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European Journal of Physics

On the analogy between the stationary states of a particle bound in a one-dimensional well and standing waves on a string

V. Černý, Ján Pišút, Peter Prešnajder

Published Apr 1, 1986• 1 citation
Publisher

Abstract

The analogy between standing waves on a string and the stationary states of a particle in an infinitely deep potential well, frequently used in elementary presentations of the ideas of quantum mechanics, is critically analysed. The author suggest a way to avoid the incorrect intermediate steps sometimes used in this analogy. In contrast to the conclusions of a recent paper by Lacina (1985), they find the analogy viable.

DOI
10.1088/0143-0807/7/2/012

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The analogy between standing waves on a string and the stationary states of a particle in an infinitely deep potential well, frequently used in elementary presentations of the ideas of quantum mechanics, is critically analysed. The author suggest a way to avoid the incorrect intermediate steps sometimes used in this analogy. In contrast to the conclusions of a recent paper by Lacina (1985), they find the analogy viable.
The motion of a particle in a potential well is studied when the particle is attached to an infinite elastic string. This is generic with the problem of dissipative quantum mechanics investigated by Caldeira and Leggett (Ann. Phys. 149, 374 (1983). doi: 10.1016/0003-4916(83)90202-6 ). Besides the dissipative motion there is another scenario of interaction of the string with the particle attached. Stationary particle–string states exist with string deformations accompanying the particle. This is like polaronic states in solids. Our polaronic states in the well are non-decaying and have a continuous energy spectrum. These states may have a link to quantum electrodynamics.
The study of the behaviour of microscopic systems using Schrodinger's wave equation is a branch of quantum mechanics known as wave mechanics. This approach is inadequate to deal with all problems of interest, but a remarkably wide range of processes can successfully be investigated solely with the theory. The physics of a particle which moves in a static potential is of particular interest. Classically, the energy of such a particle is conserved. The wave function is reminiscent of the classical problem of standing waves on a stretched string with fixed endpoints, with the stationary states corresponding to the normal modes of vibration. Just as there are discrete frequencies of vibration on the string, so there are discrete energy levels. The energy, which is a continuous parameter in classical mechanics, is 'quantized' to certain discrete energy levels.