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I'm thinking about the wavelength of a freely propagating photon vs. a freely propagating electron.

For the photon, we have the classical picture of oscillating [itex]E[/itex] and [itex]B[/itex] fields perpendicular to the direction of propagation, and we call the wavelength of the photon, which can be identified as its de Broglie wavelength, as the length between successive crests of either of those fields.

For an electron, we can use the simple quantum picture of an oscillating probability function, and we can identify the de Broglie wavelength as the length between successive crests of the probability field.

As far as I know there is no transverse oscillating EM field associated with a freely propagating electron, and the Schrodinger equation is inadequate to analyze the wavefunction of a photon - so, the classical picture of the dB wavelength fails for the electron, and the quantum picture fails for the photon.

Probably something like QFT is needed to really understand what is going on, but does anyone know of a model or a conceptual picture that makes sense for a direct comparison for the wavelengths of electrons or photons that doesn't require jumping between two theories? For instance,cana freely propagating electron be modeled as an oscillating EM field in the same manner as a photon (this doesn't seem correct to me, but I'm wanting to give an idea of the kind of thig I'm looking for)? Similarly, can we picture the photon as a propagating probability oscillation (or more what I'm looking for, if so how does this then lead to the classical oscillating EM fields)? Or is there a good semi-classical model that fits both well, and reveals some interesting stuff as a result/gives a good intuition of why the concept of a dB wavelength is applicable to both photons and electrons as a result?

Thanks very much.

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# De Broglie wavelength - model for comparing photons and electrons

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