66 research outputs found
Symmetry of the Atomic Electron Density in Hartree, Hartree-Fock, and Density Functional Theory
The density of an atom in a state of well-defined angular momentum has a
specific finite spherical harmonic content, without and with interactions.
Approximate single-particle schemes, such as the Hartree, Hartree-Fock, and
Local Density Approximations, generally violate this feature. We analyze, by
means of perturbation theory, the degree of this violation and show that it is
small. The correct symmetry of the density can be assured by a
constrained-search formulation without significantly altering the calculated
energies. We compare our procedure to the (different) common practice of
spherically averaging the self-consistent potential. Kohn-Sham density
functional theory with the exact exchange-correlation potential has the correct
finite spherical harmonic content in its density; but the corresponding exact
single particle potential and wavefunctions contain an infinite number of
spherical harmonics.Comment: 11 pages, 6 figures. Expanded discussion of spherical harmonic
expansion of Hartree density. Some typos corrected, references adde
Non-Linear Vibrations in Nuclei
We have perfomed Time Dependant Hartree-Fock (TDHF) calculations on the non
linear response of nuclei. We have shown that quadrupole (and dipole) motion
produces monopole (and quadrupole) oscillations in all atomic nuclei. We have
shown that these findings can be interpreted as a large coupling between one
and two phonon states leading to large anharmonicities.Comment: 4 pages, 3 figure
Alfv\'en Reflection and Reverberation in the Solar Atmosphere
Magneto-atmospheres with Alfv\'en speed [a] that increases monotonically with
height are often used to model the solar atmosphere, at least out to several
solar radii. A common example involves uniform vertical or inclined magnetic
field in an isothermal atmosphere, for which the Alfv\'en speed is exponential.
We address the issue of internal reflection in such atmospheres, both for
time-harmonic and for transient waves. It is found that a mathematical boundary
condition may be devised that corresponds to perfect absorption at infinity,
and, using this, that many atmospheres where a(x) is analytic and unbounded
present no internal reflection of harmonic Alfv\'en waves. However, except for
certain special cases, such solutions are accompanied by a wake, which may be
thought of as a kind of reflection. For the initial-value problem where a
harmonic source is suddenly switched on (and optionally off), there is also an
associated transient that normally decays with time as O(t-1) or O(t-1 ln t),
depending on the phase of the driver. Unlike the steady-state harmonic
solutions, the transient does reflect weakly. Alfv\'en waves in the solar
corona driven by a finite-duration train of p-modes are expected to leave such
transients.Comment: Accepted by Solar Physic
Far-from-equilibrium quantum many-body dynamics
The theory of real-time quantum many-body dynamics as put forward in Ref.
[arXiv:0710.4627] is evaluated in detail. The formulation is based on a
generating functional of correlation functions where the Keldysh contour is
closed at a given time. Extending the Keldysh contour from this time to a later
time leads to a dynamic flow of the generating functional. This flow describes
the dynamics of the system and has an explicit causal structure. In the present
work it is evaluated within a vertex expansion of the effective action leading
to time evolution equations for Green functions. These equations are applicable
for strongly interacting systems as well as for studying the late-time
behaviour of nonequilibrium time evolution. For the specific case of a bosonic
N-component phi^4 theory with contact interactions an s-channel truncation is
identified to yield equations identical to those derived from the 2PI effective
action in next-to-leading order of a 1/N expansion. The presented approach
allows to directly obtain non-perturbative dynamic equations beyond the widely
used 2PI approximations.Comment: 20 pp., 6 figs; submitted version with added references and typos
corrected
Weakly-Bound Three-Body Systems with No Bound Subsystems
We investigate the domain of coupling constants which achieve binding for a
3-body system, while none of the 2-body subsystems is bound. We derive some
general properties of the shape of the domain, and rigorous upper bounds on its
size, using a Hall--Post decomposition of the Hamiltonian. Numerical
illustrations are provided in the case of a Yukawa potential, using a simple
variational method.Comment: gzipped ps with 11 figures included. To appear in Phys. Rev.
