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.

Energy spectrum of a relativistic two-dimensional hydrogen-like atom in a constant magnetic field of arbitrary strength

“…It is worth mentioning that angular parameter l takes half integer values [27] and therefore it can be related to m as follows…”

Relativistic electron wave packets featuring persistent quantum backflow

“…The separability of Dirac's equation in axially symmetric potentials has been widely studied [38,39], [40, p. 134], even with a nonminimal (or anomalous moment) coupling [41][42][43], driven by applications in, e.g., electron vortex beam experiments [44][45][46][47], Kumakhov (or channeling) radiation studies [48][49][50], the Aharonov-Bohm (-Casher) effect, and neutron confinement [51]. There are only two axisymmetric (electrostatic) potentials for which the Dirac equation is solvable, viz., the constant potential and the −1/ρ potential [52][53][54]. The former opens the possibility of piecewiseconstant potentials capable of modelling a cylindrical WG, e.g., the finite square well [55,Sec.…”

A Square-Integrable Spinor Solution to Non-Interacting Dirac Equations

“…One of the most important problems in mathematical physics is to study the properties of the Dirac spinor field equations by assigning a specific potential and finding its exact solutions, according to various methods [1][2][3][4][5][6][7][8][9][10][11][12][13].…”