Semiclassical Distributions Research Articles

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

ABSTRACTThis is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we construct the high energy resolvent on general non-trapping asymptotically hyperbolic manifolds, using semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, together with the 0-calculus of Mazzeo-Melrose.Our results generalize recent work of Melrose, Sá Barreto and Vasy, which applies to metrics close to the exact hyperbolic metric. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent.

Global solutions and asymptotic behavior for two dimensional gravity water waves

This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof is based on a bootstrap argument involving $L^2$ and $L^\infty$ estimates. The $L^2$ bounds are proved in a companion paper of this article. They rely on a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. We give here the proof of the uniform bounds, interpreting the equation in a semi-classical way, and combining Klainerman vector fields with the description of the solution in terms of semi-classical lagrangian distributions. This, together with the $L^2$ estimates of the companion paper, allows us to deduce our main global existence result.

Scattering by an oscillating barrier: Quantum, classical, and semiclassical comparison

We present a detailed study of scattering by an amplitude-modulated potential barrier using three distinct physical frameworks: quantum, classical, and semiclassical. Classical physics gives bounds on the energy and momentum of the scattered particle, while also providing the foundation for semiclassical theory. We use the semiclassical approach to selectively add quantum-mechanical effects such as interference and diffraction. We find good agreement between the quantum and semiclassical momentum distributions. Our methods and results can be used to understand quantum and classical aspects of transport mechanisms involving time-varying potentials, such as quantum pumping.

Quantum dynamics with fermion coupled coherent states: Theory and application to electron dynamics in laser fields

We present an alternate version of the coupled-coherent-state method, specifically adapted for solving the time-dependent Schr\odinger equation for multielectron dynamics in atoms and molecules. This theory takes explicit account of the exchange symmetry of fermion particles, and it uses fermion molecular dynamics to propagate trajectories. As a demonstration, calculations in the He atom are performed using the full Hamiltonian and accurate experimental parameters. Single- and double-ionization yields by 160-fs and 780-nm laser pulses are calculated as a function of field intensity in the range 10${}^{14}$--10${}^{16}$ W/cm${}^{2}$, and good agreement with experiments by Walker et al. is obtained. Since this method is trajectory based, mechanistic analysis of the dynamics is straightforward. We also calculate semiclassical momentum distributions for double ionization following 25-fs and 795-nm pulses at 1.5\ifmmode\times\else\texttimes\fi{}10${}^{15}$ W/cm${}^{2}$, in order to compare them with the detailed experiments by Rudenko et al. For this more challenging task, full convergence is not achieved. However, major effects such as the fingerlike structures in the momentum distribution are reproduced.

Long quantum transition times due to unstable semiclassical dynamics

Quantum transitions are described semiclassically as motions of systems along (complex) trajectories. We consider the cases when the semiclassical trajectories are unstable and find that durations of the corresponding transitions are large. In addition, we show that the probability distributions over transition times have unusual asymmetric form in cases of unstable trajectories. We investigate in detail three types of processes related to unstable semiclassical dynamics. First, we analyze recently proposed mechanism of multidimensional tunneling where transitions proceed by formation and subsequent decay of classically unstable ``states.'' The second class of processes includes one-dimensional activation transitions due to energy dispersion. In this case the semiclassical transition-time distributions have universal form. Third, we investigate long-time asymptotics of transition-time distributions in the case of overbarrier wave packet transmissions. We show that behavior of these asymptotics is controlled by unstable semiclassical trajectories which linger near the barrier top.

Note on semiclassical uncertainty relations

An important manifestation of the Uncertainty Principle, one of the cornerstones of our present understanding of Nature, is that related to semiclassical localization in phase-space. We wish here to add some notes on the subject with reference to the canonical harmonic oscillator problem, with emphasis in the concepts of semiclassical Husimi distributions, the associated Wehrl entropy, escort distributions, and Fisher's information measure.

Information, Deformed қ-Wehrl Entropies and Semiclassical Delocalization

Semiclassical delocalization in phase space constitutes a manifestation of the Uncertainty Principle, one indispensable part of the present understanding of Nature and the Wehrl entropy is widely regarded as the foremost localization-indicator. We readdress the matter here within the framework of the celebrated semiclassical Husimi distributions and their associatedWehrl entropies, suitably қ-deformed. We are able to show that it is possible to significantly improve on the extant phase-space classical-localization power.

Semiclassical localization and uncertainty principle

Semiclassical localizability problems in phase space constitute a manifestation of the uncertainty principle, one of the cornerstones of our present understanding of Nature. We revisit the subject here within the framework of the celebrated semiclassical Husimi distributions and their associated Wehrl entropy. By recourse to the concept of escort distributions, a well-established statistical concept, we show that it is possible to significantly improve on the current phase-space classical-localization power, thus approaching more closely than before the bounds imposed by Husimi's thermal uncertainty relation.

