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Title: Time operator and quantum projection evolution

Abstract

In this paper, we consider time as a dynamical variable. In particular, we present the explicit realization of the time operator within four-dimensional nonrelativistic spacetime. The approach assumes including events as a part of the evolution. The evolution is not driven by the physical time, but it is based on the causally related physical events. The usual Schroedinger unitary evolution can be easily derived as a special case of the three-dimensional projection onto the space of simultaneous events. Also the time-energy uncertainty relation makes clear and mathematically rigorous interpretation.

Authors:
;  [1]
  1. University of Marie Curie-Sklodowska, Institute of Physics (Poland), E-mail: mdebicki@kft.umcs.lublin.pl
Publication Date:
OSTI Identifier:
21075918
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Atomic Nuclei; Journal Volume: 70; Journal Issue: 3; Other Information: DOI: 10.1134/S106377880703012X; Copyright (c) 2007 Nauka/Interperiodica; Article Copyright (c) 2007 Pleiades Publishing, Ltd; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; EVOLUTION; FOUR-DIMENSIONAL CALCULATIONS; MATHEMATICAL SPACE; SCHROEDINGER EQUATION; SPACE-TIME; THREE-DIMENSIONAL CALCULATIONS

Citation Formats

Gozdz, A., E-mail: Andrzej.Gozdz@umcs.lublin.pl, and Debicki, M. Time operator and quantum projection evolution. United States: N. p., 2007. Web. doi:10.1134/S106377880703012X.
Gozdz, A., E-mail: Andrzej.Gozdz@umcs.lublin.pl, & Debicki, M. Time operator and quantum projection evolution. United States. doi:10.1134/S106377880703012X.
Gozdz, A., E-mail: Andrzej.Gozdz@umcs.lublin.pl, and Debicki, M. Thu . "Time operator and quantum projection evolution". United States. doi:10.1134/S106377880703012X.
@article{osti_21075918,
title = {Time operator and quantum projection evolution},
author = {Gozdz, A., E-mail: Andrzej.Gozdz@umcs.lublin.pl and Debicki, M.},
abstractNote = {In this paper, we consider time as a dynamical variable. In particular, we present the explicit realization of the time operator within four-dimensional nonrelativistic spacetime. The approach assumes including events as a part of the evolution. The evolution is not driven by the physical time, but it is based on the causally related physical events. The usual Schroedinger unitary evolution can be easily derived as a special case of the three-dimensional projection onto the space of simultaneous events. Also the time-energy uncertainty relation makes clear and mathematically rigorous interpretation.},
doi = {10.1134/S106377880703012X},
journal = {Physics of Atomic Nuclei},
number = 3,
volume = 70,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}
  • We present a Monte Carlo algorithm suitable for the calculation of excited state energies of multidimensional quantum systems. Energies are extracted from a maximum entropy analysis of the imaginary time evolution of a state prepared by application of a projection operator on an initial wave function. The imaginary time evolution is computed with a pure diffusion Monte Carlo algorithm. The method is demonstrated on a harmonic oscillator and several Morse oscillator test problems. {copyright} {ital 1997} {ital The American Physical Society}
  • The coordinate matrix element of the time evolution operator, exp(-iH-italic-circumflext/(h/2..pi..)), is determined by expanding (its exponent) in a power series in t. Recursion relations are obtained for the expansion coefficients which can be analytically evaluated for any number of degrees of freedom. Numerical application to the tunneling matrix element in a double well potential and to the reactive flux correlation function for a barrier potential show this approach to be a dramatic improvement over the standard short time approximation for the propagator. Its use in a Feynman path integral means that fewer ''time slices'' in the matrix product exp((-i/(h/2..pi..))..delta..tH-italic-circumflex)/sup N/,more » ..delta..t = t/N, will be required. The first few terms in the present expansion constitute a fully quantum version of the short time propagator recently obtained by us using semiclassical methods (Chem. Phys. Lett. 151, 1 (1988)).« less
  • No abstract prepared.
  • A method is developed for treating complex molecular collision processes through the application of stochastic reduction formalisms. We begin by describing a projection operator method for decomposing a complicated collision system into two (or more) subsystems, each of which is assumed to be weakly correlated with the others. Approximations to this correlation are then introduced, resulting in a set of coupled equations for the reduced density operators (or classical phase space distributions) associated with each subsystem. We then examine the classical mechanical application of this theory to the forced oscillator model of V--T energy transfer. Arguments of multiple time scalesmore » are used to uncouple the stochastically reduced equations of motion to evaluate the memory kernel analytically. This leads to a single diffusion equation for the time evolution of the action in the oscillator during the collision. Comparison with the corresponding exact results indicates excellent agreement of low order moments of the classical distributions of action in the limit of small energy transfer (i.e., ..delta..E/h..omega..<1). Of particular note is the fact that our stochastic theory predicts an average energy transfer (first moment) in exact agreement with the exact result independent of magnitude of the energy transfer. In a related application of our general stochastic formalism, we consider the quantum mechanical forced oscillator model. This problem is treated: (a) through the use of reduced density matrices (which leads to master equations), and (b) through the Wigner equivalent formalism. The transition probabilities obtained from these two equivalent applications are shown to be identical. Comparison of stochastic and exact quantum results indicates quantitative agreement of the probabilities for ..delta..E/h..omega..< or =0.1, and average agreement of the probabilities for larger values of the energy transfer.« less