Berry's Phase

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Berry's phase [1] is a quantum phase effect arising in systems that undergo a slow, cyclic evolution. It is a remarkable correction to the quantum adiabatic theorem and to the closely related Born-Oppenheimer approximation [2]. Berry's elegant and general analysis has found application to such diverse fields as atomic, condensed matter, nuclear and elementary ► particle physics, and optics. In this brief review, we first derive Berry's phase in the context of the quantum adiabatic theorem and then in the context of the Born-Oppenheimer approximation. We mention generalizations of Berry's phase and analyze its relation to the ► Aharonov-Bohm effect. Consider a Hamiltonian H f (R) that depends on parameters R 1, R 2, …, R n, components of a vector R. Let us assume that H f (R) has at least one discrete and nondegenerate eigenvalue E i (R) with | Ψ i (R)⟩ its eigenstate; E i (R) and | Ψ i (R)⟩ inherit their dependence on R from H(R). If the vectorR changes in time, then | Ψ i (R)⟩ is not an exact solution to the time-dependent ► Schrödinger equation. But if R changes slowly enough, the system does not ► quantum jump to another eigenstate. Instead, it adjusts itself to the changing Hamiltonian. A heavy weight hanging on a string illustrates such adiabaticity. Pull the string quickly — it snaps and the weight falls. Pull the string slowly — the weight comes up with it.
Original languageEnglish
Title of host publicationCompendium of Quantum Physics
Editors D. Greenberger, K. Hentschel, F. Weinert
Place of PublicationBerlin, Heidelberg
PublisherSpringer Berlin Heidelberg
Edition1st ed.
ISBN (Print)1-283-00370-8
StatePublished - 25 Jul 2009


  • Quantum physics
  • Electron Wave Function
  • Berry Phase
  • Adiabatic Limit
  • Topological Phase
  • Cyclic Evolution


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