Given only a finite ensemble of identically prepared particles, how precisely can one determine their states? We describe optimal measurement procedures in the case of spin $1/2$ particles. Furthermore, we prove that optimal measurement procedures must necessarily view the ensemble as a single composite system rather than as the sum of its components, i.e., optimal measurements cannot be realized by separate measurements on each particle.

The study of wave-packet revivals is extended to the case of Hamiltonians which are made time dependent through the adiabatic cycling of some parameters. It is shown that the quantal geometric phase (Berry's phase) causes the revived packet to be displaced along the classical trajectory by an amount equal to the classical geometric phase (Hannay's angle), in one degree of freedom. A physical example illustrating this effect in three degrees of freedom is mentioned.

We study dynamics of nearly elastic particles constrained to move on a line with energy input from the boundaries. We find that for typical initial conditions, the system evolves to an “extraordinary” state with particles separated to two groups: The majority of the particles get clamped into a small region of space and move with very slow velocities; a few remaining particles travel between the boundaries at much higher speeds. Such a state clearly violates equipartition of energy. The simplest hydrodynamic approach fails to give a correct description of the system.

A Comment on the Letter by Eric Lewin Altschuler et al., Phys. Rev. Lett. 72, 2671 (1994).

A Reply to the Comment by Erber, T. and Hockney, G. M.

By inserting a gas cell in one arm of an atom interferometer, we have measured both the attenuation and the phase shift of a sodium matter wave that passes through monatomic (He, Ne, Ar, Kr, and Xe) or molecular gases (${\mathrm{N}}_2$, C${\mathrm{O}}_2$, N${\mathrm{H}}_3$, and ${\mathrm{H}}_2$O). This determines the complex index of refraction for Na matter waves and, more accurately, the ratio of the real to the imaginary part of the forward scattering amplitude. These measurements are compared with several semiclassical scattering models.

We obtain the von Neumann entropy per state of the Haldane exclusion statistics with parameter $g$ in terms of the mean occupation number $\overline{n}$ as $S_H=-\overline{n}\{w\ln w-(1+w\left)\mathrm{}\ln \mathrm{}\right(1+w\left)\right\}$, where $w=\frac{(1-\overline{n}g)}{\overline{n}}$. This reduces correctly to the well known expressions in the limiting cases of Bose ($g=0\mathrm{}$) and Fermi ($g=1\mathrm{}$) statistics. We have derived the second and third order fluctuations in the occupation numbers for arbitrary $g$. An elegant general duality relationship between the $w$ factor associated with the particle and that associated with the hole at the reciprocal $g$ is deduced along with the attendant relationship between the two respective entropies.

Stochastic resonance in a bistable potential is fully characterized as a synchronization effect of the hopping mechanism induced by the external periodic bias. Most notably, synchronization is shown to attain a maximum by increasing the forcing frequency close to the relevant switching rate, thus revealing a bona fide resonant process.

For $1\le \alpha \le 2$, the family of fast diffusions, $u_t={\left[u^{-\alpha }{u}_x\right]}_x$, $0<\mathrm{}\alpha <2\mathrm{}$, coexists with superfast processes which, prior to termination within a finite time, assume a time-space separable form. The remarkable properties of $u_t{\left[\ln \mathrm{}\right(u\left)\right]}_{\mathrm{xx}}$ guide the understanding of these processes; two interacting kinks form a superfast shrinking pattern or a persisting motion of two poles. For $\alpha =1\mathrm{}$, the superfast axisymmetric diffusion coexists with a modified fundamental diffusion: a response to a singular core and a ring of $\delta$ sources. In 3D, fast ($0<\mathrm{}\alpha <\frac{2}{3}$) and the separable ($\frac{4}{5}\le \alpha <1\mathrm{}$) superfast processes are distinct. Inhomogeneity of the medium is considered.

With the ansatz that a data set's correlation matrix has a certain parametrized from (one general enough, however, to allow the arbitrary specification of a slowly varying decorrelation distance and population variance), the general machinery of Wiener or optimal filtering can be reduced from $O\left(n^3\right)$ to $O\left(n\right)\mathrm{}$ operations, where $n$ is the size of the data set. The implied vast increase in computational speed can allow many common suboptimal or heuristic data analysis methods to be replaced by fast, relatively sophisticated, statistical algorithms. Three examples are given: data rectification, high- or low-pass filtering, and linear least-squares fitting to a model with unaligned data points.

