Dirac equation in external vector fields: New exact solutions

Abstract

New exact solutions are searched for on the basis of the method of separation of variables proposed in earlier work by the present authors [J. Math. Phys. 30, 2132 (1989)]. The essence of this method consists of constructing first-order matrix differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with the operator of the equation or between them is not assumed.The classical problems are considered as possibilities, namely, electrons in the field of plane monochromatic electromagnetic waves (Volkov’s problem) and electrons in the Coulomb field (hydrogen atom). Then ‘‘plane’’ external electromagnetic fields are considered for which some new exact solutions are obtained in terms of special functions. Four new exact solutions of the Dirac equation in the fields with axial cylindrical symmetry are also shown, and lastly one ‘‘free’’ solution with exotic geometry is demonstrated, namely, ‘‘free’’ parabolic cylindrical spinor waves.