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One-dimensional motion

Victor M. Galitski, Boris Mikhailovich Karnakov, Vladimir I Kogan, Victor M. Galitski

Published Feb 28, 2013• 0 citations
Publisher

Abstract

Abstract This chapter analyzes problems that deal with stationary states in the discrete spectrum; the Schrödinger equation in momentum space (the Green function and integral form of the Schrödinger equation); the continuous spectrum (reflection from and transmission through potential barriers); and systems with several degrees of freedom (particle in a periodic potential).

DOI
10.1093/acprof:oso/9780199232710.003.0002

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Abstract This chapter analyzes problems that deal with stationary states in the discrete spectrum; the Schrödinger equation in momentum space (the Green function and integral form of the Schrödinger equation); the continuous spectrum (reflection from and transmission through potential barriers); and systems with several degrees of freedom (particle in a periodic potential).
In the previous chapter we have analyzed the peculiarities of quantized electron motion in layered structures with one-dimensional potential wells. From the mathematical point of view, the solution of the Schrödinger equation for one-dimensional potential profiles is much simpler. However, many quantum objects, such as atoms, molecules, and quantum dots, are three-dimensional objects. Thus, in order to analyze electron motion in such objects we need to find solutions of the Schrödinger equation for three-dimensional potential profiles. The electron motion in spaces with dimensionality higher than one, especially for rectangular potential profiles with infinite potential barriers, is not so difficult to analyze. At the same time we have to keep in mind that such potential profiles frequently represent some approximation of the more complex, real potential profiles. Depending on the type of structure and on the form of the potential profile, the electron motion may be limited in two directions (two-dimensional quantization) or in three directions (three-dimensional quantization). In this chapter we will show that the existence of discrete energy levels in the electron spectrum is an intrinsic feature of electron motion in potential wells of any form and dimensionality. An electron in a rectangular potential well (quantum box) In the previous chapter we studied the electron motion in one-dimensional potential wells. An electron's motion along one direction was confined by the potential profile and the momentum in this direction was quantized.
In the past few chapters, the Schrödinger equation has been developed and applied to the solution of a number of quantum mechanical problems, mainly to develop experience with the results of simple and common systems. The results from these examples can be summarized relatively simply. In general, quantization enters the problem through the non-commuting nature of conjugate operators such as position and momentum as well as from classical boundary quantization. This has led to the Schrödinger equation itself being the primary equation of motion for the wave function solution to the problem. In essence, duality is incredibly important and is the basic method by which the electron is treated as a wave, rather than as a particle, and the wave equation of interest is the Schrödinger equation. When boundary conditions are applied, either through potential barriers or through the form of the potential (as in the harmonic oscillator), the time-independent Schrödinger equation yields solutions that are often special functions. Examples of this are the sines and cosines in the rectangular-barrier case, the Airy functions in the triangular-well case, and the Hermite polynomials in the harmonic oscillator case.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.