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Title: Quantum mechanics without potential function

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.
Authors:
 [1] ;  [2]
  1. Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438 (Saudi Arabia)
  2. Department of Mathematics, University of Central Florida, Orlando, Florida 32816 (United States)
Publication Date:
OSTI Identifier:
22479578
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; BOUND STATE; ENERGY SPECTRA; PHASE SHIFT; POLYNOMIALS; POTENTIALS; QUANTUM MECHANICS; QUANTUM SYSTEMS; SCATTERING; SCHROEDINGER EQUATION; SPACE; WAVE FUNCTIONS