- SM358_1The quantum world
Scattering and tunnelling
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978-1-4730-1445-9 (.kdl)IntroductionIn this course we shall consider two physical phenomena of fundamental importance: *scattering* and *tunnelling*. Each will be treated using both a stationary-state approach *and* a wave-packet approach.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 *stationary-state* approach can be used. In other cases, where probabilities are time-dependent and motion is really taking place, a *wave-packet* 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.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 *Modelling with Fourier series* and M337_1 *Introduction to complex numbers*.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 Summary.This OpenLearn course is an adapted extract from the Open University course : SM358 *The quantum world*.After studying this course, you should be able to:explain the meanings of key terms and use them appropriatelydescribe the behaviour of wave packets when they encounter potential energy steps, barriers and wellsdescribe how stationary-state solutions of the Schrödinger equation can be used to analyse scattering and tunnelling for a range of simple potential energy functions obtain the solution of the time-independent Schrödinger equation and use continuity boundary conditions to find reflection and transmission coefficientspresent information about solutions of the time-independent Schrödinger equation in graphical terms.1 What are scattering and tunnelling?The phenomenon of **scattering** 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 (Figure 1). 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.It can be argued that almost everything we know about the world is learnt as a result of scattering. When we look at a non-luminous 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 (Figure 2). 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.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 *surface of last scattering* observed all around us at microwave wavelengths (Figure 3).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 *quantum field theory*, a part of quantum physics beyond the scope of this course. Rather, we shall mainly restrict ourselves to one-dimensional problems in which an incident beam or particle is either *transmitted* (allowed to pass) or *reflected* (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 Figure 4. Despite these restrictions, our discussion of quantum-mechanical scattering will contain many surprises. For instance, you will see that a finite potential energy barrier of height *V*_{0} can reflect a particle of energy *E*_{0}, even when *E*_{0} > *V*_{0}. Perhaps even more amazingly, you will see that unbound particles can be reflected when they encounter a finite well.The phenomenon of **tunnelling** is entirely quantum-mechanical with no analogue in classical physics. It is an extension of the phenomenon of *barrier penetration*, which may be familiar in the context of particles bound in potential energy wells. Barrier penetration involves the appearance of particles in *classically forbidden regions*. In cases of tunnelling, such as that shown in Figure 5, a particle with energy *E*_{0} < *V*_{0} can penetrate a potential energy barrier of height *V*_{0}, 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.Tunnelling phenomena are common in many areas of physics. In this course you will see how tunnelling provides an explanation of the *alpha decay* of radioactive nuclei, and is also an essential part of the *nuclear fusion* processes by which stars produce light. Finally you will see how quantum tunnelling has allowed the development of instruments called *scanning tunnelling microscopes* (STMs) that permit the positions of individual atoms on a surface to be mapped in stunning detail.2 Scattering: a wave-packet approach2.1 OverviewSession 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 formThis is a superposition of de Broglie waves, with the function corresponding to momentum *p*_{x} = *k* and energy *E*_{k} = ^{2}k^{2}/2*m*, where *k* = 2/λ is the wave number. The momentum amplitude function *A*(*k*) 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.2.2 Wave packets and scattering in one dimension
Figure 6 shows the scattering of a wave packet, incident from the left, on a target represented by a potential energy function of the formPotential energy functions of this type are called **finite square barriers**. They are simple idealisations of the more general kind of finite barrier shown in Figure 4a. The de Broglie waves that make up the wave packet extend over a range of energy and momentum values. In the case illustrated in Figure 6, the expectation value of the energy, 〈*E*〉, has a value *E*_{0} that is greater than the height of the potential energy barrier *V*_{0}. The classical analogue of the process illustrated in Figure 6 would be a particle of energy *E*_{0} 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 *E*_{0} is greater than *V*_{0}, 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.The quantum analysis tells a different story. Based on numerical solutions of Schrödinger's equation, the computer-generated results in Figure 6 show successive snapshots of ||^{2}, 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 *x*-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 ||^{2}. 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.It's important to realise that the splitting of the wave packet illustrated in Figure 6 does *not* 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 Figure 6 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.Exercise 1Simply judging by eye, what are the respective probabilities of reflection and transmission as the final outcome of the scattering process shown in Figure 6?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.The probability that a given incident particle is reflected is called the **reflection coefficient**, *R*, while the probability that it is transmitted is called the **transmission coefficient**, *T*. Since one or other of these outcomes must occur, we always haveThe values of *R* and *T* that apply in any particular case of one-dimensional 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 *R* and *T* are found by measuring the areas under the graph of ||^{2} that are located to the left and to the right of the target at that time.Reflection and transmission coefficients can also describe scattering from an attractive target. The idealised case of a wave packet with 〈*E*〉 = *E*_{0} > 0 encountering a finite square well of depth *V*_{0} and width *L* is shown in Figure 7. 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.