In quantum mechanics, the Berry phase is a phase acquired by quantum states when subjected to adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. It was named after Sir Michael Berry, and is also known as Pancharatnam-Berry or geometric phase. It appears in particular in the theory of the Aharonov-Bohm effect and of the conical intersection of potential energy surfaces. In the case of the Aharonov-Bohm effect, the adiabatic parameter is the magnetic field inside the solenoid. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. Generally speaking, it occurs whenever one can externally control at least two parameters affecting a wave. Fig. ... Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ... In quantum mechanics, an adiabatic process is an infinitely slow change in the Hamiltonian of a system. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... Michael Berry is also the name of a writer. ... The Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded, proposed by Aharonov and Bohm in 1959. ... In quantum chemistry, a conical intersection of two potential energy surfaces of the same spatial and spin symmetries is the set of molecular geometry points where the two potential energy surfaces are degenerate (intersect). ... A potential energy surface is generally used within the adiabatic or Born-Oppenheimer approximation in quantum mechanics and statistical mechanics to model reactions and interactions in simple chemical and physical systems. ... Current flowing through a wire produces a magnetic field (M) around the wire. ... In engineering, a solenoid is a mechanical device that converts energy into linear motion. ... Geometry of the water molecule Molecules have fixed equilibrium geometries--bond lengths and angles--that are dictated by the laws of quantum mechanics. ... A wave crashing against the shore A wave is a disturbance that propagates. ... See also list of optical topics. ...

Waves are characterized by amplitude and phase, and both may vary as a function of those parameters. The Berry phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could e.g. involve rotations but also translations of particles, which are apparently undone at the end. Intuitively one expects that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursion is a cyclic loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the Berry phase, and its occurrence typically indicates that the system's parameter dependence is singular (undefined) for some combination of parameters. Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation. ... Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ... In communications or computer systems, a configuration is an arrangement of functional units according to their nature, number, and chief characteristics. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

To measure the Berry phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the Berry phase. This mechanics analogue of the Berry phase is known as the Hannay angle. Measure can mean: To perform a measurement. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... From Latin ex- + -periri (akin to periculum attempt). ... Foucaults Pendulum in the Panthéon, Paris A Foucault pendulum, or Foucaults pendulum, named after the French physicist Léon Foucault, was conceived as an experiment to demonstrate the rotation of the Earth; its action is a result of the Coriolis force. ... In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ...

See also

• For the connection to mathematics, see curvature tensor,
• The Aharonov-Bohm effect,
• Conical intersections of potential energy surfaces.

In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... The Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded, proposed by Aharonov and Bohm in 1959. ... In quantum chemistry, a conical intersection of two potential energy surfaces of the same spatial and spin symmetries is the set of molecular geometry points where the two potential energy surfaces are degenerate (intersect). ... A potential energy surface is generally used within the adiabatic or Born-Oppenheimer approximation in quantum mechanics and statistical mechanics to model reactions and interactions in simple chemical and physical systems. ...

References

• Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9. (See chapter 13 for a mathematical treatment)

• Connections to other physical phenomena (such as the Jahn-Teller effect) are discussed here: [1]