2,679 research outputs found
From Heisenberg matrix mechanics to EBK quantization: theory and first applications
Despite the seminal connection between classical multiply-periodic motion and
Heisenberg matrix mechanics and the massive amount of work done on the
associated problem of semiclassical (EBK) quantization of bound states, we show
that there are, nevertheless, a number of previously unexploited aspects of
this relationship that bear on the quantum-classical correspondence. In
particular, we emphasize a quantum variational principle that implies the
classical variational principle for invariant tori. We also expose the more
indirect connection between commutation relations and quantization of action
variables. With the help of several standard models with one or two degrees of
freedom, we then illustrate how the methods of Heisenberg matrix mechanics
described in this paper may be used to obtain quantum solutions with a modest
increase in effort compared to semiclassical calculations. We also describe and
apply a method for obtaining leading quantum corrections to EBK results.
Finally, we suggest several new or modified applications of EBK quantization.Comment: 37 pages including 3 poscript figures, submitted to Phys. Rev.
Remarks on a Proposed Super-Kamiokande Test for Quantum Gravity Induced Decoherence Effects
Lisi, Marrone, and Montanino have recently proposed a test for quantum
gravity induced decoherence effects in neutrino oscillations observed at
Super-Kamiokande. We comment here that their equations have the same
qualitative form as the energy conserving objective state vector reduction
equations discussed by a number of authors. However, using the Planckian
parameter value proposed to explain state vector reduction leads to a neutrino
oscillation effect many orders of magnitude smaller than would be detectable at
Super-Kamiokande. Similar estimates hold for the Ghirardi, Rimini, and Weber
spontaneous localization approach to state vector reduction, and our remarks
are relevant as well to proposed meson and meson tests of gravity
induced decoherence.Comment: 10 pages, plain Tex, no figure
Quantum interference and sub-Poissonian statistics for time-modulated driven dissipative nonlinear oscillator
We show that quantum-interference phenomena can be realized for the
dissipative nonlinear systems exhibiting hysteresis-cycle behavior and quantum
chaos. Such results are obtained for a driven dissipative nonlinear oscillator
with time-dependent parameters and take place for the regimes of long time
intervals exceeding dissipation time and for macroscopic levels of oscillatory
excitation numbers. Two schemas of time modulation: (i) periodic variation of
the strength of the {\chi}(3) nonlinearity; (ii) periodic modulation of the
amplitude of the driving force, are considered. These effects are obtained
within the framework of phase-space quantum distributions. It is demonstrated
that the Wigner functions of oscillatory mode in both bistable and chaotic
regimes acquire negative values and interference patterns in parts of
phase-space due to appropriately time-modulation of the oscillatory nonlinear
dynamics. It is also shown that the time-modulation of the oscillatory
parameters essentially improves the degree of sub-Poissonian statistics of
excitation numbers
Problems and Aspects of Energy-Driven Wavefunction Collapse Models
Four problematic circumstances are considered, involving models which
describe dynamical wavefunction collapse toward energy eigenstates, for which
it is shown that wavefunction collapse of macroscopic objects does not work
properly. In one case, a common particle position measuring situation, the
apparatus evolves to a superposition of macroscopically distinguishable states
(does not collapse to one of them as it should) because each such
particle/apparatus/environment state has precisely the same energy spectrum.