Control of recollision wave packets for molecular orbital tomography using short laser pulses

The tomographic imaging of arbitrary molecular orbitals via high-order harmonic generation requires that electrons recollide from one direction only. Within a semi-classical model, we show that extremely short phase-stabilized laser pulses offer control over the momentum distribution of the returning electrons. By adjusting the carrier–envelope phase, recollisions can be forced to occur from mainly one side, while retaining a broad energy spectrum. The signatures of the semi-classical distributions are observed in harmonic spectra obtained by the numerical solution of the time-dependent Schrödinger equation.

Semiclassical information from deformed and escort information measures

Escort distributions are a well established but (for physicists) a relatively new concept that is rapidly gaining wide acceptance in world. In this work we wish to revisit the concept within the strictures of the celebrated semiclassical Husimi distributions (HDs) and thereby investigate the possibility of extracting new semiclassical information contained, not in the HD themselves, but in their associated escort Husimi distributions. We will also establish relations, for various information measures, between their deformed versions [J. Naudts, Physica A 316 (2002) 323] and those built up with escort HDs. Bounds on the concomitant power exponents will be determined.

Convolutions of semi-classical spectral distributions and periodic-orbit theory

We study the convolution of semi-classical spectral distributions associated to h-pseudodifferential operators on R n . Under standard assumptions the micro-support of this object can be characterized via families of periodic orbits correlated simultaneously by energy and periods. When all the orbits are non-degenerate the convolution admits, as h tends to 0, an explicit asymptotic expansion in term of the respective dynamical systems. In this setting, this result validates the theory of orbits pairs used by physicists in quantum chaos. Some new contributions, related to the crossing of the period functions, are also analyzed.

Statistical sampling of semiclassical distributions: Calculating quantum mechanical effects using Metropolis Monte Carlo

A statistical sampling method is proposed for computing oscillatory integrals associated with the semiclassical initial value representation. The semiclassical expression is rewritten as an integral over a phase distribution P(s). The phase distribution is obtained from Metropolis sampling of trajectories according to a properly chosen weight function. The averaging of oscillatory integrals is converted into a Monte Carlo algorithm where one diffuses through trajectory space. A histogram of phases is collect from importance sampling. Techniques of Metropolis Monte Carlo such as umbrella (or biased) sampling are generalized to the present context. From example calculations, phase distributions are seen to be multi-peaked, thus clearly demonstrating the origin of quantum interference. Trajectories that are responsible for the interference patterns can be collected using this method.

SEMICLASSICAL MOTION OF DRESSED ELECTRONS

We consider an electron coupled to the quantized radiation field and subject to a slowly varying electrostatic potential. We establish that over sufficiently long times radiation effects are negligible and the dressed electron is governed by an effective one-particle Hamiltonian. In the proof only a few generic properties of the full Pauli–Fierz Hamiltonian H PF enter. Most importantly, H PF must have an isolated ground state band for |p|<p c ≤∞ with p the total momentum and p c indicating that the ground state band may terminate. This structure demands a local approximation theorem, in the sense that the one-particle approximation holds until the semiclassical dynamics violates |p|<p c . Within this framework we prove an abstract Hilbert space theorem which uses no additional information on the Hamiltonian away from the band of interest. Our result is applicable to other time-dependent semiclassical problems. We discuss semiclassical distributions for the effective one-particle dynamics and show how they can be translated to the full dynamics by our results.

Quantum, classical and semiclassical momentum distributions: I. Theory and elementary examples

We discuss the basic properties of momentum distributions in quantum mechanics for elementary systems as well as their classical analogue. Semiclassical approximations can show a quantitative connection between the classical and quantum cases. We believe that such distributions provide a useful tool to improve the understanding of elementary quantum mechanics. Important differences between distributions in coordinate and momentum space are pointed out. Elementary examples (free and uniformly accelerating particle, harmonic oscillator and square-well potential) are discussed.

Radial probability distributions of atoms in diatomic molecules represented by the rotating Morse and Tietz-Hua oscillators

Radial probability distributions of diatomic molecules in excited rotational-vibrational states are derived using the Bohr-Sommerfeld quantization rule and the Hamilton-Jacoby theory. The obtained semiclassical distributions for the rotating Morse and Tietz-Hua oscillators are compared, in a broad range of rotational and vibrational quantum numbers, with one another and with quantum-mechanical distributions obtained from numerical solution of the Schrödinger equation using an RKR potential.

