On the quantum mechanical scattering from a potential step

The problem of finding the exact spacetime particle's propagator in the presence of a potential step (interface between different materials) is revisited. In contrast to the conventional Feynman path-integral approach, integration over all energy values of the particle's spectral density matrix (discontinuity of the energy-dependent Green's function across the real energy axis) is suggested for obtaining the exact spacetime propagator. The energy-dependent Green's functions are found in the framework of the multiple scattering theory (MST). The problem of finding the step-localized energy-dependent potentials responsible for the particle's reflection from and transmission through a potential step, which are needed for MST application, is solved. The obtained exact result for the particle's propagator is expressed in terms of integrals of elementary functions and has a significantly simpler form than that reported earlier. The obtained expressions allow easy evaluation of all limiting cases, including the case of the infinitely large potential step, as well as simple numerical visualization. The square of the absolute value of the propagator, which represents the relative transition probability density between two spacetime points, is plotted and discussed in detail for the cases of particle reflection and transmission.