In the case in exercise 4, calculate the S-matrix and the Hermitian matrix \(\hat{\Delta}\), with \(\hat{S}=e^{i \hat{\Delta}}\).

Data From Exercise 4:-

Consider a barrier potential, \(V=V_{0}\) for \(|x| \leq L / 2\) and \(V=0\) for \(|x|>L / 2\), and an energy \(E \leq V_{0}\). Calculate \(\psi_{1}(x), \psi_{2}(x)\), and the corresponding transfer matrix \(K\).