Completeness of scattering states of the Dirac Hamiltonian with a step potential

TL;DR

This paper explicitly demonstrates the completeness and orthonormality of eigenfunctions of the Dirac Hamiltonian with a step potential, providing a basis for relativistic quantum field theoretical treatments of scattering processes.

Contribution

It provides an explicit proof of the completeness and orthonormality of scattering eigenfunctions for the Dirac Hamiltonian with a step potential, facilitating future field theory analyses.

Findings

Eigenfunctions form a complete basis for the system

Explicit summation confirms the resolution of the identity

Proper momentum integration is crucial for proof

Abstract

The completeness, together with the orthonormality, of the eigenfunctions of the Dirac Hamiltonian with a step potential is shown explicitly. These eigenfunctions describe the scattering process of a relativistic fermion off the step potential and the resolution of the identity in terms of them (completeness) is shown by explicitly summing them up, where appropriate treatments of the momentum integrations are crucial. The result would bring about a basis on which a field theoretical treatment for such a system can be developed.