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I've seen both classical and quantized electromagnetic fields in quantum mechanics problems - for example, classical in linear response and the Coulomb interaction, and quantum in photon absorption - but I've never read or heard a clear characterization of when the quantum treatment is needed. What are the criteria that necessitate a quantum treatment?

This came to mind when thinking of cyclotrons and synchrotrons. Despite being particle accelerators, i.e. accelerating distinctly quantum payloads, and using high-frequency pulses to accelerate their payloads, they can be understood with simple classical physics. Perhaps it's more of a surprise that the particle can be understood classically than that the field can, though.

Usually, a few quantum particles behave quantum-ly and many quantum particles behave classically. However, there are exceptions, such as superconductors and the Bose-Einstein condensate, so number of particles isn't a sufficient condition for classical treatment to be valid. Therefore I'd expect that in most cases it would be fine to use classical electromagnetic fields as long as the intensity (photon density) is high.

Is that generally the right idea? What are the exceptions for photons?

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    $\begingroup$ You're basically asking when we can afford to be semiclassical in a specific way. You may be interested in this. $\endgroup$
    – J.G.
    Commented Oct 22, 2021 at 15:56
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    $\begingroup$ This is discussed in Landau volume 4 section 5, yes - as they discuss there - there is indeed more to it than just that the occupation numbers are large. $\endgroup$
    – bolbteppa
    Commented Oct 22, 2021 at 16:03
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    $\begingroup$ @bobteppa Interesting, can you give a more precise reference within those 60 pages? $\endgroup$
    – my2cts
    Commented Oct 22, 2021 at 17:48
  • $\begingroup$ There is a class of states of light that can’t be described classically. But are you disregarding those and working only with “classical” states of light? $\endgroup$
    – Superfast Jellyfish
    Commented Oct 22, 2021 at 19:50

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