On Epstein's trajectory model of non-relativistic quantum mechanics

In 1952 Bohm presented a theory about non-relativistic point-particles moving along deterministic trajectories and showed how it reproduces the predictions of standard quantum theory. This theory was actually presented before by de Broglie in 1926, but Bohm's particular formulation of the theory inspired Epstein to come up with a different trajectory model. The aim of this paper is to examine the empirical predictions of this model. It is found that the trajectories in this model are in general very different from those in the de Broglie-Bohm theory. In certain cases they even seem bizarre and rather unphysical. Nevertheless, it is argued that the model seems to reproduce the predictions of standard quantum theory (just as the de Broglie-Bohm theory).Comment: 12 pages, no figures, LaTex; v2 minor improvement

Fluctuation, time-correlation function and geometric Phase

We establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change to the time integral of the quantum correlation function between the projection operator and force operator of the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalised susceptibility. Relation between the open-path geometric phase, diagonal elements of the quantum metric tensor and the force-force correlation function is provided and the classical limit of the fluctuation-correlation theorem is also discussed.Comment: Latex, 12 pages, no figures, submitted to J. Phys. A: Math & Ge

Localization properties of groups of eigenstates in chaotic systems

In this paper we study in detail the localized wave functions defined in Phys. Rev. Lett. {\bf 76}, 1613 (1994), in connection with the scarring effect of unstable periodic orbits in highly chaotic Hamiltonian system. These functions appear highly localized not only along periodic orbits but also on the associated manifolds. Moreover, they show in phase space the hyperbolic structure in the vicinity of the orbit, something which translates in configuration space into the structure induced by the corresponding self--focal points. On the other hand, the quantum dynamics of these functions are also studied. Our results indicate that the probability density first evolves along the unstable manifold emanating from the periodic orbit, and localizes temporarily afterwards on only a few, short related periodic orbits. We believe that this type of studies can provide some keys to disentangle the complexity associated to the quantum mechanics of these kind of systems, which permits the construction of a simple explanation in terms of the dynamics of a few classical structures.Comment: 9 pages, 8 Postscript figures (low resolution). For high resolution versions of figs http://www.tandar.cnea.gov.ar/~wisniack/ To appear in Phys. Rev.

Geometric phase and angle for noncyclic adiabatic change, revivals and measures of quantal instability

Within the adiabatic theorem we must explicitly add to the instantaneous adiabatic vectors, parametrized by an external parameter following an open curve, a dynamical as well as a geometrical phase contribution to be able to predict the quantum observables. We give an alternative to Berry's proof for cyclic paths without use of Stokes's theorem that is then generalized to open paths. Recent analyses that argued, for an adiabatic open path case, that there is only dynamical phase change or that the situation is undefined are shown to be incorrect. The noncyclic geometric phase correction is discussed in detail for the spin in a magnetic field, an Aharonov-Bohm experiment and a resonator. Properties of the geometric phase contribution for open paths are studied. First, it is shown that degeneracies play an important role. Second, it is shown to be semiclassically related to a geometric angle shift on the final torus. The revival structure of wavepackets is shown to be affected by the noncyclic geometric phases of the, contributing instantaneous vectors that make the revived wavepacket to be shifted on the final torus by the classical geometric angle. We also propose two measures of quantum sensitivity to initial conditions. The first one is valid for quantum and classical Liouville states and it is given by the different distances of two close states to a third fixed state. The second measure is given by the divergence of the flow lines of suitable quantum phase space representations and reduces to the classical Lyapunov exponent. (author)SIGLEAvailable from British Library Document Supply Centre- DSC:D210288 / BLDSC - British Library Document Supply CentreGBUnited Kingdo