Exact Solutions of the Schrödinger Equation for the Kratzer Potential

The Kratzer potential was originally intended to approximate the interatomic interaction in diatomic molecules [1]. This has long since been superseded by superior alternatives, such as the Morse potential. However, the Kratzer potential belongs to the small number of problems for which the Schrödinger equation is exactly solvable, and is thus of intrinsic interest. The solution is outlined in the Details below.

For selected parameters and , you can display an energy diagram, showing the first seven 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 a Whittaker function,

and

.

The quantization is determined by the condition that the eigenfunctions must approach 0 as . The asymptotic behavior of the Whittaker functions is given by [2]:

as .

Clearly this is divergent as , unless the parameters and produce a singularity in the gamma function of the denominator, which requires

,

with .

The corresponding eigenvalues are thereby determined:

.

References

[1] A. Kratzer, "Die ultraroten Rotationsspektren der Halogenwasserstoffe," Zeitschrift für Physik, 3(5), 1920 pp. 289–307. doi:10.1007/BF01327754.

[2] "Section 13.19 Asymptotic Expansions for Large Argument." NIST Digital Library of Mathematical Functions. (May 10, 2019) dlmf.nist.gov/13.19.