7.8 Consider the potential illustrated in Fig. 7.19, where there is a finite potential barrier of height \(V_{0}\) and width \(a\), that terminates at \(x=0\) at which the potential becomes infinite for all \(x>0\). Consider a plane wave of momentum \(p\) incident on this potential from the left, toward \(x \rightarrow-\infty\).

(a) Write down the S-matrix for scattering off this potential. What is the transmission amplitude?

(b) What is the phase accumulated from the reflection of an incident wave as it passes over the finite barrier and returns traveling to the left?