985 research outputs found
On the decrease of the number of bound states with the increase of the angular momentum
For the class of central potentials possessing a finite number of bound
states and for which the second derivative of is negative, we prove,
using the supersymmetric quantum mechanics formalism, that an increase of the
angular momentum by one unit yields a decrease of the number of bound
states of at least one unit: . This property is used
to obtain, for this class of potential, an upper limit on the total number of
bound states which significantly improves previously known results
The Ground States of Large Quantum Dots in Magnetic Fields
The quantum mechanical ground state of a 2D -electron system in a
confining potential ( is a coupling constant) and a homogeneous
magnetic field is studied in the high density limit , with fixed. It is proved that the ground state energy and
electronic density can be computed {\it exactly} in this limit by minimizing
simple functionals of the density. There are three such functionals depending
on the way varies as : A 2D Thomas-Fermi (TF) theory applies
in the case ; if the correct limit theory
is a modified -dependent TF model, and the case is described
by a ``classical'' continuum electrostatic theory. For homogeneous potentials
this last model describes also the weak coupling limit for arbitrary
. Important steps in the proof are the derivation of a new Lieb-Thirring
inequality for the sum of eigenvalues of single particle Hamiltonians in 2D
with magnetic fields, and an estimation of the exchange-correlation energy. For
this last estimate we study a model of classical point charges with
electrostatic interactions that provides a lower bound for the true quantum
mechanical energy.Comment: 57 pages, Plain tex, 5 figures in separate uufil
Bound States in one and two Spatial Dimensions
In this paper we study the number of bound states for potentials in one and
two spatial dimensions. We first show that in addition to the well-known fact
that an arbitrarily weak attractive potential has a bound state, it is easy to
construct examples where weak potentials have an infinite number of bound
states. These examples have potentials which decrease at infinity faster than
expected. Using somewhat stronger conditions, we derive explicit bounds on the
number of bound states in one dimension, using known results for the
three-dimensional zero angular momentum. A change of variables which allows us
to go from the one-dimensional case to that of two dimensions results in a
bound for the zero angular momentum case. Finally, we obtain a bound on the
total number of bound states in two dimensions, first for the radial case and
then, under stronger conditions, for the non-central case.Comment: Latex, 27pp no figure
Stability and Related Properties of Vacua and Ground States
We consider the formal non relativistc limit (nrl) of the :\phi^4:_{s+1}
relativistic quantum field theory (rqft), where s is the space dimension.
Following work of R. Jackiw, we show that, for s=2 and a given value of the
ultraviolet cutoff \kappa, there are two ways to perform the nrl: i.) fixing
the renormalized mass m^2 equal to the bare mass m_0^2; ii.) keeping the
renormalized mass fixed and different from the bare mass m_0^2. In the
(infinite-volume) two-particle sector the scattering amplitude tends to zero as
\kappa -> \infty in case i.) and, in case ii.), there is a bound state,
indicating that the interaction potential is attractive. As a consequence,
stability of matter fails for our boson system. We discuss why both
alternatives do not reproduce the low-energy behaviour of the full rqft. The
singular nature of the nrl is also nicely illustrated for s=1 by a rigorous
stability/instability result of a different nature.Comment: Late
Onsager's Inequality, the Landau-Feynman Ansatz and Superfluidity
We revisit an inequality due to Onsager, which states that the (quantum)
liquid structure factor has an upper bound of the form (const.) x |k|, for not
too large modulus of the wave vector k. This inequality implies the validity of
the Landau criterion in the theory of superfluidity with a definite, nonzero
critical velocity. We prove an auxiliary proposition for general Bose systems,
together with which we arrive at a rigorous proof of the inequality for one of
the very few soluble examples of an interacting Bose fluid, Girardeau's model.
