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Quantizing the EM field gives you the standard sum of all quantum harmonic oscillators as given by this hamiltonian:

$$ \hat{H} = \sum_k \hbar\omega_k( \hat{a}^\dagger_k \hat{a}_k + 1/2) $$

Conceptually it makes most sense when discussing a cavity, as each harmonic oscillator corresponds to a mode with a different number of excitations in it. However, I'm having trouble understanding this formulation when applying it to a free field. A free field is just a cavity with infinitely large boundaries, right? That would seem to imply that for each frequency, there is only one harmonic oscillator.

I have also heard from many sources, including colleagues, that you can see the field as composed of a collection of harmonic oscillators separated in space. However, I find that difficult to reconcile with the above equation. What is the best way to visually interpret this?

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    $\begingroup$ Best way to visualize quantum fields: Don't. $\endgroup$ – ACuriousMind Mar 4 '15 at 14:47
  • $\begingroup$ @mactud The harmonic oscillators in your formula don't need to be separated in space, but separated in other features, e.g. frequency of oscillation (see my answer). $\endgroup$ – Sofia Mar 6 '15 at 21:46
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    $\begingroup$ @ACuriousMind oh come on, man. You can think of it almost exactly like a vibrating string. $\endgroup$ – DanielSank Jun 26 '15 at 7:19
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    $\begingroup$ @ACuriousMind, if you can't visualise it, you don't understand it. Many are reluctant to visualise precisely because it brings into sharp relief absurdities or conceptual ambiguities and deficits. $\endgroup$ – Steve Feb 7 '18 at 9:00
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What oscillates in the e.m. field is not the amplitude of some chord, but the electric field vector and the magnetic field vector. For visualizing your formula, see first of all the animated graph in this site. You see there an e.m. field propagating, and the electric and magnetic field oscillating in time between a maximally positive amplitude to a maximally negative amplitude. The oscillations have a certain frequency.

Now, imagine yourself another such field, oscillating with a different frequency, superimposed on the field in the animated picture. And then, one more field, with another frequency, and one more field, etc. The total oscillation will be wild, but there will be oscillation.

This is as far as visualizing the oscillations. However, your formula does not describe the electric and magnetic field of each oscillator. Each term in the summation represents something else, the energy of the respective oscillator. To make an association with something that you know, then, if you heard of the photoelectric effect, then you know that from a certain type of atom, an electron can be kicked out by bombarding the atom with photons of a given frequency. The frequency is proportional with the photon energy. Well your formula says that the e.m. field energy is the sum of the energies of all the single-frequency e.m. fields components.

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However, I'm having trouble understanding this formulation when applying it to a free field. A free field is just a cavity with infinitely large boundaries, right? That would seem to imply that for each frequency, there is only one harmonic oscillator.

A free field is a field that obeys homogeneous Maxwell equations (with charge and current density vanishing).

The sum over modes in your formula makes sense only for field in finite region. For whole space, the boundary conditions are different and one is lead to an integral rather than sum.

However, with such an integral over whole space there are problems, especially for thermal radiation (the integral would be divergent).

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Visualizing such a Quantum field is indeed possible, it is described in the article "A new way of visualizing Quantum fields", which you can find here:

https://www.researchgate.net/publication/321684743_A_new_way_of_visualizing_quantum_fields

Best regards Helmut

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  • $\begingroup$ Proper visualisation involves what on paper is called animation. That is, things have to move or change, or be capable of moving or changing. It is unfortunate that that paper gives little indication of how anything depicted moves. $\endgroup$ – Steve Feb 7 '18 at 9:07
  • $\begingroup$ Most of the charts in the paper visualize energy eigenstates which are essentially time-independent, so they don't really 'move' (except for a trivial rotation through the complex plane). The only exception are the states with one or two localized particles, respectively, which would evolve quickly into a more spread-out oscillation pattern as indicated in the caption to figure 6. $\endgroup$ – Helmut Feb 12 '18 at 7:38
  • $\begingroup$ Indeed. On a positive note however, I agree that good visualisation is what is needed in physics - the inability to paint an integrated picture of something, is normally a sign that the concepts being employed are in bad order, and more work of this kind ought to be done on creating convincing visualisations. I wasn't responsible for the downvote and that person appears to have left no remarks, so to counter that I've upvoted you (back to zero!). $\endgroup$ – Steve Feb 12 '18 at 17:54

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