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John Baez: A Rather Odd Question

Say we have a mono-chromatic beam of light, photon density N photons per square meter per second, angular frequency w. [Pronounce that w as if it were spelled omega.] Well, this is also a plane electromagnetic wave with amplitude A and angular frequency w. Ought to be possible just using dimensional analysis to figure out the relationship of A to N and vice versa.
Hmm. Who wanted to know that, anyway? The maniacal Weiss? The judge? (What judge? I seem to remember a judge.) Anyway, it's a rather odd question if the goal is to learn quantum field theory, as Weiss apparently wanted. While it should indeed be possible to answer it using only dimensional analysis, that would sort of miss the interesting aspect: namely, that electromagnetic field strength and photon number are probably noncommuting observables, so an electromagnetic plane wave with a precisely measured amplitude probably doesn't have a precise photon density, and conversely. Maybe Weiss meant the average photon density.

Let's see if I can work out the answer using just dimensional analysis, the way Weiss wanted. Then if he kidnaps me again I can tell him about coherent states and all that if he promises to pay for it.

This is not gonna be very precise, after being dosed with nitrous and then knocked over the head.

First let's think of light as being classical. Then the energy density is (E2 + B2)/2, modulo a possible factor of 4 pi that comes from using funny units. In a plane wave of light E and B keep wiggling back and forth and turning into each other, but the overall size of either of them is the amplitude A. So the energy density is something like A2. Maybe there's supposed to be a 1/2 or something somewhere, since neither the E nor the B field is equal to its maximum A all the time, but let's not worry about that...we'll be glad if we get anything close to the right answer. Okay, so A2 is the energy density, measured in something like joules per cubic meter.

On the other hand, let's think of light as being made of photons. The energy of a photon is E = hbar w where hbar is Planck's constant and w is the angular frequency, not the number of full wiggles per second but that times 2 pi (which is why hbar = h/2pi). So the energy density should be hbar w d, where d is the average number of photons per cubic meter. So we should have

A2 = hbar w d

Does that make sense? The lower the frequency, the more photons we need to get a particular energy, so that part makes sense. It's sort of odd how the number of photons is proportional to the square of the amplitude; you mighta thunk it would just be proportional.

Hmm, this is sort of like a physics qualifying exam...which is one of the reasons I didn't go into physics...all those questions like ``Say you drop a pion from a height of 2 meters in an external magnetic field of 40 Gauss pointing along the y axis. How high will the pion bounce and what is its charge?'' So full of weird kinds of intuitive reasoning, so much harder than proving theorems.

Anyway, let's see, for some reason Weiss wanted to know the density N of photons per square meter per second, instead of per cubic meter. Since photons move at the speed of light, I guess ``per second'' is the same as ``c times per meter'' here, so N = cd, so

A2 = hbar w N/c.

Hmm. So what might be fun is to remember how coherent states work and see why the state that looks like a plane wave of amplitude A has average photon density proportional to sqrt(A), if that's actually true.

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Michael Weiss