## SHM p.7

### Some Details on QM Oscillation

As previously stated, energy eigenfunctions have
position probabilities that do not depend on time (although
the energy eigenfunctions themselves do have a [complex]
time dependence). Thus
to see motion (i.e., positions that depend on time),
we must not be in *a* energy eigenfunction.
The alternative is to be in several energy eigenfunctions, i.e.,
to be in a superposition of energy eigenfunctions.
Thus the idea is to create a superposition of energy eigenfunctions
which initially has a narrow range of position possibilities, and
watch that range of position possibilities move.
The game then is a version of Fourier analysis: producing a
sum-of-eigenfunctions that adds up to a given function. Orthogonality
is what makes this easy; in particular, that our energy eigenfunctions
are mutually orthogonal.

Our normalized energy eigenfunctions are gaussian-damped Hermite polynomials:

where *N*_{n} is given by:

These eigenfunctions are orthogonal:

So if we try to express a given function *f(x')* in terms
of a sum of these functions:

orthogonality allows us to solve for the *b*_{n}:

In this particular example, I choose *f(x')* to have the shape
of the *n*=52 state, but with *f(x')*=0 for
*x'*<~9.4 (The first zero of the wavefunction is at ~9.4,
thus this *f(x')* is the solution to the Schrödinger's
equation with a quadratic potential for *x'*>~9.4 and
an infinite potential for *x'*<~9.4.)
Of course, we can't sum all the terms of the infinite sum.
I've limited the sum to the 17 eigenfunctions near the *n*=52
eigenfunction. Thus the sum using these 17 *b*_{n}
does not exactly equal the initial *f(x')*. The below plot displays
(in black) *f(x')* and (in red) the sum using these 17 *b*_{n}.

The missing terms would contribute only 5% additional probability,
so it's a fair approximation. To see how this probability lump moves
with time, we need only use the time dependence of the energy eigenfunctions:

The previous page shows the results: the lump of probability moves
just like a classical particle (with broadening of the lump caused
by Heisenberg-required uncertainty in initial velocity).

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