Say we have a mono-chromatic beam of light, photon densityHmm. 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 theNphotons per square meter per second, angular frequencyw. [Pronounce thatwas if it were spelledomega.] Well, this is also a plane electromagnetic wave with amplitudeAand angular frequencyw. Ought to be possible just using dimensional analysis to figure out the relationship ofAtoNand vice versa.

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 *( E^{2} + B^{2})/2*, modulo a possible
factor of 4

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/*2*pi*).
So the energy density should be *hbar w d*, where *d* is the
average number of photons per cubic meter. So we should have

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

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.