Rigorous Derivation of the Phase Shift Formula for the Hilbert Space Scattering Operator of a Single Particle

For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.