Interbasis expansions for the isotropic 3D harmonic oscillator and bivariate Krawtchouk polynomials

TL;DR

This paper derives explicit formulas for bivariate Krawtchouk polynomials using representation theory of SO(3) and connects them to harmonic oscillator energy states, expanding the mathematical tools for quantum systems analysis.

Contribution

It provides a new explicit expression for bivariate Krawtchouk polynomials in terms of classical orthogonal polynomials, linking them to SO(3) representations and harmonic oscillator states.

Findings

Explicit formula for bivariate Krawtchouk polynomials

Connection between polynomials and SO(3) representations

Overlap coefficients derived from su(1,1) Clebsch-Gordan problem

Abstract

An explicit expression for the general bivariate Krawtchouk polynomials is obtained in terms of the standard Krawtchouk and dual Hahn polynomials. The bivariate Krawtchouk polynomials occur as matrix elements of the unitary reducible representations of SO(3) on the energy eigenspaces of the 3-dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components. The decomposition entails expanding the Cartesian basis states in the spherical bases that span irreducible SO(3) representations. The overlap coefficients are obtained from the Clebsch-Gordan problem for the su(1,1) Lie algebra.