Time-dependent variational principle for the expectation value of an observable: mean-field applications
This paper is devoted to the study of an algorithm providing a quantitative answer to the question: ''Is it possible, in the framework of some approximation, to obtain a best estimate for the expectation value of some observable A at the time \(t_ 1\) from the knowledge of \(D(t_ 0)\), the density operator at a time \(t_ 0<t_ 2?''\) For that purpose, the authors construct an action-like functional whose stationary value is precisely the above mentioned expectation value. The varational objects are the time dependent density operator and observable associated with the measurement, appearing as conjugate variables. By restricting the variational spaces, various approximations are derived, which provide equations of motion best suited to the quantity to be measured: the approximate dynamics will depend upon the nature of the measurement to be performed at time \(t_ 1.\) In this context, time-dependent Hartree-Fock equation appears as the best mean-field equation for predicting averages of single-particle observables, and results naturally from the variational principle by choosing the trial observables among the single particle ones and the density operators among the exponentials of single particle operators. Hence the TDHF approximation is obtained only because one wants to make predictions for a single particle observable. For a more complicated one, the TDHF equations must be abandonned. Other mean field formalisms are derived, providing insights into the approach to the fluctuation of a single particle operator or to transition probabilities. This variational principle also provides with a class of generalized mean fields by choosing time-dependent states with maximum entropy as trial states.
time-dependent states with maximum entropy
scientific article; zbMATH DE number 3988018