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Michael Weiss: Roll over Beethoven

John Baez writes:
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:

e-icp |0>

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

exp(-D2/2) SUMn (Dn/ sqrt(n!)) |n>

where D=c/ sq2  [sq2 being our symbol for sqrt(2)]. This time I've included the factor exp(-D2/2)  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 D2:

Prob(we're in state |n> ) = exp(-D2)  ((D2)n/n!)

Mean value = SUMnProb(we're in state |n>)

(To do the mean-value sum, notice that the n=0 term politely disappears, and if you pull out a factor of D2, 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 amplitude A and angular frequency w, how many photons does it contain per unit volume, on the average?''
and dimensional analysis said: A2/w. Well, presumably c and D will end up being proportional to A, once we actually start talking about electromagnetic waves, and not this mickey-mouse-dot racing around a circle! And the mean value should be proportional to the photon density. So we have our explanation for the A2 factor.

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 e-icp cos t|0>, since that's the formula for simple harmonic motion.

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:

eibq e-icp |0>

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 |n> (if we factor out the common e-it/2, i.e., redefine the energy zero-point).

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

[q,pn] = inpn-1

Now this looks a lot like a derivative formula: [q,pn] = i (dpn/dp). (Ignore the fact that I haven't defined d/dp.) Actually, this shouldn't be surprising: we know that

[p,A] = -i (dA/dx)

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:

[p, f(q)] = -i df/dq

[q, g(p)] = i dg/dp

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.)

[s, AB] = sAB - ABs

[s, A] B + A [s, B] = (sA - As) B + A (sB - Bs)

                          = sAB - AsB + AsB - ABs

It works!


[q, e-icp] = i(-ic) e-icp = c e-icp

[p, eibq] = -i(ib) eibq = b eibq

[(q+ip), eibq e-icp] = (c+ib) eibq e-ipc

You have to be a little bit careful here. eA eB is not generally equal to eA+B if A and B don't commute.

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):

Coh1(c+ib) = iota e-(c2+b2)/2 SUMn ((c+ib)n/ sqrt(n!)) |n>

Remember now that we want to time-evolve this. (So much physics, so little e-iHt...) As I said earlier, |n> evolves to e-int |n> (if choose our energy zero-point so as to get rid of the vacuum energy).

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 c+ib with (c+ib)e-it. So:

e-iHt Coh1(c+ib) = Coh1( e-it (c+ib) )

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]

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Michael Weiss