= Solution
In a finite potential barrier with $E<V$, the stationary <Schrödinger equation> has exponentially decaying and growing solutions, with local decay constant $\kappa=\sqrt{2m(V-E)}/\hbar$. Matching the <wavefunction> and its <derivative> at finite steps produces a transmitted <probability current> even though a classical particle at that <energy> could not cross. <energy> remains conserved. For a smooth thick barrier, the leading WKB transmission factor is
$$
\boxed{T\sim\exp\left[-2\int_{x_1}^{x_2}\frac{\sqrt{2m[V(x)-E]}}\hbar\,dx\right]}.
$$
For a rectangular barrier of height $V_0$, width $d$ and $0<E<V_0$, exact matching gives $T=[1+V_0^2\sinh^2(\kappa d)/(4E(V_0-E))]^{-1}$. Thickness and <energy> therefore affect transmission exponentially in the opaque limit; prefactors and near-turning-point corrections require more than the leading exponent.
<alpha decay> illustrates escape from a nuclear well through a <Coulomb barrier>. A rough decay rate is an internal encounter <frequency> times barrier penetrability, multiplied by the <probability> of forming the outgoing cluster; penetrability alone is not a complete nuclear lifetime calculation. In a <scanning tunnelling microscope>, <electron> transmission through the vacuum gap gives an approximately exponential distance dependence $I\propto e^{-2\kappa d}$ at fixed conditions. A feedback loop adjusts tip height to maintain current, providing surface information. Bias, tip electronic structure and local electronic density of states also affect the current, so constant-current height is not purely geometric topography. Both applications exploit the same sensitivity of transmission to a classically forbidden region.
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