <i>N</i><sub>0</sub>e<sup>−<i>λ</i><i>t</i></sup>, where <i>λ</i>, the decay constant for a particular kind of nucleus, determines the rate at which the nuclei decay. The halflife is the time needed for <i>half</i> of any sufficiently large sample to decay. It is related to the decay constant by <i>T</i><sub>1/2</sub> = (ln2)/<i>λ</i>.</p><p>We shall now consider an important type of radioactive decay called <b>alpha decay</b> in which an atomic nucleus emits an energetic alpha particle. The emitted alpha particle consists of two protons and two neutrons and is structurally identical to a helium4 nucleus (<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d87ed3a8/sm358_1_ie030i.gif" alt="" width="30" height="24" style="maxwidth:30px;" class="oucontentinlinefigureimage"/></span>). Alpha decay is the dominant decay mode for a number of heavy nuclei (typically those with atomic numbers greater than 84); a specific example is the decay of uranium to thorium represented by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ebb34a2/sm358_1_ue023i.gif" alt=""/></div><p>where <i>α</i> denotes the alpha particle. Note that the atomic number of the parent nucleus decreases by two and its mass number decreases by four.</p><p>Alpha decay was discovered and named by Rutherford in 1898. It was soon established that each type of alphadecaying nucleus emits an alpha particle with a characteristic energy, <i>E</i><sub><i>α</i></sub>. While these alpha emission energies cover a fairly narrow range of values (from about 2 MeV to 8 MeV), the halflives of the corresponding nuclei cover an enormous range (from 10<sup>−12</sup> s to 10<sup>17</sup> s). Experiments showed that, within certain families of alphaemitting nuclei, the halflives and alpha emission energies were related to one another. Written in terms of the decay constant, <i>λ</i> = (ln 2)/<i>T</i><sub>1/2</sub>, this relationship can be expressed in the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_054"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/811176e6/sm358_1_e054i.gif" alt=""/></div><p>where <i>A</i> is a constant that characterises the particular family of nuclei, and <i>B</i> depends on the charge of the individual nucleus. We shall refer to this empirical law as the <b>GeigerNuttall relation</b>.
</p><p>Despite all this information, by the early 1920s alpha decay had become a major puzzle to physicists. The cause of the observed GeigerNuttall relation was not understood. Attempts to explain it on the basis of classical physics, with the alpha particle initially confined within the nucleus by an energy barrier that it eventually manages to surmount, did not work. In some cases the observed emission energies were too low to be consistent with surmounting the energy barrier at all. So, how could the alpha particles escape, why did their emission lead to such a staggering range of halflives, and what was the origin of the GeigerNuttall relation?</p><p>Answering these questions was one of the early triumphs of wave mechanics. In 1928 the RussianAmerican physicist George Gamow, and then, independently, Gurney and Condon, proposed a successful theory of alpha decay based on quantum tunnelling. In a simplified version of their approach, the potential energy function responsible for alpha decay has the form shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#fig007020">Figure 23</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig007_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2d8517c0/sm358_1_020i.jpg" alt="Figure 23 The potential energy function V(r) for an alpha particle in the vicinity of an atomic nucleus of atomic number Z" width="270" height="171" style="maxwidth:270px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8541184"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 23 The potential energy function <i>V</i>(<i>r</i>) for an alpha particle in the vicinity of an atomic nucleus of atomic number <i>Z</i></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8541184&clicked=1">Long description</a></div><a id="back_longdesc_idp8541184"></a></div><p>Note that <i>V</i>(<i>r</i>) is a function of a radial coordinate <i>r</i>; this is because we are dealing with a threedimensional problem in which the potential energy function is spherically symmetric, and <i>r</i> represents the distance from an origin at the centre of the nucleus. Initially, an alpha particle of energy <i>E</i><sub><i>α</i></sub> is confined within a distance <i>r</i> = <i>r</i><sub>0</sub> of the origin by the welllike part of the potential energy function. This well is due to the powerful but shortrange interaction known as the <i>strong nuclear force</i>. In addition, a longrange <i>electrostatic force</i> acts between the positively charged alpha particle and the remainder of the positively charged nucleus, and has the effect of repelling the alpha particle from the nucleus. The electrostatic force corresponds to the potential energy function</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_055"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/185cf265/sm358_1_e055i.gif" alt=""/></div><p>where <i>Z</i> is the atomic number of the nucleus, 2<i>e</i> is the charge of the alpha particle, (<i>Z</i> − 2)<i>e</i> is the charge of the nucleus left behind after the decay and ε<sub>0</sub> is a fundamental constant called <i>the permittivity of free space</i>. This potential energy function is often called the <b>Coulomb barrier</b>. Notice that the Coulomb barrier exceeds the energy of the alpha particle in the region between <i>r</i> = <i>r</i><sub>0</sub> and <i>r</i> = <i>r</i><sub>1</sub> (defined by <i>V</i>(<i>r</i><sub>1</sub>) = <i>E</i><sub><i>α</i></sub>). In classical physics, the alpha particle does not have enough energy to enter this region, but in quantum physics it may tunnel through. Once beyond the point <i>r</i> = <i>r</i><sub>1</sub>, the alpha particle is electrostatically repelled from the nucleus.