Sums over histories

As for waves on a string, we can define ``transmission'' and ``reflection'' coefficients at a boundary between regions (in our case, at $x=a$) with different potential energies and thus different wave vectors $k_1$ and $k_2$. For a wave going from Region 1 to Region 2 the coefficients are¹

$$R_{1 \rightarrow 2} = \frac{k_1-k_2}{k_1+k_2}$$

$$T_{1 \rightarrow 2} = \frac{2 k_1}{k_1+k_2}$$

$$R_{2 \rightarrow 1} = \frac{k_2-k_1}{k_1+k_2}$$

$$T_{2 \rightarrow 1} = \frac{2 k_2}{k_1+k_2}$$

(a)

What is the wavefunction at $x=a$ of the wave that was reflected from the boundary of the potential well without entering into it? What about the wave that entered the potential well, was reflected from the infinite potential wall at $x=0$, and then escaped from the well?

(b)

Generalizing your result from part (a), obtain the wavefunction at $x=a$ of the wave moving from left to right in the region outside the potential well by summing over all possible back-and-forth reflections within the well. Use this result to obtain $\underline{r}$.