615 research outputs found
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 , where 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
Spatial interference from well-separated condensates
We use magnetic levitation and a variable-separation dual optical plug to
obtain clear spatial interference between two condensates axially separated by
up to 0.25 mm -- the largest separation observed with this kind of
interferometer. Clear planar fringes are observed using standard (i.e.
non-tomographic) resonant absorption imaging. The effect of a weak inverted
parabola potential on fringe separation is observed and agrees well with
theory.Comment: 4 pages, 5 figures - modified to take into account referees'
improvement
One-dimensional semirelativity for electrons in carbon nanotubes
It is shown that the band structure of single-wall semiconducting carbon
nanotubes (CNT) is analogous to relativistic description of electrons in
vacuum, with the maximum velocity = cm/s replacing the light velocity.
One-dimensional semirelativistic kinematics and dynamics of electrons in CNT is
formulated. Two-band k.p Hamiltonian is employed to demonstrate that electrons
in CNT experience a Zitterbewegung (trembling motion) in absence of external
fields. This Zitterbewegung should be observable much more easily in CNT than
its analogue for free relativistic electrons in vacuum.Comment: 4 pages no figure
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
.Comment: 7 pages, two figure
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
Recommended from our members
The Intersections of Race, Gender, Age, and Socioeconomic Status: Implications for Reporting Discrimination and Attributions to Discrimination.
This study employed an intersectional approach (operationalized as the combination of more than one social identity) to examine the relationship between aspects of social identity (i.e., race, gender, age, SES), self-reported level of mistreatment, and attributions for discrimination. Self-reported discrimination has been researched extensively and there is substantial evidence of its association with adverse physical and psychological health outcomes. Few studies, however, have examined the relationship of multiple demographic variables (including social identities) to overall levels self-reported mistreatment as well the selection of attributions for discrimination. A diverse community sample (N = 292; 42.12% Black; 47.26% male) reported on experiences of discrimination using the Everyday Discrimination Scale. General linear models were used to test the effect of sociodemographic characteristics (i.e., race, gender, age, SES) on total discrimination score and on attributions for discrimination. To test for intersectional relationships, we tested the effect of two-way interactions of sociodemographic characteristics on total discrimination score and attributions for discrimination. We found preliminary support for intersectional effects, as indicated by a significant race by age interaction on the selection of the race attribution for discrimination; gender by SES on the age attribution; age by gender on the education attribution; and race by SES on the economic situation attribution. Our study extends prior work by highlighting the importance of testing more than one factor as contributing to discrimination, particularly when examining to what sources individuals attribute discrimination
Zitterbewegung (trembling motion) of electrons in narrow gap semiconductors
Theory of trembling motion [Zitterbewegung (ZB)] of charge carriers in
various narrow-gap materials is reviewed. Nearly free electrons in a periodic
potential, InSb-type semiconductors, bilayer graphene, monolayer graphene and
carbon nanotubes are considered. General features of ZB are emphasized. It is
shown that, when the charge carriers are prepared in the form of Gaussian wave
packets, the ZB has a transient character with the decay time of femtoseconds
in graphene and picoseconds in nanotubes. Zitterbewegung of electrons in
graphene in the presence of an external magnetic field is mentioned. A
similarity of ZB in semiconductors to that of relativistic electrons in a
vacuum is stressed. Possible ways of observing the trembling motion in solids
are mentioned.Comment: 8 pages, 5 figure
Zitterbewegung of electrons and holes in III-V semiconductor quantum wells
The notion of zitterbewegung is a long-standing prediction of relativistic
quantum mechanics. Here we extend earlier theoretical studies on this
phenomenon for the case of III-V zinc-blende semiconductors which exhibit
particularly strong spin-orbit coupling. This property makes nanostructures
made of these materials very favorable systems for possible experimental
observations of zitterbewegung. Our investigations include electrons in n-doped
quantum wells under the influence of both Rashba and Dresselhaus spin-orbit
interaction, and also the two-dimensional hole gas. Moreover, we give a
detailed anaysis of electron zitterbewegung in quantum wires which appear to be
particularly suited for experimentally observing this effect.Comment: 10 pages, 3 figures include
- …