Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 35D / c / Solution

PutAn even solution is proportional to $\cosh (\kappa x)$ inside the potential barrier. Normalize the right exterior region asContinuity of $\psi$ and ${\psi }^{\prime }$ at $x=a/2$ gives, with
$\gamma =\kappa \tanh (\kappa a/2)$,Solving,Its modulus is one, as expected for the even-parity eigenvalue of the unitary S-matrix.