OK, last time I figured out that the coherent state you get by sliding the Gaussian bumpVery nice. I would like to understand this better and think more about the best way of deriving it. The way I suggested to you was stupid and grungy, but you seem to have fought your way through and then discovered some much nicer approaches. I haven't thought about this stuff enough, so I'd like to polish your work to a fine sheen before moving on.cunits to the right ...

So...let me go back to our general picture of coherent state.

What's the best quantum approximation to a classical particle on the line with specified position and momentum? A Gaussian bump with a corkscrew twist! We will only be interested here in bumps that have equal uncertainty in position and momentum:

The simplest case is when

Then we use the ground state of the harmonic oscillator:

where I left out the normalization factor to reduce clutter.

To get coherent states with other expectation values of position and momentum, say

we can take our ground state, translate it in position space by an amount
*c*, and then translate it in momentum space by an amount *b*:

where I have attempted to make an even number of sign errors. [An allusion
to Dirac's comment on the first presentation of the Klein-Nishina formula
(by Nishina). See Gamow, *Thirty Years That Shook Physics*. --*ed.*]

But wait! We could also have translated it *first* in momentum
space and *then* in position space, getting

How does this answer fit with the other? Well, it differs only by a phase.

``Only a phase''...ah, what an understatement! When physicists and mathematicians mutter darkly about ``cocycles'', ``projective representations'', ``double covers'', ``central extensions'', and even more intimidating things like ``anomalies'', ``the Virasoro algebra'' and ``affine Lie algebras'', they are secretly complaining about the many subtleties that can caused by a mere phase!

So let us think about this a little bit. The two coherent states above
differ by the phase *e*^{-ibc}. That should be no surprise;
the Heisenberg commutations relations

lead directly -- with a dose of mathematical optimism -- to the exponentiated version called the ``Weyl commutation relations''

which describe how translations in position space and momentum space commute only up to a phase. Actually, mathematical physicists of the rigorous variety prefer to take the Weyl relations as basic and derive the Heisenberg relations as a consequence. But we are being relaxed here so we can think of them as two ways of saying the same thing.

Now, Michael has taken

as his definition of a coherent state with expected momentum *b*
and expected position c. This is fine...up to a phase...but it's slightly
annoying how one needs to ``break the symmetry'' between momentum and position
in this definition. Why not

Or even better, how about some choice that treats position and momentum
even-handedly! ``Mind your *p*'s and *q*'s!'' There's much wisdom
in that phrase...

Here's a nice way to mind our *p*'s and *q*'s; we make the
following new definition:

Here we ``simultaneously translate in position and momentum space''
instead of favoring one or the other. This state is not equal to *either*
of the two choices listed above, but again it differs only by a phase.

Why?

Well, one can show that

at least if I've not made a sign error. So this new definition ``steers a middle course'' between the other two choices, phase-wise.

(Also, fans of symplectic geometry will appreciate the funny skew- symmetric
quality of the expression -*icp* + *ibq* in our new definition.
But let's not get into that.)

Here's a little assignment for Michael, or any other students willing
to pitch in! Remember that *c* represents *position* and *b*
above represents *momentum*, so (*c*,*b*) represents a point
in phase space. Also remember that it's good to think of phase space here
as the *complex plane*. So let's define

(Don't confuse this lower-case *z* with the upper-case *Z* we
had before. The upper-case *Z* was an exhalted operator; the lower-case
*z* is just a lowly complex number.)

Now: take the expression

and rewrite it in terms of *z* and its complex conjugate *z ^{*}*,
while simultaneously rewriting

Some nice stuff should happen.

If we do this, we will get a cool expression for our coherent state
in terms of annihilation and creation operators applied to the vacuum state.
This won't immediately solve all our problems, but it should help us understand
a lot about how our coherent states of the harmonic oscillator *evolve
in time*.