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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?

user2757771
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    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. – Cosmas Zachos Jul 25 '19 at 19:11
  • Possibly useful. – Cosmas Zachos Jul 25 '19 at 19:13
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    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. – dmckee --- ex-moderator kitten Jul 25 '19 at 19:42
  • @CosmasZachos thanks for the link. I'm interested in an elaboration of the last point of the first answer. "the Standard Model of particle physics, is ultimately based on quantizing classical fields (like electromagnetic fields) and realizing that particles basically just emerge from excitations of these fields, and these excitations are mathematically modeled as an infinite system of coupled, quantum harmonic oscillators." – user2757771 Jul 25 '19 at 21:11
  • @dmckee I'm familiar with the formalism for the QHO, I just don't see how this is an equivalent description for a quantized electromagnetic field – user2757771 Jul 25 '19 at 21:14
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    @CosmasZachos , what about non-abelian fields?) and what about quantization of fields for non-stationary situation? – Artem Alexandrov Jul 25 '19 at 21:33
  • @CosmasZachos , I mean only that your first comment seems very "general" – Artem Alexandrov Jul 25 '19 at 21:33
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    @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! – Cosmas Zachos Jul 25 '19 at 21:46
  • @Artem Nonabelian fields are accommodated trivially by multiplets of oscillators through the Jordan map; no subtlety there. Time dependence is easily worked out as in the Heisenberg picture or interaction picture, when it comes to realistic applications. Remember: "oscillator" is just the basic bit of operator in the Fourier split of the quantum field operator--the sand grain to the dune. – Cosmas Zachos Jul 26 '19 at 14:10
  • @CosmasZachos , thank you for interesting comments! – Artem Alexandrov Jul 27 '19 at 08:59

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