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A general procedure is first reviewed for constructing variational principles (v.p.) suited to the optimization of a quantity of interest. The method is enlightened by a geometric interpretation. 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-t~l~ is built, generalizing to observables A # 1 the usual free energy v.p. To evaluate both the expectation value of an observable Q and its fluctuations (as well as correlations between Q1 and Q2), the characteristic function, determined variationally by letting A = exp[-~Q], is expanded in powers of ~. 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 a known time-dependent v.p. for a state and an observable [R. Balian and M. V6n6roni, Ann. Phys. (N.Y.) 16

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Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0003-4916)

Published

1988

Language

EN

Field

Physics and Astronomy (Physical Sciences)