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 *R* or *T* 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 *collapse of the wave function* – 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.3 Scattering: a stationary-state approach3.1 OverviewScattering 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.Session 3 introduces the stationary-state approach to scattering. The discussion is mainly confined to one dimension, so a stationary-state solution to the Schrödinger equation can be written in the formwhere (*x*) satisfies the appropriate time-independent Schrödinger equation at energy *E*. The first challenge is to find a way of interpreting stationary-state solutions that makes them relevant to an inherently time-dependent phenomenon like scattering.3.2 Stationary states and scattering in one dimensionThe key idea of the stationary-state 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 *E*_{0}. 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 set-up taken at one time would be identical to a similar snapshot taken at another time. In contrast to the wave-packet approach, there are no moving ‘blobs of probability density’, so the whole process can be described in terms of stationary states.For a one-dimensional beam, we define the **intensity** *j* to be the number of beam particles that pass a given point per unit time. We also define the **linear number density** *n* 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 *v*, the beam intensity is given by *j* = *vn*. Specialising this relationship to the incident, reflected and transmitted beams, we haveIn the stationary-state approach, the reflection and transmission coefficients can be expressed in terms of beam intensity ratios, as follows:If all the incident particles are scattered, and no particles are created or destroyed, it must be the case that *j*_{inc} = *j*_{ref} + *j*_{trans}. Dividing both sides by *j*_{inc} and rearranging gives *R* + *T* = 1, as expected from our earlier discussions.We now need to relate these steady beam intensities to stationary-state 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 one-dimensional scattering target.3.3 Scattering from a finite square stepThe kind of one-dimensional scattering target we shall be concerned with in this section is called a **finite square step**. It can be represented by the potential energy functionThe finite square step (Figure 8) 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 *x* = 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 *two* regions of the *x*-axis: Region 1 where *x* ≤ 0, and Region 2 where *x* > 0.Classically, when a finite square step of height *V*_{0} scatters a rightward moving beam in which each particle has energy *E*_{0} > *V*_{0}, each of the particles will continue moving to the right but will be suddenly slowed as it passes the point *x* = 0. The transmitted particles are slowed because, in the region *x* > 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.Exercise 2In general terms, how would you expect the outcome of the quantum scattering process to differ from the classical outcome?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 *E*_{0} > *V*_{0}.We start our analysis by writing down the relevant Schrödinger equation:where *V*(*x*) is the finite square step potential energy function given in Equations 7.6 and 7.7. We seek stationary-state solutions of the form
, where *E*_{0} is the fixed energy of each beam particle. The task of solving Equation 7.8 then reduces to that of solving the time-independent Schrödinger equationsA simple rearrangement givesand it is easy to see that these equations have the general solutionswhere *A, B, C* and *D* are arbitrary complex constants, and the wave numbers in Region 1 and Region 2 are respectivelyYou may wonder why we have expressed these solutions in terms of complex exponentials rather than sines and cosines (recall the identity *e*^{ix} = cos *x* + i sin *x*). The reason is that the individual terms in Equations 7.11 and 7.12 have simple interpretations in terms of the incident, reflected and transmitted beams. To see how this works, it is helpful to note thatwhere
is the momentum operator in the *x* direction.It therefore follows that terms proportional to e^{ikx} are associated with particles moving rightward at speed , while terms proportional to e^{−ikx} are associated with particles moving leftward at speed .These directions of motion can be confirmed by writing down the corresponding stationary-state solutions, which take the formwhere *ω* = *E*_{0}/. We can then identify terms of the form e^{i(kx−ωt)} as plane waves travelling in the positive *x*-direction, while terms of the form e^{−i(kx+ωt)} are plane waves travelling in the negative *x*-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.In most applications of wave mechanics, the wave function (*x, t*) describes the state of a single particle, and |(*x, t*)|^{2} represents the probability density for that particle. In the steady-state approach to scattering, however, it is assumed that the wave function (*x, t*) describes steady beams of particles, with |(*x, t*)|^{2} 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.Looking at Equation 7.14, and recalling that the first term *A*e^{i(k1x−ωt)} represents a wave travelling in the positive *x*-direction for *x* ≤ 0, we identify this term as representing the incident wave in Region 1 (*x* ≤ 0). We can say that each particle in the beam travels to the right with speed , and that the linear number density of particles in the beam is(You will find further justification of this interpretation in Section 3.4.)Similarly, the second term on the right of Equation 7.14 can be interpreted as representing the reflected beam in Region 1 (*x* ≤ 0). This beam travels to the left with speed *v*_{ref} = *k*_{1}/*m* and has linear number density *n*_{ref} = |*B*|^{2}.The first term on the right of Equation 7.15 represents the transmitted beam in Region 2 (*x* > 0). This beam travels to the right with speed *v*_{trans} = *k*_{2}/*m* and has linear number density *n*_{trans} = |*C*|^{2}. The second term on the right of Equation 7.15 would represent a leftward moving beam in the region *x* > 0. On physical grounds, we do not expect there to be any such beam, so we ensure its absence by setting *D* = 0 in our equations.Using these interpretations, we see that the beam intensities are:Expressions for the reflection and transmission coefficients then follow from Equation 7.5:It is worth noting that the expression for the transmission coefficient includes the wave numbers *k*_{1} and *k*_{2}, 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.To calculate *R* and *T*, we need to find the ratios *B/A* and *C/A*. To achieve this, we must eliminate unwanted arbitrary constants from our solutions to the time-independent Schrödinger equation. This can be done by requiring that the solutions satisfy continuity boundary conditions:
*(x)* is continuous everywhere.
d*(x)*/d*x* is continuous where the potential energy function is finite.
