Static and dynamic variational principles for expectation values of observables

Abstract A general procedure is first reviewed for constructing variational principles (v.p.) suited to the optimization of a quantity of interest. A geometric interpretation of the method is given. Conditions under which the solution may be obtained as a maximum rather than as a saddle-point are examined. Applications are then worked out, providing v.p. adapted to the evaluation of expectation values, fluctuations, or correlations of quantum observables, whether the system is at thermal equilibrium or whether it has evolved. In particular, a v.p. suited to the optimization of Tr Ae−sH is built, generalizing the usual free energy v.p. to observables A ≠ 1. To evaluate both the expectation value of an observable Q and its fluctuations (as well as correlations between Q1 and Q2), the characteristic function is determined variationally by letting A = exp[−ξQ], and then expanded in powers of ξ. Within a given class of trial states, the best approximate state depends in general on the question raised. Another class of v.p. concerns dynamical problems. The general method allows one to recover time-dependent v.p. for a state and an observable which was previously proposed and which deals with the following question: Given the state of the system at the time t0, what is the expectation value 〈A〉 at a later time t1? A more general v.p. applies to situations in which the initial state is too complicated to be handled exactly. Both the approximate initial conditions and the approximate evolution are then determined so as to optimize 〈A〉. Finally, analogous v.p. are constructed in classical statistical mechanics and Hamiltonian dynamics. The recent formulation of classical mechanics in terms of covariant Poisson brackets due to Marsden et. al. comes out naturally in this context from the v.p. for the evaluation of a classical expectation value.

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