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American Journal of Physics

An Instructive Example of the Sudden Approximation in Quantum Mechanics

D. W. Schlitt, Conley Stutz

Published Jan 1, 1970• 21 citations
Publisher

Abstract

Some features of the sudden approximation are examined and applied to the example of a particle in a suddenly expanded box. Among the points illustrated are the form of the work energy relation in quantum mechanics, the application of the energy-time uncertainty relation, some pitfalls in the use of theorems that apply only to Hermitian operators, and the periodic time-dependence of expectation values.

DOI
10.1119/1.1976231

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Some features of the sudden approximation are examined and applied to the example of a particle in a suddenly expanded box. Among the points illustrated are the form of the work energy relation in quantum mechanics, the application of the energy-time uncertainty relation, some pitfalls in the use of theorems that apply only to Hermitian operators, and the periodic time-dependence of expectation values.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
By means of a one-dimensional model of a particle in an infinite square-well potential with one wall moving at a constant speed, we examine aspects of time-dependent phenomena in quantum mechanics such as adiabatic and sudden processes. The particle is assumed to be initially in the ground state of the potential with its initial width. The time dependence of the wavefunction of the particle in the well is generally more complicated when the potential well is compressed than when it is expanded. We are particularly interested in the case in which the potential well is suddenly compressed. The so-called sudden approximation is not applicable in this case. We also study the energy of the particle in the changing well as a function of time for expansion and contraction as well as for expansion followed by contraction and vice versa.
The one-dimensional square quantum well of varying width is shown to be an instructive example for testing the appropriateness of approximate solutions in quantum mechanics. The range of wall speeds over which the adiabatic approximation is applicable is investigated and related to the initial energy of the quantum system for both expanding and contracting wells. However, of more interest is the applicability of the sudden approximation to these systems. It is shown that this approximation is appropriate for expanding wells at quite modest wall speeds, whereas the approximation cannot be applied for contracting wells no matter how rapidly the well width contracts. The physical reasons for the inapplicability of the sudden approximation for contracting wells is investigated.
The time-dependent Schrödinger equation forms the chapter's starting point. For a time-independent Hamiltonian, a quantum state may be time evolved with the standard, unitary time evolution operator. After obtaining this operator, an algorithm is provided by which a state may be time evolved, and measurement probabilities determined. A spin 1/2 example illustrates time evolution concretely. All this makes close contact with the discussion of unitary operators in Chapter 10. The time dependence of expectation values forms the next topic, followed by the concept of a constant of the motion, which is carefully contrasted with that of a stationary state. The energy-time uncertainty relations, which rest on a quite different conceptual footing than the ‘standard’ uncertainty relations, occupy the rest of the chapter. An example from spin 1/2 illustrates the energy-time uncertainty relations.
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.