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Journal of Shanxi Teachers University

On the Expansion Problem of the Wavefunctions of a Stationary State for a Particle in One-Dimensinal Potentional Well with Infinitely High Walls

Liu Deng

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

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.

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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.
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.
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.
The quantum mechanical problem of a particle in a infinite potential well with slanted walls is studied. The energy eigenvalues are determined by a transcendental equation involving the Airy function of the first kind. Two limiting cases are discussed. Next, the allowed energies are found by graphical and numerical methods. The graphical analysis hints that for large quantum numbers the consecutive energy levels get arbitrarily close together. This asymptotic behavior of the energy spectrum actually holds, as shown by means of an intuitive argument and confirmed by resorting to basic properties of Airy functions. In a sense, Bohr's correspondence principle is more accurately fulfilled than in the cases of the standard particle in a box or the harmonic oscillator. A state is regarded the more confined the less the probability of finding the particle in the classically forbidden region. For some stationary states the confinement degree is calculated and a comparison is made with the harmonic oscillator, for which the ground state is the least confined of all stationary states. In the symmetric case --- right and left walls of equal slope --- something unexpected happens: beyond a certain critical slope the ground state is no longer the least confined eigenstate. Over and above its interesting physical features, this problem provides students with the opportunity to get acquainted with Airy functions, which do not belong to the standard mathematical repertoire of physics undergraduates.
We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called ∗\\*∗-genvalue (``stargenvalue'') equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding ∗\\*∗-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schr\\"odinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.
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.