4 research outputs found
Uniform Approximation from Symbol Calculus on a Spherical Phase Space
We use symbol correspondence and quantum normal form theory to develop a more
general method for finding uniform asymptotic approximations. We then apply
this method to derive a result we announced in an earlier paper, namely, the
uniform approximation of the -symbol in terms of the rotation matrices. The
derivation is based on the Stratonovich-Weyl symbol correspondence between
matrix operators and functions on a spherical phase space. The resulting
approximation depends on a canonical, or area preserving, map between two pairs
of intersecting level sets on the spherical phase space.Comment: 18 pages, 5 figure
Quantum dynamics and breakdown of classical realism in nonlinear oscillators
The dynamics of a quantum nonlinear oscillator is studied in terms of its
quasi-flow, a dynamical mapping of the classical phase plane that represents
the time-evolution of the quantum observables. Explicit expressions are derived
for the deformation of the classical flow by the quantum nonlinearity in the
semiclassical limit. The breakdown of the classical trajectories under the
quantum nonlinear dynamics is quantified by the mismatch of the quasi-flow
carried by different observables. It is shown that the failure of classical
realism can give rise to a dynamical violation of Bell's inequalities.Comment: RevTeX 4 pages, no figure
Semiclassical analysis of Wigner -symbol
We analyze the asymptotics of the Wigner -symbol as a matrix element
connecting eigenfunctions of a pair of integrable systems, obtained by lifting
the problem of the addition of angular momenta into the space of Schwinger's
oscillators. A novel element is the appearance of compact Lagrangian manifolds
that are not tori, due to the fact that the observables defining the quantum
states are noncommuting. These manifolds can be quantized by generalized
Bohr-Sommerfeld rules and yield all the correct quantum numbers. The geometry
of the classical angular momentum vectors emerges in a clear manner. Efficient
methods for computing amplitude determinants in terms of Poisson brackets are
developed and illustrated.Comment: 7 figure file
Moyal star product approach to the Bohr-Sommerfeld approximation
The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional
quantum Hamiltonian is derived through order (i.e., including the
first correction term beyond the usual result) by means of the Moyal star
product. The Hamiltonian need only have a Weyl transform (or symbol) that is a
power series in , starting with , with a generic fixed point in
phase space. The Hamiltonian is not restricted to the kinetic-plus-potential
form. The method involves transforming the Hamiltonian to a normal form, in
which it becomes a function of the harmonic oscillator Hamiltonian.
Diagrammatic and other techniques with potential applications to other normal
form problems are presented for manipulating higher order terms in the Moyal
series.Comment: 27 pages, no figure