Unravelling quantum carpets: a travelling wave approach

Quantum carpets are generic spacetime patterns formed in the probability distributions P(x,t) of one-dimensional quantum particles, first discovered in 1995. For the case of an infinite square well potential, these patterns are shown to have a detailed quantitative explanation in terms of a travelling-wave decomposition of P(x,t). Each wave directly yields the time-averaged structure of P(x,t) along the (quantised)spacetime direction in which the wave propagates. The decomposition leads to new predictions of locations, widths depths and shapes of carpet structures, and results are also applicable to light diffracted by a periodic grating and to the quantum rotator. A simple connection between the waves and the Wigner function of the initial state of the particle is demonstrated, and some results for more general potentials are given.Comment: Latex, 26 pages + 6 figures, submitted to J. Phys. A (connections with prior literature clarified

Self-interference of a single Bose-Einstein condensate due to boundary effects

A simple model wavefunction, consisting of a linear combination of two free-particle Gaussians, describes many of the observed features seen in the interactions of two isolated Bose-Einstein condensates as they expand, overlap, and interfere. We show that a simple extension of this idea can be used to predict the qualitative time-development of a single expanding BEC condensate produced near an infinite wall boundary, giving similar interference phenomena. We also briefly discuss other possible time-dependent behaviors of single BEC condensates in restricted geometries,such as wave packet revivals.Comment: 8 pages, no figures, to appear in Physica Script

Optimum spectral window for imaging of art with optical coherence tomography

Optical Coherence Tomography (OCT) has been shown to have potential for important applications in the field of art conservation and archaeology due to its ability to image subsurface microstructures non-invasively. However, its depth of penetration in painted objects is limited due to the strong scattering properties of artists’ paints. VIS-NIR (400 nm – 2400 nm) reflectance spectra of a wide variety of paints made with historic artists’ pigments have been measured. The best spectral window with which to use optical coherence tomography (OCT) for the imaging of subsurface structure of paintings was found to be around 2.2 μm. The same spectral window would also be most suitable for direct infrared imaging of preparatory sketches under the paint layers. The reflectance spectra from a large sample of chemically verified pigments provide information on the spectral transparency of historic artists’ pigments/paints as well as a reference set of spectra for pigment identification. The results of the paper suggest that broadband sources at ~2 microns are highly desirable for OCT applications in art and potentially material science in general

Comparison of age-specific cataract prevalence in two population-based surveys 6 years apart

BACKGROUND: In this study, we aimed to compare age-specific cortical, nuclear and posterior subcapsular (PSC) cataract prevalence in two surveys 6 years apart. METHODS: The Blue Mountains Eye Study examined 3654 participants (82.4% of those eligible) in cross-section I (1992–4) and 3509 participants (75.1% of survivors and 85.2% of newly eligible) in cross-section II (1997–2000, 66.5% overlap with cross-section I). Cataract was assessed from lens photographs following the Wisconsin Cataract Grading System. Cortical cataract was defined if cortical opacity comprised ≥ 5% of lens area. Nuclear cataract was defined if nuclear opacity ≥ Wisconsin standard 4. PSC was defined if any present. Any cataract was defined to include persons who had previous cataract surgery. Weighted kappa for inter-grader reliability was 0.82, 0.55 and 0.82 for cortical, nuclear and PSC cataract, respectively. We assessed age-specific prevalence using an interval of 5 years, so that participants within each age group were independent between the two surveys. RESULTS: Age and gender distributions were similar between the two populations. The age-specific prevalence of cortical (23.8% in 1(st), 23.7% in 2(nd)) and PSC cataract (6.3%, 6.0%) was similar. The prevalence of nuclear cataract increased slightly from 18.7% to 23.9%. After age standardization, the similar prevalence of cortical (23.8%, 23.5%) and PSC cataract (6.3%, 5.9%), and the increased prevalence of nuclear cataract (18.7%, 24.2%) remained. CONCLUSION: In two surveys of two population-based samples with similar age and gender distributions, we found a relatively stable cortical and PSC cataract prevalence over a 6-year period. The increased prevalence of nuclear cataract deserves further study

Analytic results for Gaussian wave packets in four model systems: II. Autocorrelation functions

The autocorrelation function, A(t), measures the overlap (in Hilbert space) of a time-dependent quantum mechanical wave function, psi(x,t), with its initial value, psi(x,0). It finds extensive use in the theoretical analysis and experimental measurement of such phenomena as quantum wave packet revivals. We evaluate explicit expressions for the autocorrelation function for time-dependent Gaussian solutions of the Schrodinger equation corresponding to the cases of a free particle, a particle undergoing uniform acceleration, a particle in a harmonic oscillator potential, and a system corresponding to an unstable equilibrium (the so-called `inverted' oscillator.) We emphasize the importance of momentum-space methods where such calculations are often more straightforwardly realized, as well as stressing their role in providing complementary information to results obtained using position-space wavefunctions.Comment: 18 pages, RevTeX, to appear in Found. Phys. Lett, Vol. 17, Dec. 200

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

Amblyopia and quality of life: a systematic review

Background/Aims Amblyopia is a common condition which can affect up to 5% of the general population. The health-related quality of life (HRQoL) implications of amblyopia and/or its treatment have been explored in the literature. Methods A systematic literature search was undertaken (16th-30th January 2007) to identify the HRQoL implications of amblyopia and/or its treatment. Results A total of 25 papers were included in the literature review. The HRQoL implications of amblyopia related specifically to amblyopia treatment, rather than the condition itself. These included the impact upon family life; social interactions; difficulties undertaking daily activities; and feelings and behaviour. The identified studies adopted a number of methodologies. The study populations included; children with the condition; parents of children with amblyopia; and adults who had undertaken amblyopia treatment as a child. Some studies developed their own measures of HRQoL, and others determined HRQoL through proxy measures. Conclusions The reported findings of the HRQoL implications are of importance when considering the management of cases of amblyopia. Further research is required to assess the immediate and long-term effects of amblyopia and/or its treatment upon HRQoL using a more standardised approach

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]

Quantum carpets woven by Wigner functions

The dynamics of many different quantum systems is characterized by a regular net of minima and maxima of probability stretching out in a spacetime representation. We offer an explanation to this phenomenon in terms of the Wigner function. This approach illustrates very clearly the crucial role played by interference