Piecewise constant potential (25 min)

We consider a 1d piecewise constant potential and a stationary wave function at energy $E$.

$$V(x)=\left\{ \begin{array}{cc} V_1, & \\ V_2, & \\ V_3, & \\ ... ... ...a_{N+1}e^{ik_{N+1}x}+b_{N+1}e^{-ik_{N+1}x}, & x_N<x< \infty \end{array} \right.$$ (14)

a) Show that $k_j=\sqrt{({2m}/{\hbar^2})\left(E-V_j\right)}$. Discuss the behaviour of the wave functions in regions with $V_j<E$ and $V_j>E$.

b) We consider the case $E>V_1,V_{N+1}$ such that $k_1$ and $k_{N+1}$ are real wave vectors and $\psi(x)$ describes running waves outside the `scattering region' $[x_1,x_N]$. Prove the matrix equation

$${\bf u}_1=T^1{\bf u}_2,\quad {\bf u}_i= \left( \begin{array}{c} a_i \\ b_i \end{array} \right),\quad i=1,2,$$ (15)

with

$$T^1=\frac{1}{2k_1}\left( \begin{array}{cc} (k_1+k_2)e^{i(k_2-k_1)... ...\ (k_1-k_2)e^{i(k_2+k_1)x_1} & (k_1+k_2)e^{-i(k_2-k_1)x_1} \end{array} \right).$$ (16)