763 research outputs found
Flux-Across-Surfaces Theorem for a Dirac Particle
We consider the asymptotic evolution of a relativistic spin-1/2-particle.
i.e. a particle whose wavefunction satisfies the Dirac equation with external
static potential. We prove that the probability for the particle crossing a
(detector) surface converges to the probability, that the direction of the
momentum of the particle lies within the solid angle defined by the (detector)
surface, as the distance of the surface goes to infinity. This generalizes
earlier non relativistic results, known as flux across surfaces theorems, to
the relativistic regime
Time delay for one-dimensional quantum systems with steplike potentials
This paper concerns time-dependent scattering theory and in particular the
concept of time delay for a class of one-dimensional anisotropic quantum
systems. These systems are described by a Schr\"{o}dinger Hamiltonian with a potential converging to different limits
and as and respectively. Due to the
anisotropy they exhibit a two-channel structure. We first establish the
existence and properties of the channel wave and scattering operators by using
the modern Mourre approach. We then use scattering theory to show the identity
of two apparently different representations of time delay. The first one is
defined in terms of sojourn times while the second one is given by the
Eisenbud-Wigner operator. The identity of these representations is well known
for systems where vanishes as (). We show
that it remains true in the anisotropic case , i.e. we prove
the existence of the time-dependent representation of time delay and its
equality with the time-independent Eisenbud-Wigner representation. Finally we
use this identity to give a time-dependent interpretation of the
Eisenbud-Wigner expression which is commonly used for time delay in the
literature.Comment: 48 pages, 1 figur
Independent electrons model for open quantum systems: Landauer-Buettiker formula and strict positivity of the entropy production
A general argument leading from the formula for currents through an open
noninteracting mesoscopic system given by the theory of non-equilibrium steady
states (NESS) to the Landauer-Buettiker formula is pointed out. Time reversal
symmetry is not assumed. As a consequence it follows that, as far as the system
has a nontrivial scattering theory and the reservoirs have different
temperatures and/or chemical potentials, the entropy production is strictly
positive.Comment: 12 pages. Submitted for publication in J. Math. Phys. on 2006-06-05.
Revision and extension of: G. Nenciu, A general proof of Landauer-Buettiker
formula, [math-ph/0603030
On the exit statistics theorem of many particle quantum scattering
We review the foundations of the scattering formalism for one particle
potential scattering and discuss the generalization to the simplest case of
many non interacting particles. We point out that the "straight path motion" of
the particles, which is achieved in the scattering regime, is at the heart of
the crossing statistics of surfaces, which should be thought of as detector
surfaces. We sketch a proof of the relevant version of the many particle flux
across surfaces theorem and discuss what needs to be proven for the foundations
of scattering theory in this context.Comment: 15 pages, 4 figures; to appear in the proceedings of the conference
"Multiscale methods in Quantum Mechanics", Accademia dei Lincei, Rome,
December 16-20, 200
Scattering into Cones and Flux across Surfaces in Quantum Mechanics: a Pathwise Probabilistic Approach
We show how the scattering-into-cones and flux-across-surfaces theorems in
Quantum Mechanics have very intuitive pathwise probabilistic versions based on
some results by Carlen about large time behaviour of paths of Nelson
diffusions. The quantum mechanical results can be then recovered by taking
expectations in our pathwise statements.Comment: To appear in Journal of Mathematical Physic
A microscopic derivation of the quantum mechanical formal scattering cross section
We prove that the empirical distribution of crossings of a "detector''
surface by scattered particles converges in appropriate limits to the
scattering cross section computed by stationary scattering theory. Our result,
which is based on Bohmian mechanics and the flux-across-surfaces theorem, is
the first derivation of the cross section starting from first microscopic
principles.Comment: 28 pages, v2: Typos corrected, layout improved, v3: Typos corrected.
Accepted for publication in Comm. Math. Phy
Limiting absorption principle for the dissipative Helmholtz equation
Adapting Mourre's commutator method to the dissipative setting, we prove a
limiting absorption principle for a class of abstract dissipative operators. A
consequence is the resolvent estimates for the high frequency Helmholtz
equation when trapped trajectories meet the set where the imaginary part of the
potential is non-zero. We also give the resolvent estimates in Besov spaces
On the Localization of One-Photon States
Single photon states with arbitrarily fast asymptotic power-law fall-off of
energy density and photodetection rate are explicitly constructed. This goes
beyond the recently discovered tenth power-law of the Hellwarth-Nouchi photon
which itself superseded the long-standing seventh power-law of the Amrein
photon.Comment: 7 pages, tex, no figure
Secondary literacy across the curriculum: Challenges and possibilities
This paper discusses the challenges and possibilities attendant upon successfully implementing literacy across the curriculum initiatives â or âschool language policiesâ as they have come to be known - particularly at the secondary or high school level. It provides a theoretical background to these issues, exploring previous academic discussions of school language policies, and highlights key areas of concern as well as opportunity with respect to school implementation of such policies. As such, it provides a necessary conceptual background to the subsequent papers in this special issue, which focus upon the Secondary Schoolsâ Literacy Initiative (SSLI) â a New Zealand funded programme that aims to establish cross-curricular language and literacy policies in secondary schools
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