# Proofs that every professional physicist should know

There's been a lot of questions about books that every physicist should read, or what are the most important papers in physics. I would say there are also proofs in physics that have that wow factor about them; proofs which suddenly switch the light on. An elementary example is Euler's laws of motion that are proven from Newton's laws of motion.

What other proofs should every professional physicist know?

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I'm not sure how much I like this question, but if it is to be on the site it's a perfect candidate for community wiki, so I'm wikifying it. – David Z Nov 4 '11 at 15:50

You have to interpret the question restrictively to get a reasonable answer-domain. If you include mathematics, there are too many to list. I will ignore any theorem which is typically taught in graduate or undergraduate classes, and the result also has to be relatively easy to make completely rigorous, and the proof will have to involve a deep qualitative idea.

Here is a very partial list, based on whim:

• The Hawking area theorem, because the theorem and the proof both provide exceptional physical insight. This is detailed here: Second Law of Black Hole Thermodynamics . The Penrose theorem on gravitational collapse is similarly physical and similarly essential. There are many other results in gravitational physics that are phrased as theorems, although perhaps not with so elegant a proof.
• In statistical physics, there is another beautiful proof I think is simple and elegant, and has deep consequences: this is the no-passing theorem for elastic depinning due to Alan Middleton. The theorem says that if you start two interfaces in a depinning model with one behind the other, then the one behind will never overtake the one in front. The reason is that, in depinning, the rule for moving forward is determined by elastic forces, and when two interfaces, A and B, collide at point x, and A is behind B everywhere else, it is easy to see that the elastic forces on A hold the point x on A back more than the elastic forces hold the point x on B back. So if A moves forward, then B moves forward at least as much, and A cannot overtake B. This is extremely significant, because it means that the transition from no-motion to motion in depinning models must be second order.
• In high energy physics, there is the Froissart theorem: in a theory with a mass-gap, the total cross section can only grow as the squared logarithm of the center of mass energy. This is an essnetial theorem for the development of physics, which is unfairly left out of the curriculum, because of its associations with S-matrix theory.
• In thermodynamics, there is the Onsager Reciprocity theorems for near-equilbrium transport coefficients with time-reversal. This is foundational, it won the nobel prize, and it is not in the curriculum.
• In fluid dynamics, there is Helmholtz's theorem on the advection of vorticity in inviscid flow.
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I have devised a simple geometric proof that shows why the effective cross-sectional area of a resonant absorber is independent of the physical size of the absorber. I worked this out originally to explain how a crystal radio can absorb power from a distant transmitter, but it has obvious implications for quantum mechanics: it makes nonsense out of most of the arguments whereby the photo-electric effect cannot be explained by the wave theory of light, especially those arguments based on the cross-sectional area of a single atom. The proof is based on the following diagram which is almost self explanatory:

This explanation should be part of any undergraduate physics degree.

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tanks for your clarifying paper There Are No Pea-Shooters for Photon. It sould be carefully read by physicists and candidates to (and dont worry with the downvotes). – Helder Velez Nov 6 '11 at 18:29
Thanks, Helder. I'm not used to people agreeing with me. – Marty Green Nov 6 '11 at 19:04
@Marty: the reason people say that the photoelectric effect cannot be explained by wave theory is not because atoms cannot absorb classical radiation, but the emitted electron has an energy which is gotten from the wave, but is independent of the wave amplitude. This is classically impossible--- the electron classically shakes in response to the local wave amplitude. – Ron Maimon Nov 6 '11 at 20:43
Did you read my article which Helder referenced? The photoelectric effect is classically impossible only if you insist on using a tiny charged ping-pong ball as your model of the electron. If you use a proper quantum model of the electron, namely Schroedinger's equation, then Maxwell's equations are sufficient to complete the picture. If the photoelectric effect is classically impossible, then the crystal radio is also classically impossible. – Marty Green Nov 6 '11 at 20:57
@Marty: If you use a Schrodinger electron, but a semiclassical EM field, you are in the domain of Bohr Kramers Slater theory. The major problem with this idea (which was historically viable in 1924) is the conservation of energy--- if you have a wave with only a few photons in it, and it washes over atoms, the atoms have to absorb the photons locally and zero out the wave globally, which is not compatible with a finite speed of light. BKS proposed that energy is only statistically conserved, but now we know better. Anyway, this is not the explanation of the photoelectric effect for sure. – Ron Maimon Nov 6 '11 at 21:45