7.5 We introduced an operator \(\hat{U}_{p}(x, x+\Delta x)\) in this chapter that represented generalized translations for which momentum was not assumed to be conserved. We used that operator to determine the reflection and transmission amplitudes for

scattering off the step potential, but we only needed the first few orders of its Taylor expansion for our purposes there. In this problem, we will identify some more properties of this translation operator and explicitly evaluate it for a finite translation and a given potential.

(a) From the definition of \(\hat{U}_{p}(x, x+\Delta x)\) as a translation operator and its connection to momentum, determine the differential equation in the displacement \(\Delta x\) that \(\hat{U}_{p}(x, x+\Delta x)\) satisfies, for a fixed total energy \(E\) and a spatially varying potential \(V(x)\).

(b) Let's consider the step potential of this chapter, with general width \(a\) and height \(V_{0}\). Determine the translation operator \(\hat{U}_{p}\left(x_{0}, x_{1}\right)\) in position space that translates the wavefunction across the entire potential, so that \(x_{0}<0\) and \(x_{1}>a\).

Hint: Remember how translation operators compose.

(c) Now, let's consider the narrow-potential limit we introduced in this chapter for which \(a \rightarrow 0\) and \(V_{0} \rightarrow \infty\), but with \(a V_{0}=\alpha=\) constant. What is the translation operator \(\hat{U}_{p}\left(x_{0}, x_{1}\right)\) in this case? Is the limit sensible?