WKB theory provides simple approximate solutions for the energy eigenfunctions when the conditions are almost classical, like for the wave packets of chapter 7.11. The approximation is named after Wentzel, Kramers, and Brillouin, who refined the ideas of Liouville and Green. The bandit scientist Jeffreys tried to rob WKB of their glory by doing the same thing two years earlier, and is justly denied all credit.
The WKB approximation is based on the rapid spatial variation of
energy eigenfunctions with almost macroscopic energies. As an
example, figure A.16 shows the harmonic oscillator energy
eigenfunction . Its energy is hundred times
the ground state energy. That makes the kinetic energy quite
large over most of the range, and that in turn makes the linear
momentum large. In fact, the classical Newtonian linear momentum
is given by
The WKB approximation is most appealing in terms of the classical
momentum as defined above. To find its form, in the
Hamiltonian eigenvalue problem
Now under almost classical conditions, a single period of oscillation
of the wave function is so short that normally is almost
constant over it. Then by approximation the solution of the
eigenvalue problem over a single period is simply an arbitrary
combination of two exponentials,
It turns out that to make the above expression work over more than one
period, it is necessary to replace by the antiderivative
. Furthermore, the
constants
and must be
allowed to vary from period to period proportional to
1.
In short, the WKB approximation of the wave function is,
{D.46}:
If you ever glanced at notes such as {D.12}, {D.14}, and {D.15}, in which the eigenfunctions for the harmonic oscillator and hydrogen atom were found, you recognize what a big simplification the WKB approximation is. Just do the integral for and that is it. No elaborate transformations and power series to grind down. And the WKB approximation can often be used where no exact solutions exist at all.
In many applications, it is more convenient to write the WKB
approximation in terms of a sine and a cosine. That can be done by
taking the exponentials apart using the Euler formula
(2.5). It produces
As an application, consider a particle stuck between two impenetrable
walls at positions and . An example would be the
particle in a pipe that was studied way back in chapter
3.5. The wave function must become zero at both
and , since there is zero possibility of finding the
particle outside the impenetrable walls. It is now smart to chose the
integration constant in so that 0. In that
case, must be zero for to be zero at ,
(A.210). The wave function must be just the sine term.
Next, for also to be zero at ,
must be a whole multiple of , because that are the only
places where sines are zero. So
, which means that
It does get a bit more tricky for a case like the harmonic oscillator where the particle is not caught between impenetrable walls, but merely prevented to escape by a gradually increasing potential. Classically, such a particle would still be rigorously constrained between the so called “turning points” where the potential energy becomes equal to the total energy , like the points 1 and 2 in figure A.16. But as the figure shows, in quantum mechanics the wave function does not become zero at the turning points; there is some chance for the particle to be found somewhat beyond the turning points.
A further complication arises since the WKB approximation becomes inaccurate in the immediate vicinity of the turning points. The problem is the requirement that the classical momentum can be approximated as a nonzero constant on a small scale. At the turning points the momentum becomes zero and that approximation fails.
However, it is possible to solve the Hamiltonian eigenvalue problem near the turning points assuming that the potential energy is not constant, but varies approximately linearly with position, {A.29}. Doing so and fixing up the WKB solution away from the turning points produces a simple result. The classical WKB approximation remains a sine, but at the turning points, stays an angular amount 4 short of becoming zero. (Or to be precise, it just seems to stay 4 short, because the classical WKB approximation is no longer valid at the turning points.) Assuming that there are turning points with gradually increasing potential at both ends of the range, like for the harmonic oscillator, the total angular range will be short by an amount 2.
Therefore, the expression for the energy eigenvalues becomes:
The WKB approximation works fine in regions where the total energy
is less than the potential energy . The classical momentum
is imaginary in such regions,
reflecting the fact that classically the particle does not have enough
energy to enter them. But, as the nonzero wave function beyond the
turning points in figure A.16 shows, quantum mechanics does
allow some possibility for the particle to be found in regions where
is less than . It is loosely said that the particle can
tunnel
through, after a popular way for criminals to
escape from jail. To use the WKB approximation in these regions, just
rewrite it in terms of the magnitude
of the classical momentum:
Key Points
- The WKB approximation applies to situations of almost macroscopic energy.
- The WKB solution is described in terms of the classical momentum and in particular its antiderivative .
- The wave function can be written as (A.209) or (A.210), whatever is more convenient.
- For a particle stuck between impenetrable walls, the energy eigenvalues can be found from (A.212).
- For a particle stuck between a gradually increasing potential at both sides, the energy eigenvalues can be found from (A.213).
- The
tunnelingwave function in regions that classically the particle is forbidden to enter can be approximated as (A.214). It is in terms of the antiderivative .
Use the equation
In this case, the WKB approximation produces the exact result, since the classical momentum really is constant. If there was a force field in the pipe, the solution would only be approximate.
Use the equation
In this case too, the WKB approximation produces the exact energy eigenvalues. That, however, is just a coincidence; the classical WKB wave functions are certainly not exact; they become infinite at the turning points. As the example above shows, the true wave functions most definitely do not.