</p><p>To apply the quantummechanical theory of tunnelling to alpha decay, we first note that a classically confined particle would oscillate back and forth inside the well; the combination of its energy (<i>E</i><sub><i>α</i></sub>) and the nuclear diameter (2<i>r</i><sub>0</sub>) implying that it is incident on the barrier about 10<sup>21</sup> times per second. Taking this idea over into quantum mechanics, we shall regard each of these encounters as an escape attempt. The small probability of escape at each attempt is represented by the transmission coefficient for tunnelling, <i>T</i>. To estimate <i>T</i> we must take account of the precise shape of the Coulomb barrier. We shall not go through the detailed arguments used to estimate <i>T</i> in this case, but we shall note that they involve the approximation</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_056"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cb556e56/sm358_1_e056i.gif" alt=""/></div><p>where <i>r</i><sub>0</sub> and <i>r</i><sub>1</sub> are the minimum and maximum values of <i>r</i> for which <i>V</i>(<i>r</i>) > <i>E</i><sub><i>α</i></sub>. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007056">Equation 7.56</a> is closely related to the expression for tunnelling through a finite square barrier given in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4">Section 4.4</a>. If the potential energy function <i>V</i>(<i>r</i>) happened to be constant over a region of length <i>L</i>, then Equation 7.56 would reproduce the exponential term of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#eqn007053">Equation 7.53</a>. The other factors in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#eqn007053">Equation 7.53</a> are not reproduced, but they vary so slowly compared with the exponential factor that they can be ignored for present purposes.</p><p>For given values of <i>E</i><sub><i>α</i></sub>, <i>Z</i> and <i>r</i><sub>0</sub>, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007056">Equation 7.56</a> can be evaluated using the Coulomb barrier potential energy function of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007055">Equation 7.55</a>. After a lengthy calculation, including some approximations, the final result is of the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_057"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/89ab0f90/sm358_1_e057i.gif" alt=""/></div><p>where <i>a</i> and <i>b</i> are constants. Multiplying <i>T</i> by the number of escape attempts per second gives the rate of <i>successful</i> escape attempts, and this can be equated to the decay constant, λ. So, according to the quantum tunnelling theory of alpha decay, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_058"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bd90a1f4/sm358_1_e058i.gif" alt=""/></div><p>This agrees with the Geiger–Nuttall relation and a detailed comparison with experimental data is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#fig007021">Figure 24</a>. The exponentialdependence on <i>E</i><sub><i>α</i></sub><sup>−1/2</sup> implies that a very wide range of decay constants is associated with a small range of emission energies. The sensitivity to energy is far greater than for a square barrier because of the shape of the Coulomb barrier; increasing the energy of the alpha particle decreases the effective width that must be tunnelled through.</p><div class="oucontentfigure" style="width:472px;" id="fig007_021"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7300a545/sm358_1_021i.jpg" alt="Figure 24 A comparison of the Geiger–Nuttall relation with experimental data for different families of nuclei. In this plot, the straight lines confirm the exponential dependence of T1/2 (and hence λ) on Eα−1/2" width="472" height="363" style="maxwidth:472px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp8600928"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">Figure 24 A comparison of the Geiger–Nuttall relation with experimental data for different families of nuclei. In this plot, the straight lines confirm the exponential dependence of <i>T</i><sub>1/2</sub> (and hence <i>λ</i>) on <i>E</i><sub><i>α</i></sub><sup>−1/2</sup></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8600928&clicked=1">Long description</a></div><a id="back_longdesc_idp8600928"></a></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2
5.2 Alpha decaySM358_1<p>You have probably met the law of radioactive decay, which says that, given a sample of <i>N</i><sub>0</sub> similar nuclei at time <i>t</i> = 0, the number remaining at time <i>t</i> is <i>N</i>(<i>t</i>) = <i>N</i><sub>0</sub>e<sup>−<i>λ</i><i>t</i></sup>, where <i>λ</i>, the decay constant for a particular kind of nucleus, determines the rate at which the nuclei decay. The halflife is the time needed for <i>half</i> of any sufficiently large sample to decay. It is related to the decay constant by <i>T</i><sub>1/2</sub> = (ln2)/<i>λ</i>.</p><p>We shall now consider an important type of radioactive decay called <b>alpha decay</b> in which an atomic nucleus emits an energetic alpha particle. The emitted alpha particle consists of two protons and two neutrons and is structurally identical to a helium4 nucleus (<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/d87ed3a8/sm358_1_ie030i.gif" alt="" width="30" height="24" style="maxwidth:30px;" class="oucontentinlinefigureimage"/></span>). Alpha decay is the dominant decay mode for a number of heavy nuclei (typically those with atomic numbers greater than 84); a specific example is the decay of uranium to thorium represented by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/4ebb34a2/sm358_1_ue023i.gif" alt=""/></div><p>where <i>α</i> denotes the alpha particle. Note that the atomic number of the parent nucleus decreases by two and its mass number decreases by four.</p><p>Alpha decay was discovered and named by Rutherford in 1898. It was soon established that each type of alphadecaying nucleus emits an alpha particle with a characteristic energy, <i>E</i><sub><i>α</i></sub>. While these alpha emission energies cover a fairly narrow range of values (from about 2 MeV to 8 MeV), the halflives of the corresponding nuclei cover an enormous range (from 10<sup>−12</sup> s to 10<sup>17</sup> s). Experiments showed that, within certain families of alphaemitting nuclei, the halflives and alpha emission energies were related to one another. Written in terms of the decay constant, <i>λ</i> = (ln 2)/<i>T</i><sub>1/2</sub>, this relationship can be expressed in the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_054"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/811176e6/sm358_1_e054i.gif" alt=""/></div><p>where <i>A</i> is a constant that characterises the particular family of nuclei, and <i>B</i> depends on the charge of the individual nucleus. We shall refer to this empirical law as the <b>GeigerNuttall relation</b>.
