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Quantal phase factors accompanying adiabatic changes. (English) Zbl 1113.81306
Summary: A quantal system in an eigenstate, slowly transported round a circuit $$C$$ by varying parameters $$R$$ in its Hamiltonian $$\hat H(R)$$, will acquire a geometrical phase factor $$e^{i\gamma(C)}$$ in addition to the familiar dynamical phase factor. An explicit general formula for $$\gamma(C)$$ is derived in terms of the spectrum and eigenstates of $$\hat H(R)$$ over a surface spanning $$C$$. If $$C$$ lies near a degeneracy of $$\hat H$$, $$\gamma(C)$$ takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration $$\gamma(C)$$ is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

##### MSC:
 81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
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