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arXiv:1702.00856

The infinite square well in a reformulation of Quantum mechanics without potential function

Abdulaziz D Alhaidari, T. J. Taiwo

Published Feb 2, 2017• 0 citations
arXivHTMLPDF

Abstract

Using a recent reformulation of quantum mechanics where the potential function is not required, we are able to obtain the energy spectrum and wave function associated with the infinite square well analytically. Therefore, this work constitutes an example of how to establish the correspondence between this approach and the standard formulation of quantum mechanics.

DOI
10.48550/arxiv.1702.00856

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Using a recent reformulation of quantum mechanics where the potential function is not required, we are able to obtain the energy spectrum and wave function associated with the infinite square well analytically. Therefore, this work constitutes an example of how to establish the correspondence between this approach and the standard formulation of quantum mechanics.
In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.
The finite square well The infinite square well potential energy rigorously restricts the associated wavefunction to an exact region of space: it is infinitely “hard.” Potential energies encountered in more realistic physical scenarios are “softer” in that they permit wavefunctions to spread throughout less well-defined regions. An important toy example of the latter is the finite square well. In this problem, the potential energy function is U(x) = 0, if 0 < x < L, and U0 otherwise.