** A more general definition of the transfer matrix $M$ ($>30$ min)

We consider a one-dimensional potential of the form

$$V(x)=\left\{ \begin{array}{cc} 0, & \\ v(x), & \\ 0 & \end{array}... ...(x), & x_1<x\le x_2 \\ c e^{ikx}+de^{-ikx}, & x_2<x< \infty \end{array} \right.$$ (21)

Here, $v(x)$ is an arbitrary real potential. The central part $\phi(x)$ of the wave function $\psi(x)$ therefore in general is very difficult to calculate. We can, however, relate the coefficients $a$, $b$ (left side) with the coefficients $c$, $d$ (right side): if some fixed values for $c$ and $d$ are chosen, this determines the solution $\psi(x)$ everywhere on the $x$-axis and therefore in particular $a$ and $b$. We write this relation as

$$\left( \begin{array}{c} a \\ b \end{array}\right)= \left( \b... ...ray}\right) \left( \begin{array}{c} c \\ d \end{array}\right).$$ (22)

a) With $\psi(x)$ also the conjugate complex $\psi^*(x)$ must be a solution of the stationary Schrödinger equation $\hat{H}\psi(x)=E\psi(x)$. Why ?

b) Take the conjugate complex $\psi^*(x)$ in (2.21) and show that this leads to the exchange $a \leftrightarrow b^*$ and $c \leftrightarrow d^*$ in (2.22).

c) Take the conjugate complex of the whole equation (2.22) and compare with the equation you obtain from part b). Show that

$$M_{11}^*=M_{22},\quad M_{12}^*=M_{21}.$$ (23)

d) Consider the current density and show that

$$\vert a\vert^2 - \vert b\vert^2 = \vert c\vert^2 - \vert d\vert^2.$$ (24)

Write this equation as a scalar product of vectors in the form

$$(a^* b^*) \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{arra... ...ray}\right) \left( \begin{array}{c} c \\ d \end{array}\right).$$ (25)

Use the matrix $M$ to derive from this

$$\det(M)=1.$$ (26)