</p><p>Despite all this information, by the early 1920s alpha decay had become a major puzzle to physicists. The cause of the observed GeigerNuttall relation was not understood. Attempts to explain it on the basis of classical physics, with the alpha particle initially confined within the nucleus by an energy barrier that it eventually manages to surmount, did not work. In some cases the observed emission energies were too low to be consistent with surmounting the energy barrier at all. So, how could the alpha particles escape, why did their emission lead to such a staggering range of halflives, and what was the origin of the GeigerNuttall relation?</p><p>Answering these questions was one of the early triumphs of wave mechanics. In 1928 the RussianAmerican physicist George Gamow, and then, independently, Gurney and Condon, proposed a successful theory of alpha decay based on quantum tunnelling. In a simplified version of their approach, the potential energy function responsible for alpha decay has the form shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#fig007020">Figure 23</a>.</p><div class="oucontentfigure oucontentmediamini" id="fig007_020"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/2d8517c0/sm358_1_020i.jpg" alt="Figure 23 The potential energy function V(r) for an alpha particle in the vicinity of an atomic nucleus of atomic number Z" width="270" height="171" style="maxwidth:270px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8541184"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 23 The potential energy function <i>V</i>(<i>r</i>) for an alpha particle in the vicinity of an atomic nucleus of atomic number <i>Z</i></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8541184&clicked=1">Long description</a></div><a id="back_longdesc_idp8541184"></a></div><p>Note that <i>V</i>(<i>r</i>) is a function of a radial coordinate <i>r</i>; this is because we are dealing with a threedimensional problem in which the potential energy function is spherically symmetric, and <i>r</i> represents the distance from an origin at the centre of the nucleus. Initially, an alpha particle of energy <i>E</i><sub><i>α</i></sub> is confined within a distance <i>r</i> = <i>r</i><sub>0</sub> of the origin by the welllike part of the potential energy function. This well is due to the powerful but shortrange interaction known as the <i>strong nuclear force</i>. In addition, a longrange <i>electrostatic force</i> acts between the positively charged alpha particle and the remainder of the positively charged nucleus, and has the effect of repelling the alpha particle from the nucleus. The electrostatic force corresponds to the potential energy function</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_055"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/185cf265/sm358_1_e055i.gif" alt=""/></div><p>where <i>Z</i> is the atomic number of the nucleus, 2<i>e</i> is the charge of the alpha particle, (<i>Z</i> − 2)<i>e</i> is the charge of the nucleus left behind after the decay and ε<sub>0</sub> is a fundamental constant called <i>the permittivity of free space</i>. This potential energy function is often called the <b>Coulomb barrier</b>. Notice that the Coulomb barrier exceeds the energy of the alpha particle in the region between <i>r</i> = <i>r</i><sub>0</sub> and <i>r</i> = <i>r</i><sub>1</sub> (defined by <i>V</i>(<i>r</i><sub>1</sub>) = <i>E</i><sub><i>α</i></sub>). In classical physics, the alpha particle does not have enough energy to enter this region, but in quantum physics it may tunnel through. Once beyond the point <i>r</i> = <i>r</i><sub>1</sub>, the alpha particle is electrostatically repelled from the nucleus.</p><p>To apply the quantummechanical theory of tunnelling to alpha decay, we first note that a classically confined particle would oscillate back and forth inside the well; the combination of its energy (<i>E</i><sub><i>α</i></sub>) and the nuclear diameter (2<i>r</i><sub>0</sub>) implying that it is incident on the barrier about 10<sup>21</sup> times per second. Taking this idea over into quantum mechanics, we shall regard each of these encounters as an escape attempt. The small probability of escape at each attempt is represented by the transmission coefficient for tunnelling, <i>T</i>. To estimate <i>T</i> we must take account of the precise shape of the Coulomb barrier. We shall not go through the detailed arguments used to estimate <i>T</i> in this case, but we shall note that they involve the approximation</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_056"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/cb556e56/sm358_1_e056i.gif" alt=""/></div><p>where <i>r</i><sub>0</sub> and <i>r</i><sub>1</sub> are the minimum and maximum values of <i>r</i> for which <i>V</i>(<i>r</i>) > <i>E</i><sub><i>α</i></sub>. <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007056">Equation 7.56</a> is closely related to the expression for tunnelling through a finite square barrier given in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4">Section 4.4</a>. If the potential energy function <i>V</i>(<i>r</i>) happened to be constant over a region of length <i>L</i>, then Equation 7.56 would reproduce the exponential term of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#eqn007053">Equation 7.53</a>. The other factors in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.4#eqn007053">Equation 7.53</a> are not reproduced, but they vary so slowly compared with the exponential factor that they can be ignored for present purposes.</p><p>For given values of <i>E</i><sub><i>α</i></sub>, <i>Z</i> and <i>r</i><sub>0</sub>, <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007056">Equation 7.56</a> can be evaluated using the Coulomb barrier potential energy function of <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007055">Equation 7.55</a>. After a lengthy calculation, including some approximations, the final result is of the form</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_057"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/89ab0f90/sm358_1_e057i.gif" alt=""/></div><p>where <i>a</i> and <i>b</i> are constants. Multiplying <i>T</i> by the number of escape attempts per second gives the rate of <i>successful</i> escape attempts, and this can be equated to the decay constant, λ. So, according to the quantum tunnelling theory of alpha decay, we have</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="eqn007_058"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/bd90a1f4/sm358_1_e058i.gif" alt=""/></div><p>This agrees with the Geiger–Nuttall relation and a detailed comparison with experimental data is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#fig007021">Figure 24</a>. The exponentialdependence on <i>E</i><sub><i>α</i></sub><sup>−1/2</sup> implies that a very wide range of decay constants is associated with a small range of emission energies. The sensitivity to energy is far greater than for a square barrier because of the shape of the Coulomb barrier; increasing the energy of the alpha particle decreases the effective width that must be tunnelled through.</p><div class="oucontentfigure" style="width:472px;" id="fig007_021"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7300a545/sm358_1_021i.jpg" alt="Figure 24 A comparison of the Geiger–Nuttall relation with experimental data for different families of nuclei. In this plot, the straight lines confirm the exponential dependence of T1/2 (and hence λ) on Eα−1/2" width="472" height="363" style="maxwidth:472px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp8600928"/><div class="oucontentfiguretext"><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">Figure 24 A comparison of the Geiger–Nuttall relation with experimental data for different families of nuclei. In this plot, the straight lines confirm the exponential dependence of <i>T</i><sub>1/2</sub> (and hence <i>λ</i>) on <i>E</i><sub><i>α</i></sub><sup>−1/2</sup></span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8600928&clicked=1">Long description</a></div><a id="back_longdesc_idp8600928"></a></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

