An incoming quantum particle, with energy \(E\), is governed by the Hamiltonian \[\hat{H} = \hat{H}_0 + V(\hat{\mathbf{r}}), \;\;\; \hat{H}_0 = \frac{\hat{\mathbf{p}}^2}{2m}.\] Here, \(\hat{H}_0\) des...An incoming quantum particle, with energy \(E\), is governed by the Hamiltonian \[\hat{H} = \hat{H}_0 + V(\hat{\mathbf{r}}), \;\;\; \hat{H}_0 = \frac{\hat{\mathbf{p}}^2}{2m}.\] Here, \(\hat{H}_0\) describes the particle’s kinetic energy, \(m\) is the particle’s mass, \(\hat{\mathbf{r}}\) and \(\hat{\mathbf{p}}\) are position and momentum operators, and \(V\) is a scattering potential describing how the scatterer affects the quantum particle.