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arXiv:1711.06561 · Quantum Physics Letters

Meixner - Pollaczek Quantum System

T. J. Taiwo

Published Apr 1, 2018• 1 citation
arXivHTMLPDF

Abstract

In an effort to achieve our aims (getting larger quantum system) in the recent reformulation of quantum mechanics without potential function [1-5], we obtained a new quantum system associated with Meixner Pollaczek Orthogonal polynomial class (hereby called Meixner - Pollaczek Quantum system). The energy spectrum and wave function of the quantum system were given. To the best of our knowledge, this quantum system is not found in physics literatures.

DOI
10.18576/qpl/070103

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In an effort to achieve our aims (getting larger quantum system) in the recent reformulation of quantum mechanics without potential function [1-5], we obtained a new quantum system associated with Meixner Pollaczek Orthogonal polynomial class (hereby called Meixner - Pollaczek Quantum system). The energy spectrum and wave function of the quantum system were given. To the best of our knowledge, this quantum system is not found in physics literatures.
Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. 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 quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.
Abstract We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner–Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials is {ℓD,ℓD+1,ℓD+2,…}\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}{ℓD​,ℓD​+1,ℓD​+2,…}, where ℓD\ell_{\mathcal{D}}ℓD​ is an even positive integer (D\mathcal{D}D: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.
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.
Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,ℜ\Reℜ). The reproducing kernel for the wavefunctions of these models is also found.