The first of these conditions tells us that our two expressions for *(x)* must match at their common boundary *x* = 0. From Equations 7.11 and 7.12, we therefore obtainTaking the derivatives of Equations 7.11 and 7.12,so requiring the continuity of d/d*x* at *x* = 0 implies thatAfter some manipulation, Equations 7.19 and 7.20 allow us to express *B* and *C* in terms of *A*
Combining these expressions with Equations 7.17 and 7.18, we finally obtainSince
and
, where *E*_{0} is the incident particle energy and *V*_{0} is the height of the step, we have now managed to express *R* and *T* entirely in terms of given quantities. The transmission coefficient, *T*, is plotted against *E*_{0}/*V*_{0} in Figure 9.The above results have been derived by considering a rightward moving beam incident on an upward step of the kind shown in Figure 8. However, almost identical calculations can be carried out for leftward moving beams or downward steps. Equations 7.21 and 7.22 continue to apply in all these cases, provided we take *k*_{1} to be the wave number of the incident beam and *k*_{2} to be the wave number of the transmitted beam.The formulae for *R* and *T* are symmetrical with respect to an interchange of *k*_{1} and *k*_{2}, so a beam of given energy, incident on a step of given magnitude, is reflected to the same extent *no matter whether the step is upwards or downwards*. 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.Another surprising feature of Equation 7.21 is that *R* is independent of *m* and so does not vanish as the particle mass *m* becomes very large. However, we know from experience that macroscopic objects are *not* 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?This puzzle can be resolved by noting that our calculation assumes an *abrupt* step. Detailed quantum-mechanical calculations show that Equation 7.21 provides a good approximation to reflections from a diffuse step *provided that* the wavelength of the incident particles is much longer than the distance over which the potential energy function varies. For example, Equation 7.21 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.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 *R* and *T*, and we have done so using relatively simple stationary-state methods based on the time-independent 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 one-dimensional scattering problems.Exercise 3(a) Use Equations 7.21 and 7.22 to show that *R* + *T* = 1.(b) Evaluate *R* and *T* in the case that *E*_{0} = 2*V*_{0}, and confirm that their sum is equal to 1 in this case.(a) From Equations 7.21 and 7.
22,
(b) From Equation 7.13, we have
So, in this case,
. Therefore
So we can check that *R* + *T* = 1 in this case.Exercise 4Consider the case where *k*_{2} = *k*_{1}/2.(a) Express *B* and *C* in terms of *A*.(b) Show that in the region *x* > 0, we have ||^{2} = 16|A|^{2}/9 = constant, while in the region *x* ≤ 0, we have
.(c) What general physical phenomenon is responsible for the spatial variation of ||^{2} to the left of the step?(d) If the linear number density in the incident beam is 1.00 × 10^{24} m^{−1}, what are the linear number densities in the reflected and transmitted beams?(a) When *k*_{2} = *k*_{1}/2, we have
(b) From Equation 7.15 with *D* = 0,
Similarly, from Equation 7.14,
Since
, we have
Multiplying out the brackets, we find
(c) The variation indicated by the cosine-dependence to the left of the step is a result of *interference* 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 stationary-state approach interference is a permanent feature.(d) The linear number densities in the incident, reflected and transmitted beams are given by |*A*|^{2}, |*B*|^{2} and |*C*|^{2}. The question tells us that |*A*|^{2} = 1.00 × 10^{24} m^{−1}, so the linear number density in the reflected and transmitted beams are
Note that the transmitted beam is denser than the incident beam: |*C*|^{2} > |*A*|^{2}. However, since *k*_{2} = *k*_{1}/2, we have *j*_{trans} < *j*_{inc}. The transmitted beam is less intense than the incident beam because it travels much more slowly.Exercise 5Based on the solution to Exercise 4, sketch a graph of ||^{2} that indicates its behaviour both to the left and to the right of a finite square step.A suitable graph is shown in Figure 10.3.4 Probability currentsThe 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 quantum-mechanical 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.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.Let us first consider the one-dimensional flow of a fluid along the *x*-axis. At each point, we define a fluid current *j*_{x}(*x, t*) that represents the rate of flow of fluid particles along the *x*-axis. If the fluid is compressible, like air, this fluid current may vary in space and time.
