Chapter 7: One Dimensional Barrier Scattering

One of the most dramatic effects in scattering theory is that of resonances - sudden changes in phase shifts and/or reflection/transmission coefficients as a function of energy. This sharp dependence of the reflection and transmission amplitude on energy generally corresponds to a matching of the energy of the free incoming or outgoing wave with an almost bound, or 'quasibound' state of the scattering potential. From a time dependent point of view, portions of the wavepacket are reflected or transmitted immediately, while other portions, with energy contents approximately equal to that of the quasibound state, spend a non-negligible amount of time in the region of the interaction potential before reaching the asymptotic region of reactants or products. The program below calculates the scattering eigenstate as a function of energy, as the wavenumber k passes from below to above the first resonance at k = π/a The phase shift, χ, is calculated from the equation: