## SHM p.14

### WKB approximation

You should have noticed that the wavefunctions presented
in the 2-d and 3-d cases did not follow the 1-d wavefunctions rules
(e.g., that where the classical velocity is slow, the wavefunction
is large). For example, recall that in order to see the
*n*_{r}=6,*l*=0,*m*=0
wavefunction near its classical turning point we needed to magnify:

Qualitatively speaking, the ever smaller wavefunction as
you move away from the origin
is the result of flux conservation: an outgoing wave's probability amplitude
is spread out over ever larger surface as *r* increases.

Mathematically we can see this as a breakdown of the assumptions
of the WKB approximation:

We can quickly solve the first problem:

Thus the new "wavefunction" *u*=*r*
satisfies the WKB-like differential equation.

In order to solve the second problem we must perform a
change of variables to move the zero of *u* at *r*=0
out to minus infinity. The transformation *s*=ln(*r*)
does just what we want.

However, the differential equation for *u* in terms of
*s* is no longer in the WKB form:

Thus is solving second problem, we've screwed up the first
and must now work to again get the differential equation into WKB form:

Putting these results into our differential equation finally gives
us a WKB form in *s* (and hence with WKB behavior at
infinity).

We can now read off the position dependent wavenumber *k*

Note that essentially what has happened is the centrifugal
barrier term *l*(*l*+1)/*r*^{2}
has become (*l*+½)^{2}/*r*^{2},
a small correction of order *l*^{-2}.

The wavefunction *v* is given by:

The integral of *k* can be rearranged to give:

Now to get the WKB estimates for the eigenenergies
we must integrate *k* through the allowed region.
The allowed region should include
*n*+½ half-waves, i.e., a phase of
(*n*+½). With a little
work on the complex plane we can evaluate that integral exactly:

exactly reproducing the correct result:

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

So the bottom line is that if you plot *r*
along with the effective potential with (*l*+½)^{2}
rather than *l*(*l*+1) things should look as in the
1-d case. In fact, this is how you will find most books displaying
3-d wavefunctions. Here is a repeat stacked wavefunction plot
(for *l*=0,1,3) following this
formula.

Here is the *n*_{r}=32 *l*=0 (probability
density)×*r*^{2} plotted along with the
WKB result (in red):

Note that while these plots look more "normal" they
hide the fact that the probability density is highest near
the origin, rather than at the far *r* turning point.

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