In the solution of the Klein-Gordon equation for the potential of a homogeneously charged sphere, one can find the energy is determined by specifying that the wave functions and their first derivatives should be continuous at (r=a). In the region (r>=a) the wave functions are given by the Whittaker function. Now, I can’t understand what’s the Whittaker function's role in this problem? And where I can find more information about this function?
Dear Farzad,
consult Exercise 1.15 on pp. 59-61 in
Walter Greiner (2000). Relativistic Quantum Mechanics. Wave Equations. Third edition. Springer Verlag.
The KG equation one reduces by a set of subsequent substitutions to Eq. (6) , which is a Whittaker differential equation having particular solution (8) in a form of confluent hypergeometric ${}_1F_1$ function. Also see Exercise 1.16, pp. 61-64, where the Kummer typer confluent hypergeometric DE is presented with the solution as ${}_1F_1$ too. The steps between confluent hyergeometric function, Kummer's functions and Whittaker $W$ functions you can find e.g. on
functions.wolfram.com/HypergeometricFunctions/WhittakerW/
So the Whittaker function form of the solution of KG equation is only one of equivalent forms in expressing it.
The Greiner book's PDF is on the www - it is downladable.
Regards,
Tibor K. Pogany
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