Confluent Heun functions and the Coulomb problem for spin 1/2 particle in Minkowski space
Abstract
In the paper, the wellknown quantum mechanical problem of a spin 1/2 particle in external Coulomb potential, reduced to a system of two firstorder differential equations, is studied from the point of view of possible applications of the Heun function theory to treat this system. It is shown that in addition to the standard way to solve the problem in terms of the confluent hypergeometric functions (proposed in 1928 by G. Darvin and W. Gordon), there are possible several other possibilities which rely on applying the confluent Heun functions. Namely, in the paper there are elaborated two combined possibilities to construct solutions: the first applies when one equation of the pair of relevant functions is expressed trough hypergeometric functions, and another constructed in terms of confluent Heun functions. In this respect, certain relations between the two classes of functions are established. It is shown that both functions of the system may be expressed in terms of confluent Heun functions. All the ways to study this problem lead us to a single energy spectrum, which indicates their correctness.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.8300
 Bibcode:
 2014arXiv1410.8300B
 Keywords:

 Mathematical Physics;
 Quantum Physics;
 33E30;
 34B30
 EPrint:
 22 pages, submitted to: Nonlinear Phenomena in Complex Systems