## SHM p.11

### Seeing QM Oscillation: 2-d

It is quite easy to produce visible *x-y* oscillation:
just find the sum of *y* wavefunctions that oscillates
in *y* (working exactly as in the 1-d case) and
find the sum of *x* wavefunctions that oscillates
in *x* and multiply them together. Most of the probability
will be in a lump at a particular (moving) (*x,y*).
To build up an orbiting probability packet, we must use
wavefunctions with minimal inward/outward motion (i.e., *n*_{r}=0)
and large angular momentum (i.e., large positive *m*). We will use
as our model wavefunction a function that is localized in
but with lots of oscillation in ,
i.e., exp[-100^{2}+200*i*]).
Our |*n*_{r}=0, *m*> wavefunctions do, of course,
depend on *r*: basically restricting the wavefunction to
*r* near *m*^{½}, i.e., the bottom of the
effective potential. But for small variations in *m*, this will be much
the same *r*. Thus we focus on localizing in .
We start by straightforward Fourier expansion of our model wavefunction
in terms of exp(*im*), the
behavior of the |*n*_{r}=0, *m*> wavefunctions.

Using these same *b*_{m} to expand our wavefunction should
achieve similar localization in while localization in
*r'* is automatic for large *m*, *n*_{r}=0
wavefunctions. In practice one can't sum over an infinity of *m*,
but the equation for *b*_{m} shows that *b*_{m}
is large only for *m* near 200 (so we restrict our sum to between
180 and 220).

Below is displayed the resulting lump of probability for *t'*=0.

We need not display similar plots for other times because the equation
for shows that plot for any other time are
identical to the above except rotated by 2*t'* radians. Thus
the lump will make one complete revolution in a period of

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