Avodyne said:
strangerep, I thought the Heisenberg algebra was [x,p]=c; how does that help us see that the harmonic oscillator has coherent states?
Sorry for taking so long to reply. I'm severely pressed for time right now.
The short (probably inadequate) answer is that it's much easier to see in
Fock representation. From the Heisenberg algebra (expressed in terms of
the usual annihilation and creation operators) one can look for
eigenstates of the annihilation operator, ie: a |v\rangle = v<br />
|v\rangle, and it then takes only a few lines to show that such
states have the required minimum uncertainty product.
With a little more work, one can show that such states can be
expressed as
<br />
|v\rangle = e^{va^\dagger - v^* a} |0> ~~(1)<br />
See, eg, Mandel & Wolf, "Optical Coherence & Quantum Optics" ch11,
for more details.
Enlarging the basic Heisenberg algebra to include the Hamiltonian (or
in this case the number operator) H \propto a^\dagger a),
we get the "oscillator algebra" and the important feature here is that
[H,a] \propto a and [H,a^\dagger] \propto<br />
a^\dagger. This means that the action of H on a coherent state
|v\rangle always yield some other coherent state
|w\rangle (which follows from (1) above).
Standing back a bit, the key point of all this is that we have a
basic algebra (Heisenberg) which the Hamiltonian preserves, meaning
that the adjoint action of the Hamiltonian on this algebra (both
considered as elements of a larger enveloping algebra) yields
another element of that same basic algebra. In the present case,
this means that although coherent states are not eigenstates of
the Hamiltonian, we
can say that
coherent states evolve into (other)
coherent states and we already know that all coherent states have
minimum uncertainty product.
There is a generalization of all this: given a dynamical algebra "A",
we find the largest commuting subalgebra "C" containing the
Hamiltonian, and then construct states by cyclically acting on a vacuum
state with the (exponentiated) remaining generators in A/C. Details can
be found in the review paper:
Zhang, Feng, Gilmore:
"Coherent states: Theory and some applications"
Rev Mod Phys 62:867-927, 1990.
Meanwhile, let me try to answer the OP's original question a bit better, i.e.,
what is special about a quadratic potential that causes the shifted
ground state to oscillate like a classical particle without dispersing
so as to saturate the uncertainty principle?
It's because there are certain states constructable from the Heisenberg
algebra alone, all of which have minimum uncertainty product. We call
these "coherent states". Then, when we consider the Hamiltonian
together with the Heisenberg algebra, we find the latter is preserved
by the action of the Hamiltonian and that coherent states evolve into
other coherent states. So it's not so much the "quadratic potential"
that's at work here, but more how the total Hamiltonian preserves
the Heisenberg algebra.
