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The concept was first introduced by S. Pancharatnam [1] as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 [2] emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics.
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.
Trigonal bipyramidal molecular shape ax = axial ligands (on unique axis) eq = equatorial ligand (in plane perpendicular to unique axis). The Berry mechanism, or Berry pseudorotation mechanism, is a type of vibration causing molecules of certain geometries to isomerize by exchanging the two axial ligands (see the figure) for two of the equatorial ones.
In this way, phase retrieval allows for the conversion of a diffraction pattern into an image without an optical lens. Using phase retrieval algorithms, it is possible to characterize complex optical systems and their aberrations. [6] For example, phase retrieval was used to diagnose and repair the flawed optics of the Hubble Space Telescope ...
The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...
Sir Michael Victor Berry (born 14 March 1941) is a British theoretical physicist. He is the Melville Wills Professor of Physics (Emeritus) at the University of Bristol . He is known for the Berry phase , a phenomenon observed in both quantum mechanics and classical optics , as well as Berry connection and curvature .
The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization.
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and ...