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 wave function, 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 there-parameter systems.

25 pages, 1 table, and 3 figures

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Cited by in corpus (25)

• Solution of the nonrelativistic wave equation in the tridiagonal representation approach
• Tridiagonal Representation Approach in Quantum Mechanics
• Open problem in orthogonal polynomials
• Establishing correspondence between the reformulation of quantum mechanics without a potential function and the conventional formulation
• The Wilson-Racah Quantum System
• Quantum mechanics with orthogonal polynomials
• Orthogonal polynomials derived from the tridiagonal representation approach
• Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications
• Solution of an Open Problem about Two Families of Orthogonal Polynomials
• The Hahn Quantum System
• Series solution of a ten-parameter second order differential equation with three regular and one irregular singularities
• Bound states and the potential parameter spectrum
• Bound states of a short-range potential with inverse cube singularity
• Energy Density Bands Engineering
• Construction of potential functions associated with a given energy spectrum -- An inverse problem. II
• Series solutions of Bessel-type differential equation in terms of orthogonal polynomials and physical applications
• Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials
• Construction of potential functions associated with a given energy spectrum
• Energy spectrum design and potential function engineering
• Structural Algebraic Quantum Field Theory
• Five-parameter potential box with inverse square singular boundaries
• Extending the class of solvable potentials. IV Inverse square potential with a rich spectrum
• Confined systems associated with the discrete Meixner polynomials
• Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity
• Progressive approximation of bound states by finite series of square-integrable functions