Approach to Equilibrium in Quantal Systems: Magnetic Resonance

Abstract

The present paper presents a derivation of the "master" or Boltzmann "gain-loss" equation from the Schrödinger equation, i.e., a derivation of the equation for the evolution in time of the probabilities of finding a physical system in its various states from the equation for the corresponding probability amplitudes. The "master" equation is derived for an, in effect completely self-enclosed, "supersystem," [$A+B$], consisting of a "system of interest," [$A$], and a "surroundings," [$B$], in relatively weak mutual interaction. A discussion is given of the range of validity of the "master" equation for [$A+B$] and it is shown that the random phase assumption is required for the state vector of [$A+B$] at the initial time only. The normally microcanonical character of the equilibrium statistical configuration of [$A+B$] is demonstrated and a treatment is given of exceptional, "extremely quantal-coherent," initial statistical distributions of [$A+B$] which may evolve away from equilibrium. Derivations are also presented of the "master" equation for [$A$] and of the "master" equation for an individual particle or quasi-particle [$q$], within [$A$]; a discussion of the range of validity of these "master" equations is given and the normally canonical character of the equilibrium statistical configuration of [$A$] is deduced. General solutions of the "master" equations for [$A+B$], [$A$], and [$q$] are worked out and the relation between the principles of "microscopic reversibility" and "detailed balance" and the nonoscillatory character of the approach to equilibrium are exhibited. A theorem is presented regarding the time variation of the entropy of [$A$].