arXiv subjects

Stefan Keppeler

Publications and source records attributed to Stefan Keppeler.

25 records · Page 2Linked to original sources

arXiv2001

Level spacings and periodic orbits

Starting from a semiclassical quantization condition based on the trace formula, we derive a periodic-orbit formula for the distribution of spacings of eigenvalues with k intermediate levels. Numerical tests verify the validity of this representation for the nearest-neighbor level spacing (k=0). In a second part, we present an asymptotic evaluation for large spacings, where consistency with random matrix theory is achieved for large k. We also discuss the relation with the method of Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472] for two-point correlations.

nlin.CD↗

arXiv2001

Quantum and classical ergodicity of spinning particles

We give a formulation of quantum ergodicity for Pauli Hamiltonians with arbitrary spin in terms of a Wigner-Weyl calculus. The corresponding classical phase space is the direct product of the phase space of the translational degrees of freedom and the two-sphere. On this product space we introduce a combination of the translational motion and classical spin precession. We prove quantum ergodicity under the condition that this product flow is ergodic.

nlin.CD↗

arXiv2000

Quantum cat maps with spin 1/2

We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a two-spinor precessing in a magnetic field. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE) of random matrices. In particular, we show that the diagonal approximation of the spectral form factor for small arguments agrees with the CSE prediction. The results are confirmed by numerical investigations.

nlin.CD↗

arXiv2000

Two-point correlations of the Gaussian symplectic ensemble from periodic orbits

We determine the asymptotics of the two-point correlation function for quantum systems with half-integer spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the leading terms of the two-point correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the role of Kramers' degeneracy.

nlin.CD↗

arXiv1999

Semiclassical form factor for chaotic systems with spin 1/2

We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the so-called diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory.

chao-dyn↗

arXiv1998

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 to dynamical and geometric phases every orbit is weighted by. We also present an analogous approach to the Pauli equation which we analyse in two different limits.

quant-ph↗

arXiv1998

Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

We investigate the Dirac equation in the semiclassical limit \hbar --> 0. A semiclassical propagator and a trace formula are derived and are shown to be determined by the classical orbits of a relativistic point particle. In addition, two phase factors enter, one of which can be calculated from the Thomas precession of a classical spin transported along the particle orbits. For the second factor we provide an interpretation in terms of dynamical and geometric phases.

quant-ph↗