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The normalization and the boundary condition of a particle wave function moving in a field of an arbitrary one-dimensional potential

A. Zh, N. A. Aleksanyan

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

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

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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
A new approach to the problem of description of the stationary motion of a quantum particle in the field of a one-dimensional arbitrary potential is developed. It is shown that the wave function of infinite motion, with the accuracy to two arbitrary constants, can be expressed by means of an arbitrary single solution for some set of linear differential equations of the first order. It is shown that one general property of the Schrödinger equation solutions lies in the basis of many known methods of the problem consideration, such as the method of integral equations, transfer-matrix method, imbedding method, and method of combination of scattering parameters. Within the framework of the proposed approach, the connection between the above-mentioned methods becomes more transparent.
A consistent approach to the description of a stationary motion of a quantum particle in an arbitrary one-dimensional potential has been developed. It has been proved that the wave function of an infinite motion can be expressed accurate to up two arbitrary constants with the use of one particular solution to the system of first-order linear differential equations. It has been shown that many well-known methods, such as the integral equation method, the transfer matrix method, the embedding method, and the method of combination of scattering parameters, are based on a general property of the solutions to the Schrödinger equation. Within the proposed approach, the relation between these methods becomes more transparent and their description can be well within a unified context.
A new method for consideration of an electron stationary motion in the field of an arbitrary one-dimensional potential. It is shown that for a case of an electron infinite motion, the problem of determination of wave functions can be represented as Cauchy problem for the one-dimensional Schrodinger equation. For the case of finite motion the equation determining the energy spectrum is found. It is proved, when the spectrum of bound states is know, the problem of building the wave functions of the discrete spectrum can be formulated as Cauchy problem for wave equation as well.
In this work a wave process evolution problem in a field of an arbitrary one-dimensional potential is considered. We investigate the wave packets derived based on linear combinations of scattering wave functions taken with magnitudes of state indexes near to resonance tunneling case. It is shown that the wave processes with different parities of asymptotes describe a wave process where the group and phase velocities have opposite directions. The conducted consideration allows to state that the bound states can be interpreted as wave packets of the scattering wave functions to be derived under a certain way.
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