Berry's Phase

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.