Tunneling Through a Barrier

A particle with energy $E$ below a barrier of height $V_0$ has a non-zero probability of being found on the other side — quantum tunneling.

Transmission coefficient

For a rectangular barrier of width $a$ and height $V_0 > E$:

$T \approx 16 \, \frac{E}{V_0}\left(1 - \frac{E}{V_0}\right) e^{-2\kappa a}, \quad \kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}$