5.3 Stellar astrophysics
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.3
Wed, 13 Apr 2016 23:00:00 GMT
<p>If tunnelling out of nuclei is possible then so is tunnelling in! As a consequence it is possible to trigger nuclear reactions with protons of much lower energy than would be needed to climb over the full height of the Coulomb barrier. This was the principle used by J.D. Cockcroft and E.T.S. Walton in 1932 when they caused lithium7 nuclei to split into pairs of alpha particles by bombarding them with highenergy protons. Their achievement won them the 1951 Nobel prize for physics. The same principle is also at work in stars, such as the Sun, where it facilitates the nuclear reactions that allow the stars to shine. Indeed, were it not for the existence of quantum tunnelling, it's probably fair to say that the Sun would not shine and that life on Earth would never have arisen.</p><p>The nuclear reactions that allow stars to shine are predominantly <b>fusion reactions</b> in which low mass nuclei combine to form a nucleus with a lower mass than the total mass of the nuclei that fused together to form it. It is the difference between the total nuclear masses at the beginning and the end of the fusion process that (via <i>E</i> = <i>mc</i><sup>2</sup>) is ultimately responsible for the energy emitted by a star. The energy released by each individual fusion reaction is quite small, but in the hot dense cores of stars there are so many fusing nuclei that they collectively account for the prodigious energy output that is typical of stars (3.8 × 10<sup>26</sup> W in the case of the Sun).</p><p>In order to fuse, two nuclei have to overcome the repulsive Coulomb barrier that tends to keep them apart. The energy they need to do this is provided by the kinetic energy associated with their thermal motion. This is why the nuclear reactions are mainly confined to a star's hot central core. In the case of the Sun, the core temperature is of the order of 10<sup>7</sup> K. Multiplying this by the Boltzmann constant indicates that the typical thermal kinetic energy of a proton in the solar core is about 1.4 × 10<sup>−16</sup> J ≈ 1 keV. However, the height of the Coulomb barrier between two protons is more than a thousand times greater than this. Fortunately, as you have just seen, the protons do not have to climb over this barrier because they can tunnel through it. Even so, and despite the hectic conditions of a stellar interior, where collisions are frequent and aboveaverage energies not uncommon, the reliance on tunnelling makes fusion a relatively slow process.</p><p>Again taking the Sun as an example, its energy comes mainly from a process called the <b>proton–proton chain</b> that converts hydrogen to helium. The first step in this chain involves the fusion of two protons and is extremely slow, taking about 10<sup>9</sup> years for an average proton in the core of the Sun. This is one of the reasons why stars are so longlived. The Sun is about 4.6 × 10<sup>9</sup> years old, yet it has only consumed about half of the hydrogen in its core. So, we have quantum tunnelling to thank, not only for the existence of sunlight, but also for its persistence over billions of years.</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.3
5.3 Stellar astrophysicsSM358_1<p>If tunnelling out of nuclei is possible then so is tunnelling in! As a consequence it is possible to trigger nuclear reactions with protons of much lower energy than would be needed to climb over the full height of the Coulomb barrier. This was the principle used by J.D. Cockcroft and E.T.S. Walton in 1932 when they caused lithium7 nuclei to split into pairs of alpha particles by bombarding them with highenergy protons. Their achievement won them the 1951 Nobel prize for physics. The same principle is also at work in stars, such as the Sun, where it facilitates the nuclear reactions that allow the stars to shine. Indeed, were it not for the existence of quantum tunnelling, it's probably fair to say that the Sun would not shine and that life on Earth would never have arisen.</p><p>The nuclear reactions that allow stars to shine are predominantly <b>fusion reactions</b> in which low mass nuclei combine to form a nucleus with a lower mass than the total mass of the nuclei that fused together to form it. It is the difference between the total nuclear masses at the beginning and the end of the fusion process that (via <i>E</i> = <i>mc</i><sup>2</sup>) is ultimately responsible for the energy emitted by a star. The energy released by each individual fusion reaction is quite small, but in the hot dense cores of stars there are so many fusing nuclei that they collectively account for the prodigious energy output that is typical of stars (3.8 × 10<sup>26</sup> W in the case of the Sun).</p><p>In order to fuse, two nuclei have to overcome the repulsive Coulomb barrier that tends to keep them apart. The energy they need to do this is provided by the kinetic energy associated with their thermal motion. This is why the nuclear reactions are mainly confined to a star's hot central core. In the case of the Sun, the core temperature is of the order of 10<sup>7</sup> K. Multiplying this by the Boltzmann constant indicates that the typical thermal kinetic energy of a proton in the solar core is about 1.4 × 10<sup>−16</sup> J ≈ 1 keV. However, the height of the Coulomb barrier between two protons is more than a thousand times greater than this. Fortunately, as you have just seen, the protons do not have to climb over this barrier because they can tunnel through it. Even so, and despite the hectic conditions of a stellar interior, where collisions are frequent and aboveaverage energies not uncommon, the reliance on tunnelling makes fusion a relatively slow process.</p><p>Again taking the Sun as an example, its energy comes mainly from a process called the <b>proton–proton chain</b> that converts hydrogen to helium. The first step in this chain involves the fusion of two protons and is extremely slow, taking about 10<sup>9</sup> years for an average proton in the core of the Sun. This is one of the reasons why stars are so longlived. The Sun is about 4.6 × 10<sup>9</sup> years old, yet it has only consumed about half of the hydrogen in its core. So, we have quantum tunnelling to thank, not only for the existence of sunlight, but also for its persistence over billions of years.</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

5.4 The scanning tunnelling microscope
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.4
Wed, 13 Apr 2016 23:00:00 GMT
