## SHM p.13

### Visualizing the Wavefunctions

We have found the following wavefunction for our
3-d oscillator:

with energy *E'*=4*n*_{r}+2*l*+3.

What do these wavefunctions look like?

If *l*=0 the wavefunctions are spherically symmetric.

If *l*>0,
the *probability densities* for these wavefunctions (but not the wavefunctions
themselves)
are independent of ("cylindrically symmetric" for a cylinder
whose axis is aligned with the *z* axis) so we only need to concern
ourselves with their dependence on and *r* (i.e.,
for example the *x-z* plane).
(The exp(*im*) dependence in the wavefunction cancels out in
^{*}.)
The probability density for +*m* and -*m*
are identical (since
*Y*_{l,-m}=(-1)^{m}Y^{*}_{l,m} );
the difference is only which way the particle is going
around the *z* axis. Note that except for *l*=0 the wavefunction goes to
zero as you approach the origin; the larger the value of *l* the faster
goes to zero. Note that unless *m*=0 the wavefunction
is zero on the *z* axis; again larger |*m*| means a more
disallowed *z* axis. For the largest possible |*m*| (i.e.,
*m*=±*l*) the probability is largely near the
*x-y* plane. Finally, there are always *n*_{r} radial
nodes (zeros) [not counting the origin].

Below we plot the probability density for *n*_{r}=2, *l*=2
("*d*" wave) for *m*=0,1,2. For several of these plots
the probability density is also plotted along a particular line.

*m*=0

*m*=1

*m*=2

Note that the probability is mostly confined to the *z*=0 plane (i.e.,
the *x-y* plane)
Here is what the probability density looks like in that plane; like all
of these solutions it is symmetric

All the the above solutions have *E'*=15. Here is a collection
of solutions with *E'*=27.

*n*_{r}=2,*l*=8,*m*=0

*n*_{r}=2,*l*=8,*m*=8

*n*_{r}=6,*l*=0,*m*=0

This *l*=0 solution is spherically symmetric, so all planes going through
the origin have the same probability density. For this particular case there
is such a large difference between the probability density at the origin and
for the last few oscillations near the edge of the classically allowed region,
that I have had to "stretch" the scales. In the case of the density plot,
contours are spaced equally in logarithm (here a factor of 2 separates adjacent
contours) rather than arithmetically (constant differences) spaced; in the case of
the plot of probability density vs *r*, I've included a "50×" magnified
plot of the large *r* probability density.

Because the angular dependence is "simple" we can usefully plot the
wavefunction just as a function of *r*. Here are
"stacked wavefunction" plots for *l*=0,1,3:

The red line is the classical "effective potential"=
*l*(*l*+1)/r^{2}+*r*^{2}
which includes the "centrifugal barrier" *l*(*l*+1)/r^{2}
for a particle with *l* total angular momentum (essentially) *l*.
Because of the centrifugal barrier, non-zero *l* wavefunctions
are excluded from the origin.

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