Tunnelling in the WKB approximation · Quantum Mechanics

Intuition

Under a barrier the WKB solution decays, and across the forbidden region its amplitude falls by the exponential of the integral of $|p|/\hbar$ . The transmission is the square of that: an exponential of twice the integral, the Gamow factor. For a rectangular barrier it gives back the $e^{-2\kappa a}$ of the chapter on one dimension; for a real barrier of changing height it gives what that chapter could not — the rates of alpha decay, of field emission from a metal tip, of nuclear fusion in the Sun.

Light through a thick fog fades exponentially with the depth of fog it crosses; where the fog is thicker it fades faster. A particle under a barrier fades the same way, with $|p|$ playing the fog’s thickness.

A barrier $V=2-x^{2}/2$ and an energy $E=1$ below its top, with $\hbar=m=1$ . Between the turning points and the particle is classically forbidden; WKB gives .

Tunnelling through a smooth barrier

For a barrier between turning points $a$ and $b$ , WKB gives a transmission probability dominated by one exponential. Prefactors of order one are dropped.

T\approx\exp\Bigl(-\frac{2}{\hbar}\int_{a}^{b}\sqrt{2m\bigl(V(x)-E\bigr)}\,dx\Bigr)

Properties

For a rectangular barrier the integral is $\kappa a\hbar$ , and $T\approx e^{-2\kappa a}$ : the thick-barrier result without its prefactor.
It is valid when the exponent is large, $T\ll1$ , and the barrier is smooth on the scale of .

The Gamow factor

\psi\approx\frac{C}{\sqrt{|p|}}e^{-\frac1\hbar\int_{a}^{x}|p|\,dx'}\ \Longrightarrow\ T\approx e^{-\frac2\hbar\int_{a}^{b}|p|\,dx}

Inside the barrier the decaying WKB solution dominates. Across the forbidden region its size falls by the exponential of the integral of $|p|/\hbar$ . Transmission compares probabilities, the squares of amplitudes, so the exponent doubles; the amplitude factors are of order one and are dropped.

Proof steps

\begin{aligned}&\psi(x)\approx\frac{C}{\sqrt{|p(x)|}}\exp\Bigl(-\frac1\hbar\int_{a}^{x}|p|\,dx'\Bigr)\\ &\text{for}\ a<x<b\end{aligned}

The decaying WKB solution under the barrier.

\frac{|\psi(b)|}{|\psi(a)|}\approx\exp\Bigl(-\frac1\hbar\int_{a}^{b}|p|\,dx\Bigr)

Compare the two ends, keeping only the exponential.

T\approx\frac{|\psi(b)|^{2}}{|\psi(a)|^{2}}\approx\exp\Bigl(-\frac2\hbar\int_{a}^{b}|p|\,dx\Bigr)

Transmission is a ratio of probabilities.

\begin{aligned}&V=V_{0}\ \text{on}\ (0,a):\ \ \frac1\hbar\int_{0}^{a}|p|\,dx=\kappa a\\ &\Rightarrow\ T\approx e^{-2\kappa a}\end{aligned}

A rectangular barrier recovers the thick-barrier exponential.

Applications

Practice

The Gamow Exponent

The transmission through a smooth barrier is about $e^{-\frac2\hbar\int|p|\,dx}$ , the integral taken across the forbidden region.

T\approx e^{-\frac2\hbar\int_{a}^{b}|p|\,dx}

Try it

With $\hbar=m=1$ , the barrier $V=2-x^{2}/2$ at $E=1$ has $\int_{a}^{b}|p|\,dx=\pi$ . What does WKB give for $T$ ? Give four decimal places.