<p>The <b>scanning tunnelling microscope</b> (STM) is a device of such extraordinary sensitivity that it can reveal the distribution of individual atoms on the surface of a sample. It can also be used to manipulate atoms and even to promote chemical reactions between specific atoms. The first STM was developed in 1981 at the IBM Laboratories in Zurich by Gerd Binnig and Heinrich Rohrer. Their achievement was recognised by the award of the 1986 Nobel prize for physics.</p><p>In an STM the sample under investigation is held in a vacuum and a very fine tip, possibly only a single atom wide, is moved across its surface (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.4#fig007022">Figure 25</a>). Things are so arranged that there is always a small gap between the tip and the surface being scanned. An applied voltage between the tip and the sample tends to cause electrons to cross the gap, but the gap itself constitutes a potential energy barrier that, classically, the electrons would not be able to surmount. However, thanks to quantum physics, they can tunnel through the barrier and thereby produce a measurable electric current. Since the current is caused by a tunnelling process, the magnitude of the current is very sensitive to the size of the gap (detailed estimates can again be obtained using <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007056">Equation 7.56</a>). This sensitivity is the key to finding the positions of tiny irregularities in the surface, including individual atoms.</p><div class="oucontentfigure" style="width:463px;" id="fig007_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7b928de9/sm358_1_022i.jpg" alt="Figure 25 (a) A scanning tunnelling microscope (STM) surrounded by vacuum chambers. (b) A schematic diagram showing an STM tip in operation" width="463" height="276" style="maxwidth:463px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp8623840"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 25 (a) A scanning tunnelling microscope (STM) surrounded by vacuum chambers. (b) A schematic diagram showing an STM tip in operation</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8623840&clicked=1">Long description</a></div><a id="back_longdesc_idp8623840"></a></div><p>In practice, the STM can operate in two different ways. In <i>constantheight mode</i>, the tip moves at a constant height and the topography of the surface is revealed by changes in the tunnelling current. In the more common <i>constantcurrent mode</i> the height of the tip is adjusted throughout the scanning process to maintain a constant current and the tiny movements of the tip are recorded. In either mode the structure of the sample's surface can be mapped on an atomic scale, though neither mode involves imaging of the kind that takes place in a conventional optical or transmission electron microscope.</p><p>STMs have now become a major tool in the developing field of nanotechnology. This is partly because of the images they supply, but even more because of their ability to manipulate individual atoms and position them with great accuracy. One of the products of this kind of nanoscale manipulation is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.4#fig007023">Figure 26</a>, the famous ‘quantum corral’ formed by positioning iron atoms on a copper surface.</p><div class="oucontentfigure oucontentmediamini" id="fig007_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f2029f51/sm358_1_023i.jpg" alt="Figure 26 Iron atoms on a copper surface forming a ring called a ‘quantum corral’. Standing waves of surface electrons trapped inside the corral are also visible" width="286" height="193" style="maxwidth:286px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8633360"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Courtesy of Don Eigler, IBM Research Division
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Courtesy of Don Eigler, IBM Research Division</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 26 Iron atoms on a copper surface forming a ring called a ‘quantum corral’. Standing waves of surface electrons trapped inside the corral are also visible</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8633360&clicked=1">Long description</a></div><a id="back_longdesc_idp8633360"></a></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.4
5.4 The scanning tunnelling microscopeSM358_1<p>The <b>scanning tunnelling microscope</b> (STM) is a device of such extraordinary sensitivity that it can reveal the distribution of individual atoms on the surface of a sample. It can also be used to manipulate atoms and even to promote chemical reactions between specific atoms. The first STM was developed in 1981 at the IBM Laboratories in Zurich by Gerd Binnig and Heinrich Rohrer. Their achievement was recognised by the award of the 1986 Nobel prize for physics.</p><p>In an STM the sample under investigation is held in a vacuum and a very fine tip, possibly only a single atom wide, is moved across its surface (see <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.4#fig007022">Figure 25</a>). Things are so arranged that there is always a small gap between the tip and the surface being scanned. An applied voltage between the tip and the sample tends to cause electrons to cross the gap, but the gap itself constitutes a potential energy barrier that, classically, the electrons would not be able to surmount. However, thanks to quantum physics, they can tunnel through the barrier and thereby produce a measurable electric current. Since the current is caused by a tunnelling process, the magnitude of the current is very sensitive to the size of the gap (detailed estimates can again be obtained using <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.2#eqn007056">Equation 7.56</a>). This sensitivity is the key to finding the positions of tiny irregularities in the surface, including individual atoms.</p><div class="oucontentfigure" style="width:463px;" id="fig007_022"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7b928de9/sm358_1_022i.jpg" alt="Figure 25 (a) A scanning tunnelling microscope (STM) surrounded by vacuum chambers. (b) A schematic diagram showing an STM tip in operation" width="463" height="276" style="maxwidth:463px;" class="oucontentfigureimage oucontentmediawide" longdesc="view.php?id=2680&extra=longdesc_idp8623840"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 25 (a) A scanning tunnelling microscope (STM) surrounded by vacuum chambers. (b) A schematic diagram showing an STM tip in operation</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8623840&clicked=1">Long description</a></div><a id="back_longdesc_idp8623840"></a></div><p>In practice, the STM can operate in two different ways. In <i>constantheight mode</i>, the tip moves at a constant height and the topography of the surface is revealed by changes in the tunnelling current. In the more common <i>constantcurrent mode</i> the height of the tip is adjusted throughout the scanning process to maintain a constant current and the tiny movements of the tip are recorded. In either mode the structure of the sample's surface can be mapped on an atomic scale, though neither mode involves imaging of the kind that takes place in a conventional optical or transmission electron microscope.</p><p>STMs have now become a major tool in the developing field of nanotechnology. This is partly because of the images they supply, but even more because of their ability to manipulate individual atoms and position them with great accuracy. One of the products of this kind of nanoscale manipulation is shown in <a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.4#fig007023">Figure 26</a>, the famous ‘quantum corral’ formed by positioning iron atoms on a copper surface.</p><div class="oucontentfigure oucontentmediamini" id="fig007_023"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/f2029f51/sm358_1_023i.jpg" alt="Figure 26 Iron atoms on a copper surface forming a ring called a ‘quantum corral’. Standing waves of surface electrons trapped inside the corral are also visible" width="286" height="193" style="maxwidth:286px;" class="oucontentfigureimage" longdesc="view.php?id=2680&extra=longdesc_idp8633360"/><div class="oucontentfiguretext"><div class="oucontentsourcereference">Courtesy of Don Eigler, IBM Research Division
