Noncommuting Coordinates in the Landau Problem

Physical Instances of Noncommuting Coordinates

Noncommuting spatial coordinates and fields can be realized in actual physical situations. Plane wave solutions to noncommuting photodynamics exhibit violaton of Lorentz invariance (special relativity).

Two-particle model with noncommuting operators of coordinates and momenta

A nonrelativistic equation for the system of two interacting particles within the framework of a model with noncommuting operators of coordinates and momenta of different particles is proposed, and a self-consistent system of equations for the wave function of every quantum state is deduced. Solutions for the lowest states of a hydrogenlike atom are found, and the comparison with analogous solutions of the Klein-Gordon equation for the relativistic spinless problem is performed. In the case where the size of a two-particle system and the Compton wavelengths of particles forming it are of the s…

Spin Hall Effect in Noncommutative Coordinates

A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncommutativity parameter theta . The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and θdeformed spin Hall conductivities which they provide are acquired. It…

On the Landau system in noncommutative phase-space

We consider a charged particle moving in a two dimensional plane in the presence of a background magnetic field perpendicular to the plane, i.e. the Landau system in a phase-space where the coordinates and momenta both follow canonical noncommutative algebra. A set of generalized transformations is derived in this paper which maps the NC problem to an equivalent commutative problem. In this set up, we study the Aharonov-Bohm effect and the Landau levels. For the Aharonov-Bohm effect, the phase-shift is found to contain corrections due to phase-space noncommutativity and also depends on the sca…

Wigner functions, coherent states, one-dimensional marginal probabilities and uncertainty structures of Landau levels

Following an approach based on generating function method phase space characteristics of Landau system are studied in the autonomous framework of deformation quantization. Coherent state property of generating functions is established and marginal probability densities along canonical coordinate lines are derived. Well defined analogs of inner product, Cauchy-Bunyakowsy-Schwarz inequality and state functional have been defined in phase space and they have been used in analyzing the uncertainty structures. The general form of the uncertainty relation for two real-valued functions is derived and…

朗道能级及其具有更高陈数类似物的唯一性

Uniqueness of Landau levels and their analogs with higher Chern numbers

本文证明了具有全纯波函数和平坦几何的唯一能量本征态是朗道能级及其更高陈数类似物，从而为构造可稳定分数量子霍尔相的所有全纯本征态提供了理论基础。