Quantum tunnelling

Remarks

Example: the rectangular barrier $V = V_0$ on $[0, d]$ and $V = 0$ elsewhere, with $0 < E < V_0$. Let $\kappa = \sqrt{2m(V_0 - E)}/\hbar$. Inside the barrier $\psi$ is a combination of $e^{\kappa x}$ and $e^{-\kappa x}$, and matching $\psi$ and $\psi'$ at $0$ and $d$ gives $$T(E) = \left(1 + \frac{V_0^2 \sinh^2(\kappa d)}{4E(V_0 - E)}\right)^{-1} > 0.$$ For $\kappa d \gg 1$, $T(E) \approx \frac{16 E (V_0 - E)}{V_0^2} e^{-2\kappa d}$.

Non-example: in classical mechanics, a particle with energy $E$ cannot enter the region where $V(x) > E$, because its kinetic energy $E - V(x)$ would be negative; its transmission is $0$.

Conservation of probability gives $\abs{r}^2 + \abs{t}^2 = 1$. For a smooth barrier, the WKB approximation gives $T(E) \approx \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2} \sqrt{2m(V(x) - E)}\, dx\right)$, where $x_1 < x_2$ are the turning points with $V = E$.