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Quantum Adiabatic Approximation, Quantum Action, and Berry’s Phase

Ali Mostafazadeh

Published Jan 1, 1996• 4 citations

Abstract

An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry’s adiabatic phase and its semi-classical limit are also discussed. Probably one of the best recognized applications of the quantum adiabatic approximation [1, 2] is in Berry’s derivation of the adiabatic geometrical phases [3]. Following Berry’s, by now, classical article on the adiabatic geometric phase [3], Hannay proposed a classical analogue of Berry’s phase [4] and Berry [5] and Anandan [6] explored the semiclassical limit of Berry’s phase and the classical analogue of the general, non-adiabatic geometric phase [7], respectively. The purpose of this note is to provide an alternative interpretation of the quantum adiabatic approximation which yields a natural approach to study the semi-classical limit of this approximation and consequently Berry’s phase.

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An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry’s adiabatic phase and its semi-classical limit are also discussed. Probably one of the best recognized applications of the quantum adiabatic approximation [1, 2] is in Berry’s derivation of the adiabatic geometrical phases [3]. Following Berry’s, by now, classical article on the adiabatic geometric phase [3], Hannay proposed a classical analogue of Berry’s phase [4] and Berry [5] and Anandan [6] explored the semiclassical limit of Berry’s phase and the classical analogue of the general, non-adiabatic geometric phase [7], respectively. The purpose of this note is to provide an alternative interpretation of the quantum adiabatic approximation which yields a natural approach to study the semi-classical limit of this approximation and consequently Berry’s phase.

We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.

The quantum dynamics of time-dependent Hamiltonians of the kind H(t)=U(t)H(0)U(t)° is studied in the framework of generalized coherent states. It is shown that the quantum and the classical dynamics are isomorphic and that the phase of ‖ψ(t)〉 is simply the classical action. In adiabatic approximation it is easy to extract Berry’s phase from this. A discussion of generalized coherent states (CS’s) necessary to deal with excited states is given; in particular for SU(2), SU(1,1) and E(2) their structure is completely characterized, showing that the deep connection with geometric quantization extends to these generalized CS’s. This technique may be employed to estimate higher-order correction to the adiabatic approximation.

We propose a new formula for the adiabatic Berry phase which is based on phase-space formulation of quantum mechanics. This approach sheds a new light into the correspondence between classical and quantum adiabatic phases -- both phases are related with the averaging procedure: Hannay's angle with averaging over the classical torus and Berry's phase with averaging over the entire classical phase space with respect to the corresponding Wigner function. Generalizations to the non-abelian Wilczek--Zee case and mixed states are also included.

The geometric phase in quantum mechanics is formulated for charged particles in a gauge-invariant, geometric manner. It is then extended to an evolution resulting from a sequence of measurements as in the work of Pancharatnam and Aharonov and Vardi. Its close connection to the Feynman formulation of quantum mechanics is pointed out. The geometric angles, which are generalizations of the classical, adiabatic angles introduced by Hannay and the quantum, adiabatic angles introduced by Anandan and Stodolsky in their group-theoretic treatment of Berry's phase, are studied in quantum and classical physics. The geometric phase for a quantum spin in a magnetic field due to a second particle is obtained using the quantum reference frame defined by the latter. The question of whether the geometric phase and angles are local or nonlocal and their relationship to the electromagnetic and gravitational phases are also discussed.

We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-

1∕2

problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.