A.29 WKB solution near the turning points

Both the classical and tunneling WKB approximations of addendum
{A.28} fail near so-called turning points

where the classical kinetic energy becomes zero. This note
explains how the problem can be fixed.

The trick is to use a different approximation near turning points. In
a small vicinity of a turning point, it can normally be assumed that
the -derivative of the potential is about constant, so that
the potential varies linearly with position. Under that condition,
the exact solution of the Hamiltonian eigenvalue problem is known to
be a combination of two special functions Ai and Bi that are called
the Airy

functions. These functions are shown in
figure A.17. The general solution near a turning point
is:

Note that is the -position measured from the point where , so that is a local, stretched -coordinate.

The second step is to relate this solution to the normal WKB approximations away from the turning point. Now from a macroscopic point of view, the WKB approximation follows from the assumption that Planck’s constant is very small. That implies that the validity of the Airy functions normally extends to region where is relatively large. For example, if you focus attention on a point where is a finite multiple of , is small, so the value of will deviate little from its value at the turning point: the assumption of linearly varying potential remains valid. Still, if is a finite multiple of , will be proportional to 1/, and that is large. Such regions of large, but not too large, are called “matching regions,” because in them both the Airy function solution and the WKB solution are valid. It is where the two meet and must agree.

It is graphically depicted in figures A.18 and A.19. Away from the turning points, the classical or tunneling WKB approximations apply, depending on whether the total energy is more than the potential energy or less. In the vicinity of the turning points, the solution is a combination of the Airy functions. If you look up in a mathematical handbook like [1] how the Airy functions can be approximated for large positive respectively negative , you find the expressions listed in the bottom lines of the figures. (After you rewrite what you find in table books in terms of useful quantities, that is!)

The expressions in the bottom lines must agree with what the
classical, respectively tunneling WKB approximation say about the
matching regions. At one side of the turning point, that relates the
coefficients and of the tunneling
approximation to the coefficients of and of
the Airy functions. At the other side, it relates the coefficients
and (or and ) of
the classical WKB approximation to and
. The net effect of it all is to relate,
connect,

the coefficients of the classical WKB
approximation to those of the tunneling one. That is why the formulae
in figures A.18 and A.19 are called
the “connection formulae.”

You may have noted the appearance of an additional constant in
figures A.18 and A.19. This nasty
constant is defined as

As an example of how the connection formulae are used, consider a right turning point for the harmonic oscillator or similar. Near such a turning point, the connection formulae of figure A.18 apply. In the tunneling region towards the right, the term better be zero, because it blows up at large , and that would put the particle at infinity for sure. So the constant will have to be zero. Now the matching at the right side equates to so will have to be zero. That means that the solution in the vicinity of the turning point will have to be a pure Ai function. Then the matching towards the left shows that the solution in the classical WKB region must take the form of a sine that, when extrapolated to the turning point , stops short of reaching zero by an angular amount 4. Hence the assertion in addendum {A.28} that the angular range of the classical WKB solution should be shortened by 4 for each end at which the particle is trapped by a gradually increasing potential instead of an impenetrable wall.

As another example, consider tunneling as discussed in chapter 7.12 and 7.13. Figure A.20 shows a sketch. The WKB approximation may be used if the barrier through which the particle tunnels is high and wide. In the far right region, the energy eigenfunction only involves a term with a forward wave speed. To simplify the analysis, the constant can be taken to be one, because it does not make a difference how the wave function is normalized. Also, the integration constant in can be chosen such that 4 at turning point 2; then the connection formulae of figure A.19 along with the Euler formula (2.5) show that the coefficients of the Airy functions at turning point 2 are 1 and . Next, the integration constant in can be taken such that 0 at turning point 2; then the connection formulae of figure A.19 imply that and 1.

Next consider the connection formulae for turning point 1 in figure
A.18. Note that can be written as
, where
, because the integration constant in
was chosen such that 0. The advantage of
using instead of is that it is
independent of the choice of integration constant. Furthermore, under
the typical conditions that the WKB approximation applies, for a high
and wide barrier, will be a very large number. It
is then seen from figure A.18 that near turning point
1, which is large while
is small and will be ignored. And that then implies,
using the Euler formula to convert Ai’s sine into exponentials,
that . As
discussed in chapter 7.13, the transmission coefficient
is given by

and plugging in 1 and , the transmission coefficient is found to be .