Scattering with longitudinally coherent matter beams

I present a derivation of the many qualitative changes to basic scattering theory that result when a ~partially! coherent matter beam inelastically interacts with a quantum target. The weak beam limit, where only one object in the beam interacts with the quantum target, is developed. Some of the basic changes to scattering theory are as follows. ~1! The transition probability can increase as the square of the number of particles in the beam. ~2! The transition probability is not proportional to the convolution of the total inelastic cross section with the momentum distribution of the beam. ~3! The transition probability to different final states of the target does not depend on the population in the initial states but depends on the amplitudes for the initial states. ~4! The total transition probability depends on the differential cross section integrated over all final angles and on the differential cross section only in the forward scattering direction. ~5! The transition probability can be nonzero even when the beam misses the target. ~6! In the perturbative limit, the transition probability can be related to classical, large scale properties of the beam ~such as the average density!.