Finite square barrier transmission at half barrier height
= Finite square barrier transmission at half barrier height
For a square <potential barrier> of width $2a$ and height $U_0=2E$, put
$$
k=\frac{\sqrt{2mE}}{\hbar}.
$$
The oscillatory wavenumber outside and decay constant inside are both $k$. Matching the <wavefunction> and its <derivative> at both faces gives transmission amplitude
$$
t=\frac{e^{-2ika}}{\cosh(2ka)}
$$
and transmission probability $\operatorname{sech}^2(2ka)$.