On 3D and 1D Weyl particles in a 1D box

TL;DR

This paper derives the most general self-adjoint boundary conditions for 3D and 1D Weyl particles confined in a 1D box, extending previous Dirac boundary condition results to Weyl Hamiltonians.

Contribution

It provides a comprehensive characterization of boundary conditions for Weyl particles in a 1D box, connecting 3D and 1D cases through representation changes.

Findings

Derived general self-adjoint boundary conditions for Weyl Hamiltonians.

Connected Weyl boundary conditions to Dirac boundary conditions via representation changes.

Discussed wave functions and representations for Weyl and Dirac equations in various dimensions.

Abstract

We construct the most general families of self-adjoint boundary conditions for three (equivalent) Weyl Hamiltonian operators, each describing a three-dimensional Weyl particle in a one-dimensional box situated along a Cartesian axis. These results are essentially obtained by using the most general family of self-adjoint boundary conditions for a Dirac Hamiltonian operator that describes a one-dimensional Dirac particle in a box, in the Weyl representation, and by applying simple changes of representation to this operator. Likewise, we present the most general family of self-adjoint boundary conditions for a Weyl Hamiltonian operator that describes a one-dimensional Weyl particle in a one-dimensional box. We also obtain and discuss throughout the article distinct results related to the Weyl equations in (3+1) and (1+1) dimensions, in addition to their respective wave functions, and present certain key results related to representations for the Dirac equation in (1+1) dimensions.