Source boundary conditions in the Keldysh formalism

ERNESTO P. DANIELI; LUIS E. F. FOA TORRES; GONZALO A. ÁLVAREZ; HORACIO M. PASTAWSKI

San Carlos de Bariloche, Argentina

Workshop; Pan American Advanced Studied Institute (PASI) on Physics at the Nanometer Scale - PASI 2003; 2003

Pan American Advanced Studied Institute

A situation which is quite overlooked in the literature is the Schrödinger equation with a time dependent source. Examples of this are the electrodes in an a.c. electrical conductivity experiment, the non-equilibrium chemical kinetics for slow rates of laser pumping into the excited state; the cross-polarization transfer in NMR, etc. We use the keldysh formalism to study the solutions of the Schrödinger equation with asource. We solve the two extreme cases in which the boundary condition results in either a coherent or an incoherent injection of quantum waves. For the coherent case, we find a relation between the source boundary condition and the initial value problem that enables proper normalization and establishes their equivalence. Possible realization in NMR are discussed. The incoherent case was used to represent the action of the environment in the system under study. We discuss the simplest case in this category: the problem of cross-polarization transfer in NMR in which the nuclear spin system 13C-1H is in contact with a spin reservoir. For a solvable model of this system, the result of the Keldysh equation with a decoherent source is shown to give the exact dynamics.