Figure 11, above, shows a small one-dimensional region between *x* and *x* + *δ**x*. The number of particles in this region can be written as *n*(*x, t*) *δ**x*, where *n*(*x, t*) is the linear number density of particles. The *change* in the number of particles in the region during a small time interval *δ**t* is thenwhere, for flow in the positive *x*-direction, the first term on the right-hand side represents the number of particles *entering* the region from the left and the second term represents the number of particles *leaving* the region to the right. Rearranging Equation 7.23 givesand, on taking the limit as *δ**x* and *δ**t* tend to zero we see thatThis result is called the **equation of continuity** in one dimension. With a little care, it can be extended to quantum mechanics.For a single-particle wave packet in quantum mechanics, the flowing quantity is *probability density*. This is evident from images of wave packets as ‘blobs’ of moving probability density (e.g. Figure 6). Now we know that probability density is represented in quantum mechanics by * , so we should be able to construct the appropriate equation of continuity by examining the time derivative of this quantity.Obviously, we havewhere Schrödinger's equation dictates thatDividing through by , the rate of change of the wave function isSubstituting this equation, and its complex conjugate, into Equation 7.25, we then obtain(since the potential energy function *V*(*x*) is real and cancels out), and a further manipulation givesThis equation can be written in the form of an equation of continuity:provided that we interpret *n*(*x, t*) as the probability density * , with a corresponding currentIn the one-dimensional situations we are considering, *j*_{x}(*x, t*) is called the **probability current**. In one dimension, the probability density *n* = * is a probability per unit length, and therefore has the dimensions of [L]^{−1}. It follows from Equation 7.27 that the probability current has dimensions ofand therefore has SI unit ‘per second’, as expected for a current of particles.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, * represents the linear number density of particles and the probability current is the rate of flow of particles in the positive *x*-direction. For a steady beam, described by a stationary-state wave function,
, the time-dependent phase factors cancel out in Equation 7.28, and the probability current can be written more simply aswhich is independent of time. It is important to realise that *j*_{x} is a *signed* quantity; it is positive for a beam travelling in the positive *x*-direction, and negative for a beam travelling in the negative *x*-direction. This is unlike the beam intensity *j* introduced earlier, which is always positive. It is natural to define the beam intensity of a steady beam to be the *magnitude* of the probability current: *j* = |*j*_{x}|. It is this definition that gives us a way of calculating beam intensities without making unwarranted classical assumptions.In fact, each beam intensity calculated using Equation 7.29 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 Exercise 7 below. So our analysis adds rigour, but contains no surprises. However, the really significant feature of Equation 7.28 is its generality; it applies to single-particle 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.Exercise 6Show that *j*_{x}(*x, t*) as defined in Equation 7.28 is a real quantity.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 *j*_{x}, we obtainSince *j**_{x} = *j*_{x}, we conclude that *j*_{x} is real.Exercise 7Using the solutions to the Schrödinger equation that were obtained in the stationary-state approach to scattering from a finite square step, evaluate the probability current in the regions *x* > 0 and *x* ≤ 0. Interpret your results in terms of the beam intensities in these two regions.In the region *x* > 0,so Equation 7.29 gives the probability currentIn the region *x* ≤ 0,andso Equation 7.29 gives the probability currentSimplifying this expression, we obtainThis can be interpreted as the sum of an incident probability current,
, and a reflected probability current,
. 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).3.5 Scattering from finite square wells and barriersThe 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:
Divide the *x*-axis into the minimum possible number of regions of constant potential energy.
Write down the general solution of the relevant time-independent Schrödinger equation in each of these regions, remembering to use the appropriate value of the wave number *k* in each region and introducing arbitrary constants as necessary.
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.
Obtain expressions for all the beam intensities relative to the intensity of the incident beam.
Determine the reflection and transmission coefficients from ratios of beam intensities.
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 Figure 12.Worked Example 1A particle of mass *m* with positive energy *E*_{0} is scattered by a one-dimensional finite square well of depth *V*_{0} and width *L* (shown in Figure 12). Derive an expression for the probability that the particle will be transmitted across the well.SolutionSuppose the particle concerned to be part of an intense steady beam of identical particles, each having energy *E*_{0} and each incident from the left on a well located between *x* = 0 and *x* = *L*.Divide the x-axis into three regions: Region 1, (*x* < 0), where *V* = 0; Region 2, (0 ≤ *x* ≤ *L*), where *V* = −*V*_{0}; and Region 3, (*x* > *L*), where *V* = 0. In each region the time-independent Schrödinger equation takes the general formso the solution in each region is:where *A, B, C, D, F* and *G* are arbitrary constants, and the wave numbers in Region 1 and Region 2 areNote 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.There is no leftward moving beam in Region 3, so we set *G* = 0. Since the potential energy function is finite everywhere, (*x*) and d/d*x* must be continuous everywhere. Continuity of (*x*) and d/d*x* at *x* = 0 implies thatwhile continuity of (*x*) and d/d*x* at *x* = *L* givesSince the wave numbers in Regions 1 and 3 are both equal to *k*_{1}, the intensities of the incident and transmitted beams areand the required transmission coefficient is given by *T* = |*F*|^{2}/|*A*|^{2}.The mathematical task is now to eliminate *B, C* and *D* from Equations 7.35 to 7.38 in order to find the ratio *F/A*. To achieve this, we note that the constant *B* only appears in the first two equations, so we take the opportunity of eliminating it immediately. Multiplying Equation 7.35 by i*k*_{1} and adding the result to Equation 7.36 we obtainNow we must eliminate *C* and *D* from the remaining equations. To eliminate *D*, we multiply Equation 7.37 by i*k*_{2} and add the result to Equation 7.38. This givesSimilarly, multiplying Equation 7.37 by i*k*_{2} and subtracting the result from Equation 7.38 we see thatFinally, substituting Equations 7.41 and 7.42 into Equation 7.40 and rearranging slightly we obtainso the transmission coefficient is given byThis is the expression we have been seeking; it only involves *k*_{1}, *k*_{2} and *L*, and both *k*_{1} and *k*_{2} can be written in terms of *m*, *E*_{0} and *V*_{0}.Although Equation 7.43 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:Treating *E*_{0} as an independent variable and *V*_{0} as a given constant, this function can be displayed as a graph of *T* against *E*_{0}/*V*_{0}, as shown in Figure 13.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 (*R* =1 − *T* ≠ 0), and for some energies the reflection probability may be quite high. However, Figure 13 also shows that there are some special incident particle energies at which *T* = 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 **transmission resonances**. Equation 7.44 shows that transmission resonances occur when sin(*k*_{2}*L*) = 0, that is whenRecalling the relationship *k* = 2/*λ* between wave number and wavelength, we can also express this condition as *Nλ*_{2} = 2*L*; in other words:A transmission resonance occurs when a whole number of wavelengths occupies the full path length 2*L* of a wave that crosses the width of the well and is reflected back again.

