Matching on the boundary

(a)

Show that the function

$$\Psi_{\rm in}(x) = Ce^{-i k_0 x} \,+\, De^{i k_0 x}$$

satisfies the Schr $\ddot{\rm o}$dinger's equation for $0<x<a$ for any values of $C, D$. Determine $k_0$.

(b)

What is the boundary condition that should be satisfied by $\Psi_{\rm in}(x)$ at $x=0$? Use this boundary condition to simplify $\Psi_{\rm in}(x)$.

(c)

Write down the equations that should be satisfied by the wavefunctions at $x=a$. Solve these equations to obtain $\underline{r}$.