107 research outputs found
Semiclassical approximations for Hamiltonians with operator-valued symbols
We consider the semiclassical limit of quantum systems with a Hamiltonian
given by the Weyl quantization of an operator valued symbol. Systems composed
of slow and fast degrees of freedom are of this form. Typically a small
dimensionless parameter controls the separation of time
scales and the limit corresponds to an adiabatic limit, in
which the slow and fast degrees of freedom decouple. At the same time
is the semiclassical limit for the slow degrees of freedom.
In this paper we show that the -dependent classical flow for the
slow degrees of freedom first discovered by Littlejohn and Flynn, coming from
an \epsi-dependent classical Hamilton function and an -dependent
symplectic form, has a concrete mathematical and physical meaning: Based on
this flow we prove a formula for equilibrium expectations, an Egorov theorem
and transport of Wigner functions, thereby approximating properties of the
quantum system up to errors of order . In the context of Bloch
electrons formal use of this classical system has triggered considerable
progress in solid state physics. Hence we discuss in some detail the
application of the general results to the Hofstadter model, which describes a
two-dimensional gas of non-interacting electrons in a constant magnetic field
in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been
strengthened with only minor changes to the proofs. A section on the
Hofstadter model as an application of the general theory was added and the
previous section on other applications was remove
The Screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and Hamiltonian dynamics
This paper treats 6j symbols or their orthonormal forms as a function of two
variables spanning a square manifold which we call the "screen". We show that
this approach gives important and interesting insight. This two dimensional
perspective provides the most natural extension to exhibit the role of these
discrete functions as matrix elements that appear at the very foundation of the
modern theory of classical discrete orthogonal polynomials. Here we present 2D
and 1D recursion relations that are useful for the direct computation of the
orthonormal 6j, which we name U. We present a convention for the order of the
arguments of the 6j that is based on their classical and Regge symmetries, and
a detailed investigation of new geometrical aspects of the 6j symbols.
Specifically we compare the geometric recursion analysis of Schulten and Gordon
with the methods of this paper. The 1D recursion relation, written as a matrix
diagonalization problem, permits an interpretation as a discrete
Schr\"odinger-like equations and an asymptotic analysis illustrates
semiclassical and classical limits in terms of Hamiltonian evolution.Comment: 14 pages,9 figures, presented at ICCSA 2013 13th International
Conference on Computational Science and Applicatio
The Screen representation of spin networks. Images of 6j symbols and semiclassical features
This article presents and discusses in detail the results of extensive exact
calculations of the most basic ingredients of spin networks, the Racah
coefficients (or Wigner 6j symbols), exhibiting their salient features when
considered as a function of two variables - a natural choice due to their
origin as elements of a square orthogonal matrix - and illustrated by use of a
projection on a square "screen" introduced recently. On these screens, shown
are images which provide a systematic classification of features previously
introduced to represent the caustic and ridge curves (which delimit the
boundaries between oscillatory and evanescent behaviour according to the
asymptotic analysis of semiclassical approaches). Particular relevance is given
to the surprising role of the intriguing symmetries discovered long ago by
Regge and recently revisited; from their use, together with other newly
discovered properties and in conjunction with the traditional combinatorial
ones, a picture emerges of the amplitudes and phases of these discrete
wavefunctions, of interest in wide areas as building blocks of basic and
applied quantum mechanics.Comment: 16 pages, 13 figures, presented at ICCSA 2013 13th International
Conference on Computational Science and Applicatio
Collective multipole expansions and the perturbation theory in the quantum three-body problem
The perturbation theory with respect to the potential energy of three
particles is considered. The first-order correction to the continuum wave
function of three free particles is derived. It is shown that the use of the
collective multipole expansion of the free three-body Green function over the
set of Wigner -functions can reduce the dimensionality of perturbative
matrix elements from twelve to six. The explicit expressions for the
coefficients of the collective multipole expansion of the free Green function
are derived. It is found that the -wave multipole coefficient depends only
upon three variables instead of six as higher multipoles do. The possible
