Search results
Results from the WOW.Com Content Network
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 one-dimensional integrals can be generalized to multiple dimensions. [2] (+) = ()Here A is a real positive definite symmetric matrix.. This integral is performed by diagonalization of A with an orthogonal transformation = = where D is a diagonal matrix and O is an orthogonal matrix.
It is the most widely accepted mechanism for pseudorotation and most commonly occurs in trigonal bipyramidal molecules such as PF 5, though it can also occur in molecules with a square pyramidal geometry. [1] The Berry mechanism is named after R. Stephen Berry, who first described this mechanism in 1960. [2] [3]
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. [1] Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature.
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). [ 1 ] [ page needed ] On the other hand, since a sphere of radius R has constant positive curvature R −2 and a flat plane has constant curvature 0, these two surfaces are not isometric, not even locally.