= Solution
<quantum tunnelling> is transmission through a finite region in which the classical kinetic <energy> $E-V$ is negative. The <wavefunction> need not vanish there: the <Schrödinger equation> permits evanescent solutions. The transmitted particle retains its conserved <energy>; it does not borrow <energy> to surmount the barrier.
For a rectangular barrier of height $V_b$ on $0<x<d$, with $0<E<V_b$ and zero exterior <potential energy>, define $k=\sqrt{2mE}/\hbar$ and $\kappa=\sqrt{2m(V_b-E)}/\hbar$. A left-incident scattering solution has the three pieces
$$
\psi_{\rm L}=e^{ikx}+re^{-ikx},\qquad
\psi_{\rm B}=b e^{\kappa x}+c e^{-\kappa x},\qquad
\psi_{\rm R}=te^{ikx}.
$$
Continuity of $\psi,\psi'$ at both boundaries imposes
$$
1+r=b+c,\quad ik(1-r)=\kappa(b-c),\quad
be^{\kappa d}+ce^{-\kappa d}=te^{ikd},\quad
\kappa(be^{\kappa d}-ce^{-\kappa d})=ik te^{ikd}.
$$
Eliminating $b,c,r$ gives
$$
t=\frac{e^{-ikd}}{\cosh\kappa d+i\frac{\kappa^2-k^2}{2k\kappa}\sinh\kappa d},\qquad
\boxed{T=|t|^2=\left[1+\frac{V_b^2\sinh^2(\kappa d)}{4E(V_b-E)}\right]^{-1}}.
$$
Equal incident and exit velocities justify $T=|t|^2$; different exterior <potential energies> require the velocity ratio. Probability-current conservation gives $R+T=1$. Although either real exponential alone has zero current, their complex-coefficient combination can carry the current crossing the barrier.
For $\kappa d\gg1$, $T\simeq16E(V_b-E)V_b^{-2}e^{-2\kappa d}$. A slowly varying barrier generalizes this to the <WKB approximation> $T\sim\exp[-2\int_{x_1}^{x_2}\sqrt{2m(V-E)}\,dx/\hbar]$, with matching prefactors and turning-point limitations. This exponential sensitivity explains why very small geometric or <energy> changes can produce large changes in a tunnelling rate.
In <alpha decay>, a quasibound alpha particle penetrates the nuclear <Coulomb barrier>; penetration <probability>, attempt <frequency> and nuclear preformation determine the rate. In a <scanning tunnelling microscope>, a bias between a conducting tip and sample gives current approximately proportional to $e^{-2\kappa d}$. Piezoelectric positioning and constant-current feedback probe surfaces, while vibration isolation and electronic-state dependence limit a purely geometric interpretation. Atomic-scale steps and surface reconstruction were resolved in the original surface-imaging study.<Surface Studies by Scanning Tunneling Microscopy by Gerd Binnig and others>[Abstract and pp. 57–61, atomic-scale surface images]{ref} In a <Josephson junction>, coherent tunnelling of <Cooper pairs> permits $I=I_c\sin\varphi$, a phase-dependent supercurrent through a weak link. Thus the same barrier physics explains natural decay and underpins sensitive surface and superconducting devices.
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