<a class="oucontentrightslink" title="Show rights info">©</a><div class="oucontentrightsinfo">Courtesy of Don Eigler, IBM Research Division</div>
</div><div class="oucontentcaption oucontentnonumber"><span class="oucontentfigurecaption">
Figure 26 Iron atoms on a copper surface forming a ring called a ‘quantum corral’. Standing waves of surface electrons trapped inside the corral are also visible</span></div></div><div class="oucontentlongdesclink oucontentlongdesconly"><a href="https://www.open.edu/openlearn/ocw/mod/oucontent/view.php?id=2680&extra=longdesc_idp8633360&clicked=1">Long description</a></div><a id="back_longdesc_idp8633360"></a></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

Conclusion
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection6
Wed, 13 Apr 2016 23:00:00 GMT
<p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1">Session 1</a></b></p><p>Scattering is a process in which incident particles interact with a target and are changed in nature, number, speed or direction of motion as a result. Tunnelling is a quantum phenomenon in which particles that are incident on a classically impenetrable barrier are able to pass through the barrier and emerge on the far side of it.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.1">Session 2</a></b></p><p>In one dimension, wave packets scattered by finite square barriers or wells generally split into transmitted and reflected parts, indicating that there are nonzero probabilities of both reflection and transmission. These probabilities are represented by the reflection and transmission coefficients <i>R</i> and <i>T</i>. The values of <i>R</i> and <i>T</i> generally depend on the nature of the target and the properties of the incident particles. If there is no absorption, creation or destruction of particles, <i>R</i> + <i>T</i> = 1.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.1">Session 3</a></b></p><p>Unnormalisable stationarystate solutions of Schrödinger's equation can be interpreted in terms of steady beams of particles. A term such as <i>A</i>e<sup>i(<i>kx − ωt</i>)</sup> can be associated with a beam of linear number density <i>n</i> = <i>A</i><sup>2</sup> travelling with speed <i>v</i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span>k/<i>m</i> in the direction of increasing <i>x</i>. Such a beam has intensity <i>j</i> = <i>nv</i>. In this approach, <i>T</i> = <i>j</i><sub>trans</sub>/<i>j</i><sub>inc</sub> and <i>R</i> = <i>j</i><sub>ref</sub>/<i>j</i><sub>inc</sub>.</p><p>For particles of energy <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>, incident on a finite square step of height <i>V</i><sub>0</sub>, the transmission coefficient is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_024"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ed787a6c/sm358_1_ue024i.gif" alt=""/></div><p>where</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_025"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8afed14d/sm358_1_ue025i.gif" alt=""/></div><p>are the wave numbers of the incident and transmitted beams. For a finite square well or barrier of width <i>L</i>, the transmission coefficient can be expressed as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_026"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a0498b41/sm358_1_ue026i.gif" alt=""/></div><p>where
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/02f25a15/sm358_1_ie012i.gif" alt="" width="140" height="23" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span>, with the plus signs being used for a well and the minus signs for a barrier. Transmission resonances, at which <i>T</i> = 1 and the transmission is certain, occur when <i>k</i><sub>2</sub><i>L</i> = <i>N</i><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span> where <i>N</i> is an integer.</p><p>Travelling wave packets and steady beams of particles can both be thought of as representing flows of probability. In one dimension such a flow is described by the probability current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_027"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/22e4e4a0/sm358_1_ue027i.gif" alt=""/></div><p>In three dimensions, scattering is described by the total crosssection, <i>σ</i>, which is the rate at which scattered particles emerge from the target per unit time per unit incident flux. For any chosen direction, the differential crosssection tells us the rate of scattering into a small cone of angles around that direction. At very high energies, total crosssections are dominated by inelastic effects due to the creation of new particles.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.1">Session 4</a></b></p><p>Wave packets with a narrow range of energies centred on <i>E</i><sub>0</sub> can tunnel though a finite square barrier of height <i>V</i><sub>0</sub> > <i>E</i><sub>0</sub>. In a stationarystate approach, solutions of the timeindependent Schrödinger equation in the classically forbidden region contain exponentially growing and decaying terms of the form <i>C</i>e<sup>−α<i>x</i></sup> and <i>D</i>e<sup>α<i>x</i></sup>, where
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/80cb5fa2/sm358_1_ie013i.gif" alt="" width="133" height="23" style="maxwidth:133px;" class="oucontentinlinefigureimage"/></span> is the attenuation coefficient. The transmission coefficient for tunnelling through a finite square barrier of width <i>L</i> and height <i>V</i><sub>0</sub> is approximately</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_028"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7822d5a5/sm358_1_ue028i.gif" alt=""/></div><p>Such a transmission probability is small and decreases rapidly as the barrier width <i>L</i> increases.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.1">Session 5</a></b></p><p>Square barriers and wells are poor representations of the potential energy functions found in Nature. However, if the potential <i>V</i>(<i>x</i>) varies smoothly as a function of <i>x</i> the transmission coefficient for tunnelling of energy <i>E</i><sub>0</sub>can be roughly represented by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_029"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/1849fbc7/sm358_1_ue029i.gif" alt=""/></div><p>This approximation can be used to provide a successful theory of nuclear alpha decay as a tunnelling phenomenon. It can also account for the occurrence of nuclear fusion in stellar cores, despite the relatively low temperatures there. In addition, it explains the operation of the scanning tunnelling microscope which can map surfaces on the atomic scale.</p><p><b>Glossary and Physics Toolkit</b></p><p>Attached below are PDFs of the original glossary and the Physics Toolkit, you may find it useful to refer to these documents as you work through the course.</p><p>Click to view <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=27243">glossary</a></span>. (16 pages, 0.4 MB)</p><p>Click to view <a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=27244">toolkit</a>. (17 pages, 0.4 MB)</p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection6
