350 research outputs found
Reconciling Semiclassical and Bohmian Mechanics: II. Scattering states for discontinuous potentials
In a previous paper [J. Chem. Phys. 121 4501 (2004)] a unique bipolar
decomposition, Psi = Psi1 + Psi2 was presented for stationary bound states Psi
of the one-dimensional Schroedinger equation, such that the components Psi1 and
Psi2 approach their semiclassical WKB analogs in the large action limit.
Moreover, by applying the Madelung-Bohm ansatz to the components rather than to
Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the
correspondence principle. As a result, the bipolar quantum trajectories are
classical-like and well-behaved, even when Psi has many nodes, or is wildly
oscillatory. In this paper, the previous decomposition scheme is modified in
order to achieve the same desirable properties for stationary scattering
states. Discontinuous potential systems are considered (hard wall, step, square
barrier/well), for which the bipolar quantum potential is found to be zero
everywhere, except at the discontinuities. This approach leads to an exact
numerical method for computing stationary scattering states of any desired
boundary conditions, and reflection and transmission probabilities. The
continuous potential case will be considered in a future publication.Comment: 18 pages, 8 figure
Two state scattering problem to Multi-channel scattering problem: Analytically solvable model
Starting from few simple examples we have proposed a general method for
finding an exact analytical solution for the two state scattering problem in
presence of a delta function coupling. We have also extended our model to deal
with general one dimensional multi-channel scattering problems
Reconciling Semiclassical and Bohmian Mechanics: III. Scattering states for continuous potentials
In a previous paper [J. Chem. Phys. 121 4501 (2004)] a unique bipolar
decomposition, Psi = Psi1 + Psi2 was presented for stationary bound states Psi
of the one-dimensional Schroedinger equation, such that the components Psi1 and
Psi2 approach their semiclassical WKB analogs in the large action limit. The
corresponding bipolar quantum trajectories, as defined in the usual Bohmian
mechanical formulation, are classical-like and well-behaved, even when Psi has
many nodes, or is wildly oscillatory. A modification for discontinuous
potential stationary stattering states was presented in a second paper [J.
Chem. Phys. 124 034115 (2006)], whose generalization for continuous potentials
is given here. The result is an exact quantum scattering methodology using
classical trajectories. For additional convenience in handling the tunneling
case, a constant velocity trajectory version is also developed.Comment: 16 pages and 14 figure
The Stokes Phenomenon and Schwinger Vacuum Pair Production in Time-Dependent Laser Pulses
Particle production due to external fields (electric, chromo-electric or
gravitational) requires evolving an initial state through an interaction with a
time-dependent background, with the rate being computed from a Bogoliubov
transformation between the in and out vacua. When the background fields have
temporal profiles with sub-structure, a semiclassical analysis of this problem
confronts the full subtlety of the Stokes phenomenon: WKB solutions are only
local, while the production rate requires global information. Incorporating the
Stokes phenomenon, we give a simple quantitative explanation of the recently
computed [Phys. Rev. Lett. 102, 150404 (2009)] oscillatory momentum spectrum of
e+e- pairs produced from vacuum subjected to a time-dependent electric field
with sub-cycle laser pulse structure. This approach also explains naturally why
for spinor and scalar QED these oscillations are out of phase.Comment: 5 pages, 4 figs.; v2 sign typo corrected, version to appear in PR
Low-energy fusion caused by an interference
Fusion of two deuterons of room temperature energy is studied. The nuclei are
in vacuum with no connection to any external source (electric or magnetic
field, illumination, surrounding matter, traps, etc.) which may accelerate
them. The energy of the two nuclei is conserved and remains small during the
motion through the Coulomb barrier. The penetration through this barrier, which
is the main obstacle for low-energy fusion, strongly depends on a form of the
incident flux on the Coulomb center at large distances from it. In contrast to
the usual scattering, the incident wave is not a single plane wave but the
certain superposition of plane waves of the same energy and various directions,
for example, a convergent conical wave. As a result of interference, the wave