This condition can be interpreted in terms of interference between waves reflected at *x* = 0 and *x* = L. The interference turns out to be destructive because reflection at the *x* = 0 interface is accompanied by a phase change of . 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 anti-reflective coatings on lenses and mirrors.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 (*x*) in the three regions we identified earlier. The value of |*A*|^{2} is equal to the linear number density of particles in the incident beam. If we regard *A* as being known, the values of the constants *B, C, D* and *F* can be evaluated using the continuity boundary conditions and graphs of (*x*) can be drawn.Since (*x*) is generally complex there are several possible graphs that might be of interest, including the real part of , the imaginary part of , and ||^{2}. Figure 14, above, shows a typical plot of the real part of for chosen values of *m*, *E*_{0}, *V*_{0} and *L*; the following points should be noted:
In each region, (*x*) is a periodic function of *x*.
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.
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.
Exercise 8With one modification, the stationary-state method that was applied to scattering from a finite square well can also be applied to scattering from a finite square barrier of height *V*_{0} and width *L*, when *E*_{0} > *V*_{0}. Describe the modification required, draw a figure analogous to Figure 14, but for the case of a square barrier, and comment on any differences between the two graphs.The main difference is that in the case of a finite square barrier *V*_{0} must be replaced by −*V*_{0} throughout the analysis. The wave number in Region 2 isAdapting Equation 7.44, the transmission coefficient is then given byA typical graph of the real part of (*x*) for scattering by a finite square barrier is shown in Figure 15, 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 *T* = 1).Exercise 9Draw two sketches to illustrate the form of the function |(*x, t*)|^{2} in the case of the stationary-state approach to scattering from (a) a finite square well, and (b) a finite square barrier. Comment on the time-dependence of |(*x,t*)|^{2} in each case.The graphs of |(*x, t*)|^{2} plotted against *x* for each case are shown in Figure 16.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 |(*x, t*)|^{2} and the corresponding plots of the real part of (*x, t*). This is because |(*x, t*)|^{2} is partly determined by the imaginary part of (*x,t*). In both cases |(*x, t*)|^{2} is independent of time. This must be the case, despite the time-dependence of (*x,t*), because we are dealing with *stationary states*.3.6 Scattering in three dimensionsSophisticated methods have been developed to analyse scattering in three-dimensions. 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.In three dimensions, we are obliged to think in terms of scattering at a given angle, rather than in terms of one-dimensional 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 **flux**; this is the rate of flow of particles *per unit time per unit area perpendicular to the beam*.If we consider a particular scattering experiment (electron-proton scattering, for example), one of the main quantities of interest is the **total cross-section**, *σ*. This is the total rate at which scattered particles emerge from the target, *per unit time per unit incident flux*. The total cross-section 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 cross-sections vary markedly with the energy of the incident particles. An acceptable SI unit for the measurement of total cross-sections would be m^{2}, but measured cross-sections are generally so small that physicists prefer to use a much smaller unit called the **barn**, defined by the relation 1 barn = 1 × 10^{−28} m^{2}. The name is intended as a joke, 1 barn being such a large cross-section in particle and nuclear physics that it can be considered to be ‘as big as a barn door’. Many cross-sections are measured in millibarn (mb), microbarn (μb) or even nanobarn (nb).Scattering processes that conserve the total kinetic energy of the colliding particles are said to be examples of **elastic scattering**. They may be contrasted with cases of **inelastic scattering** 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 cross-sections often contain both elastic and inelastic contributions.Scattering experiments are often analysed in great detail. The total cross-section arises as a sum of contributions from particles scattered in different directions. For each direction, we can define a quantity called the **differential cross-section**, which tells us the rate of scattering in a small cone of angles around the given direction. The integral of the differential cross-section, taken over all directions, is equal to the total cross-section. We can also vary the energy of the incident beam. Both the total cross-section and the differential cross-section depend on the energies of the incident particles. There is therefore a wealth of experimental information to collect, interpret and explain.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 cross-section at an energy of about 1 eV. The experimental discovery of this **Ramsauer-Townsend effect** 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 three-dimensional analogue of the transmission resonance we met earlier.At the much higher collision energies made available by modern particle accelerators, such as those at the CERN laboratory in Geneva, total cross-sections become dominated by inelastic effects, as new particles are produced. As an example, Figure 17 shows some data concerning the scattering of K^{−} mesons by protons. The upper curve shows the variation of the total cross-section over a very wide range of energies, up to several GeV (1 GeV = 10^{9} 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.4 Tunnelling: wave packets and stationary states4.1 OverviewOne 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 quantum-mechanical tunnelling that was mentioned in Session 1.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 stationary-state methods, similar to those used in our earlier discussion of scattering from wells and barriers.4.2 Wave packets and tunnelling in one dimension
