Zitterbewegung of relativistic electrons in a magnetic field and its simulation by trapped ions

One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion (Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron Zitterbewegung: it introduces intraband frequency components into the motion, influences all the frequencies and makes the motion stationary (not decaying in time) in case of the 2+1 Dirac equation. Finally, simulations of the 2+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.Comment: 19 pages, 9 figures, published versio

Non-locality of Foldy-Wouthuysen and related transformations for the Dirac equation

Non-localities of Foldy-Wouthuysen and related transformations, which are used to separate positive and negative energy states in the Dirac equation, are investigated. Second moments of functional kernels generated by the transformations are calculated, the transformed functions and their variances are computed. It is shown that all the transformed quantities are smeared in the coordinate space by the amount comparable to the Compton wavelength $\lambda_c=\hbar/mc$.Comment: 7 pages, two figure

Redshifts of CLASS Radio Sources

Spectroscopic observations of a sample of 42 flat-spectrum radio sources from the Cosmic Lens All-Sky Survey (CLASS) have yielded a mean redshift of $= 1.27$ with an RMS spread of 0.95, at a completeness level of 64%. The sample consists of sources with a 5-GHz flux density of 25-50 mJy, making it the faintest flat-spectrum radio sample for which the redshift distribution has been studied. The spectra, obtained with the Willam Herschel Telescope (WHT), consist mainly of broad-line quasars at $z>1$ and narrow-line galaxies at $z<0.5$. Though the mean redshift of flat-spectrum radio sources exhibits little variation over more than two orders of magnitude in radio flux density, there is evidence for a decreasing fraction of quasars at weaker flux levels. In this paper we present the results of our spectroscopic observations, and discuss the implications for constraining cosmological parameters with statistical analyses of the CLASS survey.Comment: 10 pages, AJ accepte

Anesthetic care during posterior spinal fusion in a patient with Prader-Willi syndrome.

Prader-Willi syndrome (PWS), originally described in 1956, is a genomic imprinting disorder affecting chromo- some 15. Three genetic subtypes have been recognized, the most common of which is the paternal chromosome 15q11-q13 deletion accounting for 65-75% of cases. Pa- tients with PWS manifest several distinguishing charac- teristics including infantile hypotonia, cognitive dysfunc- tion, hyperphagia leading to obesity, short stature, ortho- pedic deformities, and hypothalamic dysfunction. Other important clinical manifestations include short stature, developmental delay, sleep disturbances including ob- structive sleep apnea, cognitive disabilities, seizures, be- havioral problems, and hypothalamic dysfunction. Due to associated end-organ involvement, surgical procedures are often required in PWS patients. We present a 12-year- old adolescent with Prader-Willi syndrome who required anesthetic care during a posterior spinal fusion to treat scoliosis. The potential periopera- tive implications of these patients are reviewed and op- tions for anesthetic care presented

Zitterbewegung of nearly-free and tightly bound electrons in solids

We show theoretically that nonrelativistic nearly-free electrons in solids should experience a trembling motion (Zitterbewegung, ZB) in absence of external fields, similarly to relativistic electrons in vacuum. The Zitterbewegung is directly related to the influence of periodic potential on the free electron motion. The frequency of ZB is $\omega\approx E_g/\hbar$, where $E_g$ is the energy gap. The amplitude of ZB is determined by the strength of periodic potential and the lattice period and it can be of the order of nanometers. We show that the amplitude of ZB does not depend much on the width of the wave packet representing an electron in real space. An analogue of the Foldy-Wouthuysen transformation, known from relativistic quantum mechanics, is introduced in order to decouple electron states in various bands. We demonstrate that, after the bands are decoupled, electrons should be treated as particles of a finite size. In contrast to nearly-free electrons we consider a two-band model of tightly bound electrons. We show that also in this case the electrons should experience the trembling motion. It is concluded that the phenomenon of Zitterbewegung of electrons in crystalline solids is a rule rather than an exception.Comment: 22 pages, 6 figures Published version, minor changes mad

High resolution observations and mass modelling of the CLASS gravitational lens B1152+199

We present a series of high resolution radio and optical observations of the CLASS gravitational lens system B1152+199 obtained with the Multi-Element Radio-Linked Interferometer Network (MERLIN), Very Long Baseline Array (VLBA) and Hubble Space Telescope (HST). Based on the milliarcsecond-scale substructure of the lensed radio components and precise optical astrometry for the lensing galaxy, we construct models for the system and place constraints on the galaxy mass profile. For a single galaxy model with surface mass density Sigma(r) propto r^-beta, we find that 0.95 < beta < 1.21 at 2-sigma confidence. Including a second deflector to represent a possible satellite galaxy of the primary lens leads to slightly steeper mass profiles.Comment: 7 pages, post-referee revision for MNRA

Measuring Cosmological Parameters with the JVAS and CLASS Gravitational Lens Surveys

The JVAS (Jodrell Bank-VLA Astrometric Survey) and CLASS (Cosmic Lens All-Sky Survey) are well-defined surveys containing about ten thousand flat-spectrum radio sources. For many reasons, flat-spectrum radio sources are particularly well-suited as a population from which one can obtain unbiased samples of gravitational lenses. These are by far the largest gravitational (macro)lens surveys, and particular attention was paid to constructing a cleanly-defined sample for the survey itself and for the underlying luminosity function. Here we present the constraints on cosmological parameters, particularly the cosmological constant, derived from JVAS and combine them with constraints from optical gravitational lens surveys, `direct' measurements of $\Omega_{0}$, $H_{0}$ and the age of the universe, and constraints derived from CMB anisotropies, before putting this final result into the context of the latest results from other, independent cosmological tests.Comment: LaTeX, 9 pages, 6 PostScript figures, uses texas.sty. To appear in the Proceedings of the 19th Texas Symposium on Relativistic Astrophysics and Cosmology (CD-ROM). Paper version available on request. Actual poster (A0 and A4 versions) available from http://multivac.jb.man.ac.uk:8000/helbig/research/publications/info/ texas98.htm