Quantum mechanics with orthogonal polynomials

Abstract

We present a formulation of quantum mechanics based on orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis in configuration space where the expansion coefficients are orthogonal polynomials in the energy. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable non-relativistic quantum mechanical problems becomes larger than in the conventional formulation (see, for example, Table 1 in the text).

18 pages, 1 table; The polynomial class of section 4, in this version, is now identified with the Wilson-Racah polynomial class

References in corpus (7)

• Solution of the nonrelativistic wave equation in the tridiagonal representation approach
• Charged particle in the field an electric quadrupole in two dimensions
• 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
• Orthogonal polynomials derived from the tridiagonal representation approach
• The Hahn Quantum System

Cited by in corpus (11)

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• Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials
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• Energy spectrum design and potential function engineering
• Structural Algebraic Quantum Field Theory
• Revisiting the Coulomb problem: A novel representation of the confluent hypergeometric function as an infinite sum of discrete Bessel functions
• Progressive approximation of bound states by finite series of square-integrable functions
• Confined systems associated with the discrete Meixner polynomials
• Nonlinear extension of the J-matrix method of scattering: A toy model