ConclusionSM358_1<p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection1">Session 1</a></b></p><p>Scattering is a process in which incident particles interact with a target and are changed in nature, number, speed or direction of motion as a result. Tunnelling is a quantum phenomenon in which particles that are incident on a classically impenetrable barrier are able to pass through the barrier and emerge on the far side of it.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection2.1">Session 2</a></b></p><p>In one dimension, wave packets scattered by finite square barriers or wells generally split into transmitted and reflected parts, indicating that there are nonzero probabilities of both reflection and transmission. These probabilities are represented by the reflection and transmission coefficients <i>R</i> and <i>T</i>. The values of <i>R</i> and <i>T</i> generally depend on the nature of the target and the properties of the incident particles. If there is no absorption, creation or destruction of particles, <i>R</i> + <i>T</i> = 1.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection3.1">Session 3</a></b></p><p>Unnormalisable stationarystate solutions of Schrödinger's equation can be interpreted in terms of steady beams of particles. A term such as <i>A</i>e<sup>i(<i>kx − ωt</i>)</sup> can be associated with a beam of linear number density <i>n</i> = <i>A</i><sup>2</sup> travelling with speed <i>v</i> = <span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/b18bbd39/hbar.gif" alt="" width="6" height="12" style="maxwidth:6px;" class="oucontentinlinefigureimage"/></span>k/<i>m</i> in the direction of increasing <i>x</i>. Such a beam has intensity <i>j</i> = <i>nv</i>. In this approach, <i>T</i> = <i>j</i><sub>trans</sub>/<i>j</i><sub>inc</sub> and <i>R</i> = <i>j</i><sub>ref</sub>/<i>j</i><sub>inc</sub>.</p><p>For particles of energy <i>E</i><sub>0</sub> > <i>V</i><sub>0</sub>, incident on a finite square step of height <i>V</i><sub>0</sub>, the transmission coefficient is</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_024"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/ed787a6c/sm358_1_ue024i.gif" alt=""/></div><p>where</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_025"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/8afed14d/sm358_1_ue025i.gif" alt=""/></div><p>are the wave numbers of the incident and transmitted beams. For a finite square well or barrier of width <i>L</i>, the transmission coefficient can be expressed as</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_026"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/a0498b41/sm358_1_ue026i.gif" alt=""/></div><p>where
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/02f25a15/sm358_1_ie012i.gif" alt="" width="140" height="23" style="maxwidth:140px;" class="oucontentinlinefigureimage"/></span>, with the plus signs being used for a well and the minus signs for a barrier. Transmission resonances, at which <i>T</i> = 1 and the transmission is certain, occur when <i>k</i><sub>2</sub><i>L</i> = <i>N</i><span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/3200b4b1/pi.gif" alt="" width="7" height="7" style="maxwidth:7px;" class="oucontentinlinefigureimage"/></span> where <i>N</i> is an integer.</p><p>Travelling wave packets and steady beams of particles can both be thought of as representing flows of probability. In one dimension such a flow is described by the probability current</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_027"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/22e4e4a0/sm358_1_ue027i.gif" alt=""/></div><p>In three dimensions, scattering is described by the total crosssection, <i>σ</i>, which is the rate at which scattered particles emerge from the target per unit time per unit incident flux. For any chosen direction, the differential crosssection tells us the rate of scattering into a small cone of angles around that direction. At very high energies, total crosssections are dominated by inelastic effects due to the creation of new particles.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection4.1">Session 4</a></b></p><p>Wave packets with a narrow range of energies centred on <i>E</i><sub>0</sub> can tunnel though a finite square barrier of height <i>V</i><sub>0</sub> > <i>E</i><sub>0</sub>. In a stationarystate approach, solutions of the timeindependent Schrödinger equation in the classically forbidden region contain exponentially growing and decaying terms of the form <i>C</i>e<sup>−α<i>x</i></sup> and <i>D</i>e<sup>α<i>x</i></sup>, where
<span class="oucontentinlinefigure"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/80cb5fa2/sm358_1_ie013i.gif" alt="" width="133" height="23" style="maxwidth:133px;" class="oucontentinlinefigureimage"/></span> is the attenuation coefficient. The transmission coefficient for tunnelling through a finite square barrier of width <i>L</i> and height <i>V</i><sub>0</sub> is approximately</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_028"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/7822d5a5/sm358_1_ue028i.gif" alt=""/></div><p>Such a transmission probability is small and decreases rapidly as the barrier width <i>L</i> increases.</p><p><b><a class="oucontentcrossref" href="https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection5.1">Session 5</a></b></p><p>Square barriers and wells are poor representations of the potential energy functions found in Nature. However, if the potential <i>V</i>(<i>x</i>) varies smoothly as a function of <i>x</i> the transmission coefficient for tunnelling of energy <i>E</i><sub>0</sub>can be roughly represented by</p><div class="oucontentequation oucontentequationequation oucontentnocaption" id="ueqn001_029"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/a189af32/1849fbc7/sm358_1_ue029i.gif" alt=""/></div><p>This approximation can be used to provide a successful theory of nuclear alpha decay as a tunnelling phenomenon. It can also account for the occurrence of nuclear fusion in stellar cores, despite the relatively low temperatures there. In addition, it explains the operation of the scanning tunnelling microscope which can map surfaces on the atomic scale.</p><p><b>Glossary and Physics Toolkit</b></p><p>Attached below are PDFs of the original glossary and the Physics Toolkit, you may find it useful to refer to these documents as you work through the course.</p><p>Click to view <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=27243">glossary</a></span>. (16 pages, 0.4 MB)</p><p>Click to view <a class="oucontenthyperlink" href="https://www.open.edu/openlearn/ocw/mod/resource/view.php?id=27244">toolkit</a>. (17 pages, 0.4 MB)</p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

Keep on learning
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection7
Wed, 13 Apr 2016 23:00:00 GMT
