Before I dig into the business of working out what coherent states look like in the basis of eigenstates of the harmonic oscillator Hamiltonian, let me comment on one thing:Uh-oh, better get the rest of my homework in before the professor goes over the assignment in class.

OK, last time I figured out that the coherent state you get by sliding
the Gaussian bump *c* units to the right:

is proportional to this, in the basis of eigenstates of the Hamiltonian, aka ``particle representation'':

where *D=c/ *sq2 [sq2 being our symbol for sqrt(2)]. This time
I've included the factor exp(-*D*^{2}/2)* * so as
to get a normalized state-vector. Since *e*^{-icp}
is a unitary operator, by some theorem or other, *e ^{-icp} *|0>
is also normalized. So we've expressed the coherent state in the particle
representation, up to a phase.

The probability distribution of this state-vector is a Poisson distribution with mean value
*D ^{2}*:

(To do the mean-value sum, notice that the *n*=0 term politely
disappears, and if you pull out a factor of *D ^{2}*, what's
left is just the sum of all the probabilities-- which had better add up
to 1.)

Now this is nice. The original question was:

``If an electromagnetic wave has amplitudeand dimensional analysis said:Aand angular frequencyw, how many photons does it contain per unit volume, on the average?''

But to make this convincing, we have to know that *D* really *does*
correspond to the radius of the dot's racetrack. So we need the time-evolution
of the coherent state.

Now here I got stuck for a while, for I ignored the old adage: ``Never
compute anything in physics unless you already know the answer!'' (Who
said that, Wheeler?) I was trying to show that *e ^{-icp} *|0>
would evolve to

No good! The coherent state wavefunction has both position *and*
momentum encoded in it. You can't expect the position to change without
the momentum *also* changing.

OK, let's start over. First, we have to consider a more general coherent
state, say one with momentum *b* and position *c*, thus:

which I'll call Coh1*(c+ib)*. [ Coh1 because it's ``coherent state,
take 1'': John Baez will shortly introduce a better choice. --*ed.*]

Now we want to express this in the particle representation, because
we know how the states |*n*> evolve: |*n*> evolves to *e ^{-int
}*|

I did this for the special coherent state Coh1*(c)* a while back,
but I was working entirely too hard. I used the fact:

Now this looks a lot like a derivative formula: [*q,p ^{n}*]

i.e., bracketing with *p* is just about the same as taking derivatives
with respect to *x*. Now *q* and *x* are pretty closely
related, and what holds for *q* ought to hold for *p*, using
Fourier transforms and all.

So we should have:

at least if *f*(*q*) is a power series in *q*, and *g*(*p*)
is a power series in *p*.

As a check, let's verify the product rule. If that works, then we should
have our result for all powers of *p* and *q* just by induction,
and then for all power series by continuity arguments. The continuity arguments
might take up a chapter or two in a functional analysis textbook, but hey,
I'm sure the kindly moderator will cut us some slack. *(Pause for thunderbolts
to dissipate.)*

It works!

So:

You have to be a little bit careful here. *e ^{A}*

If that last equation doesn't leap out at you from the previous two, well it's just a bit of straightforward grinding.

And so:

So Coh1*(c+ib)* is an eigenvector of *q*+*ip* with eigenvalue
*c*+*ib*. It then follows that, up to a phase (call it *iota*):

Remember now that we want to time-evolve this. (So much physics, so
little *e ^{-iHt}...*) As I said earlier, |

So the *n*-th term of our formula for Coh1*(c+ib)* will acquire
the factor *e*^{-int} in *t*-seconds. But we can
absorb this into the factor (*c*+*ib*)* ^{n}*, just
by replacing

Roll over Beethoven, how classical can you get! If I told you that Coh1*(c+ib)*
was my symbol for a dot in the complex plane at position *c*+*ib*,
you'd say the equation I just wrote is obvious.

[Moderator's note: The aphorism, ``Never calculate anything until you
know the answer,'' is indeed due to Wheeler. It appears in Taylor and Wheeler's
book *Spacetime Physics* under the name of ``Wheeler's First Moral
Principle.'' No other moral principles are mentioned, so maybe it's Wheeler's
Only Moral Principle. --Ted Bunn]