Oh professor, I think you took too much off!Oh, sorry. So you replaced the Hamiltonian

by its normal-ordered form, where all annihilation operators are pushed to the right:

Somehow I hadn't noticed that. This will come in handy in full-fledged
quantum field theory, but it's not necessary here, and it's sort of enlightening
*not* to do it.

Hmm, maybe you're saying the formulas are trying to tell me something-- that IMessing around with the vacuum energy, eh? You may unleash powerful forces -- forces that mankind was never meant to meddle with! For example, you have a perfectly nice representation of the Lie algebra of the symplectic group on your hands; getting rid of the vacuum energy will turn it into a nastyshouldn'tmonkey around with the zero-point?

In simpler terms: there are lots of interesting classical observables
built using quadratic expressions in the *p*'s and *q*'s, of
which the Hamiltonian is one. When we replace the classical *p*'s
and *q*'s by operators, we'd like Poisson brackets to go over to commutators.
If we try to do this for general polynomials in the *p*'s and *q*'s,
it doesn't work very well. However, for quadratic expressions in the *p*'s
and *q*'s it does, *if* we don't mess with them by normal-ordering.

As for the rest of your post...

Great. You got a very nice formula:

implying

which is a very precise way of stating what you noted quite a while
ago: *the number of quanta in a coherent state is given by a Poisson
distribution*.

But now let's see what happens if we evolve our coherent state in time. We'll see something nice, a cute relation between the harmonic oscillator and the spin-1/2 particle, which we discussed once upon a time...