We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a sp
ecific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.
We study the problem of wave transport in a one-dimensional disordered system, where the scatterers of the chain are $n$ barriers and wells with statistically independent intensities and with a spatial extension $l_c$ which may contain an arbitrary n
umber $delta/2pi$ of wavelengths, where $delta = k l_c$. We analyze the average Landauer resistance and transmission coefficient of the chain as a function of $n$ and the phase parameter $delta$. For weak scatterers, we find: i) a regime, to be called I, associated with an exponential behavior of the resistance with $n$, ii) a regime, to be called II, for $delta$ in the vicinity of $pi$, where the system is almost transparent and less localized, and iii) right in the middle of regime II, for $delta$ very close to $pi$, the formation of a band gap, which becomes ever more conspicuous as $n$ increases. In regime II, both the average Landauer resistance and the transmission coefficient show an oscillatory behavior with $n$ and $delta$. These characteristics of the system are found analytically, some of them exactly and some others approximately. The agreement between theory and simulations is excellent, which suggests a strong motivation for the experimental study of these systems. We also present a qualitative discussion of the results.
Measurements on classical systems are usually idealized and assumed to have infinite precision. In practice, however, any measurement has a finite resolution. We investigate the theory of non-ideal measurements in classical mechanics using a measurem
ent probe with finite resolution. We use the von Neumann interaction model to represent the interaction between system and probe. We find that in reality classical systems are affected by measurement in a similar manner as quantum systems. In particular, we derive classical equivalents of Luders rule, the collapse postulate, and the Lindblad equation.
The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in Quantum Mechanics (QM). We investigate whether time averages over one realization of a single system are rela
ted to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of von Neumann model of measurement, coupling the system to $N$ probes --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.
We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in
a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Luders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called Wigners formula emerges in the strong-coupling limit, while Kirkwoods quasi-probability distribution is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.
Wigner function is a quasi-distribution that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between Wigner function and appropriate measurements invo
lving the system position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space, and then turn to the case of a discrete, finite-dimensional Hilbert space.
Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show th
at this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.
We observed undulated smectic textures for some compounds of the 4,n-alkyloxybenzoic (nOBAC) acid series, at transitions between the smectic and the isotropic phase and between the smectic and nematic phase. Studied compounds were 12OBAC, 16OBAC and
a binary mixture of 12- and 16OBAC. The undulations are dressing a usual Schlieren texture. In the case of the binary mixture, an interesting fingerprint pattern is observed too.
We study successive measurements of two observables using von Neumanns measurement model. The two-pointer correlation for arbitrary coupling strength allows retrieving the initial system state. We recover Luders rule, the Wigner formula and the Kirkw
ood-Dirac distribution in the appropriate limits of the coupling strength.
In a previous paper (Phase Transitions, 80(9), 987, 2007) we discussed the smectic and nematic textures of some mesomorphic oxadiazole compounds with terminal Cl-substituent. Optical microscope investigations showed a very interesting behaviour of sm
ectic and nematic phases; the smectic phase has fan-shaped and toric textures and the nematic phase has spherulitic domains, which disappear as the sample is further heated, the texture changing into a smooth one. Here, we investigate four oxadiazole compounds with the same structure but terminal Br-substituent. The behaviour of the smectic and nematic phases is like that observed in the compounds with Cl. The focus of the paper is on the growth of toric domains from the nematic melt and on the role of defects in the domain structure.
