Expectation Values $<r^p>$ for Harmonic Oscillator in $R^n$

TL;DR

This paper derives explicit formulas and recurrence relations for the expectation values of powers of the radius in an n-dimensional harmonic oscillator, connecting them with dual Hahn polynomials and comparing with hydrogen atom results.

Contribution

It introduces a novel method to evaluate $<r^p>$ for the harmonic oscillator using dual Hahn polynomials and establishes new recurrence and reflection relations.

Findings

Derived explicit expressions for $<r^p>$ in terms of dual Hahn polynomials.

Established a three-term recurrence relation for these expectation values.

Compared results with those for the hydrogen atom.

Abstract

We evaluate the matrix elements $<r^{p}>$ for the $n$ -dimensional harmonic oscillator in terms of the dual Hahn polynomials and derive a corresponding three-term recurrence relation and a Pasternack-type reflection relation. A short review of similar results for nonrelativistic hydrogen atom is also given.