Time-Dependent Variational Principle for the Expectation Value of an Observable: Mean-Field Applications

✍ Scribed by Roger Balian; Marcel Vénéroni

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 402 KB
• Volume: 281
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1006/aphy.2000.6008

✦ Synopsis

Given the state of a system at time t 0 , the expectation value of an observable at a later time t 1 is expressed as the stationary value of an action-like functional, in which a time-dependent state and an observable are the conjugate variables. By restricting the variational spaces, various approximations are derived. They provide equations of motion best suited to the quantity to be measured. In particular, time-dependent Hartree Fock (TDHF) appears as the best mean-field equation for predicting averages of single-particle observables. Other meanfield formalisms are derived, which are fitted to the prediction of other observables, such as fluctuations or transition probabilities. In the later case, known coupled equations are recovered and discussed. The variational principle for a state and an observable also provides an alternative class of generalized mean-fields, when time-dependent states with maximum entropy are chosen as trial states. Linear response theory is incorporated naturally in this variational framework.