Journal of Physics A Mathematical and General

Measurement description by means of a nonlinear Schrodinger equation

Alexander N. Grigorenko

Published Mar 7, 1995• 14 citations

Abstract

A new general formulation for nonlinear quantum mechanics is discussed. It is shown that a nonlinear deterministic evolution with random variables can model a measurement as a dynamical process. Phases of state vectors are treated as random variables of the theory.

DOI: 10.1088/0305-4470/28/5/028

Related papers

Stochastic Reduction Models in Nonlinear Quantum Mechanics

Dorje C. Brody, Lane P. Hughston

Stochastic extensions of the Schrödinger equation have attracted attention recently as plausible models for state reduction in quantum mechanics. Here we formulate a general approach to stochastic Schrödinger dynamics in the case of a nonlinear state space of the type proposed by Kibble. We derive a number of new identities for observables in the nonlinear theory, and establish general criteria on the curvature of the state space sufficient to ensure collapse of the wave function.

Jan 1, 20020 citations

Stochastic reduction in nonlinear quantum mechanics

Dorje C. Brody, Lane P. Hughston

arXiv:quant-ph/0011125Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesMay 8, 200216 citations

Stochastic Reduction in Nonlinear Quantum Mechanics

Brody, D C, Lane P. Hughston

Stochastic extensions of the Schrodinger equation have attracted attention recently as plausible models for state reduction in quantum mechanics. Here we formulate a general approach to stochastic Schrodinger dynamics in the case of a nonlinear state space of the type proposed by Kibble. We derive a number of new identities for observables in the nonlinear theory, and establish general criteria on the curvature of the state space sufficient to ensure collapse of the wave function.

Nov 30, 200013 citations

Nonlinear Generalisation of Quantum Mechanics

Alireza Jamali

According to our current understanding of quantum mechanics, a `measurement' violates unitarity. In other words as the act of measurement `resets' the probabilities described by the Schrödinger equation, in the very `moment' of the collapse of the wavefunction, conservation of probability does not hold. To make progress in our understanding of the measurement problem it is thus expected that one must encompass in a single equation both otherwise mutually-exclusive processes of measurement and unitary evolution. In this light, turning to the continuity equation, we realize the possibility that if we allow for existence of sources and sinks in the `fluid of probability' we get closer to such a theory by arriving at nonlinear generalizations of Schrödinger and Klein-Gordon equations. The generalized equations derived are then shown to satisfy all conditions that are expected from a proper generalization: simplification to their linear counterparts by a well-defined dynamical condition, and Galilean and Lorentz invariance.

Jul 1, 20221 citation

Mobility and measurements in nonlinear wave mechanics

Jacek Waniewski

The theory based on the nonlinear Schrödinger equation with an additional term λ(ψ̄ψ)αψ is investigated. The standard quantum mechanical interpretation of ψ is assumed at the beginning of the considerations. It turns out that every finite set of pure states can be transformed with the aid of an adequate time sequence of external potentials into a set of pairwise nearly orthogonal states. As a consequence, there exist measurements more selective than quantum ones. In particular, it is possible to discriminate between various mixtures of states that are equivalent in quantum mechanics. The possibility of existence of deterministic measurements is also discussed.

Journal of Mathematical PhysicsJul 1, 19864 citations

Stochastic Dynamics In Hilbert Space

Heinz‐Peter Breuer, Francesco Petruccione

Abstract This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.

Jan 25, 20070 citations