Semiclassical quantisation rules for the Dirac and Pauli equations

A semiclassical approach to the Dirac equation

We derive a semiclassical time evolution kernel and a trace formula for the Dirac equation. The classical trajectories that enter the expressions are determined by the dynamics of relativistic point particles. We carefully investigate the transport of the spin degrees of freedom along the trajectories which can be understood geometrically as parallel transport in a vector bundle with SU(2) holonomy. Furthermore, we give an interpretation in terms of a classical spin vector that is transported along the trajectories and whose dynamics, dictated by the equation of Thomas precession, gives rise t…

Semiclassical trajectory-coherent approximation in quantum mechanics: II. High order corrections to the Dirac operators in external electromagnetic field

High approximations of semiclassical trajectory-coherent states (TCS) and of semiclassical Green function (in the class of semiclassically concentrated states) for the Dirac operator with anomalous Pauli interaction are obtained. For Schrodinger and Dirac operators trajectory-coherent representations are constructed up to any precision with respect to h, h-->0.

Erlangen Programme at Large 3.1: Hypercomplex Representations of the Heisenberg Group and Mechanics

In the spirit of geometric quantisation we consider representations of the Heisenberg(--Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit h->0. Keywords: Heisenberg group, Kirillov'…

Semiclassical limit for almost fermionic anyons

In two-dimensional space there are possibilities for quantum statistics continuously interpolating between the bosonic and the fermionic one. Quasi-particles obeying such statistics can be described as ordinary bosons and fermions with magnetic interactions. We study a limit situation where the statistics/magnetic interaction is seen as a "perturbation from the fermionic end". We vindicate a mean-field approximation, proving that the ground state of a gas of anyons is described to leading order by a semi-classical, Vlasov-like, energy functional. The ground state of the latter displays anyonic…

The semiclassical resonance spectrum of Hydrogen in a constant magnetic field

We present the first purely semiclassical calculation of the resonance spectrum in the Diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with $L_z =0$. The classical system is unbound and completely chaotic for a scaled energy $ε\sim E B^{-2/3}$ larger than a critical value $ε_c >0$. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the poten…

Trapping neutral particles endowed with a magnetic moment by an electromagnetic wave carrying orbital angular momentum: Semiclassical theory

The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schrödinger-Pauli wave equation and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the …

费米子的统计势：涌现的吸引力与保罗i晶体形成

Attractive statistical forces and Pauli crystal formation in trapped Fermi gases

研究揭示了N个相同费米子的热力学可通过集体统计势解释，该势能解释了德泽压力的微观起源，且对于N≥3时产生吸引力贡献，形成保罗i晶体结构。

Effect of a strong laser on spin precession

The semiclassical dynamics of a charged spin-1/2 particle in an intense electromagnetic plane wave is analyzed beyond the electric dipole approximation and taking into account the leading relativistic corrections to the Pauli equation. It is argued that the adiabatic spin evolution driven by a low intensity radiation changes its character drastically as an intensity of a laser is increasing. Particularly, it is shown that a charged particle exposes a spin flip resonance at a certain pick value of a laser field strength, which is determined by a particle's gyromagnetic ratio.

Topological spin transport of relativistic electron

We derive the semiclassical (with accuracy of $\hbar$) motion equation for relativistic electron, which follow from the Dirac equation. We determine both the evolution equation for electron polarization, which takes the non-Abelian Berry phase into account, and Hamiltonian equations for trajectories of the particle (wave packet) center in the phase space. The equations have covariant form with respect to U(2) gauge transformations and contain topological spin terms that are connected to the Berry gauge field arising during the diagonalization of the Dirac equation. The trajectory equations obt…