Figure 18 shows a sequence of images captured from a wave packet simulation program. The sequence involves a Gaussian wave packet, with energy expectation value 〈*E*〉 = *E*_{0}, incident from the left on a finite square barrier of height *V*_{0}. The sequence is broadly similar to that shown in Figure 6, which involved a similar wave packet and a similar barrier, but with one important difference; in the earlier process *E*_{0} was greater than *V*_{0}, so transmission was classically allowed, but in the case of Figure 18 *E*_{0} is less than *V*_{0} and transmission is classically forbidden. The bottom image shows that transmission can occur in quantum mechanics.In the case shown in Figure 18, 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 *tunnel through* 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 stationary-state methods to investigate this phenomenon.4.3 Stationary states and barrier penetrationThe example of tunnelling we have just been examining can be regarded as a special case of scattering; it just happens to have *E*_{0} < *V*_{0}. As long as we keep this energy range in mind, we can apply the same stationary-state methods to the study of tunnelling that we used earlier when studying scattering.As before, we shall start by considering the finite square step, whose potential energy function was defined in Equations 7.6 and 7.7. This is shown for the case *E*_{0} < *V*_{0} in Figure 19. The potential energy function divides the *x*-axis into two regions: Region 1 (*x* ≤ 0) which contains the incident and reflected beams, and Region 2 (*x* > 0) which contains what is effectively an infinitely wide barrier. There is no possibility of tunnelling *through* 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.Proceeding as before, we seek stationary-state solutions of the Schrödinger equation of the form
, where (*x*) is a solution of the corresponding time-independent Schrödinger equation. In this case we might try to use exactly the same solution as in Section 3.3 but doing so would complicate the analysis since we would find that in the region *x* > 0 the wave number
would be imaginary. In view of this, it is better to recognise that *E*_{0} < *V*_{0} generally implies a combination of exponentially growing and exponentially decaying terms in the region *x* > 0, and write the solutions aswhere, *A, B, C* and *D* are arbitrary complex constants, while *k*_{1} and α are *real* quantities given byWe require *D* to be zero on physical grounds, to avoid having any part of the solution that grows exponentially as *x* approaches infinity. To determine the values of *B* and *C* relative to that of *A* we impose the usual requirement (for a finite potential energy function) that both (*x*), and its derivative d/d*x*, must be continuous everywhere. Applying these conditions at *x* = 0 we find:from which it follows thatThe reflection coefficient is given by(*Note:* For any complex number, *z*, )So, if particles of energy *E*_{0} < *V*_{0} encounter a finite square step of height *V*_{0}, 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 (*x*) is not zero inside the step (see Figure 20). Rather, it decreases exponentially over a length scale determined by the quantity
, which is usually called the **attenuation coefficient**. This is an example of the phenomenon of *barrier penetration*. 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.Exercise 10Show that the stationary-state probability density |(*x*)|^{2} in Region 1 of Figure 20 is a periodic function of *x* with minima separated by /*k*_{1}.From Equation 7.46
andsoThis is a periodic function that runs through the same range of values each time *x* increases by 2/2*k*_{1}, so its minima are separated by /*k*_{1}.Exercise 11Show that the probability current in Region 2 of Figure 20 is zero.Using Equation 7.47, with *D* = 0 and *α* real, and recalling the definition of probability current given in Equation 7.28, we have4.4 Stationary states and tunnelling in one dimensionWe will now use the stationary-state approach to analyse the tunnelling of particles of energy *E*_{0} through a finite square barrier of width *L* and height *V*_{0} when *E*_{0} < *V*_{0} (see Figure 21).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 time-independent Schrödinger equation:where *A, B, C, D* and *F* are arbitrary constants andNote 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.Requiring the continuity of (*x*) and d/d*x* at *x* = 0 and at *x* = *L* leads to the following four relations:After some lengthy algebra, similar to that in Worked Example 1, these four equations can be reduced to a relationship between *F* and *A*, from which it is possible to obtain the following expression for the transmission coefficient *T* = |*F*/*A*|^{2}.When *αL* ≫ 1, the denominator of this expression can be approximated by
, and the transmission coefficient through the barrier is well-described by the useful relationship(Remember: sinh *x* = (*e*^{x} − *e*^{−x})/2.)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 *αL* ≫ 1, and will decrease rapidly as the barrier width *L* increases.A graph of |^{2}| plotted against *x* for a finite square barrier in the case *E*_{0} < *V*_{0} will look something like Figure 22. Note that because the incident and reflected beams have different intensities, the minimum value of |^{2}| 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 (*x*) in that region.Worked Example 2 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 *T* and hence the probability that any particular electron will tunnel through the barrier.SolutionIn this case, *V*_{0} − *E*_{0} = (10 − 5) eV = 5 × 1.6 × 10^{−19}J. It follows thatWith *L* = 5 × 10^{−10} m, it follows thatSince this is much larger than 1, we can use Equation 7.53 to estimateThis is the probability that any particular electron will tunnel through the barrier. Exercise 12Determine 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.From Equation 7.29,In Region 1, where
