quantum tunnelling

Basic idea

A particle has nonzero probability of crossing a potential barrier higher than its kinetic energy — classically forbidden, quantum-mechanically routine.

Key formulas

• Transmission coefficient (rectangular barrier, $E<V$): $T \approx e^{-2\kappa L}$
• Decay constant inside barrier: $\kappa = \dfrac{\sqrt{2m(V-E)}}{\hbar}$
• WKB approximation: $T \approx \exp\!\left(-\dfrac{2}{\hbar}\int_a^b \sqrt{2m(V(x)-E)}\,dx\right)$

Interesting

• Quantum tunelling points to the fact that particles may travel faster than light, see “Hartman effect”, https://www.quantamagazine.org/quantum-tunnel-shows-particles-can-break-the-speed-of-light-20201020/

Resources

• https://www.quantamagazine.org/quantum-tunnel-shows-particles-can-break-the-speed-of-light-20201020/
• https://en.wikipedia.org/wiki/Hartman_effect