Michael Weiss: Okay, thanks,  Schmotons  John Baez: My lysdexia

John Baez: Michael has roamed ergodically

Okay, after having caught various slips by the absentminded professor, Michael has shown the following:

Let z = c+ib be a point in the phase space of a particle on the line, corresponding to position c and momentum b.

The coherent state with average position c and momentum b is

Coh(z) = e^{-icp + ibq} |0>

                    = e^{(za* - z*a)/ sq2} |0>

Very good!

Here's some more homework. Actually, looking back over this thread, I see that Michael has roamed ergodically over the space of ways of thinking of this stuff, and has come very close to almost all possible ways, so this homework is not terribly novel.

A) Use the formula

Coh(z) = e^{(za* - z*a)/ sq2} |0>

to get a curiously similar formula involving an exponential of only creation operators, applied to the vacuum.

The formula is something like

Coh(z) = exp(-|z|²) e^za* |0>

but you'll need to stick in a couple of constant factors here and there.

(Actually Michael has already done something like this, starting from a different angle.)

B) Use the commutation relations between H and a^* to work out

e^itH a^* e^-itH

and then

e^itH e^za* e^-itH

Together with A), use this to work out the time evolution of the coherent state Coh(z).

 C) Show that if we evolve a coherent state over one period of our oscillator -- i.e., take t = 2 pi -- it does not return to the same wavefunction, unlike for the classical oscillator.

This corrects a little mistake of Michael's [HA! see below. --ed.] where he claimed that

e^-itH Coh(z) = Coh(e^-it z).

It's not quite so simple and nice. Hint: vacuum energy.

3/10/1998