OK, so let's start with a classical dot racing around in a circle. Projecting onto theOf course, by ``x-axis gives simple harmonic motion. We'll useqfor thex-coordinate andpfor the y-coordinate, following Hamiltonian tradition.

Now as complex number,Indeed. Ever wonder why they call it ``phase space''? I don't know the history, but here we see a damn good reason: as our pointq+ipis that racing dot. Everything flows fromq+ip.

The whole point of coherent states is to see very clearly what happens
to this picture when we go into quantum mechanics. Of course, everybody
knows that when *p* and *q* become operators, we can make *Z*
and its complex conjugate into operators too, which are basically just
the creation and annihilation operators:

But the cool part is that there are also *states* that are quantum
analogues of points circling the origin: the coherent states. As you note,
these are just the eigenstates of the annihilation operator. But I prefer
to visualize them as Gaussian wavefunctions: a kind of blurred-out version
of a state in which a particle has a definite position and momentum. If
you start with such a state, and evolve it in time, its (expectation value
of) position and momentum oscillate just like that of a classical particle,
and if I remember correctly, they maintain a basically Gaussian shape,
though probably with some funny complex phase factor stuff thrown in...

...nothing races around at all, instead we have a bunch of wavefunctions that justYeah, sure, thesit there...

``Take this eraser. [I brandish my eraser threateningly as I stand before
the blackboard.] Put it into an eigenstate of the Hamiltonian. Now it's
in a stationary state! It doesn't *do anything* as time passes. It
just sits there, except for a time-dependent phase! [I demonstrate an eraser
nonchalantly hovering in midair, only its phase wiggling slightly.] So
what does this mean, that you can levitate an eraser just by putting it
into an eigenstate?''

And of course they eventually get the point: it's not so easy in practice
to put anything big into an eigenstate of the Hamiltonian. It's the *coherent
states* that are close to the classical physics we know and love.

Could these utterly different kinematics really come from the same cinematographer?That should be true. Remember, the expected value of the Hamiltonian isSure! Just use coherent states. As

Dincreases continuously, the average energy increases continuously, as does the range of motion of the bump.Hmm. Guess I should check that last assertion.

As we take our basic Gaussian bump (the ground state) and translate
it, <*p ^{2}*> stays the same, since its shape stays the
same. <