= Solution
<Quantum tunnelling> is transmission through a finite classically forbidden region, where $E<V(x)$. The stationary <Schrödinger equation> permits an evanescent <wavefunction> inside such a barrier rather than forcing it to vanish at its entrance. Matching to propagating states beyond a barrier of finite width gives a nonzero transmitted <probability current>.
For a rectangular barrier of height $V_0$ and width $a$, with equal exterior potentials and $0<E<V_0$, write $\kappa=\sqrt{2m(V_0-E)}/\hbar$. The exact <rectangular-barrier tunnelling probability> is
$$
T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}.
$$
For an opaque barrier its dominant dependence is $e^{-2\kappa a}$. For a slowly varying barrier, the <WKB approximation> generalizes this to $T\sim\exp[-(2/\hbar)\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx]$, apart from matching prefactors. Barrier width, particle <mass> and <energy> therefore have a strong influence. <Energy> is conserved in a static barrier: tunnelling does not require the particle to borrow <energy> or acquire a classical path with negative <kinetic energy>.
In <alpha decay>, a nuclear alpha-particle configuration is confined by the short-range nuclear interaction but separated from the exterior by a <Coulomb barrier>. Its <energy> can be positive relative to infinity while still lying below the barrier maximum. Escape by tunnelling produces alpha particles at that conserved <energy>, with an exponentially small decay <probability> per nuclear encounter. The pronounced dependence of half-life on alpha <energy> is a characteristic consequence of the barrier exponent, unlike a simple classical escape threshold.
In <scanning tunneling microscopy>, a conducting tip is brought within a nanometer-scale vacuum gap of a conducting surface and biased weakly. <Electron> tunnelling gives a current that depends approximately exponentially on separation, $I\propto Ve^{-2\kappa s}$ in a simple low-bias model. Feedback adjusts the tip height to keep current fixed while scanning, producing a surface map sensitive to both geometry and electronic states. Finite current through the non-contact gap, and its steep dependence on separation, are the direct tunnelling signatures; current is not a direct measurement of topographic height alone.
A <Josephson junction> provides a collective superconducting example. Two superconductors separated by a thin barrier can carry a coherent pair current $I=I_c\sin\varphi$ even at zero <voltage>. At constant <voltage> the <AC Josephson effect> gives $d\varphi/dt=2eV/\hbar$ and oscillation <frequency> $2eV/h$. The coupled condensate <quantum phases> and pair transfer distinguish this from independent normal-electron transmission. These examples involve different microscopic barriers, but all require quantum coupling across a region that suppresses classical passage. Comparing weak-temperature tunnelling behavior with thermally activated escape helps distinguish the mechanisms experimentally.
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