= Solution
In a <classically forbidden region>, the <Schrödinger equation> has evanescent solutions rather than forcing the <wavefunction> to vanish. Matching at both sides of a finite barrier permits nonzero transmission. For a broad barrier the <WKB approximation> gives
$$
T\simeq C\exp\left[-2\int_{x_1}^{x_2}\frac{\sqrt{2m(V(x)-E)}}{\hbar}\,dx\right],
$$
where $C$ is a matching prefactor. \b[Transmission is exponentially sensitive to barrier width and height.] A single decaying <exponential function> carries no <probability current>; the matched scattering solution includes the combination needed for transmitted current. <energy> is conserved throughout.
In <alpha decay>, a quasibound alpha particle penetrates the nuclear <Coulomb barrier>. Multiplying the penetration <probability> by an assault <frequency> estimates the decay rate; the strong dependence on alpha <energy> explains the wide range of lifetimes. In a <scanning tunnelling microscope>, a <voltage> drives <electron> <quantum tunnelling> between a conducting tip and sample. Approximately $I\propto V e^{-2\kappa d}$, so constant-current feedback resolves changes in gap width, with electronic <density of states> also affecting the image. Piezoelectric motion, vibration isolation and stable <voltage> are necessary; a bright feature is not automatically a taller atom. Atomic-scale imaging was demonstrated with a vacuum-tunnelling microscope in the original surface study.<Surface Studies by Scanning Tunneling Microscopy by Gerd Binnig and others>[Abstract and pp. 57–61: atomic-scale steps and surface reconstruction]{ref} In a <Josephson junction>, <Cooper pair> <quantum tunnelling> through a weak link gives $I=I_c\sin\varphi$, allowing supercurrent without an applied <voltage> and phase-sensitive interference. These are distinct applications of barrier penetration and coupling with <quantum coherence>, not particles borrowing <energy> to climb over a barrier.
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