56 research outputs found
Zeta Function Zeros, Powers of Primes, and Quantum Chaos
We present a numerical study of Riemann's formula for the oscillating part of
the density of the primes and their powers. The formula is comprised of an
infinite series of oscillatory terms, one for each zero of the zeta function on
the critical line and was derived by Riemann in his paper on primes assuming
the Riemann hypothesis. We show that high resolution spectral lines can be
generated by the truncated series at all powers of primes and demonstrate
explicitly that the relative line intensities are correct. We then derive a
Gaussian sum rule for Riemann's formula. This is used to analyze the numerical
convergence of the truncated series. The connections to quantum chaos and
semiclassical physics are discussed
Haldane exclusion statistics and second virial coefficient
We show that Haldanes new definition of statistics, when generalised to
infinite dimensional Hilbert spaces, is equal to the high temperature limit of
the second virial coefficient. We thus show that this exclusion statistics
parameter, g , of anyons is non-trivial and is completely determined by its
exchange statistics parameter . We also compute g for quasiparticles in
the Luttinger model and show that it is equal to .Comment: 11 pages, REVTEX 3.
Some exact results for a trapped quantum gas at finite temperature
We present closed analytical expressions for the particle and kinetic energy
spatial densities at finite temperatures for a system of noninteracting
fermions (bosons) trapped in a d-dimensional harmonic oscillator potential. For
d=2 and 3, exact expressions for the N-particle densities are used to calculate
perturbatively the temperature dependence of the splittings of the energy
levels in a given shell due to a very weak interparticle interaction in a
dilute Fermi gas. In two dimensions, we obtain analytically the surprising
result that the |l|-degeneracy in a harmonic oscillator shell is not lifted in
the lowest order even when the exact, rather than the Thomas-Fermi expression
for the particle density is used. We also demonstrate rigorously (in two
dimensions) the reduction of the exact zero-temperature fermionic expressions
to the Thomas-Fermi form in the large-N limit.Comment: 14 pages, 4 figures include
Exact first-order density matrix for a d-dimensional harmonically confined Fermi gas at finite temperature
We present an exact closed form expression for the {\em finite temperature}
first-order density matrix of a harmonically trapped ideal Fermi gas in any
dimension. This constitutes a much sought after generalization of the recent
results in the literature, where exact expressions have been limited to
quantities derived from the {\em diagonal} first-order density matrix. We
compare our exact results with the Thomas-Fermi approximation (TFA) and
demonstrate numerically that the TFA provides an excellent description of the
first-order density matrix in the large-N limit. As an interesting application,
we derive a closed form expression for the finite temperature Hartree-Fock
exchange energy of a two-dimensional parabolically confined quantum dot. We
numerically test this exact result against the 2D TF exchange functional, and
comment on the applicability of the local-density approximation (LDA) to the
exchange energy of an inhomogeneous 2D Fermi gas.Comment: 12 pages, 3 figures included in the text, RevTeX4. Text before
Eq.(25) corrected. Additional equation following Eq.(25) has been adde
Applications of the Collective Field Theory for the Calogero-Sutherland Model
We use the collective field theory known for the Calogero-Sutherland model to
study a variety of low-energy properties. These include the ground state energy
in a confining potential upto the two leading orders in the particle number,
the dispersion relation of sound modes with a comparison to the two leading
terms in the low temperature specific heat, large amplitude waves, and single
soliton solutions. The two-point correlation function derived from the
dispersion relation of the sound mode only gives its nonoscillatory asymptotic
behavior correctly, demonstrating that the theory is applicable only for the
low-energy and long wavelength excitations of the system.Comment: LaTeX, 31 page
Exact Multiplicities in the Three-Anyon Spectrum
Using the symmetry properties of the three-anyon spectrum, we obtain exactly
the multiplicities of states with given energy and angular momentum. The
results are shown to be in agreement with the proper quantum mechanical and
semiclassical considerations, and the unexplained points are indicated.Comment: 16 pages plus 3 postscript figures, Kiev Institute for Theoretical
Physics preprint ITP-93-32
Anomalous particle-number fluctuations in a three-dimensional interacting Bose-Einstein condensate
The particle-number fluctuations originated from collective excitations are
investigated for a three-dimensional, repulsively interacting Bose-Einstein
condensate (BEC) confined in a harmonic trap. The contribution due to the
quantum depletion of the condensate is calculated and the explicit expression
of the coefficient in the formulas denoting the particle-number fluctuations is
given. The results show that the particle-number fluctuations of the condensate
follow the law and the fluctuations vanish when
temperature approaches to the BEC critical temperature.Comment: RevTex, 4 page
Heavy Meson Description with a Screened Potential
We perform a quark model calculation of the and spectra
from a screened funnel potential form suggested by unquenched lattice
calculations. A connection between the lattice screening parameter and an
effective gluon mass directly derived from QCD is established. Spin-spin energy
splittings, leptonic widths and radiative decays are also examined providing a
test for the description of the states.Comment: 17 pages, no figures, to appear in Phys. Rev.
On low temperature kinetic theory; spin diffusion, Bose Einstein condensates, anyons
The paper considers some typical problems for kinetic models evolving through
pair-collisions at temperatures not far from absolute zero, which illustrate
specific quantum behaviours. Based on these examples, a number of differences
between quantum and classical Boltzmann theory is then discussed in more
general terms.Comment: 25 pages, minor updates of previous versio
Semiclassical evaluation of average nuclear one and two body matrix elements
Thomas-Fermi theory is developed to evaluate nuclear matrix elements averaged
on the energy shell, on the basis of independent particle Hamiltonians. One-
and two-body matrix elements are compared with the quantal results and it is
demonstrated that the semiclassical matrix elements, as function of energy,
well pass through the average of the scattered quantum values. For the one-body
matrix elements it is shown how the Thomas-Fermi approach can be projected on
good parity and also on good angular momentum. For the two-body case the
pairing matrix elements are considered explicitly.Comment: 15 pages, REVTeX, 6 ps figures; changed conten
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