Photon wave mechanics and position eigenvectors

TL;DR

This paper develops a photon wave function framework using a photon position operator, linking different wave function forms, and providing a quantum description of photon localization and angular momentum.

Contribution

It introduces a photon wave function based on a photon position operator that couples spin and orbital angular momentum, unifying various forms in the literature.

Findings

Photon wave functions derived from a photon position operator.

Biorthogonal field-potential pair preserves eigenvalues and expectation values.

Framework compatible with quantum mechanics rules and describes optical angular momentum.

Abstract

One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.