Exact Solutions of the Schrödinger Equation for Pseudoharmonic Potential

The pseudoharmonic potential belongs to the small number of problems for which the Schrödinger equation is exactly solvable [1] and is thus of intrinsic interest. The solution is outlined in the Details below.

For selected parameters and , you can either display an energy diagram, showing the first several eigenvalues superposed on the potential energy curve, or a plot of the radial function for a selected value of the quantum number .

The Schrödinger equation for the radial function , in the case of zero angular momentum, is given by

.

The eigenfunctions for bound states are found to be

,

where is an associated Laguerre polynomial. The solution of the differential equation initially gave associated Laguerre functions , with

and

.

The quantization is determined by the condition that the eigenfunctions must approach 0 as . The asymptotic behavior of Laguerre functions is given by:

as .

This would overtake the converging factor in the solution as , unless is an integer , which would then produce a singularity in a gamma function of the denominator. The corresponding eigenvalues are thereby determined:

with .

Reference

[1] I. I. Gol'dman and V. D. Krivchenkov, Problems in Quantum Mechanics (B. T. Geǐlikman, ed., E. Marquit and E. Lepa, trans.), Reading, MA: Addison-Wesley, 1961.