Introduction: Computing Tunneling for Any Barrier
WKB provides a simple formula for the tunneling probability through arbitrary barriers—essential for understanding alpha decay, field emission, and chemical reaction rates.
The WKB Tunneling Formula
For a barrier between classical turning points \(x_1\) and \(x_2\):
\[T \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x) - E)}\,dx\right)\]The argument is called the Gamow factor.
Physical Interpretation
The tunneling probability depends exponentially on:
- Barrier height: higher barrier → smaller \(T\)
- Barrier width: wider barrier → smaller \(T\)
- Particle mass: heavier particle → smaller \(T\)
Worked Example: Alpha Decay
An alpha particle tunnels through the nuclear Coulomb barrier. The Gamow factor explains why half-lives range from microseconds to billions of years—small changes in \(Q\)-value cause exponential changes in tunneling rate.
Geiger-Nuttall law: \(\log t_{1/2} \propto 1/\sqrt{Q}\)
The Quantum Connection
WKB tunneling formulas are used for field emission (electrons from sharp tips), scanning tunneling microscopy, molecular isomerization, and nuclear fusion in stars. The exponential sensitivity to barrier parameters makes tunneling a sensitive probe of potential shapes.