Formation and control of electron molecules in artificial atoms: Impurity and magnetic-field effects
Interelectron interactions and correlations in quantum dots can lead to
spontaneous symmetry breaking of the self-consistent mean field resulting in
formation of Wigner molecules. With the use of spin-and-space unrestricted
Hartree-Fock (sS-UHF) calculations, such symmetry breaking is discussed for
field-free conditions, as well as under the influence of an external magnetic
field. Using as paradigms impurity-doped (as well as the limiting case of
clean) two-electron quantum dots (which are analogs to helium-like atoms), it
is shown that the interplay between the interelectron repulsion and the
electronic zero-point kinetic energy leads, for a broad range of impurity
parameters, to formation of a singlet ground-state electron molecule,
reminiscent of the molecular picture of doubly-excited helium. Comparative
analysis of the conditional probability distributions for the sS-UHF and the
exact solutions for the ground state of two interacting electrons in a clean
parabolic quantum dot reveals that both of them describe formation of an
electron molecule with similar characteristics. The self-consistent field
associated with the triplet excited state of the two-electron quantum dot
(clean as well as impurity-doped) exhibits symmetry breaking of the Jahn-Teller
type, similar to that underlying formation of nonspherical open-shell nuclei
and metal clusters. Furthermore, impurity and/or magnetic-field effects can be
used to achieve controlled manipulation of the formation and pinning of the
discrete orientations of the Wigner molecules. Impurity effects are futher
illustrated for the case of a quantum dot with more than two electrons.Comment: Latex/Revtex, 10 pages with 4 gif figures. Small changes to explain
the difference between Wigner and Jahn-Teller electron molecules. A complete
version of the paper with high quality figures inside the text is available
at http://shale.physics.gatech.edu/~costas/qdhelium.html For related papers,
see http://www.prism.gatech.edu/~ph274c
Majorana solutions to the two-electron problem
A review of the known different methods and results devised to study the
two-electron atom problem, appeared in the early years of quantum mechanics, is
given, with particular reference to the calculations of the ground state energy
of helium. This is supplemented by several, unpublished results obtained around
the same years by Ettore Majorana, which results did not convey in his
published papers on the argument, and thus remained unknown until now.
Particularly interesting, even for current research in atomic and nuclear
physics, is a general variant of the variational method, developed by Majorana
in order to take directly into account, already in the trial wavefunction, the
action of the full Hamiltonian operator of a given quantum system. Moreover,
notable calculations specialized to the study of the two-electron problem show
the introduction of the remarkable concept of an effective nuclear charge
different for the two electrons (thus generalizing previous known results), and
an application of the perturbative method, where the atomic number Z was
treated effectively as a continuous variable, contributions to the ground state
energy of an atom with given Z coming also from any other Z. Instead,
contributions relevant mainly for pedagogical reasons count simple broad range
estimates of the helium ionization potential, obtained by suitable choices for
the wavefunction, as well as a simple alternative to Hylleraas' method, which
led Majorana to first order calculations comparable in accuracy with well-known
order 11 results derived, in turn, by Hylleraas.Comment: amsart, 20 pages, no figure
Group theoretical analysis of symmetry breaking in two-dimensional quantum dots
We present a group theoretical study of the symmetry-broken unrestricted
Hartree-Fock orbitals and electron densities in the case of a two-dimensional
N-electron single quantum dot (with and without an external magnetic field).
The breaking of rotational symmetry results in canonical orbitals that (1) are
associated with the eigenvectors of a Hueckel hamiltonian having sites at the
positions determined by the equilibrium molecular configuration of the
classical N-electron problem, and (2) transform according to the irreducible
representations of the point group specified by the discrete symmetries of this
classical molecular configuration. Through restoration of the total-spin and
rotational symmetries via projection techniques, we show that the point-group
discrete symmetry of the unrestricted Hartree-Fock wave function underlies the
appearance of magic angular momenta (familiar from exact-diagonalization
studies) in the excitation spectra of the quantum dot. Furthermore, this
two-step symmetry-breaking/symmetry-restoration method accurately describes the
energy spectra associated with the magic angular momenta.Comment: A section VI.B entitled "Quantitative description of the lowest
rotational band" has been added. 16 pages. Revtex with 10 EPS figures. A
version of the manuscript with high quality figures is available at
http://calcite.physics.gatech.edu/~costas/uhf_group.html For related papers,
see http://www.prism.gatech.edu/~ph274c
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