A one-dimensional system has the potential

$$V(x)= \begin{cases}0 & x<0 \\ \frac{\hbar^{2} U}{2 m} & 0<x<L \\ 0 & x>L\end{cases}$$

For energy $E=\hbar^{2} \epsilon /(2 m), \epsilon<U$, the wave function has the form

$$\psi(x)= \begin{cases}a e^{i k x}+c e^{-i k x} & x<0 \\ e \cosh K x+f \sinh K x & 0<x<L \\ d e^{i k(x-L)}+b e^{-i k(x-L)} & x>L\end{cases}$$

By considering the relation between incoming and outgoing waves explain why we should expect

$$|c|^{2}+|d|^{2}=|a|^{2}+|b|^{2}$$

Find four linear relations between $a, b, c, d, e, f$. Eliminate $d, e, f$ and show that

$$c=\frac{1}{D}\left[b+\frac{1}{2}\left(\lambda-\frac{1}{\lambda}\right) \sinh K L a\right]$$

where $D=\cosh K L-\frac{1}{2}\left(\lambda+\frac{1}{\lambda}\right) \sinh K L$ and $\lambda=K /(i k)$. By using the result for $c$, or otherwise, explain why the solution for $d$ is

$$d=\frac{1}{D}\left[a+\frac{1}{2}\left(\lambda-\frac{1}{\lambda}\right) \sinh K L b\right]$$

For $b=0$ define the transmission coefficient $T$ and show that, for large $L$,

$$T \approx 16 \frac{\epsilon(U-\epsilon)}{U^{2}} e^{-2 \sqrt{U-\epsilon} L}$$