<div class="oucontentfigure oucontentmediamini"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/8ff4c822/d3c986e6/ol_skeleton_keeponlearning_image.jpg" alt="" width="300" height="200" style="maxwidth:300px;" class="oucontentfigureimage"/></div><p> </p><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Study another free course</h2><p>There are more than <b>800 courses on OpenLearn</b> for you to choose from on a range of subjects. </p><p>Find out more about all our <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">free courses</a></span>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Take your studies further</h2><p>Find out more about studying with The Open University by <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">visiting our online prospectus</a>. </p><p>If you are new to university study, you may be interested in our <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Access Courses</a> or <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Certificates</a>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">What’s new from OpenLearn?</h2><p>
<a class="oucontenthyperlink" href="http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Sign up to our newsletter</a> or view a sample.</p><p> </p></div><div class="oucontentbox oucontentshollowbox2 oucontentsbox oucontentsnoheading "><div class="oucontentouterbox"><div class="oucontentinnerbox"><p>For reference, full URLs to pages listed above:</p><p>OpenLearn – <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a>
</p><p>Visiting our online prospectus – <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses</a>
</p><p>Access Courses – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>doit/<span class="oucontenthidespace"> </span>access</a>
</p><p>Certificates – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>certificateshe</a>
</p><p>Newsletter ­– <a class="oucontenthyperlink" href=" http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>aboutopenlearn/<span class="oucontenthidespace"> </span>subscribetheopenlearnnewsletter</a>
</p></div></div></div>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsection7
Keep on learningSM358_1<div class="oucontentfigure oucontentmediamini"><img src="https://www.open.edu/openlearn/ocw/pluginfile.php/69482/mod_oucontent/oucontent/513/8ff4c822/d3c986e6/ol_skeleton_keeponlearning_image.jpg" alt="" width="300" height="200" style="maxwidth:300px;" class="oucontentfigureimage"/></div><p> </p><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Study another free course</h2><p>There are more than <b>800 courses on OpenLearn</b> for you to choose from on a range of subjects. </p><p>Find out more about all our <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">free courses</a></span>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">Take your studies further</h2><p>Find out more about studying with The Open University by <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">visiting our online prospectus</a>. </p><p>If you are new to university study, you may be interested in our <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Access Courses</a> or <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Certificates</a>.</p><p> </p></div><div class="oucontentinternalsection"><h2 class="oucontenth2 oucontentinternalsectionhead">What’s new from OpenLearn?</h2><p>
<a class="oucontenthyperlink" href="http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">Sign up to our newsletter</a> or view a sample.</p><p> </p></div><div class="oucontentbox oucontentshollowbox2 oucontentsbox
oucontentsnoheading
"><div class="oucontentouterbox"><div class="oucontentinnerbox"><p>For reference, full URLs to pages listed above:</p><p>OpenLearn – <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a>
</p><p>Visiting our online prospectus – <a class="oucontenthyperlink" href="http://www.open.ac.uk/courses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses</a>
</p><p>Access Courses – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/doit/access?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>doit/<span class="oucontenthidespace"> </span>access</a>
</p><p>Certificates – <a class="oucontenthyperlink" href=" http://www.open.ac.uk/courses/certificateshe?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.ac.uk/<span class="oucontenthidespace"> </span>courses/<span class="oucontenthidespace"> </span>certificateshe</a>
</p><p>Newsletter – <a class="oucontenthyperlink" href=" http://www.open.edu/openlearn/aboutopenlearn/subscribetheopenlearnnewsletter?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>aboutopenlearn/<span class="oucontenthidespace"> </span>subscribetheopenlearnnewsletter</a>
</p></div></div></div>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University

Acknowledgements
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsectionacknowledgements
Wed, 13 Apr 2016 23:00:00 GMT
<p>The material acknowledged below is Proprietary (see <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.ac.uk/conditions">terms and conditions</a></span>) and used under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/4.0/">licence</a> (not subject to Creative Commons licence).</p><p>The content is from SM358_1 Book 1 <i>Wave Mechanics</i> – Chapter 7 Scattering and Tunnelling, pages 178–209.</p><p>Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:</p><p>Course image: <a class="oucontenthyperlink" href="https://www.flickr.com/photos/infomastern/">Susanne Nilsson</a> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/bysa/2.0/legalcode">Creative Commons AttributionShareAlike 2.0 Licence</a>.</p><p>Figure 2 Jon Arnold Images/Alamy;</p><p>Figure 3 Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, <i>Astrophysical Journal</i> (submitted) © 2003 The American Astronomical Society;</p><p>Figure 25 Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge;</p><p>Figure 26 Courtesy of Don Eigler, IBM Research Division.</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>
https://www.open.edu/openlearn/sciencemathstechnology/science/physicsandastronomy/scatteringandtunnelling/contentsectionacknowledgements
AcknowledgementsSM358_1<p>The material acknowledged below is Proprietary (see <span class="oucontentlinkwithtip"><a class="oucontenthyperlink" href="http://www.open.ac.uk/conditions">terms and conditions</a></span>) and used under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/byncsa/4.0/">licence</a> (not subject to Creative Commons licence).</p><p>The content is from SM358_1 Book 1 <i>Wave Mechanics</i> – Chapter 7 Scattering and Tunnelling, pages 178–209.</p><p>Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:</p><p>Course image: <a class="oucontenthyperlink" href="https://www.flickr.com/photos/infomastern/">Susanne Nilsson</a> in Flickr made available under <a class="oucontenthyperlink" href="https://creativecommons.org/licenses/bysa/2.0/legalcode">Creative Commons AttributionShareAlike 2.0 Licence</a>.</p><p>Figure 2 Jon Arnold Images/Alamy;</p><p>Figure 3 Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, <i>Astrophysical Journal</i> (submitted) © 2003 The American Astronomical Society;</p><p>Figure 25 Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge;</p><p>Figure 26 Courtesy of Don Eigler, IBM Research Division.</p><p><b>Don't miss out:</b></p><p>If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University  <a class="oucontenthyperlink" href="http://www.open.edu/openlearn/freecourses?utm_source=openlearn&utm_campaign=ol&utm_medium=ebook">www.open.edu/<span class="oucontenthidespace"> </span>openlearn/<span class="oucontenthidespace"> </span>freecourses</a></p>The Open UniversityThe Open UniversityCoursetext/htmlenGBScattering and tunnelling  SM358_1Copyright © 2016 The Open University