A trick discovered by Vaidman, Aharanov, and Albert permitting retrodiction of the outcomes of more measurements than one would naively have thought possible is extended to a case in which the retrodicted observables are forbidden all to have values by a Bell-Kochen-Specker theorem. A rather peculiar analysis shows that an even better trick that retrodicts the outcomes of more informative measurements of these same observables is impossible.

It is shown that the transformations of Einstein-Podolsky-Rosen states such as those used in communication and cryptography schemes can be described as a hopping motion on a finite phase space lattice associated with a finite Heisenberg group. Quantum mechanical Hamiltonians that generate the hopping are shown to cause phase oscillations characterized by the number-theoretic Legendre symbol.

A mathematically consistent procedure for coupling quasiclassical and quantum variables through coupled Hamilton-Heisenberg equations of motion is derived from a variational principle. During evolution, the quasiclassical variables become entangled with the quantum variables with the result that the value of the quasiclassical variables depends on the quantum state. This provides a formalism to compute the backreaction of any quantum system on a quasiclassical one. In particular, it leads to a natural candidate for a theory of gravity coupled to quantized matter in which the gravitational field is not quantized.

The fully ionized (metallic) hydrogen plasma is studied by ab initio molecular dynamics simulations, for classical protons and fully degenerate electrons, in the strong coupling regime of the protons. The calculated ion-electron pair distribution function signals the breakdown of linear screening with decreasing density while the melting temperature of the proton crystal drops rapidly. The simulations reveal the remarkable persistence of a weakly damped high-frequency ion acoustic mode, even under conditions of strong electron screening.

As in quantum field theory, for some singular two-body potentials, the momentum-space perturbative treatment of the three-dimensional quantum $N$-body problem is shown to exhibit ultraviolet divergences. Renormalization of these models leads to a scale and finite observables. For a Dirac delta potential we carry on this renormalization for two- and three-body systems. A new divergence, and hence a new scale, emerges for each added particle. For $N\ge 3$ there are no divergences in one and two dimensions. The physical implications of these results are discussed.

The time evolution of quantum decay of an arbitrary state initially located within an interaction region of finite range is investigated. We compare the survival $S\left(t\right)\mathrm{}$ and nonescape $P\left(t\right)\mathrm{}$ probabilities. Our approach considers the full resonant spectra of the system using a novel representation of the time-dependent Green function. It is shown that, for an initial state near a resonance, $S\left(t\right)\mathrm{}$ and $P\left(t\right)\mathrm{}$ exhibit at long times a different behavior; not only the onset to a power law decay occurs at a different time, but also instead of the well known $S\left(t\right)\mathrm{}\sim t^{-3\mathrm{}}$ we obtain $P\left(t\right)\mathrm{}\sim t^{-1\mathrm{}}$.

We demonstrate a close connection between the decoherent histories approach to quantum mechanics and the quantum state diffusion picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the quantum state diffusion picture also define an approximately decoherent set of histories in the decoherent histories approach. The degree of localization is related to the degree of decoherence, and the probabilities for histories prescribed by each approach are essentially the same.

A simple model of a driven diffusive system which exhibits spontaneous symmetry breaking in one dimension is introduced. The model has short range interactions and unbounded noise. It is characterized by an asymmetric exclusion process of two types of charges moving in opposite directions on an open chain. The model is studied by mean field and Monte Carlo methods. Exact solutions can be found in a restricted region of its parameter space. A simple physical picture for the symmetry breaking mechanism is presented.

Inspired by the multicanonical approach to simulations of first-order phase transitions we propose for $q$-state Potts models a combination of cluster updates with reweighting of the bond configurations in the Fortuin-Kastelein-Swendsen-Wang representation of this model. Numerical tests for the two-dimensional models with $q=7, 10, \mathrm{and} 20$ show that the autocorrelation times of this algorithm grow with the system size $V$ as $\tau \propto V^{\alpha }$, where the exponent takes the optimal random walk value of $\alpha \approx 1$.

A class of Monte Carlo algorithms which incorporate absorbing Markov chains is presented. In a particular limit, the lowest order of these algorithms reduces to the $n$-fold way algorithm. These algorithms are applied to study the escape from the metastable state in the two-dimensional square-lattice nearest-neighbor Ising ferromagnet in an unfavorable applied field, and the agreement with theoretical predictions is very good. It is demonstrated that the higher-order algorithms can be many orders of magnitude faster than either the traditional Monte Carlo or $n$-fold way algorithms.