2,653 research outputs found
Ground State Energy of the One-Component Charged Bose Gas
The model considered here is the `jellium' model in which there is a uniform,
fixed background with charge density in a large volume and in
which particles of electric charge and mass move --- the
whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to
investigate the ground state energy of this system for bosonic particles in the
large limit. He found that the energy per particle is in this limit, where .
Here we prove that this formula is correct, thereby validating, for the first
time, at least one aspect of Bogolubov's pairing theory of the Bose gasComment: 38 pages latex. Typos corrected.Lemma 6.2 change
Proof of the Wehrl-type Entropy Conjecture for Symmmetric SU(N) Coherent States
The Wehrl entropy conjecture for coherent (highest weight) states in
representations of the Heisenberg group, which was proved in 1978 and recently
extended by us to the group , is further extended here to symmetric
representations of the groups for all . This result gives further
evidence for our conjecture that highest weight states minimize group integrals
of certain concave functions for a large class of Lie groups and their
representations.Comment: 15 pages. To appear in Commun. Math. Phy
Ground-State Energy of a Dilute Fermi Gas
Recent developments in the physics of low density trapped gases make it
worthwhile to verify old, well known results that, while plausible, were based
on perturbation theory and assumptions about pseudopotentials. We use and
extend recently developed techniques to give a rigorous derivation of the
asymptotic formula for the ground state energy of a dilute gas of fermions
interacting with a short-range, positive potential of scattering length .
For spin 1/2 fermions, this is ,
where is the energy of the non-interacting system and is the
density.Comment: Contribution to the proceedings of the 2005 International Conference
on Differential Equations and Mathematical Physics, University of Alabama,
Birmingha
The Mathematics of the Bose Gas and its Condensation
This book surveys results about the quantum mechanical many-body problem of
the Bose gas that have been obtained by the authors over the last seven years.
These topics are relevant to current experiments on ultra-cold gases; they are
also mathematically rigorous, using many analytic techniques developed over the
years to handle such problems. Some of the topics treated are the ground state
energy, the Gross-Pitaevskii equation, Bose-Einstein condensation,
superfluidity, one-dimensional gases, and rotating gases. The book also
provides a pedagogical entry into the field for graduate students and
researchers.Comment: 213 pages. Slightly revised and extended version of Oberwolfach
Seminar Series, Vol. 34, Birkhaeuser (2005
Ground State Energy of the Two-Component Charged Bose Gas
We continue the study of the two-component charged Bose gas initiated by
Dyson in 1967. He showed that the ground state energy for particles is at
least as negative as for large and this power law was verified
by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that
the exact constant was given by a mean-field minimization problem that
used, as input, Foldy's calculation (using Bogolubov's 1947 formalism) for the
one-component gas. Earlier we showed that Foldy's calculation is exact insofar
as a lower bound of his form was obtained. In this paper we do the same thing
for Dyson's conjecture. The two-component case is considerably more difficult
because the gas is very non-homogeneous in its ground state.Comment: 49 pages, Dedicated to Freeman J. Dyson on the occasion of his 80th
birthday. Final version (only minor changes) to appear in Commun. math. Phy
There are No Unfilled Shells in Hartree-Fock Theory
Hartree-Fock theory is supposed to yield a picture of atomic shells which may
or may not be filled according to the atom's position in the periodic table. We
prove that shells are always completely filled in an exact Hartree-Fock
calculation. Our theorem generalizes to any system having a two-body
interaction that, like the Coulomb potential, is repulsive.Comment: 5 pages, VBEHLMLJPS--16/July/9
Bose-Einstein Quantum Phase Transition in an Optical Lattice Model
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an
external potential, such as that presented by an optical lattice. We present a
model of this phenomenon which we are able to analyze rigorously. The system is
a hard core lattice gas at half-filling and the optical lattice is modeled by a
periodic potential of strength . For small and temperature,
BEC is proved to occur, while at large or temperature there is no
BEC. At large the low-temperature states are in a Mott insulator
phase with a characteristic gap that is absent in the BEC phase. The
interparticle interaction is essential for this transition, which occurs even
in the ground state. Surprisingly, the condensation is always into the
mode in this model, although the density itself has the periodicity of the
imposed potential.Comment: RevTeX4, 13 pages, 2 figure
Let’s Think Secondary Science: Evaluation report and executive summary
Let’s Think Secondary Science (LTSS) aims to develop students’ scientific reasoning by teaching them scientific principles such as categorisation, probability and experimentation LTSS was evaluated using a randomised controlled trial with over 8000 students in 53 schools. Schools were randomly allocated to deliver either the programme or their ‘business as usual’ science teaching. It should be considered an effectiveness trial, as it aimed to test a scalable intervention under realistic conditions in a large number of schools. The primary outcome measure was an age-appropriate science test based on a Key Stage 3 SATs paper, and the secondary measures were English and maths tests provided by GL Assessment. The process evaluation consisted of lesson observations, surveys and interviews with staff, and surveys and focus groups with students. The trial started in September 2013 and ended in July 2015.This evaluation provided no evidence that Let’s Think Secondary Science improved the science attainment of students by the end of Year 8
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