A coherent state is supposed to be roughly a ``best quantum approximation to a classical state''. There is a big theory of coherent states which makes this a lot more precise, but let's not get into that. Instead, let's just ponder what this might mean.

Classically, a particle on the line has a definite position *q*
and momentum *p*, so it is described by a point in phase space, (*q*,*p*).
Quantum mechanically, the more we know about position, the less we know
about momentum, and vice versa. Our ability to know both at once is limited
by Heisenberg's uncertainty principle:

(Here, as always in this thread, we are working in units where *hbar
*is one, as well as the mass of our particle and the spring constant
of our harmonic oscillator.)

Which states do the best job of simultaneously minimizing the uncertainty
in position and momentum? Which states make Delta* p *Delta*
q* *equal* to 1/2? It turns out that these states are precisely
the Gaussians, possibly translated, and possibly multiplied by a complex
exponential.

Perhaps the nicest Gaussian of all is

since this is the ground state of the harmonic oscillator Hamiltonian, at least after we normalize it.

This function is its own Fourier transform (if we define our Fourier
transform right). Since you can compute the uncertainty in momentum by
taking the Fourier transform of your wavefunction and then computing the
uncertainty in position of *that*, this Gaussian must have the same
uncertainty in position as it does in momentum. If everything I've said
so far is true, we must therefore have

Of course, there are lots of other Gaussians centered at the origin
with Delta* p *Delta* q = *1/2 We can squish our Gaussian
or stretch it out like this:

The Fourier transform of a squished-in skinny Gaussian is a stretched-out squat Gaussian and vice versa. So all these Gaussians have

So: the Gaussian

is the primordial ``coherent state''. In this state, the expectation value of position is obviously zero, since the bump is symmetrically centered at the origin. The expectation value of momentum is also obviously zero, since:

1) the Fourier transform of this function is itself, so whatever applies to position applies to momentum as well,

or if you prefer,

2) the expectation value of momentum is zero for any *real-valued*
wavefunction. (Hint: to see this, just use integration by parts.)

So exp(-*x ^{2}*/2) is a coherent state with expectation
values

We can get lots more coherent states by taking this Gaussian and translating
it in position space and/or momentum space. Translating in position space
by *c*, we get a Gaussian

This is the coherent state I wanted Michael to express in terms of eigenstates of the harmonic oscillator Hamiltonian. This obviously has

and since it's real-valued it still has

Translating in momentum space by some amount *b* is the same as
multiplying by a complex exponential *e ^{ibx}*. Or, if you
prefer, just take a Fourier transform, translate by

which is our coherent state with

and

Why does it still have <*q*>* = 0*? Well, we are just multiplying
our Gaussian bump by a unit complex number or ``phase'' at each point,
and this doesn't affect the expectation value of position.

Finally, we can translate in *both* position *and* momentum
space directions. These two operations don't commute, of course, since
the position and momentum operators don't commute, and momentum is the
generator of translations in position space, while position is the generator
of translations in momentum space (possibly up to an annoying minus sign).

What do we get? Well, take our bump and first translate it in position
space by *c*:

and then in momentum space by *b*:

Or, alternatively, first translate it in momentum space by *b*:

and then in position space by *c*:

I claim that these are equal *up to a phase*...one of them is *e ^{ibc}*
times the other. This is always how translatiions in position space and
momentum space fail to commute.

While terribly important in some ways, the phase is not such a big deal in other ways. (That's the weird thing about quantum mechanics when you are first learning it: sometimes a phase is very important, while other times it doesn't matter at all. Of course, it just depends what you're doing.) A phase times a coherent state is still a coherent state in my book. So we have gotten our hands on a coherent state with

A translated Gaussian bump, with a corkscrew twist thrown in! I hope
you *visualize* this thing for various values of *b* and *c*;
it's a very pretty thing, and it will serve as our quantum ``best approximation''
to a particle with momentum *b* and position *c*.

Then, later, we will use a souped-up version of this as the quantum- field-theoretic ``best approximation'' to a particular state of a classical field theory, like electromagnetism.