Hence, \[ik_-\, \left[\psi_+(x_a) - \psi_-(x_a)\right] = ik_+\, \left[\psi_+(x_b) - \psi_-(x_b)\right].\] These two equations can be combined into a single matrix equation: \[\begin{bmatrix}1 & 1 \\ k...Hence, \[ik_-\, \left[\psi_+(x_a) - \psi_-(x_a)\right] = ik_+\, \left[\psi_+(x_b) - \psi_-(x_b)\right].\] These two equations can be combined into a single matrix equation: \[\begin{bmatrix}1 & 1 \\ k_- & - k_-\end{bmatrix}\begin{bmatrix}\psi_+(x_a) \\ \psi_-(x_a) \end{bmatrix} = \begin{bmatrix}1 & 1 \\ k_+ & - k_+\end{bmatrix} \begin{bmatrix}\psi_+(x_b) \\ \psi_-(x_b) \end{bmatrix}.\] After doing a matrix inversion, this becomes \[\Psi_b = \mathbf{M}_s(k_+,k_-) \, \Psi_a, \;\;\;\mathrm{where}\…
