Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 15D / b / i / Solution

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 $\operatorname{sech}^2(2ka)$.

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 Incoming links: Finite square barrier transmission at half barrier height