An application of the HeunB function

TL;DR

This paper explores how adding gravitational potential affects quantum harmonic oscillator solutions, showing that eigenfunctions transition from parabolic cylinder functions to biconfluent HeunB functions, with eigenvalues computed via series expansion.

Contribution

It demonstrates the application of the HeunB function to quantum systems with gravitational potential, extending the exact solutions of harmonic oscillators.

Findings

Eigenfunctions change from parabolic cylinder to HeunB functions with gravitational potential inclusion.

Eigenvalues are obtained through a series expansion in Hermite functions.

The study provides a method to incorporate gravity into quantum harmonic oscillator models.

Abstract

How does the inclusion of the gravitational potential alter an otherwise exact quantum mechanical result? This question motivates this report, with the answer determined from an edited version of problem #12 on p.273 of Ref.1. To elaborate, we begin with the Hamiltonian associated with the system of two masses in the problem obeying Hooke's law and vibrating about their equilibrium positions in one dimension; the Schrodinger equation for the reduced mass is then solved to obtain the parabolic cylinder functions as eigenfunctions and the eigenvalues of the reduced Hamiltonian are calculated exactly. Parenthetically,the quantum mechanics of a bounded linear harmonic oscillator was perhaps first studied by Auluck and Kothari[2]. The introduction of the gravitational potential in the aforesaid Schrodinger equation alters the eigenfunctions to the biconfluent HeunB function[3]; and the eigenvalues are the determined from a recent series expansion[4] in terms of the Hermite functions for the solution of the differential equation whose exact solution is the aforesaid HeunB function.