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This paper examines how best to associate quantum states of a single particleto different modes of a narrowly collimated beam of classical radiation modeledin the paraxial approximation. Our analysis stresses the importance of therelationship between two inner products naturally arising in the problem. Theseare the inner product used to expand a general beam as a superposition oforthogonal modes in the paraxial approximation, on the one hand, and thecanonical inner product on which the statistical interpretation of quantum(field) theory is founded, on the other. While several candidates for the sortof association between beam modes and single-particle quantum states have beenproposed in the literature, here we argue that one of them is uniquely wellsuited to the task. Specifically, the mapping of beam modes to the``henochromatic'' quantum states previously introduced by Sudarshan, Simon andMukunda is unique within a large class of similar mappings in that it isunitary in a mathematically precise sense.

The paraxial approximation in quantum optics I: Henochromatic modes of a scalar field