Wigner quasi-probability distribution for the infinite square well: energy eigenstates and time-dependent wave packets

We calculate the Wigner quasi-probability distribution for position and momentum, P_W^(n)(x,p), for the energy eigenstates of the standard infinite well potential, using both x- and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P_W(x,p;t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like `mini-packets' which appear at fractional multiples of the exact revival time.Comment: 45 pages, 16 embedded, low-resolution .eps figures (higher resolution, publication quality figures are available from the authors); submitted to American Journal of Physic

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

Superrevivals in the quantum dynamics of a particle confined in a finite square well potential

We examine the revival features in wave packet dynamics of a particle confined in a finite square well potential. The possibility of tunneling modifies the revival pattern as compared to an infinite square well potential. We study the dependence of the revival times on the depth of the square well and predict the existence of superrevivals. The nature of these superrevivals is compared with similar features seen in the dynamics of wavepackets in an anharmonic oscillator potential.Comment: 8 pages in Latex two-column format with 5 figures (eps). To appear in Physical Review

Quantum carpet interferometry for trapped atomic Bose-Einstein condensates

We propose an ``interferometric'' scheme for Bose-Einstein condensates using near-field diffraction. The scheme is based on the phenomenon of intermode traces or quantum carpets; we show how it may be used in the detection of weak forces.Comment: 4 figures. Submitted to Phys. Rev.

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

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

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

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

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]

Identification of Radiopure Titanium for the LZ Dark Matter Experiment and Future Rare Event Searches

The LUX-ZEPLIN (LZ) experiment will search for dark matter particle interactions with a detector containing a total of 10 tonnes of liquid xenon within a double-vessel cryostat. The large mass and proximity of the cryostat to the active detector volume demand the use of material with extremely low intrinsic radioactivity. We report on the radioassay campaign conducted to identify suitable metals, the determination of factors limiting radiopure production, and the selection of titanium for construction of the LZ cryostat and other detector components. This titanium has been measured with activities of $^{238}$U$_{e}$~$<$1.6~mBq/kg, $^{238}$U$_{l}$~$<$0.09~mBq/kg, $^{232}$Th$_{e}$~$=0.28\pm 0.03$~mBq/kg, $^{232}$Th$_{l}$~$=0.25\pm 0.02$~mBq/kg, $^{40}$K~$<$0.54~mBq/kg, and $^{60}$Co~$<$0.02~mBq/kg (68\% CL). Such low intrinsic activities, which are some of the lowest ever reported for titanium, enable its use for future dark matter and other rare event searches. Monte Carlo simulations have been performed to assess the expected background contribution from the LZ cryostat with this radioactivity. In 1,000 days of WIMP search exposure of a 5.6-tonne fiducial mass, the cryostat will contribute only a mean background of $0.160\pm0.001$(stat)$\pm0.030$(sys) counts.Comment: 13 pages, 3 figures, accepted for publication in Astroparticle Physic