Rectangular-barrier tunnelling probability ($T=[1+V_0^2{\sinh }^2(\kappa a)/(4E(V_0-E)){]}^{-1}$)

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 $k=\sqrt{2mE}/\hslash$ and interior decay constant $\kappa =\sqrt{2m(V_0-E)}/\hslash$. Write the three wavefunctions as $e^{ikx}+r{e}^{-ikx}$, $A{e}^{\kappa x}+B{e}^{-\kappa x}$, and $t{e}^{ikx}$. 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 $\kappa a\gg 1$, ${\sinh }^2(\kappa a)\simeq e^{2\kappa a}/4$, soKeeping 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.