Approach to Equilibrium in Quantal Systems: Magnetic Resonance

The present paper presents a derivation of the "master" or Boltzmann "gain-loss" equation from the Schr\"odinger 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].As illustrations of the general methods developed two important processes in magnetic resonance---the time variation of the longitudinal magnetization, 〈μ〉ₜ, and the time variation of the transverse magnetization, 〈μ^'〉ₜ---are discussed in some detail. It is shown that the variation of 〈μ〉ₜ with t and of 〈μ^'〉ₜ with t for a "nonrigid" lattice can be described by means of the "master" equation for an individual spin [q] and several special cases are discussed on the basis of the evaluation of the appropriate transition probabilities; a comparison with the "spin-temperature" procedure is also appended. On the other hand, it is demonstrated that for a "rigid" lattice no description of the variation of 〈μ^'〉ₜ with t can be given on the basis of a "master" equation; in this case, quantal coherence effects neglected in the derivation of the "master" equation from the Schr\"odinger equation are vital and 〈μ^'〉ₜ must be evaluated by a rigorous calculation of Trace {[appropriate time dependent density matrix] μ^'}.