3,766 research outputs found
Uniform analytic approximation of Wigner rotation matrices
We derive the leading asymptotic approximation, for low angle {\theta}, of
the Wigner rotation matrix elements , uniform in
and . The result is in terms of a Bessel function of integer order. We
numerically investigate the error for a variety of cases and find that the
approximation can be useful over a significant range of angles. This
approximation has application in the partial wave analysis of wavepacket
scattering.Comment: 8 pages, 5 figure
Prediction of deviations from the Rutherford formula for low-energy Coulomb scattering of wavepackets
We calculate the nonrelativistic scattering of a wavepacket from a Coulomb
potential and find deviations from the Rutherford formula in all cases. These
generally occur only at low scattering angles, where they would be obscured by
the part of the incident beam that emerges essentially unscattered. For a model
experiment, the scattering of helium nuclei from a thin gold foil, we find the
deviation region is magnified for low incident energies (in the keV range), so
that a large shadow zone of low probability around the forward direction is
expected to be measurable.
From a theoretical perspective, the use of wavepackets makes partial wave
analysis applicable to this infinite-range potential. It allows us to calculate
the everywhere finite probability for a wavepacket to wavepacket transition and
to relate this to the differential cross section. Time delays and advancements
in the detection probabilities can be calculated. We investigate the optical
theorem as applied to this special case.Comment: 21 pages, 8 figures. This version contains clarifications and
additional results compared to the previous version. The paper has been
accepted for publication in J.Phys.
Hybrid phase-space simulation method for interacting Bose fields
We introduce an approximate phase-space technique to simulate the quantum
dynamics of interacting bosons. With the future goal of treating Bose-Einstein
condensate systems, the method is designed for systems with a natural
separation into highly occupied (condensed) modes and lightly occupied modes.
The method self-consistently uses the Wigner representation to treat highly
occupied modes and the positive-P representation for lightly occupied modes. In
this method, truncation of higher-derivative terms from the Fokker-Planck
equation is usually necessary. However, at least in the cases investigated
here, the resulting systematic error, over a finite time, vanishes in the limit
of large Wigner occupation numbers. We tested the method on a system of two
interacting anharmonic oscillators, with high and low occupations,
respectively. The Hybrid method successfully predicted atomic quadratures to a
useful simulation time 60 times longer than that of the positive-P method. The
truncated Wigner method also performed well in this test. For the prediction of
the correlation in a quantum nondemolition measurement scheme, for this same
system, the Hybrid method gave excellent agreement with the exact result, while
the truncated Wigner method showed a large systematic error.Comment: 13 pages; 6 figures; references added; figures correcte
Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator
We construct ladder operators, and , for a
multi-step rational extension of the harmonic oscillator on the half plane,
. These ladder operators connect all states of the spectrum in only
infinite-dimensional representations of their polynomial Heisenberg algebra.
For comparison, we also construct two different classes of ladder operator
acting on this system that form finite-dimensional as well as
infinite-dimensional representations of their respective polynomial Heisenberg
algebras. For the rational extension, we construct the position wavefunctions
in terms of exceptional orthogonal polynomials. For a particular choice of
parameters, we construct the coherent states, eigenvectors of with
generally complex eigenvalues, , as superpositions of a subset of the energy
eigenvectors. Then we calculate the properties of these coherent states,
looking for classical or non-classical behaviour. We calculate the energy
expectation as a function of . We plot position probability densities for
the coherent states and for the even and odd cat states formed from these
coherent states. We plot the Wigner function for a particular choice of .
For these coherent states on one arm of a beamsplitter, we calculate the two
excitation number distribution and the linear entropy of the output state. We
plot the standard deviations in and and find no squeezing in the regime
considered. By plotting the Mandel parameter for the coherent states as a
function of , we find that the number statistics is sub-Poissonian.Comment: 16 pages, 11 figure
- …