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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
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2 Answers

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
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@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
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@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
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@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
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I just purchased Feynman's Thesis, which provides some insight on how Feynman saw the world, and provides some context here. One of the key issues Feynman was trying to reconcile in his Lagrangian approach was how to describe quantum mechanics without rely on a field defined by harmonic oscillators, from page 5:

"In particular, the problem of the equivalence in quantum mechanics of direct interaction and interaction through the agency of an intermediate harmonic oscillator will be discussed in detail. The solution of this problem is essential if one is going to be able to comapare a theory which considers field oscillators as real mechanical and quantized systems, with a theory which considers the field as just a mathematical construction of classical electrodynamics required to simplify the discussion of the interaction between particles."

So we have to understand that Feynman viewed matter as something different than the harmonic oscillator. Since mass is a parameter for a simple harmonic oscillator, we can see that in his discussion, Feynman didn't necessarily view matter as being the same thing as mass. I suspect, matter would be viewed as the tangible reality that we are familar with, and quantum harmonic oscillators are the abstract entities that we use to describe behavior, so it is some way necessarily to map, or couple, the real to the astract.

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Yes, I recognize that there must be subtleties, but I'm not even all that familiar with the slightly misleading ideas that you seem to be trying to correct. I am unfortunately much more ignorant about physics than that. –  Qiaochu Yuan Jan 22 '11 at 14:42
    
I hope I didn't offend, I didn't mean to imply or correct any misleading ideas, I just think that one often learns more by understanding the context and viewpoint of the author than from the mechanix of the math. Understanding why Feynman might phrase something in a particular way is as least as important as the physical picture he is referring to...but I digress. –  Humble Jan 22 '11 at 14:47
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