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Journal of Huanggang Normal University

The normalized wave functions of the particles bound in one-dimensional square potential wells

Xiaomei Wang

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Till now no book has given the normalized wave functions for the particle freely moving in a one-dimensional finite potential well. This paper acquires the normalized wave functions of one-dimensional symmetric and asymmetric wells with finite potentials. The results are in accord with those of infinite wells when the potentials tends to infinite.

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Cambridge University Press eBooks•Dec 5, 2024•0 citations
Till now no book has given the normalized wave functions for the particle freely moving in a one-dimensional finite potential well. This paper acquires the normalized wave functions of one-dimensional symmetric and asymmetric wells with finite potentials. The results are in accord with those of infinite wells when the potentials tends to infinite.
Three methods-a documental analysis, a textual method, a comprehensive analysis- are used in this article to analyze the energy levels of motional particles and their wave functions of one-dimensional finite deep symmetrical square potential wells, thus solving the eigen-values and eigen-functions at their corresponding energetic levels. Due to the three following advantages of a one-dimensional problem-its simpler mathematical processing, and the possible detailed-discussed results, and the display of the many features of quantum systems, so in the paper, the eigen-values and the exact formulas of the wave functions of moving particles in a one-dimensional finite deep symmetric square potential wells, which are not specified in the textbooks, are deviated according to the Schrodinger equation of stationary state through a case study of moving particles in a one-dimensional finite deep symmetric square potential wells.
The normalization problem of a wave function for an arbitrary one-dimensional field is considered. It is shown, that for an infinite motion type the magnitude of the normalization constant does not depend on the form of a scattering potential and it is determined by the boundary conditions imposed on the character of the investigated motion. The general formula for the normalization constant as a function of amplitudes of the waves converging to the potential is derived. Keyword: Normalization problem, wave function, one-dimensional motion, arbitrary potential PACS: 03.65.Nk, 02.30.Em
Abstract We present details of a new solution method for the bound energy states of a quantum particle in a one-dimensional symmetric finite square well potential. The solution is obtained in a clear way by introducing a simple method that relies on the use of two auxiliary functions. The approach is straightforward and leads to the well-known even and odd parity wave function solutions of the problem without having to do any a priori assumptions about the nature or symmetry of the quantum states.