and
, we haveSimplifying giveswhich can be interpreted as the sum of the probability density currents associated with the incident and reflected beams.Similarly, in Region 2, where
and
, we havewhich simplifies toFinally, in Region 3, where
The result for Region 2 is a real non-zero 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 *requires* that the probability current should be the same in all three regions.5 Applications of tunnelling5.1 OverviewThe 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.5.2 Alpha decayYou have probably met the law of radioactive decay, which says that, given a sample of *N*_{0} similar nuclei at time *t* = 0, the number remaining at time *t* is *N*(*t*) = *N*_{0}e^{−λt}, where *λ*, the decay constant for a particular kind of nucleus, determines the rate at which the nuclei decay. The half-life is the time needed for *half* of any sufficiently large sample to decay. It is related to the decay constant by *T*_{1/2} = (ln2)/*λ*.We shall now consider an important type of radioactive decay called **alpha decay** 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 helium-4 nucleus (). 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 bywhere *α* denotes the alpha particle. Note that the atomic number of the parent nucleus decreases by two and its mass number decreases by four.Alpha decay was discovered and named by Rutherford in 1898. It was soon established that each type of alpha-decaying nucleus emits an alpha particle with a characteristic energy, *E*_{α}. While these alpha emission energies cover a fairly narrow range of values (from about 2 MeV to 8 MeV), the half-lives of the corresponding nuclei cover an enormous range (from 10^{−12} s to 10^{17} s). Experiments showed that, within certain families of alpha-emitting nuclei, the half-lives and alpha emission energies were related to one another. Written in terms of the decay constant, *λ* = (ln 2)/*T*_{1/2}, this relationship can be expressed in the formwhere *A* is a constant that characterises the particular family of nuclei, and *B* depends on the charge of the individual nucleus. We shall refer to this empirical law as the **Geiger-Nuttall relation**.
Despite all this information, by the early 1920s alpha decay had become a major puzzle to physicists. The cause of the observed Geiger-Nuttall 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 half-lives, and what was the origin of the Geiger-Nuttall relation?Answering these questions was one of the early triumphs of wave mechanics. In 1928 the Russian-American 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 Figure 23.Note that *V*(*r*) is a function of a radial coordinate *r*; this is because we are dealing with a three-dimensional problem in which the potential energy function is spherically symmetric, and *r* represents the distance from an origin at the centre of the nucleus. Initially, an alpha particle of energy *E*_{α} is confined within a distance *r* = *r*_{0} of the origin by the well-like part of the potential energy function. This well is due to the powerful but short-range interaction known as the *strong nuclear force*. In addition, a long-range *electrostatic force* 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 functionwhere *Z* is the atomic number of the nucleus, 2*e* is the charge of the alpha particle, (*Z* − 2)*e* is the charge of the nucleus left behind after the decay and ε_{0} is a fundamental constant called *the permittivity of free space*. This potential energy function is often called the **Coulomb barrier**. Notice that the Coulomb barrier exceeds the energy of the alpha particle in the region between *r* = *r*_{0} and *r* = *r*_{1} (defined by *V*(*r*_{1}) = *E*_{α}). 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 *r* = *r*_{1}, the alpha particle is electrostatically repelled from the nucleus.To apply the quantum-mechanical 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 (*E*_{α}) and the nuclear diameter (2*r*_{0}) implying that it is incident on the barrier about 10^{21} 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, *T*. To estimate *T* we must take account of the precise shape of the Coulomb barrier. We shall not go through the detailed arguments used to estimate *T* in this case, but we shall note that they involve the approximationwhere *r*_{0} and *r*_{1} are the minimum and maximum values of *r* for which *V*(*r*) > *E*_{α}. Equation 7.56 is closely related to the expression for tunnelling through a finite square barrier given in Section 4.4. If the potential energy function *V*(*r*) happened to be constant over a region of length *L*, then Equation 7.56 would reproduce the exponential term of Equation 7.53. The other factors in Equation 7.53 are not reproduced, but they vary so slowly compared with the exponential factor that they can be ignored for present purposes.For given values of *E*_{α}, *Z* and *r*_{0}, Equation 7.56 can be evaluated using the Coulomb barrier potential energy function of Equation 7.55. After a lengthy calculation, including some approximations, the final result is of the formwhere *a* and *b* are constants. Multiplying *T* by the number of escape attempts per second gives the rate of *successful* escape attempts, and this can be equated to the decay constant, λ. So, according to the quantum tunnelling theory of alpha decay, we haveThis agrees with the Geiger–Nuttall relation and a detailed comparison with experimental data is shown in Figure 24. The exponential-dependence on *E*_{α}^{−1/2} 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.5.3 Stellar astrophysicsIf 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 lithium-7 nuclei to split into pairs of alpha particles by bombarding them with high-energy 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.The nuclear reactions that allow stars to shine are predominantly **fusion reactions** 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 *E* = *mc*^{2}) 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^{26} W in the case of the Sun).