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 .


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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
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 .
[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.‬
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