We point out a new class of level statistics where the level-spacing distribution follows an inverse power law p(s)∼${\mathit{s}}^{\mathrm{-}\mathrm{\beta }}$, with β=3/2. It is characteristic of level clustering rather than level repulsion and appears to be universal for systems exhibiting unbounded quantum diffusion on 1D lattices. A relaxation of this class is met in a model of Bloch electorns in a magnetic field, where we find a purely diffusive spread of wave packets without the quantum limitations known from chaotic systems like the kicked rotator.

A pure state of a quantum-mechanical system, like a completely polarized state of light, is specified by the results of a few well-chosen experiments. We present a method for selecting such experiments, with applications to low-j angular momentum states.

In Debye-Hückel theory, the free energy of an electric double layer near a colloidal (or any other) surface can be related to the statistics of random walks near that surface. We present a numerical method based on this correspondence for the calculation of the double-layer free energy for an arbitrary charged or conducting surface. For self-similar surfaces, we propose a scaling law for the behavior of the free energy as a function of the screening length and the surface dimension. This scaling law is verified by numerical computation. Capacitance measurements on rough surfaces of, e.g., colloids can test these predictions.

We calculate the effects of surface singularities, e.g., wedges, on the thermodynamics of electric double layers near a rough colloid or charged membrane. Each wedge yields an additive contribution to the free energy scrF that is a function of the angle. We give a geometrical analysis of scrF entirely similar to that of vibration eigenmodes given by Kac in ‘‘Can one hear the shape of a drum?,’’ with a probabilistic Brownian representation of scrF, our analysis yields a universal scaling law for the free energy of a fractal colloid with its Minkowski dimension. The case of a colloidal random walk is derived from conformal invariance.

When processes not conserving the double occupation number are inhibited, one obtains a model like the Kivelson-Schrieffer-Su-Heeger extension of the Hubbard model. We solve this model exactly in any number of dimensions. The system free energy is given for any fixed filling. The ground-state energy and double-occupancy order parameter are discussed for d=2.

It is argued that any quantum probability should correspond to a ratio between the number of counts (positive results) in some measurement and the number of copies of the physical system initially prepared. Then, the proofs of Bell’s theorem are criticized on the grounds that the probabilities used to show a violation of the Bell inequalities do not fulfill that condition. A hidden-variables model is proposed which reproduces the results of the optical experimental tests of the inequalities, even with perfect polarizers and detectors.

An interaction picture, applied to observables, can be used to introduce in and out constants of the motion even when a collision takes place inside a medium. The localized nature of chemical interactions makes this a practical tool. A computational example (using classical dynamics) is provided.

We study Witten’s (2+1)-dimensional Chern-Simons theory with gauge group SU(2). The partition function in this theory is a topological invariant of three-manifolds. Following the prescription given by Witten, we explicitly calculate this partition function on some three-manifolds and verify many of the predictions made by the path integral. Most strikingly, we check the perturbation theory against the exact solution.

We present the first realization of a measurement of a ‘‘weak value,’’ a concept recently introduced by Aharonov, Albert, and Vaidman (AAV). Our experiment uses a birefringent crystal to separate the two linear-polarization components of a laser beam by a distance small compared to the laser-beam waist. This ‘‘weak measurement’’ is followed by a strong measurement which translates the centroid of the beam by a distance far larger than the birefringence-induced separation. In addition, we present data corresponding to orthogonal initial and final states, for which the weak value is not defined. This interference effect may have application in the amplification and detection of weak effects.

A two-photon interference experiment has been carried out, based on an idea recently proposed by Croca, Garuccio, Lepore, and Moreira. The experiment is designed to test a prediction of the de Broglie guided-wave theory, according to which the waves have a physical reality in addition to yielding photon detection probabilities. The results clearly contradict the de Broglie theory, but are consistent with conventional quantum mechanics.

In this Letter, we present a theoretical investigation into the quantum-nondemolition properties of a system which utilizes intracavity second-harmonic generation. The results show a significant improvement over the quantum-nondemolition properties of other systems studied previously, and should be accessible to experiment.

Two quantum systems are identically prepared in different locations. An observer’s task is to determine their state. A simple example shows that a pair of measurements of the von Neumann type is less effective than a sequence of nonorthogonal probability-operator measures, alternating between the two quantum systems. However, the most efficient set of operations of that type that we were able to design falls short of a single combined measurement, performed on both system together.

Using both experimental and theoretical results, this Letter describes how low-energy, feedback control signals can be successfully utilized to suppress (laminarize) chaotic flow in a thermal convection loop.

Using an ultrasensitive, cryogenic, current-comparator bridge the quantized Hall resistance ${\mathit{R}}_{\mathit{H}}$(2) in a GaAs/AlGaAs heterostructure has been compared directly with ${\mathit{R}}_{\mathit{H}}$(4) in a silicon MOSFET. The measurements show that ${\mathit{R}}_{\mathit{H}}$(2;GaAs)/${\mathit{R}}_{\mathit{H}}$(4;Si)=2[1-0.22(3.5)×${10}^{\mathrm{-}10}$]. Within the 1σ combined uncertainty of ±3.5×${10}^{\mathrm{-}10}$ the result suggests that the quantized Hall resistance is a universal quantity, independent of the host lattice and Landau-level index, and is probably equivalent to h/${\mathit{e}}^2$, the relationship predicted theoretically.

A general analytic method for calculating finite-size corrections, central charges, and scaling dimensions of solvable lattice models is presented. The approach is to solve the special functional equations or inversion identities satisfied by the commuting row transfer matrices of these lattice models at criticality. For purposes of illustration, the method is applied to calculate the central charge c=4/5 and leading magnetic scaling dimension x=2/15 of hard hexagons. These numbers are rational due to special values of Rogers dilogarithms.

In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol’d-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, we present strong evidence that its fractal dimension is 1.

The deformation of the energy-level spectra of quantum systems as a function of a control parameter governing the strength of a nonintegrable perturbation can be determined by standard molecular-dynamics techniques. The integration of Newton’s equations of motion provides detailed information on the spectral deformation when the classical system undergoes a transition from mostly regular to mostly chaotic motion. This method is used to study the changes in the energy-level statistics and allows the direct determination of the Brody parameter in the predominantly chaotic regime.

A novel random-matrix ensemble is introduced which mimics the global structure inherent in the Hamiltonian matrices of autonomous, ergodic systems. Changes in its parameters induce a transition between a Poisson and a Wigner distribution for the level spacings, P(s). The intermediate distributions are uniquely determined by a single scaling variable. Semiclassical constraints force the ensemble to be in a regime with Wigner P(s) for systems with more than two freedoms.

By constructing action variables that are very nearly invariant in a region Ω of phase space, and by examining their residual variation, we set long-term bounds on any orbit starting in an open subregion of Ω. A new and generally applicable method for constructing the required high-precision invariants is applied. The technique is illustrated for transverse oscillations in a circular accelerator, a case with 21/2 degrees of freedom and strong nonlinearity.

It is shown that a phenomenon analogous to Berry’s phase and Hannay’s angle occurs in dissipative systems. Adiabatic transport of a dissipative oscillatory system about a closed path in parameter space produces a geometric shift in the variable parametrizing the limit cycle. This quantity is written as the integral of a two-form over a surface bounded by the parameter-space loop.