Chaotic eigenfunctions in momentum space

We study eigenstates of chaotic billiards in the momentum representation and propose the radially integrated momentum distribution as useful measure to detect localization effects. For the momentum distribution, the radially integrated momentum distribution, and the angular integrated momentum distribution explicit formulae in terms of the normal derivative along the billiard boundary are derived. We present a detailed numerical study for the stadium and the cardioid billiard, which shows in several cases that the radially integrated momentum distribution is a good indicator of localized eigenstates, such as scars, or bouncing ball modes. We also find examples, where the localization is more strongly pronounced in position space than in momentum space, which we discuss in detail. Finally applications and generalizations are discussed.Comment: 30 pages. The figures are included in low resolution only. For a version with figures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp99-2.htm

Proceedings of the Tenth Annual Beef Cattle Short Course.

91 p

On the rate of quantum ergodicity in Euclidean billiards

For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A version with all figures can be obtained from http://www.physik.uni-ulm.de/theo/qc/ (File: http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any problems contact Arnd B\"acker (e-mail: [email protected]) or Roman Schubert (e-mail: [email protected]

Bounds on Integrals of the Wigner Function

The integral of the Wigner function over a subregion of the phase-space of a quantum system may be less than zero or greater than one. It is shown that for systems with one degree of freedom, the problem of determining the best possible upper and lower bounds on such an integral, over all possible states, reduces to the problem of finding the greatest and least eigenvalues of an hermitian operator corresponding to the subregion. The problem is solved exactly in the case of an arbitrary elliptical region. These bounds provide checks on experimentally measured quasiprobability distributions.Comment: 10 pages, 1 PostScript figure, Latex file; revised following referees' comments; to appear in Physical Review Letter

Chiral tunneling and the Klein paradox in graphene

The so-called Klein paradox - unimpeded penetration of relativistic particles through high and wide potential barriers - is one of the most exotic and counterintuitive consequences of quantum electrodynamics (QED). The phenomenon is discussed in many contexts in particle, nuclear and astro- physics but direct tests of the Klein paradox using elementary particles have so far proved impossible. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment by using electrostatic barriers in single- and bi-layer graphene. Due to the chiral nature of their quasiparticles, quantum tunneling in these materials becomes highly anisotropic, qualitatively different from the case of normal, nonrelativistic electrons. Massless Dirac fermions in graphene allow a close realization of Klein's gedanken experiment whereas massive chiral fermions in bilayer graphene offer an interesting complementary system that elucidates the basic physics involved.Comment: 15 pages, 4 figure

The concurrent decline of soil lead and children’s blood lead in New Orleans

Lead (Pb) is extremely toxic and a major cause of chronic diseases worldwide. Pb is associated with health disparities, particularly within low-income populations. In biological systems, Pb mimics calcium and, among other effects, interrupts cell signaling. Furthermore, Pb exposure results in epigenetic changes that affect multigenerational gene expression. Exposure to Pb has decreased through primary prevention, including removal of Pb solder from canned food, regulating lead-based paint, and especially eliminating Pb additives in gasoline. While researchers observe a continuous decline in children’s blood lead (BPb), reservoirs of exposure persist in topsoil, which stores the legacy dust from leaded gasoline and other sources. Our surveys of metropolitan New Orleans reveal that median topsoil Pb in communities (n = 274) decreased 44% from 99 mg/kg to 54 mg/kg (P value of 2.09 × 10−08), with a median depletion rate of ∼2.4 mg·kg·y−1 over 15 y. From 2000 through 2005 to 2011 through 2016, children’s BPb declined from 3.6 μg/dL to 1.2 μg/dL or 64% (P value of 2.02 × 10−85), a decrease of ∼0.2 μg·dL·y−1 during a median of 12 y. Here, we explore the decline of children’s BPb by examining a metabolism of cities framework of inputs, transformations, storages, and outputs. Our findings indicate that decreasing Pb in topsoil is an important factor in the continuous decline of children’s BPb. Similar reductions are expected in other major US cities. The most contaminated urban communities, usually inhabited by vulnerable populations, require further reductions of topsoil Pb to fulfill primary prevention for the nation’s children

Overcoming the Challenges Associated with Image-based Mapping of Small Bodies in Preparation for the OSIRIS-REx Mission to (101955) Bennu

The OSIRIS-REx Asteroid Sample Return Mission is the third mission in NASA's New Frontiers Program and is the first U.S. mission to return samples from an asteroid to Earth. The most important decision ahead of the OSIRIS-REx team is the selection of a prime sample-site on the surface of asteroid (101955) Bennu. Mission success hinges on identifying a site that is safe and has regolith that can readily be ingested by the spacecraft's sampling mechanism. To inform this mission-critical decision, the surface of Bennu is mapped using the OSIRIS-REx Camera Suite and the images are used to develop several foundational data products. Acquiring the necessary inputs to these data products requires observational strategies that are defined specifically to overcome the challenges associated with mapping a small irregular body. We present these strategies in the context of assessing candidate sample-sites at Bennu according to a framework of decisions regarding the relative safety, sampleability, and scientific value across the asteroid's surface. To create data products that aid these assessments, we describe the best practices developed by the OSIRIS-REx team for image-based mapping of irregular small bodies. We emphasize the importance of using 3D shape models and the ability to work in body-fixed rectangular coordinates when dealing with planetary surfaces that cannot be uniquely addressed by body-fixed latitude and longitude.Comment: 31 pages, 10 figures, 2 table

Semiclassical Approximations in Phase Space with Coherent States

We present a complete derivation of the semiclassical limit of the coherent state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial value representation for the semiclassical propagator, based on an initial gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed gaussian approximation. It is very different from the Herman--Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.Comment: 80 pages, 4 figure

Quantum Tunneling in the Wigner Representation

Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one - dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semi-classical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem.Comment: 18 pages, LATEX, no figure