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In this case, the Berry phase corresponding to any given path on the unit sphere in magnetic-field space is just half the solid angle subtended by the path. The integral of the Berry curvature over the whole sphere is therefore exactly 2 π {\displaystyle 2\pi } , so that the Chern number is unity, consistent with the Chern theorem.
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 ...
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 the more common Schrödinger picture, even the states of free particles change over time: typically the phase changes at a rate that depends on their energy. In the alternative Heisenberg picture , state vectors are kept constant, at the price of having the operators (in particular the observables ) be time-dependent.
Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a Berry phase-like term stemming from spin or other internal degrees of freedom. [9] The index distinguishes the primitive periodic orbits: the shortest period orbits of a given set of initial conditions.
In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem [1] [2] [3] after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain ...
The result of Mehler can also be linked to probability. For this, the variables should be rescaled as x → x/ √ 2, y → y/ √ 2, so as to change from the 'physicist's' Hermite polynomials H (.) (with weight function exp(− x 2)) to "probabilist's" Hermite polynomials He (.) (with weight function exp(− x 2 /2)).