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Consider a single photon. Since it is not possible to create a photon with a certain frequency it can be characterized by a normalized frequency distribution $f(\nu)$ that is peaked around some mean frequency.

Now I sometimes hear or read that the Fourier transformation of $f(\nu)$ is considered as the wave function of the photon (interpreted as the probability density in space). This is especially done in quantum optics. But I don't understand that.

The reason is that in classical optics it's completely clear to consider the wave vector and the position as conjugate variables. Also in standard QM textbook this is clear due to the commutator relation of the Position and Momentum Operator (dealing with massive particles). But for a single photon, described by a creation operator, I can not find a reason to interpret the Fourier transformation of $f(\nu)$ as the spatial probability density of the photon.

AlexH
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thyme
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2 Answers2

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There is no position operator for photons, so photons do not have a spatial probability density. Associated with a photon (in a laser beam, say) one has only a probability density of hitting any given surface crossing the beam at a particular point of the surface.

See Chapter B2: Photons and Electrons (and the entries ''Particle positions and the position operator'' and ''Localization and position operators'' of Chapter B1: The Poincare group) of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html

Community
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Arnold Neumaier
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One can certainly define the wave function a photon in the representation of second quantization: $$|\mathbf{k},\lambda\rangle = b_{\mathbf{k}\lambda}^\dagger|0\rangle,$$ where $\lambda$ is the photon polarization state, and $|0\rangle$ is the photon vacuum.

In fact, one can make an even stronger claim: for massive particles the first quantization means describing them by a wave equation (Schrodinger's, Dirac, etc.), whereas the second quantization is the description by the filling numbers. For photons the Maxwell equations are already wave equations, and their description by the filling numbers is their first quantization.

Roger V.
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