= Rectangular-barrier tunnelling probability
{title2=$T=[1+V_0^2\sinh^2(\kappa a)/(4E(V_0-E))]^{-1}$}
For a <rectangular potential barrier> of height $V_0$, width $a$, and equal constant potentials outside, a particle with $0<E<V_0$ has exterior <wave number> $k=\sqrt{2mE}/\hbar$ and interior decay constant $\kappa=\sqrt{2m(V_0-E)}/\hbar$. Write the three <wavefunctions> as $e^{ikx}+re^{-ikx}$, $Ae^{\kappa x}+Be^{-\kappa x}$, and $te^{ikx}$. Continuity of the <wavefunction> and its derivative at both interfaces determines the <transmission probability> $T=|t|^2$ in the title. Equal exterior velocities are what permit the last equality.
For $\kappa a\gg1$, $\sinh^2(\kappa a)\simeq e^{2\kappa a}/4$, so
$$
T\simeq\frac{16E(V_0-E)}{V_0^2}e^{-2\kappa a}.
$$
Keeping only $e^{-2\kappa a}$ estimates the exponentially dominant <quantum tunnelling> suppression; the interface prefactor can still change a numerical answer by a factor of order one.
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