  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.
