scholarly journals Solvable discrete quantum mechanics: q-orthogonal polynomials with q=1 and quantum dilogarithm

2015 ◽  
Vol 56 (7) ◽  
pp. 073502 ◽  
Author(s):  
Satoru Odake ◽  
Ryu Sasaki
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Quantum Dilogarithm

scholarly journals HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS

2011 ◽  
Vol 26 (25) ◽  
pp. 1843-1852 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Higher Order ◽  
Wave Functions ◽  
Supersymmetric Quantum Mechanics ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials

Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

scholarly journals Special Orthogonal Polynomials in Quantum Mechanics

2019 ◽  
Author(s):  
Abdulaziz D. Alhaidari
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Solution Space ◽  
Type Equation ◽  
Algebraic Method ◽  
Recursion Relations ◽  
New Class ◽  
Continuous And Discrete Spectra ◽  
Discrete Spectra

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

scholarly journals RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM

2011 ◽  
Vol 26 (32) ◽  
pp. 5337-5347 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Supersymmetric Quantum Mechanics ◽  
Shape Invariance ◽  
Order Analysis ◽  
First Order ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials ◽  
Differential Relations

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

scholarly journals Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

2011 ◽  
Vol 369 (1939) ◽  
pp. 1301-1318 ◽  
Author(s):  
Ryu Sasaki
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Essential Role ◽  
Dynamical Symmetry ◽  
Oscillator Algebra ◽  
Discrete Variable ◽  
Shape Invariance ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Creation Operators

A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q -oscillator algebra and the Askey–Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional ‘discrete’ quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.

scholarly journals Confined systems with a linear energy spectrum

2021 ◽  
Vol 36 (10) ◽  
pp. 2150064
Author(s):  
A. D. Alhaidari ◽  
T. J. Taiwo
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Orthogonal Polynomials ◽  
Bound States ◽  
Quantum Systems ◽  
Potential Functions ◽  
Physical Parameters ◽  
Confined Systems

Using a formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters, we study quantum systems totally confined in space and with a linearly spaced energy spectrum. We present several examples of such systems, derive their corresponding potential functions, and plot some of their bound states.

Numerical quadrature

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Recurrence Relations ◽  
Classical Mechanics ◽  
Numerical Quadrature ◽  
Classical Orthogonal Polynomials ◽  
Gaussian Integration ◽  
Physical Problems ◽  
Important Means ◽  
Improper Integrals

This chapter discusses the numerous applications of numerical quadrature (integration) in classical mechanics, in semiclassical approaches to quantum mechanics, and in statistical mechanics; and then describes several ways of implementing integration in C++, for both proper and improper integrals. Various algorithms are described and analyzed, including simple classical quadrature algorithms as well as those enhanced with speedups and convergence tests. Classical orthogonal polynomials, whose properties are reviewed, are the basis of a sophisticated technique known as Gaussian integration. Practical implementations require the roots of these polynomials, so an algorithm for finding them from three-term recurrence relations is presented. On the computational side, the concept of polymorphism is introduced and exploited (prior to the detailed treatment later in the text). The nondimensionalization of physical problems, which is a common and important means of simplifying a problem, is discussed using Compton scattering and the Schrödinger equation as an example.

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