± *K*_{|m|}(*r*)
exp(*im*) where:

because *K*_{m} is the normalizable (decreasing exponential) solution to
Schrödinger's equation there. Inside of the well (i.e., *r*<*a*) we must have

*J*_{|m|}(*kr*)
exp(*im*) where:

because *J*_{m} is the normalizable (regular at zero) solution to
Schrödinger's equation there. These two solutions need to match up
at the well boundary; we require both and
to be continuous at *r*=*a*.
(The second derivative of
will have a step at *r*=*a* to match the step in potential as
required by Schrödinger's equation.) As before dividing the
-match equation by the -match equation
(i.e., matching the logarithmic derivative ) eliminates
the unknown normalization constants:

The equation is a nonlinear equation
for the unknown *k'* given the size of the potential step: *U*_{0}'.
Clearly 0<*k'*<(*U*_{0}')^{½}.
We examine the nature of the solutions by displaying the right hand side of the
equation in blue and the left hand side in red. Solutions are where the two curves
cross.

The lhs (red) starts (*k'*=0) with *y* value *m*; it crosses
the *x*-axis at the zeros of *J*_{m}' and reaches
± asymptotically at the zeros of *J*_{m}.
The rhs (blue) is always negative since the *K*_{m}(*x*) have negative
slope; in fact, since for large *x*

*K*_{m}(*x*) *e*^{-x}

so *K*_{m}'/*K*_{m} should be approximately -1. Thus
the rhs should start at nearly -(*U*_{0}')^{½}. It ends
at *k'*=(*U*_{0}')^{½} with a *y* value
of -*m*. [The above plot is for *m*=1 and *U*_{0}'=50
and shows two solutions. The wavefunctions are displayed below.] Clearly there will
typically be an intersection for *k'* between a zero of *J*_{m}'
and a zero of *J*_{m}. Since *J*_{0}' has a zero at
*k'*=0, there is always a solution for *m*=0, even for arbitrarily small
potentials. For *m*=1 a solution requires *U*_{0}'>5.78;
*m*=2, *U*_{0}'>14.7.

Once the magic value of *k'* is found, the following wavefunction
has continuous and .
The constant *A* is determined by normalizing the wavefunction.

Here are some resulting wavefunction for *U*_{0}'=50.

Note that the *m*=0 *n*_{r}=2 wavefunction is
just barely bound: *E'*=49.992. Since the region *r'*>1
is just barely disallowed, the wavefunction extends deeply into the
large *r'* region, and hence the probability of finding the particle
in the allowed region is unusually small (i.e.,
is relatively small for this wavefunction in the allowed region).

Below is a plot comparing the infinite square well energies with
those of the *U*_{0}'=50 system (each infinite square
well energy [in black] is connected by a blue line to the equivalent
finite square well energy [in red]).

It is also interesting to compare our *U*_{0}'=50 system
to a simple harmonic oscillator system (seen below in green). The shm system has an exact
*m* degeneracy crudely reflected in the *U*_{0}'=50
system where the high |*m*| states are lower than the
"equivalent" low |*m*| states. While the energy spacing is exactly
constant in the shm system,
the *U*_{0}'=50 system spacings are increasing (but not
as much as the infinite square well).