A review of the time-dependent variational principle

Abstract

The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space ℳ with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space ℳ := G/G0 under under G, they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value ℋ of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.