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^{7} 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^{−16} 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 above-average energies not uncommon, the reliance on tunnelling makes fusion a relatively slow process.Again taking the Sun as an example, its energy comes mainly from a process called the **proton–proton chain** that converts hydrogen to helium. The first step in this chain involves the fusion of two protons and is extremely slow, taking about 10^{9} years for an average proton in the core of the Sun. This is one of the reasons why stars are so long-lived. The Sun is about 4.6 × 10^{9} 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.5.4 The scanning tunnelling microscopeThe **scanning tunnelling microscope** (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.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 Figure 25). 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 Equation 7.56). This sensitivity is the key to finding the positions of tiny irregularities in the surface, including individual atoms.In practice, the STM can operate in two different ways. In *constant-height mode*, 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 *constant-current mode* 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.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 nano-scale manipulation is shown in Figure 26, the famous ‘quantum corral’ formed by positioning iron atoms on a copper surface.Conclusion**Session 1**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.**Session 2**In one dimension, wave packets scattered by finite square barriers or wells generally split into transmitted and reflected parts, indicating that there are non-zero probabilities of both reflection and transmission. These probabilities are represented by the reflection and transmission coefficients *R* and *T*. The values of *R* and *T* 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, *R* + *T* = 1.**Session 3**Unnormalisable stationary-state solutions of Schrödinger's equation can be interpreted in terms of steady beams of particles. A term such as *A*e^{i(kx − ωt)} can be associated with a beam of linear number density *n* = |*A*|^{2} travelling with speed *v* = k/*m* in the direction of increasing *x*. Such a beam has intensity *j* = *nv*. In this approach, *T* = *j*_{trans}/*j*_{inc} and *R* = *j*_{ref}/*j*_{inc}.For particles of energy *E*_{0} > *V*_{0}, incident on a finite square step of height *V*_{0}, the transmission coefficient iswhereare the wave numbers of the incident and transmitted beams. For a finite square well or barrier of width *L*, the transmission coefficient can be expressed aswhere
, with the plus signs being used for a well and the minus signs for a barrier. Transmission resonances, at which *T* = 1 and the transmission is certain, occur when *k*_{2}*L* = *N* where *N* is an integer.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 currentIn three dimensions, scattering is described by the total cross-section, *σ*, 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 cross-section tells us the rate of scattering into a small cone of angles around that direction. At very high energies, total cross-sections are dominated by inelastic effects due to the creation of new particles.**Session 4**Wave packets with a narrow range of energies centred on *E*_{0} can tunnel though a finite square barrier of height *V*_{0} > *E*_{0}. In a stationary-state approach, solutions of the time-independent Schrödinger equation in the classically forbidden region contain exponentially growing and decaying terms of the form *C*e^{−αx} and *D*e^{αx}, where
is the attenuation coefficient. The transmission coefficient for tunnelling through a finite square barrier of width *L* and height *V*_{0} is approximatelySuch a transmission probability is small and decreases rapidly as the barrier width *L* increases.**Session 5**Square barriers and wells are poor representations of the potential energy functions found in Nature. However, if the potential *V*(*x*) varies smoothly as a function of *x* the transmission coefficient for tunnelling of energy *E*_{0}can be roughly represented byThis 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.**Glossary and Physics Toolkit**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.Click to view glossary. (16 pages, 0.4 MB)Click to view toolkit. (17 pages, 0.4 MB)Keep on learning Study another free courseThere are more than **800 courses on OpenLearn** for you to choose from on a range of subjects. Find out more about all our free courses. Take your studies furtherFind out more about studying with The Open University by visiting our online prospectus. If you are new to university study, you may be interested in our Access Courses or Certificates. What’s new from OpenLearn?
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The material acknowledged below is Proprietary (see terms and conditions) and used under licence (not subject to Creative Commons licence).The content is from SM358_1 Book 1 *Wave Mechanics* – Chapter 7 Scattering and Tunnelling, pages 178–209.Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:Course image: Susanne Nilsson in Flickr made available under Creative Commons Attribution-ShareAlike 2.0 Licence.Figure 2 Jon Arnold Images/Alamy;Figure 3 Bennett, C.L. et al. ‘First Year Wilkinson Microwave Anisotrophy Probe (WMAP) Observations; Preliminary Maps and Basic Results’, *Astrophysical Journal* (submitted) © 2003 The American Astronomical Society;Figure 25 Courtesy of Dr Andrew Flewitt in the Engineering Department University of Cambridge;Figure 26 Courtesy of Don Eigler, IBM Research Division.**Don't miss out:**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 - www.open.edu/openlearn/free-courses