Details

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.\:202c

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