General conditions for vanishing the current J^{z} for Dirac field on boundaries of the domain between two planes

TL;DR

This paper derives general conditions under which the z-component of the Dirac current vanishes at boundaries between two planes, providing explicit solutions and quantization rules for the momentum parameter in a magnetic field context.

Contribution

It formulates the conditions for zero current at boundaries as a solvable algebraic system and explicitly finds solutions for the roots of the resulting algebraic equation.

Findings

Explicit solutions for the roots of the algebraic equation governing boundary conditions.

Quantization rules for the Dirac particle momentum parameter k.

Dependence of solutions on four arbitrary phase parameters.

Abstract

In connection with the Casimir effect for a spinor field in presence of external magnetic field, of special interest are solutions of the Dirac equation in the domains restricted by two planes, which have vanishing the third projection of the conserved current J^{z} on two boundaries. General conditions for vanishing the current are formulated, they reduce to linear homogeneous algebraic system, for which solutions exist only when vanishing the determinant of the linear system, that is for the roots of a 4-th order algebraic equation with respect to the variable e^{2ik a}, where a is a half-distance between the planes, and k stands for the third projection of the Dirac particle momentum. All solutions of the equation have been found explicitly, each of them provides us in principle with a special possibility to get the quantization rules for parameter k; the most of produced expression for the roots can be solved with respect to parameter k only numerically. Generally, solutions e^{2ik a} depend on 4 arbitrary phase parameters which influence the appropriate wave functions with vanishing current: J^{z}(z=-z,+a)=0.