Derivation of the phase factor and geometrical phase for an N-state degenerate system

1998 ◽  
Vol 109 (3) ◽  
pp. 891-894 ◽  
Author(s):  
Michael Baer
Keyword(s):  
Phase Factor ◽  
Degenerate System ◽  
Geometrical Phase

scholarly journals Geometrical Phase Factor for a Non-Hermitian Hamiltonian

1991 ◽  
Vol 14 (1) ◽  
pp. 91-91
Author(s):  
Ch Miniatura ◽  
C Sire ◽  
J Baudon ◽  
J Bellissard
Keyword(s):  
Phase Factor ◽  
Geometrical Phase

WEYL'S GAUGE TRANSFORM AND QUANTUM GEOMETRIC PHASE FACTORS

2007 ◽  
Vol 22 (07n10) ◽  
pp. 645-650
Author(s):  
HUA-ZHONG LI
Keyword(s):  
Gauge Theory ◽  
Gauge Transformation ◽  
Quantum State ◽  
Geometric Phase ◽  
State Vector ◽  
Parallel Transport ◽  
Phase Factor ◽  
Geometrical Phase ◽  
Phase Loop ◽  
Gauge Transform

The historical and geometrical origin of Gauge Transformation and Yang's phase loop of gauge theory are discussed. In the present talk, we present the following points: 1. Parallel transport of a vector; 2. Weyl 1918 gauge transformation 3. Concept of non-integrable phase factor; 4. Berry's quantum geometrical phase; 5. Parallel transport of quantum state vector produces the phase physics.

Quantal phase factors accompanying adiabatic changes

1984 ◽  
Vol 392 (1802) ◽  
pp. 45-57 ◽  
Keyword(s):  
Phase Factor ◽  
Symmetric Matrices ◽  
Sign Reversal ◽  
Dynamical Phase ◽  
Geometrical Phase ◽  
Sign Change ◽  
Bohm Effect ◽  
Aharonov Bohm ◽  
Special Case ◽  
Quantal System

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

Geometrical Phase Factor for a Non-Hermitian Hamiltonian

1990 ◽  
Vol 13 (3) ◽  
pp. 199-203 ◽  
Author(s):  
Ch Miniatura ◽  
C Sire ◽  
J Baudon ◽  
J Bellissard
Keyword(s):  
Phase Factor ◽  
Geometrical Phase

scholarly journals Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 71 ◽  
Author(s):  
Olga Tsekhan
Keyword(s):  
Sufficient Conditions ◽  
Singularly Perturbed ◽  
Degenerate System ◽  
Multiple Delays ◽  
Complete Controllability ◽  
Singularly Perturbed System ◽  
System With Delay ◽  
Perturbed System ◽  
Time Invariant ◽  
Controllability Conditions

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

scholarly journals Path-Integral for a Colour Spin and Path-Ordered Phase Factor

1979 ◽  
Vol 62 (2) ◽  
pp. 544-553 ◽  
Author(s):  
J. Ishida ◽  
A. Hosoya
Keyword(s):  
Path Integral ◽  
Phase Factor

"MUTUAL FRACTIONAL STATISTICS" AND "RELATIVE ANYONS"

1991 ◽  
Vol 05 (10) ◽  
pp. 1771-1778 ◽  
Author(s):  
Chia-Ren Hu
Keyword(s):  
Quantum Hall ◽  
Ground States ◽  
Two Dimensions ◽  
Statistical Properties ◽  
Phase Factor ◽  
Fractional Statistics ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
General Properties ◽  
Topological Argument

A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.

scholarly journals Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

1999 ◽  
Vol 172 ◽  
pp. 443-444
Author(s):  
Massimiliano Guzzo
Keyword(s):  
Degenerate System ◽  
Escape Time ◽  
Kam Theorem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Arbitrary Potential ◽  
Hamiltonian Perturbation Theory ◽  
Three Body ◽  
Nekhoroshev Stability ◽  
Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

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