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 $E-V$ becomes zero. This note explains how the problem can be fixed.

Figure A.17: The Airy Ai and Bi functions that solve the Hamiltonian eigenvalue problem for a linearly varying potential energy. Bi very quickly becomes too large to plot for positive values of its argument.
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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 $x$-​derivative $V'$ 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:

\begin{displaymath}
\psi = C_{\rm {A}} {\rm Ai}(\overline{x}) + C_{\rm {B}} {\...
...2}}\frac{V-E}{V'}
\quad V' \equiv \frac{{\rm d}V}{{\rm d}x}
\end{displaymath}

Note that $(V-E)$$\raisebox{.5pt}{$/$}$$V'$ is the $x$-​position measured from the point where $V$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E$, so that $\overline{x}$ is a local, stretched $x$-​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 $\hbar$ is very small. That implies that the validity of the Airy functions normally extends to region where $\vert\overline{x}\vert$ is relatively large. For example, if you focus attention on a point where $V-E$ is a finite multiple of $\hbar^{1/3}$, $V-E$ is small, so the value of $V'$ will deviate little from its value at the turning point: the assumption of linearly varying potential remains valid. Still, if $V-E$ is a finite multiple of $\hbar^{1/3}$, $\vert\overline{x}\vert$ will be proportional to 1/$\hbar^{1/3}$, and that is large. Such regions of large, but not too large, $\vert\overline{x}\vert$ 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.

Figure A.18: Connection formulae for a turning point from classical to tunneling.
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Figure A.19: Connection formulae for a turning point from tunneling to classical.
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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 $\overline{x}$, 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 $C_{\rm {p}}$ and $C_{\rm {n}}$ of the tunneling approximation to the coefficients of $C_{\rm {A}}$ and $C_{\rm {B}}$ of the Airy functions. At the other side, it relates the coefficients $C_{\rm {f}}$ and $C_{\rm {b}}$ (or $C_{\rm {c}}$ and $C_{\rm {s}}$) of the classical WKB approximation to $C_{\rm {A}}$ and $C_{\rm {B}}$. 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 $c$ in figures A.18 and A.19. This nasty constant is defined as

\begin{displaymath}
c = \frac{\sqrt{\pi}}{(2m\vert V'\vert\hbar)^{1/6}} %
\end{displaymath} (A.215)

and shows up uninvited when you approximate the Airy function solution for large $\vert\overline{x}\vert$. By cleverly absorbing it in a redefinition of the constants $C_{\rm {A}}$ and $C_{\rm {B}}$, figures A.18 and A.19 achieve that you do not have to worry about it unless you specifically need the actual solution at the turning points.

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 $C_{\rm {p}}e^{\gamma}$ better be zero, because it blows up at large $x$, and that would put the particle at infinity for sure. So the constant $C_{\rm {p}}$ will have to be zero. Now the matching at the right side equates $C_{\rm {p}}$ to $C_{\rm {B}}e^{-\gamma_t}$ so $C_{\rm {B}}$ 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 $\theta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\theta_t$, stops short of reaching zero by an angular amount $\pi$$\raisebox{.5pt}{$/$}$​4. Hence the assertion in addendum {A.28} that the angular range of the classical WKB solution should be shortened by $\pi$$\raisebox{.5pt}{$/$}$​4 for each end at which the particle is trapped by a gradually increasing potential instead of an impenetrable wall.

Figure A.20: WKB approximation of tunneling.
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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 $C^{\rm {r}}e^{{{\rm i}}\theta}$ with a forward wave speed. To simplify the analysis, the constant $C^{\rm {r}}$ can be taken to be one, because it does not make a difference how the wave function is normalized. Also, the integration constant in $\theta$ can be chosen such that $\theta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pi$$\raisebox{.5pt}{$/$}$​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 $C_{\rm {B}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $C_{\rm {A}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\rm i}$. Next, the integration constant in $\gamma$ can be taken such that $\gamma$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 at turning point 2; then the connection formulae of figure A.19 imply that $C^{\rm {m}}_{\rm {p}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12{\rm i}$ and $C^{\rm {m}}_{\rm {n}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.

Next consider the connection formulae for turning point 1 in figure A.18. Note that $e^{-\gamma_1}$ can be written as $e^{\gamma_{12}}$, where $\gamma_{12}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\gamma_2-\gamma_1$, because the integration constant in $\gamma$ was chosen such that $\gamma_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. The advantage of using $e^{\gamma_{12}}$ instead of $e^{-\gamma_1}$ 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, $e^{\gamma_{12}}$ will be a very large number. It is then seen from figure A.18 that near turning point 1, $C_{\rm {A}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2e^{\gamma_{12}}$ which is large while $C_{\rm {B}}$ is small and will be ignored. And that then implies, using the Euler formula to convert Ai’s sine into exponentials, that $\vert C^{\rm {l}}_{\rm {f}}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^{\gamma_{12}}$. As discussed in chapter 7.13, the transmission coefficient is given by

\begin{displaymath}
T = \frac{p_{\rm {c}}^{\rm {r}}}{p_{\rm {c}}^{\rm {l}}}
...
...rt C^{\rm {l}}_{\rm {f}}/\sqrt{p_{\rm {c}}^{\rm {l}}}\vert^2}
\end{displaymath}

and plugging in $C^{\rm {r}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $\vert C^{\rm {l}}_{\rm {f}}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^{\gamma_{12}}$, the transmission coefficient is found to be $e^{-2\gamma_{12}}$.