applications of the developed theory to the three-body molecular break-up
processes are discussed.Comment: 20 pages, 2 figure
Symplectic evolution of Wigner functions in markovian open systems
The Wigner function is known to evolve classically under the exclusive action
of a quadratic hamiltonian. If the system does interact with the environment
through Lindblad operators that are linear functions of position and momentum,
we show that the general evolution is the convolution of the classically
evolving Wigner function with a phase space gaussian that broadens in time. We
analyze the three generic cases of elliptic, hyperbolic and parabolic
Hamiltonians. The Wigner function always becomes positive in a definite time,
which is shortest in the hyperbolic case. We also derive an exact formula for
the evolving linear entropy as the average of a narrowing gaussian taken over a
probability distribution that depends only on the initial state. This leads to
a long time asymptotic formula for the growth of linear entropy.Comment: this new version treats the dissipative cas
Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective
A unified vision of the symmetric coupling of angular momenta and of the
quantum mechanical volume operator is illustrated. The focus is on the quantum
mechanical angular momentum theory of Wigner's 6j symbols and on the volume
operator of the symmetric coupling in spin network approaches: here, crucial to
our presentation are an appreciation of the role of the Racah sum rule and the
simplification arising from the use of Regge symmetry. The projective geometry
approach permits the introduction of a symmetric representation of a network of
seven spins or angular momenta. Results of extensive computational
investigations are summarized, presented and briefly discussed.Comment: 15 pages, 10 figures, presented at ICCSA 2014, 14th International
Conference on Computational Science and Application
Fredholm methods for billiard eigenfunctions in the coherent state representation
We obtain a semiclassical expression for the projector onto eigenfunctions by
means of the Fredholm theory. We express the projector in the coherent state
basis, thus obtaining the semiclassical Husimi representation of the stadium
eigenfunctions, which is written in terms of classical invariants: periodic
points, their monodromy matrices and Maslov indices.Comment: 12 pages, 10 figures. Submitted to Phys. Rev. E. Comments or
questions to [email protected]
Shape Space Methods for Quantum Cosmological Triangleland
With toy modelling of conceptual aspects of quantum cosmology and the problem
of time in quantum gravity in mind, I study the classical and quantum dynamics
of the pure-shape (i.e. scale-free) triangle formed by 3 particles in 2-d. I do
so by importing techniques to the triangle model from the corresponding 4
particles in 1-d model, using the fact that both have 2-spheres for shape
spaces, though the latter has a trivial realization whilst the former has a
more involved Hopf (or Dragt) type realization. I furthermore interpret the
ensuing Dragt-type coordinates as shape quantities: a measure of
anisoscelesness, the ellipticity of the base and apex's moments of inertia, and
a quantity proportional to the area of the triangle. I promote these quantities
at the quantum level to operators whose expectation and spread are then useful
in understanding the quantum states of the system. Additionally, I tessellate
the 2-sphere by its physical interpretation as the shape space of triangles,
and then use this as a back-cloth from which to read off the interpretation of
dynamical trajectories, potentials and wavefunctions. I include applications to
timeless approaches to the problem of time and to the role of uniform states in
quantum cosmological modelling.Comment: A shorter version, as per the first stage in the refereeing process,
and containing some new reference
Imprints of the Quantum World in Classical Mechanics
The imprints left by quantum mechanics in classical (Hamiltonian) mechanics
are much more numerous than is usually believed. We show Using no physical
hypotheses) that the Schroedinger equation for a nonrelativistic system of
spinless particles is a classical equation which is equivalent to Hamilton's
equations.Comment: Paper submitted to Foundations of Physic
Quantum Blobs
Quantum blobs are the smallest phase space units of phase space compatible
with the uncertainty principle of quantum mechanics and having the symplectic
group as group of symmetries. Quantum blobs are in a bijective correspondence
with the squeezed coherent states from standard quantum mechanics, of which
they are a phase space picture. This allows us to propose a substitute for
phase space in quantum mechanics. We study the relationship between quantum
blobs with a certain class of level sets defined by Fermi for the purpose of
representing geometrically quantum states.Comment: Prepublication. Dedicated to Basil Hile
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