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Time dependence in quantum mechanics : Floquet theory and the Berry phase

D J Moore

Published Feb 15, 1993• 1 citation
Publisher

Abstract

We give a short review of some aspects of the dynamics of time-dependent quantum systems. In particular we discuss how to recast time-periodic quantum problems into an equivalent time-independent form, contrasting the results with those gained for a similar reformulation in the case of general time-dependence. This approach, usually known under the name of Floquet theory, can be used to clarify the structure of, for example, invariants of the given quantum system. We also apply it to the calculation of the Berry phase and show how it enables us to understand Berry phases for time-independent systems. This involves rewriting earlier work of the author in a coordinate-free fashion

DOI
10.5169/seals-116563

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We give a short review of some aspects of the dynamics of time-dependent quantum systems. In particular we discuss how to recast time-periodic quantum problems into an equivalent time-independent form, contrasting the results with those gained for a similar reformulation in the case of general time-dependence. This approach, usually known under the name of Floquet theory, can be used to clarify the structure of, for example, invariants of the given quantum system. We also apply it to the calculation of the Berry phase and show how it enables us to understand Berry phases for time-independent systems. This involves rewriting earlier work of the author in a coordinate-free fashion
Shows that (non-adiabatic) Berry phases in time-independent systems arise for fundamentally different reasons than those for time-dependent systems. For time-independent Hamiltonians the Berry phases are seen to only depend on the chosen cyclic initial state, the Hamiltonian merely providing the appropriate period. The author also discusses two time-dependent examples where the time dependence arises from different sources, namely a two-level atom in an intense laser and a harmonic oscillator with a periodic forcing term.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.