Postcollision-interaction recapture of photoelectrons and zero-kinetic-energy electron emission: A quantum model

Near-threshold photoelectron recapture due to postcollision interaction with a single Auger electron is examined within the quantum-mechanical framework of radiationless resonant Raman scattering. Model calculations using a hydrogenic approximation are performed, revealing general features of the escape probability as a function of inner-shell decay width \ensuremath{\Gamma}. The escape probability is useful in estimating diagram and spectator Auger intensities near threshold. The process of recapture produces a delay in the onset of double ionization to energies above threshold, which scales as ${\ensuremath{\Gamma}}^{2/3}.$ The present model is also applied to the cross section for ejection of photoelectrons with zero kinetic energy (ZKE), assuming a stable final state. A comparison with semiclassical results at small \ensuremath{\Gamma} suggests that existing semiclassical distributions are formally correct, with further convolution suggested by Thomas et al. [J. Phys. B 29, 3245 (1996)] not required. Results of the present quantum ZKE model, which does not include additional cascade effects, are in agreement with existing data. Quantities that describe the escape and ZKE-emission distributions are identified and their variation with \ensuremath{\Gamma} is examined.

Quantum and semiclassical Husimi distributions for a one-dimensional resonant system

We compare exact and semiclassical Husimi distributions for the single eigenstates of a one-dimensional resonant Hamiltonian. We find that both distributions concentrate near the unstable fixed points even when these points are made complex by suitably varying a parameter.

Quantum statistical weights for reorientational correlation functions in the nuclear magnetic resonance relaxation of methyl groups

Rotationally symmetric groups containing nuclei with nonzero spin often exhibit uniquely quantum mechanical properties. Symmetry requirements on the total wave function restrict the allowable combinations of electronic, vibrational, rotational, and nuclear spin wave functions. NMR relaxation depends on operators that act on both the nuclear spin and rotational wave functions. The interdependence of the wave functions produces quantum statistical weights in the rotational correlation functions that can differ from semiclassical or classical probability distributions. The quantum statistical weights for the reorientational correlation functions in NMR relaxation have been derived for A3 and AX3 systems of spin 1/2 nuclei and for deuterium relaxation in a deuterated methyl or equivalent group. It is found that classical weights apply to deuterium relaxation. The A3 system has distinctly nonclassical weights. And the AX3 system has eight sets of weights, with different weights applying to different types of correlation functions. One of the eight is the classical weight, and two of the sets can lead to terms that do not arise in the usual, semiclassical theory. It is shown that, under certain circumstances, the predictions of this quantum mechanical theory differ from those of the semiclassical model. The present theory also predicts that changing the symmetry of the electronic/vibrational states can alter the NMR relaxation behavior. Calculations with a stochastic quantum dynamical model indicate that these differences may be experimentally important for, among other things, relaxation involving a methyl group in a biomolecule at physiological temperatures.

Comparison of semiclassical and quantum mechanical angular distributions for nuclear fission and evaporation

There is quite a variety of statistical-model equations in the literature that have been derived with different assumptions and for various classes of nuclear reactions. Some have been obtained by consideration of the quantum mechanics and some by semiclassical approximations thereto. Often it is not clear whether or not the results of certain analyses are clouded by these semiclassical approximations. We consider three statistical models that have been heavily employed in recent years: (1) fission decay from a rigid rotor (2) particle evaporation from a spherical system and (3) fission decay from two uncoupled spheres. Comparisons are made of numerical results from semiclassical and from quantum mechanical equations for certain specific reactions. In addition, some general features are displayed and discussed for the differences between these formalisms. For average spins that are often encountered in heavy ion reactions, the results are very similar. We conclude that major differences in current analyses are not due to the methodology; instead they must center on assumptions concerning the physical nature of the transition-state (or decision-point) configurations and their associated level densities.

Semiclassical description of currents in normal and superfluid rotating nuclei

First the technique of partial ħ-resummation is used to obtain semiclassical current distributions of slowly-rotating normal fluid nuclei. We find that the average intrinsic current flows mainly in the nuclear surface. The analogy with the Bohr-van Leeuven theorem of diamagnetism is pointed out. Strutinsky- and temperature-averaged calculations are in good agreement with the semiclassical results. Secondly we study rotating superfluid nuclei and find in semiclassical approximation analytic closed expressions for the currents using a deformed harmonic-oscillator potential. The currents pass with increasing gap continuously from rigid-rotation flow to irrotational flow patterns. Analogous results hold true for the moment of inertia. For a fixed value of the gap our formulas for the moment of inertia interpolate smoothly between irrotational flow and rigid-body values as a function of deformation.