I've recently been learning the basics of Quantum optics and it seems to be a fundamental concept that light is best described in the framework of the Quantum Harmonic Oscillator.

This lead to a relation for the Hamiltonian which is not clear to me $$\int \frac{\varepsilon_{0}}{2}\left(\varepsilon \hat{E}^{2}+\frac{c^{2}}{\mu} \hat{B}^{2}\right) \mathrm{d} x=\sum_{k} \hbar \omega_{k}\left(\hat{a}_{k}^{\dagger} \hat{a}_{k}+\frac{1}{2}\right)$$

Why must every particle be treated as an identical H.O., is this just a good model of is there more mathematical significance that I'm missing?

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    $\begingroup$ Related. All fields, including the EM field, are quantized into an infinity of free abstract oscillators in momentum space. These oscillators represent photons, the quanta of the respective field. $\endgroup$ – Cosmas Zachos Jul 25 at 19:11
  • $\begingroup$ Possibly useful. $\endgroup$ – Cosmas Zachos Jul 25 at 19:13
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    $\begingroup$ If you are not already familiar with the commutator/operator treatment of the QHO (that is, if the $\hat{a}_k^\dagger$ and $\hat{a}_k$ notation is unfamiliar to you) you really need to go back and bone up on that first. If you are familiar with it then the trick is to compare the RHS of the block equation to the naive Hamiltonian for a more familiar oscillator (say a mass on a spring) to see where to jump off. $\endgroup$ – dmckee Jul 25 at 19:42
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    $\begingroup$ @CosmasZachos , what about non-abelian fields?) and what about quantization of fields for non-stationary situation? $\endgroup$ – Artem Alexandrov Jul 25 at 21:33
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    $\begingroup$ @Artem & OP: of course my pointer is general. I'm just reminding you field quantization is elaborate and precise, and covered in all QFT books but cannot be done justice to in a PSE answer, a WP mini-review, or a glib video. I fear half the QFT answers on this site are remedial plugs for something settled in all decent courses. Something not all QFT courses emphasize adequately is the normal mode description of classical fields, as a repackaging of an infinite system of coupled, classical harmonic oscillators, before one even quantizes! $\endgroup$ – Cosmas Zachos Jul 25 at 21:46

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