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European Journal of Physics

Plane waves and wave packets in elementary quantum mechanics problems

Bernard Diu

Published Oct 1, 1980• 10 citations
Publisher

Abstract

In many quantum mechanics courses, plane waves and wave packets are introduced at the very beginning, and used to solve 'simple' one-dimensional problems with square potentials. The purpose of this article is to clarify and criticise the meaning of commonly used statements such as: 'an e ikx term represents a particle moving in the positive x direction' and/or 'e iks represents the incident beam of particles'. The necessary notions and results are first briefly reviewed. A simple problem (scattering by a one-dimensional square potential step) is then studied in detail in order to exhibit and discuss the difficulties in a concrete situation. The more general relevance of this discussion is finally indicated.

DOI
10.1088/0143-0807/1/4/010

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In many quantum mechanics courses, plane waves and wave packets are introduced at the very beginning, and used to solve 'simple' one-dimensional problems with square potentials. The purpose of this article is to clarify and criticise the meaning of commonly used statements such as: 'an e ikx term represents a particle moving in the positive x direction' and/or 'e iks represents the incident beam of particles'. The necessary notions and results are first briefly reviewed. A simple problem (scattering by a one-dimensional square potential step) is then studied in detail in order to exhibit and discuss the difficulties in a concrete situation. The more general relevance of this discussion is finally indicated.
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The transmission and the reflection of a wave incident on a potential in one-dimensional quantum mechanics are usually discussed by assuming that a plane wave eikx or a broad wave packet is incident. We are interested in what happens if the width of the incident wave packet is very small. In this case, the incident wave packet can spread out in such a way that part of the wave packet turns around before reaching the potential area and proceeds backward. This can clearly be visualized by means of the Bohm trajectories associated with the wave packet. We discuss a complication that arises in this case regarding the notion of the reflection probability.PACS Nos.: 03.65–w, 03.65.Nk
The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.