Time-dependent variational principle for the expectation value of an observable: Mean-field applications (Reprinted from Annals of Physics, vol 164, pg 334-410, 1985)

Given the state of a system at time t(0), the expectation value of an obser vable at a later time t(1) is expressed as the stationary Value of an actio n-like Functional, in which a time-dependent state and an observable are th e conjugate variables. By restricting the variational spaces, various appro ximations are derived. They provide equations of motion best suited to the quantity to be measured. In particular, time-dependent Hartrre-Fock (TDHF) appears as the best mean-field equation for predicting averages of single-p article observables. Other mean-field formalisms are derived, which are fit ted to the prediction of other observables, such as fluctuations or transit ion probabilities. In the later case, known coupled equations are recovered and discussed. The variational principle for a state and an observable als o provides an alternative class of generalized mean-fields, when time-depen dent states with maximum entropy are chosen as trial states. Linear respons e theory is incorporated naturally in this variational framework. (C) 1985 Academic Press.