We can think of states of the harmonic oscillator as wavefunctions,
complex functions on the line, but they have a basis given by eigenstates
of the harmonic oscillator Hamiltonian *H*. We call these states |*
n*> where *n* = 0,1,2,3, etc., and we have

As wavefunctions, |0> is a Gaussian bump centered at the origin. This
is the ``ground state'' of the harmonic oscillator, the state with least
energy. The state |*n*> is the same Gaussian bump times a polynomial
of degree *n*, giving a function whose graph crosses the *x*
axis *n* times:

We can think of |*n*> as a state with *n* ``quanta'' in it.
Quanta of what? Quanta of energy! This is a little weird, but it'll come
in handy to think of this way later. By the time we get to step 6, these
``quanta'' will be honest-to-goodness photons.

So it's nice to have operators that create and destroy quanta. We'll
use the usual annihilation operator *a* and creation operator *a ^{*}*,
given by

and

One can relate these guys to the momentum and position operators *p*
and *q*, which act on wavefunctions as follows:

In the latter equation I really mean ``*q* is multiplication by the
function *x*''; these equations make sense if you apply both sides
to some wavefunction.

So maybe Michael can remember or figure out the formulas relating the
*p*'s and *q*'s to the *a*'s and *a ^{*}*'s.

Once we have those, there's something fun we can do.

To translate a wavefunction *psi* to the right by some amount *c*,
all we need to do is apply the operator

to it. The reason is that

so the rate at which *e ^{-icp} psi* changes as we change

So we can get a wavefunction that's a Gaussian bump centered at the
point *x* = *c* by taking our ground state |0> and translating
it, getting:

This is called a ``coherent state''. In some sense it's the best quantum
approximation to a classical state of the harmonic oscillator where the
momentum is zero and the position is *c*. (We can make this more precise
later if desired.)

If we express *p* in terms of *a* and *a ^{*}*,
and write

we can expand our coherent state in terms of the eigenstates |*n*>.
What does it look like?

If we figure this out, we can see what is the expected number of ``quanta''
in the coherent state. And this will eventually let us figure out the expected
number of photons in a coherent state of the electromagnetic field: for
example, a state which is the best quantum approximation to a plane wave
solution of the classical Maxwell equations. It looks like there should
should be about *c ^{2}* ``quanta'' in the coherent state

The thing to understand is why, even when we have a whole bunch of photons presumably in phase and adding up to a monochromatic beam of light, the amplitude is only proportional to the square root of the photon number. You could easily imagine that a bunch of photons completely randomly out of phase would give an average amplitude proportional to the square root of the photon number, just as |heads - tails| grows on average like the number of coins tossed (for a fair coin).

A few more details:

Suppose we have a wavefunction *psi*. What is *e ^{-icp}
psi*? The answer is: it's just

Why? If we take *psi* and translate it *c* units to the right
we get

so we need to show that

To show this, first note that it's obviously true when *c* = 0.
Then take the derivative of both sides as a function of *c* and note
that they are equal. That does the job.

We are assuming that if two differentiable functions are equal somewhere and their derivatives agree everywhere, then they can't ``start being different'', so they must be equal everywhere.

Or if that sounds too vague:

Technically, we are just using the fundamental theorem of calculus.
Say we have two differentiable functions *f*(*s*) and *g*(*s*).
Then

It follows from this that if *f*(0) = *g*(0) and *f*'(*s*)
= *g*'(*s*) for all *s*, then *f*(*x*) = *g*(*x*)
for all *x*.