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An example of physical systems where an electron moves along a closed path is cyclotron motion (details are given in the page of Berry phase). Berry phase must be considered to obtain the correct quantization condition.
There are several important aspects of this generalization of Berry's phase: 1) Instead of the parameter space for the original Berry phase, this Ning-Haken generalization is defined in phase space; 2) Instead of the adiabatic evolution in quantum mechanical system, the evolution of the system in phase space needs not to be adiabatic.
The Hannay angle is defined in the context of action-angle coordinates.In an initially time-invariant system, an action variable is a constant. After introducing a periodic perturbation (), the action variable becomes an adiabatic invariant, and the Hannay angle for its corresponding angle variable can be calculated according to the path integral that represents an evolution in which the ...
The Berry mechanism in square pyramidal molecules (such as IF 5) is somewhat like the inverse of the mechanism in bipyramidal molecules.Starting at the "transition phase" of bipyramidal pseudorotation, one pair of fluorines scissors back and forth with a third fluorine, causing the molecule to vibrate.
[32] [73] Berry's phase arises due to chirality or dependence (locking) of the pseudospin quantum number on the momentum of low-energy electrons near the Dirac points. [34] The temperature dependence of the oscillations reveals that the carriers have a non-zero cyclotron mass, despite their zero effective mass in the Dirac-fermion formalism.
It is generally argued that the Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, in quantum mechanics.Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom are not even unique for a given electromagnetic field.
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Well-known examples are the intramolecular isomerization of trigonal bipyramidal compounds by the Berry pseudorotation mechanism, and the out-of-plane motions of carbon atoms exhibited by cyclopentane, leading to the interconversions it experiences between its many possible conformers (envelope, twist). [2]