7.4 In this chapter, we considered the scattering of momentum eigenstates off localized potentials. Of course, momentum eigenstates are not states in the Hilbert space, and so we have to consider \(L^{2}\)-normalizable linear combinations of momentum eigenstates as honest, physical states. In this exercise, we extend what was done in Example 7.2 where we determined the transmission and reflection amplitudes for an initial wave packet scattering off a potential that only has support near \(x=0\). The initial wave packet we consider in this problem has a profile in momentum space of

\[\begin{equation*}g(p)=\frac{e^{-i \frac{p x_{0}}{\hbar}}}{\left(\pi \sigma_{p}^{2}\right)^{1 / 4}} e^{-\frac{\left(pp_{0}\right)^{2}}{2 \sigma_{p}^{2}}} \tag{7.129}\end{equation*}\]

exactly as in that example.

(a) Verify that this wave packet is \(L^{2}\)-normalized.

(b) Determine the initial wavefunction in position space \(\psi(x)\) through Fourier transforming the momentum space representation.

(c) What is the group velocity, the velocity of the center-of-probability, of this wave packet in the region where there is 0 potential?

(d) What are the transmitted and reflected wavefunctions in position space?

(e) What do the transmitted and reflected wavefunctions become in the limit that the initial wave packet becomes very narrow about momentum \(p_{0}, \sigma_{p} \rightarrow\) 0 ? Does this make sense from the analysis of this chapter?