We constructed the S-matrix as encoding the reflection and transmission amplitudes for individual momentum eigenstates, but of course these are not physical states on the Hilbert space. In this example, we will use these results for the scattering of momentum eigenstates off this localized potential illustrated in Fig. 7.12 and determine the reflection and transmission amplitudes for scattering of a physical wave packet. The form of the wave packet we will consider is, in momentum space:

The center of the wave packet is located at x = x0 and as x0 → −∞, the center of the wave packet is moved infinitely far to the left of the potential. Here, p0 is an initial momentum with p0 > 0, meaning that the packet moves right and σp is a measure of the width or spread of the wave packet about p0. This wave packet is plotted in momentum space in Fig. 7.13. More about this example will be explored in Exercise 7.4.

(p)= e-MD (P-PQ) (02)1/4 e 20 (7.88)