function close to the Coulomb center is determined by a cusp caustic which is
probed by de Broglie waves. The particle flux gets away from the cusp and moves
to the Coulomb center providing a not small probability of fusion (cusp driven
tunneling). Getting away from a caustic cusp also occurs in optics and
acoustics
Cosmological particle production and the precision of the WKB approximation
Particle production by slow-changing gravitational fields is usually
described using quantum field theory in curved spacetime. Calculations require
a definition of the vacuum state, which can be given using the adiabatic (WKB)
approximation. I investigate the best attainable precision of the resulting
approximate definition of the particle number. The standard WKB ansatz yields a
divergent asymptotic series in the adiabatic parameter. I derive a novel
formula for the optimal number of terms in that series and demonstrate that the
error of the optimally truncated WKB series is exponentially small. This
precision is still insufficient to describe particle production from vacuum,
which is typically also exponentially small. An adequately precise
approximation can be found by improving the WKB ansatz through perturbation
theory. I show quantitatively that the fundamentally unavoidable imprecision in
the definition of particle number in a time-dependent background is equal to
the particle production expected to occur during that epoch. The results are
illustrated by analytic and numerical examples.Comment: 14 pages, RevTeX, 5 figures; minor changes, a clarification in Sec.
II
Landau-Zener problem for energies close to potential crossing points
We examine one overlooked in previous investigations aspect of well - known
Landau - Zener (LZ) problem, namely, the behavior in the intermediate, i.e.
close to a crossing point, energy region, when all four LZ states are coupled
and should be taken into account. We calculate the 4 x 4 connection matrix in
this intermediate energy region, possessing the same block structure as the
known connection matrices for the tunneling and in the over-barrier regions of
the energy, and continously matching those in the corresponding energy regions.Comment: 5 pages, 1 figur
Kelvin mode of a vortex in a nonuniform Bose-Einstein condensate
In a uniform fluid, a quantized vortex line with circulation h/M can support
long-wavelength helical traveling waves proportional to e^{i(kz-\omega_k t)}
with the well-known Kelvin dispersion relation \omega_k \approx (\hbar k^2/2M)
\ln(1/|k|\xi), where \xi is the vortex-core radius. This result is extended to
include the effect of a nonuniform harmonic trap potential, using a quantum
generalization of the Biot-Savart law that determines the local velocity V of
each element of the vortex line. The normal-mode eigenfunctions form an
orthogonal Sturm-Liouville set. Although the line's curvature dominates the
dynamics, the transverse and axial trapping potential also affect the normal
modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi
condensate. The leading effect of the nonuniform condensate density is to
increase the amplitude along the axis away from the trap center. Near the ends,
however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary
conditions. For a given applied frequency, the next-order correction
renormalizes the local wavenumber k(z) upward near the trap center, and k(z)
then increases still more toward the ends.Comment: 9 pages, 1 figur
Unsupervised Classifiers, Mutual Information and 'Phantom Targets'
We derive criteria for training adaptive classifier networks to perform unsupervised
data analysis. The first criterion turns a simple Gaussian classifier
into a simple Gaussian mixture analyser. The second criterion, which is
much more generally applicable, is based on mutual information. It simplifies
to an intuitively reasonable difference between two entropy functions,
one encouraging 'decisiveness,' the other 'fairness' to the alternative interpretations
of the input. This 'firm but fair' criterion can be applied
to any network that produces probability-type outputs, but it does not
necessarily lead to useful behavior
Study of a class of non-polynomial oscillator potentials
We develop a variational method to obtain accurate bounds for the
eigenenergies of H = -Delta + V in arbitrary dimensions N>1, where V(r) is the
nonpolynomial oscillator potential V(r) = r^2 + lambda r^2/(1+gr^2), lambda in
(-infinity,\infinity), g>0. The variational bounds are compared with results
previously obtained in the literature. An infinite set of exact solutions is
also obtained and used as a source of comparison eigenvalues.Comment: 16 page
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