14.1: Impact Parameter and Classical Analogies

Figure $\PageIndex{1}$: Relation between classical and quantum angular momentum

Knowing the impact parameter gives us some classical idea of whether a scattering event is likely. If the impact parameter is larger than the range of the potential, then classically the particles would miss. In the quantum case, we expect this to mean that the phase shift for that angular momentum is zero, and hence that the contribution from that term in the expansion is zero. Thus at a given incoming momentum, $\hbar k$, we can determine how many terms in the partial wave expansion to consider from $\hbar kb_{max} \approx l_{max}\hbar$, where $b_{max}$ is the maximum impact parameter for classical collision, i.e. the range of the potential.