Variational Principles for Scattering Processes. I

A systematic treatment is presented of the application of variational principles to the quantum theory of scattering. Starting from the time-dependent theory, a pair of variational principles is provided for the approximate calculation of the unitary (collision) operator that describes the connection between the initial and final states of the system. An equivalent formulation of the theory is obtained by expressing the collision operator in terms of an Hermitian (reaction) operator; variational principles for the reaction operator follow. The time-independent theory, including variational principles for the operators now used to describe transitions, emerges from the time-dependent theory by restricting the discusson to stationary states. Specialization to the case of scattering by a central force field establishes the connection with the conventional phase shift analysis and results in a variational principle for the phase shift. As an illustration, the results of Fermi and Breit on the scattering of slow neutrons by bound protons are deduced by variational methods.