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arXiv:1711.03317

Justification for the group-theoretical method as the right way to solve the infinite spherical well in quantum mechanics

Young‐Sea Huang, Chung‐Hsien Wu, Kung-Te Wu, Tzuu-Kang Chyi, Huitzu Tu

Published Nov 9, 2017• 0 citations

Abstract

Recently, the problem of the infinite spherical well was solved by the group-theoretical method to resolve all the peculiarities in the currently accepted solution [DOI: 10.13140/RG.2.2.18172.44162 (Researchgate, 2017)]. With a view to further justifying the group-theoretical method, the problem is first studied from the viewpoint of classical mechanics. Then the radial probability densities predicted by classical mechanics are compared with those predicted from solutions of the problem obtained by the group-theoretical method. The comparisons clearly indicate the convergence of predictions of quantum mechanics and classical mechanics in the limit of large eigen-energies. Therefore, the group-theoretical method is justified as the right way to solve the problem of the infinite spherical well.

DOI: 10.48550/arxiv.1711.03317

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Recently, the problem of the infinite spherical well was solved by the group-theoretical method to resolve all the peculiarities in the currently accepted solution [DOI: 10.13140/RG.2.2.18172.44162 (Researchgate, 2017)]. With a view to further justifying the group-theoretical method, the problem is first studied from the viewpoint of classical mechanics. Then the radial probability densities predicted by classical mechanics are compared with those predicted from solutions of the problem obtained by the group-theoretical method. The comparisons clearly indicate the convergence of predictions of quantum mechanics and classical mechanics in the limit of large eigen-energies. Therefore, the group-theoretical method is justified as the right way to solve the problem of the infinite spherical well.

This solves quantum physics problems using the integral Rohit transform, including the scattering of low energy particles by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. The standard calculus approach is typically used to solve these quantum mechanics issues. To solve these quantum physics challenges, this research presents a new method: the integral Rohit transform. When compared to other approaches that are currently in the literature, the obtained solutions show how accurate the suggested method is. This demonstrates the potential and efficacy of the suggested approach to overcoming quantum mechanical issues, such as low-energy particle scattering by a completely rigid sphere and particle behavior in a one-dimensional infinitely high potential box. In this paper, the successful application of the integral Rohit Transform has been demonstrated in solving the one-dimensional time-independent Schrodinger's equation. This application has yielded results that include the determination of eigen energy values and eigen functions for a particle confined within an infinitely high potential well, as well as the calculation of the total scattering cross-section for low energy particles interacting with a perfectly rigid sphere. In the case of low energy limit, the total scattering cross-section for low energy particles due to a perfectly rigid sphere, as determined through quantum mechanics, is equivalent to the geometrical cross-section of said sphere. Additionally, the energy values that the particle can possess within a 1d infinitely high potential well demonstrate that the energy of said particle, when confined within this potential well, is quantized.

We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.