Some remarks about the `free' Dirac particle in a one-dimensional box

Abstract

The problem of a relativistic `free' Dirac particle in a one-dimensional box, i.e., at the box, but not confined to the box, is considered. A four-parameter family of self-adjoint extensions of the momentum operator P = -i12(d/dx) is obtained, as well as sub-families of boundary conditions for which this operator transforms as a vector. Physical conditions (self-adjointness and not spontaneously broken C T symmetry in the subspace of positive energies) imposed upon the Hamiltonian operator, which is a function of the momentum operator, give the physical Hamiltonian operator for this problem. The physical self-adjoint extension of H corresponds to the periodic boundary condition.