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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 $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The exact rectangular-barrier tunnelling probability isFor an opaque barrier its dominant dependence is $e^{-2\kappa a}$. For a slowly varying barrier, the WKB approximation generalizes this to $T\sim \exp [-(2/\hslash ){\int }_{x_1}^{x_2}\sqrt{2m(V(x)-E)}dx]$, 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, $I\propto V{e}^{-2\kappa s}$ 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 $I=I_c\sin \phi$ even at zero voltage. At constant voltage the AC Josephson effect gives $d\phi /dt=2eV/\hslash$ 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.

Inside the barrier, the stationary Schrödinger equation gives an exponentially decaying wavefunction with inverse penetration lengthThe amplitude falls approximately as $e^{-\kappa a}$, so the leading exponential estimate for the transmission probability isFor comparison, wavefunction matching at a finite potential step at both edges gives the rectangular-barrier tunnelling probabilityThe difference is the interface prefactor: for an opaque barrier it is $16E(V_0-E)/V_0^2=2.56$. Both give a transmission of order $10^{-6}$; the first is the requested exponential estimate.

Inside the rectangular potential barrier, the electron's wavefunction is evanescent, with decay constantFor width $a=1\mathrm{nm}$, the squared exponential suppression in the quantum tunnelling probability isThis is the usual leading-exponential estimate. The rectangular-barrier tunnelling probability, retaining the interface-matching prefactor for equal potentials on the two sides, isHere that prefactor is $16(8)(2)/10^2=2.56$, so the more accurate result is $\boxed{T\simeq 1.30\times 10^{-6}}$. Both estimates show that transmission is of order one part in a million. A classical particle with this kinetic energy cannot cross the barrier at all.