How and why to think about scattering in terms of wave packets instead of plane waves

We discuss "the plane wave approximation" to quantum mechanical scattering using simple one-dimensional examples. The central points of the paper are that (a) plane waves should be thought of as infinitely wide wave packets, and (b) the calculations of reflection and transmission probabilities $R$ and $T$ in standard textbook presentations involve an approximation which is almost never discussed. We argue that it should be discussed explicitly, and that doing so provides a simple and intuitively revealing alternative way to derive and understand certain formulas. Using an under-appreciated exact expression for wave packet scattering probabilities, we calculate, for two standard examples, expressions for $R$ and $T$ for an incident Gaussian wave packet. Comparing these results to the corresponding ones calculated using the plane wave approximation helps illuminate the domain of applicability of that approximation.