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Вестник Московского государственного технического университета им. Н.Э. Баумана. Серия «Естественные науки»

Движение микрочастицы в одномерной прямоугольной потенциальной яме с подвижной стенкой

N. I. Yurasov, Л. К. Мартинсон

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

The paper considers a problem of quantum particle movement in theone-dimensional square potential well with impermeable walls. As opposed to the standard approach, the authors consider a well with a fixed wall which has a coordinate x = 0 while the other wall starts moving at the moment t = 0. The well width changes within the limits a...b. Due to the wall movement, a microparticle state becomes unsteady. Both a power spectrum and a wave function are determined for this state

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The paper considers a problem of quantum particle movement in theone-dimensional square potential well with impermeable walls. As opposed to the standard approach, the authors consider a well with a fixed wall which has a coordinate x = 0 while the other wall starts moving at the moment t = 0. The well width changes within the limits a...b. Due to the wall movement, a microparticle state becomes unsteady. Both a power spectrum and a wave function are determined for this state
The expansion problem of the wavefunctions of a stationary state for a particle in one dimensional potential well with infinitely high walls in studied. We found, in the interval ( 0≤x≤a ), that the wavefunctions of a stationary state can be expressed as the superposition of two plane waves, which traveling in the opposite directions, when the energy quantum number n is even. However, it can not be done when n is odd, because they are not the period functions in this interval. If only if the width of the well tends to infinity, or n is very large, and the expanding coefficients on the base of continuous spectrum become to the superposition of two symmetrical delta functions, all of the stationary wavefunctions can be expanded as the superposition of two plane waves.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.