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arXiv:quant-ph/0609086

Photon wave function and position eigenvectors

Margaret Hawton

Published Sep 11, 2006• 0 citations
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Abstract

One and two photon wave functions are obtained by projection onto a basis of simultaneous eigenvectors of the position and number operators.
DOI
10.48550/arxiv.quant-ph/0609086

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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 quantum mechanics, this wave function gives the probability to count a photon at any position in space.
Recently a photon position operator with commuting components and a particle density that is positive definite for both positive and negative frequency states has led to a first quantized theory of the photon. Here we extend this theory to a manifestly covariant form and show for the first time that, with invariant plane wave normalization, the position operator is Hermitian with real instantaneously localized eigenvectors. Reality of the field and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The position space wave function is the real part of the projection of the photon's state vector onto a basis of position eigenvectors. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Gupta-Bleuler gauge condition.