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
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
to get a curiously similar formula involving an exponential of only creation operators, applied to the vacuum.
The formula is something like
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
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
It's not quite so simple and nice. Hint: vacuum energy.