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Quasi-adiabatic dynamics in classical mechanics

Nikolai A. Sinitsyn, V. L. Pokrovsky

Published Feb 5, 2026• 0 citations
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Abstract

Abstract This chapter is introduction to geometric phases and nonadiabatic effects in classical mechanics. Applications are described for models of a passage through a phase transition in a scalar mean field theory.
DOI
10.1093/oso/9780198988007.003.0010

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Abstract This book is an introduction to explicitly time-dependent physics. We describe the theory of the dynamic regime that we refer to as “quasi-adiabatic”. It pertains to time-dependent evolution that, while slow, is not genuinely adiabatic. It is rich in effects that can be understood even in quantum systems with complex many-body interactions. Examples from research in quantum computing, phase transitions, Bose-Enstein condensates, and quantum control will be used to illustrate these effects. Prerequisites include undergraduate introductory courses in quantum and classical mechanics, as well as complex analysis. Our book aims to fill the gap in the literature by providing an accessible, for early graduate and advanced undergraduate students, introduction to both geometric phases and nonadiabatic transitions at a level sufficient to consider numerous applications.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
Why Is Group Theory Useful in Quantum Mechanics? Irreducible Representations of SU(2) and SO(3), Rotation Matrices The Symmetric Groups The Lie Algebra of SO(4) and the Hydrogen Atom Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole The Berry Phase in Molecular Dynamics The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions The Connection Is the Gauge Field and the Curvature Is the Force: Some Topology and Differential Geometry Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect Chern-Simons Theory: The Fractional Quantum Hall Effect and other chapters.