# What's a good reference for this classical picture Feynman's talking about?

I have a mathematics background but am trying to educate myself a little about physics. At the beginning of Feynman's QED book (not the popular one) is the following:

Suppose all of the atoms in the universe are in a box. Classically the box may be treated as having natural modes describable in terms of a distribution of harmonic oscillators with coupling between the oscillators and matter.

I guess this is something that physicists learn, but I have never heard of it. What is Feynman talking about and where can I learn more about it? The Wikipedia article on harmonic oscillators gives no indication that physicists do this.

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To me it sounds as if he is talking about classical Fourier analysis and its applications to solving PDEs. You separate the variables, get many equations of harmonic oscillators and so on. That is what you do in quantum field theory plus the extra step of quantizing them. –  MBN Jan 22 '11 at 5:01

This is a way of giving systematic meaning to the radiation continuum in the context of a set of discrete states.

You assume some set of boundary conditions on the EM fields where they hit the box {1}, derive a set of allowed modes in terms of the geometry of the box {2}, then allow the box to expand without limit. Thus you arrive at a continuum of allowed modes.

{1} Say $E = M = 0$ at the boundary as if the box were a very good conductor.

{2} If the fields go to zero at the sides of the box then a half-integer number of wavelengths must fit, so only some wavelengths are allowed.

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Cool. So you're talking about the electromagnetic wave equation, right? And the eigenvalues of the Laplacian are the "modes," and the eigenvalue equation is the harmonic oscillator associated to that mode? Are we not worrying about the rest of Maxwell's equations? –  Qiaochu Yuan Jan 21 '11 at 22:29
@Qiaochu: Yep, pretty much. Technically a harmonic oscillator is a system with a quadratic potential $U(x) \propto x^2$, but the quantum harmonic oscillator has an evenly spaced set of eigenvalues, and there is a tendency to talk about any other system with that property as if it were a harmonic oscillator. –  David Z Jan 21 '11 at 22:34
@David: thanks very much for clearing that up. Another question: what exactly does Feynman mean by "coupling"? –  Qiaochu Yuan Jan 21 '11 at 22:36
@Qiaochu: "coupling" generally refers to some sort of interaction. I'd guess that here Feynman is talking about the fact that the atoms can exchange energy with the electromagnetic field. If you model the EM field as an infinite set of harmonic oscillators, an atom interacts with an oscillator of natural frequency $\omega$ when the atom's energy decreases by $\hbar\omega$ and the oscillator's energy increases by the same amount (i.e. it jumps up by a mode), or vice versa. –  David Z Jan 21 '11 at 23:18
There is also a fantastic British comedy called "Coupling". Of course its more about coupling between humans than elementary particles ;) –  user346 Jan 22 '11 at 3:49