Quantum Mechanics Without Potential Function

Abstract

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.

Publication Date

7-1-2015

Publication Title

Journal of Mathematical Physics

Volume

56

Issue

7

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1063/1.4927262

Socpus ID

84941253941 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84941253941

This document is currently not available here.

Share

COinS