Tridiagonal Representation Approach in Quantum Mechanics

Abstract

We present an algebraic approach for finding exact solutions of the wave equation. The approach, which is referred to as the Tridiagonal Representation Approach (TRA), is inspired by the J-matrix method and based on the theory of orthogonal polynomials. The class of exactly solvable problems in this approach is larger than the conventional class. All properties of the physical system (energy spectrum of the bound states, phase shift of the scattering states, energy density of states, etc.) are obtained in this approach directly and simply from the properties of the associated orthogonal polynomials.

arXiv comment22 pages, 3 tables, 1 figure

References in corpus (11)

• Asymptotic iteration method for eigenvalue problems
• Construction of exact solutions to eigenvalue problems by the asymptotic iteration method
• An extended class of L2-series solutions of the wave equation
• Analytic solution of the Schrodinger equation for an electron in the field of a molecule with an electric dipole moment
• Quantum mechanics without potential function
• Charged particle in the field an electric quadrupole in two dimensions
• Open problem in orthogonal polynomials
• The Wilson-Racah Quantum System
• Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications
• Energy Density Bands Engineering
• Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment

Cited by in corpus (15)

• Open problem in orthogonal polynomials
• Deformed Morse-like potential
• Bound states of a quartic and sextic inverse-powerlaw potential for all angular momenta
• Bound states and the potential parameter spectrum
• 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
• Mapping Schrödinger equation into a Heun-type and identifying the corresponding potential function, energy and wavefunction
• Bound-state solutions of the Schrödinger equation for two novel potentials
• Finite series representation for the bound states of a spiked isotropic oscillator with inverse-quartic singularity
• Construction of potential functions associated with a given energy spectrum
• Structural Algebraic Quantum Field Theory
• Solutions of the scattering problem in a complete set of Bessel functions with a discrete index
• 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
• Electrostatic multipole contributions to the binding energy of electrons