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John Baez: A Gaussian bump with a corkscrew twist!

OK, last time I figured out that the coherent state you get by sliding the Gaussian bump c units to the right ...
Very 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.

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:

Delta p = Delta q = 1/2

The simplest case is when

<p> = <q> = 0

Then we use the ground state of the harmonic oscillator:

|0> = exp(-x2)

where I left out the normalization factor to reduce clutter.

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

<p> = b,    <q> = c

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:

eibq e-icp |0> = exp(ibx - (x-c)2)

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

e-icp eibq |0> = exp(ib(x-c) - (x-c)2)

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

pq-qp = -i

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

e-icp eibq = e-ibc eibq e-icp

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

Coh1(c+ib) = eibq e-icp |0>

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

Coh2(c+ib) = e-icp eibq |0>    ?

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:

Coh(c+ib) = e-icp + ibq |0>

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.


Well, one can show that

e-icp+ibq = e-ibc/2 eibq e-icp

               = e+ibc/2 e-icp eibq

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

z = c+ib.

(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

e-icp + ibq

and rewrite it in terms of z and its complex conjugate z*, while simultaneously rewriting p and q in terms of annihilation and creation operators. Remember that

a = (q+ip)/ sq2     q = (a+a*)/ sq2

a* = (q-ip)/ sq2     p = (a-a*)/i sq2

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.

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Michael Weiss