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I'm wondering when do we need fully quantum mechanically treatment of photon.(i.e. quantized photon and quantum matter.)

I kind of know that for sub-Poisson(photon statistics) light, fully quantum mechanically treatment was necessary.

However, I heard that some other experiments like photoelectric effect was shown to be able to treat semi-classically. The interference experiment was also some where semi-classical treatment was popular, but sometimes fully quantum theory was necessary.

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    $\begingroup$ Another example is spontaneous emission (although stochastic semi-classical approaches reproduce it). $\endgroup$ – Sunyam May 7 at 21:14
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You cannot treat a photon quantum mechanically. In the sense that quantum mechanics cannot describe photons.

For several reasons:

  • Quantum mechanics is just built for few-body problems, it does not extend well (in terms of tidiness) to many-body problems, such as large sources of photons/high intensities
  • Normalisation. Photons can be absorbed / emitted, i.e. (dis)apperaing from thin air. Cannot normalise the wavefunction
  • Causality. Quantum mechanics does not preserve causality. Relativistic quantum mechanics (where the Hamiltonian $H=\sqrt{p^2+m^2}$) does not either. And for photons, moving at $c$, causality is quite important.

Hence, a new framework is needed, quantum field theory, where the wavefunction is not the protagonist but rather the quantum field operators. When people talk about "quantum treatment" of light, they mean quantum field theory.

Quantising light means that you define the creation operator $a^\dagger_{\mathbf{k}}$, such that you create one photon at momentum $\mathbf{k}$ from the vacuum $|0\rangle$: $$a^\dagger_{\mathbf{k}} |0\rangle = |\mathbf{k}\rangle .$$

Semi-classical treatments usually see the atoms being treated* quantum mechanically*, meaning they have discrete energy levels and obey the Schrödinger equation, while the light is treated classically, i.e. made of a continuum of photons of given intensity I.

The semi-classical approximation is what's behind the usual treatment of light-matter interactions (look up "dipole approximation"), light absorption from a medium, photoelectric effect etc.

It breaks down for single or few photon events, where you really need to consider photons individually. It also breaks down for spontaneous emission, because this process involves the various "photon modes" of the vacuum (different $a^\dagger$) coupling to your specific system. In these cases you need to quantise both the atom and the light, see e.g. here.

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