Quantum tunnelling

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

A symmetric two-well system has nearly localized left/right states coupled by a positive quantum tunnelling magnitude $\Delta =\hslash K{e}^{-S_I/\hslash }$. Its effective Hamiltonian is $E_{\mathrm{well}}I-\Delta {\sigma }_x$, so the superposition with an even spatial wavefunction is lower, and the level splitting is $2\Delta$. The dilute instanton gas reproduces these two exponents through even/odd crossing sums.

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 ${\mathrm{sech}}^2(2ka)$.