We have found the following wavefunctions for the infinite spherical square well:

with energy *E'*=*k'*^{2}.

We looked at 1D plots of these wavefunctions, but what do the
probability densities look like in space? The *l*=0 (*s*-wave) wavefunctions
are spherically symmetric, so the 1D plots show the successive shells
around the center.

If *l* isn't zero the
*probability densities* for these wavefunctions are not spherically
symmetric. However the probability densities (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)

Note from the 1D plots that the probability density is high near the origin. In fact in the contour plots the central region is "over exposed", i.e., in order to show the details on the large outer lobes, I chose a maximum contour (pure white) that was well below the the first peak's height.

Below we plot the probability density for *n _{r}*=2,

Note that chemists call this the *p*_{z} orbital.
There is a node on the plane *z*=0.
One can use linear combinations of the *l*=1 *m*=±1
wavefunctions to produce orbitals identical to *p*_{z}, except
located around the *x*- (called *p*_{x}) and
*y*- (called *p*_{y}) axes (i.e., with nodal planes
*x*=0 and *y*=0).

Note that the probability is somewhat 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's symmetric

Below we plot the probability density for *n _{r}*=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 its symmetric

The display "problem" of high probability densities near the origin is commonly
"solved" by plotting *r* rather then .
As we explained earlier the result of this is to produce
WKB-like wavefunctions.

Here is what *r* looks like for *n _{r}*=10,

This is exactly a sine wave; If you were surprised look back at the
differential equation for (*R*)...for *l*=0
it is the famous differential equation:

*f*'' = -*f*.

Note that for this *l*=0 solution,
is spherically symmetric and has 10 nodal spheres. Don't be fooled by the above
plot: the probability density oscillations are not of uniform height.
Probability density is highest near the origin.

Here is what it like for *n _{r}*=5,

Note the exclusion of this high *l* wavefunction from the
origin.

(Don't forget that this radial solution must be matched with various
*Y*_{10m}; none is spherically symmetric.)

Finally note that in producing the above "*Y*_{lm}" solutions, we have
produced *complex* rather than real wavefunctions....
that factor of exp(i*m*). This factor
cancels out of our ^{*}
probability density (so the probability density is -symmetric),
but if we were to look at just the real part of
we would find oscillation as a function of
in addition to its oscillations in .
Oscillation indicates momentum, so these
solutions have angular momentum...in fact *m*
angular momentum in the "*z*" direction (i.e., perpendicular to the
plane) and a total angular momentum squared of
^{2}*l(l*+1*)*. One can
easily produce real valued wavefunctions (like *p*_{x},
*d*_{xy}, etc. discussed previously) but these seemly simpler
functions in fact turn out to be more difficult to calculate with.