Time‐dependent quantum‐mechanical scattering in two dimensions

While scattering phenomena are discussed in most quantum mechanics textbooks, the analysis is usually performed using time‐independent techniques. These techniques have certainly been successful in predicting the results of scattering events but fail to present an intuitive picture of what happens to the physical system during a collision. A remedy to this situation could be affected by the introduction of time‐dependent techniques, utilizing the conceptually simple idea of a wavefunction evolving over time, but these methods have traditionally been computationally unmanageable. This paper describes a novel numerical technique, the split‐operator Fourier transform method, which greatly reduces the amount of computational effort required to accurately solve the time‐dependent Schrödinger equation. This method is applied to the Rutherford scattering problem in two dimensions and it is shown that the method is not only efficient enough to yield results in less than 15 min when implemented on a standard personal computer, but is also simple enough to be employed by an advanced undergraduate. The method also has the advantage of reinforcing the quantum‐mechanical formalism in the student’s mind; both the evolution operator and the complementary relationship between operators and wavefunctions in coordinate and momentum space are exploited. The results of the calculations are presented as color graphs portraying the magnitude and phase of wave packets as they evolve through time.