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Scattering and tunnelling
Scattering is fundamental to almost everything we know about the world, such as why the...
Scattering is fundamental to almost everything we know about the world, such as why the sky is blue. Tunnelling is entirely quantum-mechanical and gives rise to such phenomena as nuclear fusion in stars. Examples and applications of both these fascinating concepts are investigated in this unit.
By the end of this unit you should be able to:
- explain the meanings of the emboldened terms and use them appropriately;
- describe the behaviour of wave packets when they encounter potential energy steps, barriers and wells;
- describe 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 coefficients;
- present information about solutions of the time-independent Schrödinger equation in graphical terms;
- evaluate probability density currents and explain their significance;
- describe and comment on applications of scattering and tunnelling in a range of situations including: three-dimensional scattering, alpha decay, nuclear fusion in stars, and the scanning tunnelling microscope.
- Current section: Introduction
- Learning outcomes
- 1 What are scattering and tunnelling?
- 2 Scattering: a wave-packet approach
- 3 Scattering: a stationary-state approach
- 4 Tunnelling: wave packets and stationary states
- 5 Applications of tunnelling
- 6 Summary
Scattering and tunnelling
In this unit 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 unit. 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 units 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 unit. PDFs of these documents have been attached in the Summary.
This unit is an adapted extract from the Open University course