The latter proof demonstrates the importance of the thermodynamic limit of the
structure factor, which must be taken initially at k different from 0. It also
substantiates very well the heuristic density functional arguments, which are
also shown to hold exactly in the limit of large wave-lengths. We also briefly
discuss which features of the proof may be present in higher dimensions, as
well as some open problems related to superfluidity of trapped gases.Comment: 28 pages, 2 figure, uses revtex
The appropriateness of using a counter app in experimental studies assessing unwanted intrusive thoughts
The reliable and valid assessment of unwanted intrusive thoughts (UITs) is crucial. The main aim of the current research was to investigate if individuals who used a counter app (a program on a mobile device that is used to count the frequency of an event by pressing the volume-up button) to assess UITs retrospectively overreported the number of UITs. The secondary aim was to establish preliminary psychometric qualities of the counter app method. A UIT was activated in N = 87 students. They were randomly allocated to one of three experimental conditions: counter app, thought monitoring, or free thinking. Retrospective descriptors of the UIT, including its frequency, were taken. The second study (N = 118) mainly aimed to replicate the results of the first study. In both studies, the retrospective frequency ratings of the UITs were 2â3 times higher in individuals who had used the counter app compared to those in the control conditions. Preliminary indicators of convergent validity and testâretest reliability were good; criterion, discriminant, and predictive validity were unsatisfactory. To conclude, using event marking such as a counter app can result in an overestimation of UITs. Alternative methods of assessment of UITs are discussed
Upper and lower limits on the number of bound states in a central potential
In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
of bound states for the -th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit with (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of
course the integer part).Comment: 44 pages, 5 figure
On the Two-Point Correlation Function for the Invariant Spin One-Half Heisenberg Chain at Roots of Unity
Using tensor calculus we compute the two-point scalar operators
(TPSO), their averages on the ground-state give the two-point correlation
functions. The TPSOs are identified as elements of the Temperley-Lieb algebra
and a recurrence relation is given for them. We have not tempted to derive the
analytic expressions for the correlation functions in the general case but got
some partial results. For , all correlation functions are
(trivially) zero, for , they are related in the continuum to the
correlation functions of left-handed and right-handed Majorana fields in the
half plane coupled by the boundary condition. In the case , one
gets the correlation functions of Mittag's and Stephen's parafermions for the
three-state Potts model. A diagrammatic approach to compute correlation
functions is also presented.Comment: 19 pages, LaTeX, BONN-HE-93-3
Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits
We calculate the chromatic polynomials and, from these, the
asymptotic limiting functions
for families of -vertex graphs comprised of repeated subgraphs
adjoined to an initial graph . These calculations of for
infinitely long strips of varying widths yield important insights into
properties of for two-dimensional lattices . In turn,
these results connect with statistical mechanics, since is the
ground state degeneracy of the -state Potts model on the lattice .
For our calculations, we develop and use a generating function method, which
enables us to determine both the chromatic polynomials of finite strip graphs
and the resultant function in the limit . From
this, we obtain the exact continuous locus of points where
is nonanalytic in the complex plane. This locus is shown to
consist of arcs which do not separate the plane into disconnected regions.
Zeros of chromatic polynomials are computed for finite strips and compared with
the exact locus of singularities . We find that as the width of the
infinitely long strips is increased, the arcs comprising elongate
and move toward each other, which enables one to understand the origin of
closed regions that result for the (infinite) 2D lattice.Comment: 48 pages, Latex, 12 encapsulated postscript figures, to appear in
Physica
Degenerate ground states and nonunique potentials: breakdown and restoration of density functionals
The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of
quantum mechanics, and constitutes the basis for the very successful
density-functional approach to inhomogeneous interacting many-particle systems.
Here we show that in formulations of density-functional theory (DFT) that
employ more than one density variable, applied to systems with a degenerate
ground state, there is a subtle loophole in the HK theorem, as all mappings
between densities, wave functions and potentials can break down. Two weaker
theorems which we prove here, the joint-degeneracy theorem and the
internal-energy theorem, restore the internal, total and exchange-correlation
energy functionals to the extent needed in applications of DFT to atomic,
molecular and solid-state physics and quantum chemistry. The joint-degeneracy
theorem constrains the nature of possible degeneracies in general many-body
systems
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