25 research outputs found
Frenkel electron on an arbitrary electromagnetic background and magnetic Zitterbewegung
We present Lagrangian which implies both necessary constraints and dynamical
equations for position and spin of relativistic spin one-half particle. The
model is consistent for any value of magnetic moment and for arbitrary
electromagnetic background. Our equations coincide with those of Frenkel in the
approximation in which the latter have been obtained by Frenkel. Transition
from approximate to exact equations yields two structural modifications of the
theory. First, Frenkel condition on spin-tensor turns into the Pirani
condition. Second, canonical momentum is no more proportional to velocity. Due
to this, even when (Frenkel case), the complete and approximate
equations predict different behavior of particle. The difference between
momentum and velocity means extra contribution into spin-orbit interaction. To
estimate the contribution, we found exact solution to complete equations for
the case of uniform magnetic field. While Frenkel electron moves around the
circle, our particle experiences magnetic {\it Zitterbewegung}, that is
oscillates in the direction of magnetic field with amplitude of order of
Compton wavelength for the fast particle. Besides, the particle has dipole
electric moment.Comment: 20 pages, 1 figure, close to published versio
Geometric Constructions Underlying Relativistic Description of Spin on the Base of Non-Grassmann Vector-Like Variable
Basic notions of Dirac theory of constrained systems have their analogs in
differential geometry. Combination of the two approaches gives more clear
understanding of both classical and quantum mechanics, when we deal with a
model with complicated structure of constraints. In this work we describe and
discuss the spin fiber bundle which appeared in various mechanical models where
spin is described by vector-like variable
Non-linear quantum effects in electromagnetic radiation of a vortex electron
There is a controversy of how to interpret interactions of electrons with a
large spatial coherence with light and matter. When such an electron emits a
photon, it can do so either as if its charge were confined to a point within a
coherence length, the region where a square modulus of a wave function
is localized, or as a continuous cloud of space charge spread over
it. This problem was addressed in a recent study R.~Remez, et al., Phys. Rev.
Lett. {\bf 123}, 060401 (2019) where a conclusion was drawn in favor of the
first (point) interpretation. Here we argue that there is an alternative
explanation for the measurements reported in that paper, which relies on purely
classical arguments and does not allow one to refute the second interpretation.
We propose an experiment of Smith-Purcell radiation from a non-relativistic
vortex electron carrying orbital angular momentum, which can unambiguously lead
to the opposite conclusion. Beyond the paraxial approximation, the vortex
packet has a non-point electric quadrupole moment, which grows as the packet
spreads and results in a non-linear -growth of the radiation intensity
with the length of the grating when is much larger than the packet's
Rayleigh length. Such a non-linear effect has never been observed for single
electrons and, if detected, it would be a hallmark of the non-point nature of
charge in a wave packet. Thus, two views on are complementary to
each other and an electron radiates either as a point charge or as a continuous
charge flow depending on the experimental conditions and on its quantum state.
Our conclusions hold for a large class of non-Gaussian packets and emission
processes for which the radiation formation length can exceed the Rayleigh
length, such as Cherenkov radiation, transition radiation, diffraction
radiation, and so forth.Comment: 25 pages; 4 figure
Eigenphase preserving two-channel SUSY transformations
We propose a new kind of supersymmetric (SUSY) transformation in the case of
the two-channel scattering problem with equal thresholds, for partial waves of
the same parity. This two-fold transformation is based on two imaginary
factorization energies with opposite signs and with mutually conjugated
factorization solutions. We call it an eigenphase preserving SUSY
transformation as it relates two Hamiltonians, the scattering matrices of which
have identical eigenphase shifts. In contrast to known phase-equivalent
transformations, the mixing parameter is modified by the eigenphase preserving
transformation.Comment: 16 pages, 1 figur
Darboux Transformation of the Green Function for the Dirac Equation with the Generalized Potential
We consider the Darboux transformation of the Green functions of the regular
boundary problem of the one-dimensional stationary Dirac equation. We obtained
the Green functions of the transformed Dirac equation with the initial regular
boundary conditions. We also construct the formula for the unabridged trace of
the difference of the transformed and the initial Green functions of the
regular boundary problem of the one-dimensional stationary Dirac equation. We
illustrate our findings by the consideration of the Darboux transformation for
the Green function of the free particle Dirac equation on an interval.Comment: 14 pages,zip. file: Latex, 1 figure. Typos corrected, the figure
replace
SUSY transformation of the Green function and a trace formula
An integral relation is established between the Green functions corresponding
to two Hamiltonians which are supersymmetric (SUSY) partners and in general may
possess both discrete and continuous spectra. It is shown that when the
continuous spectrum is present the trace of the difference of the Green
functions for SUSY partners is a finite quantity which may or may not be equal
to zero despite the divergence of the traces of each Green function. Our
findings are illustrated by using the free particle example considered both on
the whole real line and on a half line
Spectral properties of non-conservative multichannel SUSY partners of the zero potential
Spectral properties of a coupled potential model obtained with
the help of a single non-conservative supersymmetric (SUSY) transformation
starting from a system of radial Schr\"odinger equations with the zero
potential and finite threshold differences between the channels are studied.
The structure of the system of polynomial equations which determine the zeros
of the Jost-matrix determinant is analyzed. In particular, we show that the
Jost-matrix determinant has zeros which may all correspond to
virtual states. The number of bound states satisfies . The
maximal number of resonances is . A perturbation technique
for a small coupling approximation is developed. A detailed study of the
inverse spectral problem is given for the case.Comment: 17 pages, 4 figure