Second, assuming an experiment takes place involving collapse to one of two
possible outcomes which is permanently recorded, it is shown in general that
this can only happen in the unlikely case that the two apparatus states
corresponding to the two outcomes have disjoint energy spectra. Next, the
progressive narrowing of the energy spectrum due to the collapse mechanism is
considered. This has the effect of broadening the time evolution of objects as
the universe evolves. Two examples, one involving a precessing spin, the other
involving creation of an excited state followed by its decay, are presented in
the form of paradoxes. In both examples, the microscopic behavior predicted by
standard quantum theory is significantly altered under energy-driven collapse,
but this alteration is not observed by an apparatus when it is included in the
quantum description. The resolution involves recognition that the statevector
describing the apparatus does not collapse, but evolves to a superposition of
macroscopically different states.Comment: 17 page
Self-Referential Noise and the Synthesis of Three-Dimensional Space
Generalising results from Godel and Chaitin in mathematics suggests that
self-referential systems contain intrinsic randomness. We argue that this is
relevant to modelling the universe and show how three-dimensional space may
arise from a non-geometric order-disorder model driven by self-referential
noise.Comment: Figure labels correcte
Optical discrimination between spatial decoherence and thermalization of a massive object
We propose an optical ring interferometer to observe environment-induced
spatial decoherence of massive objects. The object is held in a harmonic trap
and scatters light between degenerate modes of a ring cavity. The output signal
of the interferometer permits to monitor the spatial width of the object's wave
function. It shows oscillations that arise from coherences between energy
eigenstates and that reveal the difference between pure spatial decoherence and
that coinciding with energy transfer and heating. Our method is designed to
work with a wide variety of masses, ranging from the atomic scale to
nano-fabricated structures. We give a thorough discussion of its experimental
feasibility.Comment: 2 figure
Spontaneous Collapse of Unstable Quantum Superposition State: A Single-Particle Model of Modified Quantum Dynamics
We propose a modified dynamics of quantum mechanics, in which classical
mechanics of a point mass derives intrinsically in a massive limit of a
single-particle model. On the premise that a position basis plays a special
role in wavefunction collapse, we deduce to formalize spontaneous localization
of wavefunction on the analogy drawn from thermodynamics, in which a
characteristic energy scale and a time scale are introduced to separate quantum
and classical regimes.Comment: 2figs., contribution to Xth ICQO 200
Martingale Models for Quantum State Reduction
Stochastic models for quantum state reduction give rise to statistical laws
that are in most respects in agreement with those of quantum measurement
theory. Here we examine the correspondence of the two theories in detail,
making a systematic use of the methods of martingale theory. An analysis is
carried out to determine the magnitude of the fluctuations experienced by the
expectation of the observable during the course of the reduction process and an
upper bound is established for the ensemble average of the greatest
fluctuations incurred. We consider the general projection postulate of L\"uders
applicable in the case of a possibly degenerate eigenvalue spectrum, and derive
this result rigorously from the underlying stochastic dynamics for state
reduction in the case of both a pure and a mixed initial state. We also analyse
the associated Lindblad equation for the evolution of the density matrix, and
obtain an exact time-dependent solution for the state reduction that explicitly
exhibits the transition from a general initial density matrix to the L\"uders
density matrix. Finally, we apply Girsanov's theorem to derive a set of simple
formulae for the dynamics of the state in terms of a family of geometric
Brownian motions, thereby constructing an explicit unravelling of the Lindblad
equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics
Non-Markovian Quantum Trajectories Versus Master Equations: Finite Temperature Heat Bath
The interrelationship between the non-Markovian stochastic Schr\"odinger
equations and the corresponding non-Markovian master equations is investigated
in the finite temperature regimes. We show that the general finite temperature
non-Markovian trajectories can be used to derive the corresponding
non-Markovian master equations. A simple, yet important solvable example is the
well-known damped harmonic oscillator model in which a harmonic oscillator is
coupled to a finite temperature reservoir in the rotating wave approximation.
The exact convolutionless master equation for the damped harmonic oscillator is
obtained by averaging the quantum trajectories relying upon no assumption of
coupling strength or time scale. The master equation derived in this way
automatically preserves the positivity, Hermiticity and unity.Comment: 19 pages, typos corrected, references adde
Modified Special Relativity on a fluctuating spacetime
It was recently proposed that deformations of the relativistic symmetry, as
those considered in Deformed Special Relativity (DSR), can be seen as the
outcome of a measurement theory in the presence of non-negligible (albeit
small) quantum gravitational fluctuations [1,2]. In this paper we explicitly
consider the case of a spacetime described by a flat metric endowed with
stochastic fluctuations and, for a free particle, we show that DSR-like
nonlinear relations between the spaces of the measured and classical momenta,
can result from the average of the stochastic fluctuations over a scale set be
the de Broglie wavelength of the particle. As illustrative examples we consider
explicitly the averaging procedure for some simple stochastic processes and
discuss the physical implications of our results.Comment: 7 pages, no figure
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