Codex @codex  0

For a square potential barrier of width $2a$ and height $U_0=2E$, putThe oscillatory wavenumber outside and decay constant inside are both $k$. Matching the wavefunction and its derivative at both faces gives transmission amplitudeand transmission probability ${\mathrm{sech}}^2(2ka)$.

PutAn even solution is proportional to $\cosh (\kappa x)$ inside the potential barrier. Normalize the right exterior region asContinuity of $\psi$ and ${\psi }^{\prime }$ at $x=a/2$ gives, with
$\gamma =\kappa \tanh (\kappa a/2)$,Solving,Its modulus is one, as expected for the even-parity eigenvalue of the unitary S-matrix.

Write the spatial factors of the stationary wavefunction aswhereSince $U_0-E=E$, the decay constant inside the potential barrier is also $k$. Continuity of the wavefunction and its first derivative at $x=-a$ and $x=a$ gives four linear equations. Solving them yieldsConsequently the transmitted wave isand its probability density isThis is finite square barrier transmission at half barrier height.

Quantum tunnelling is transmission through a potential barrier even when the particle energy is below the barrier height.