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arXiv:quant-ph/0512158

Nonlinear evolution as a possible explanation for quantum mechanical statevector reduction

Henrik Brusheim-Johansson

Published Dec 19, 2005• 0 citations
arXivHTMLPDF

Abstract

A non-local toy-model is proposed for the purpose of modelling the ``wave function collapse'' of a two-state quantum system. The collapse is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or a part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the indeterminacy we are experiencing for a single quantum system. Through this randomness, we no longer need the statistical ``ensemble'' behaviour to describe a single quantum system. A brief introduction to the ``measurement problem'' is also given.

DOI
10.48550/arxiv.quant-ph/0512158

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A non-local toy-model is proposed for the purpose of modelling the wave function collapse'' of a two-state quantum system. The collapse is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or a part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the indeterminacy we are experiencing for a single quantum system. Through this randomness, we no longer need the statistical ensemble'' behaviour to describe a single quantum system. A brief introduction to the ``measurement problem'' is also given.
A new model is proposed for the purpose of modelling the ``wave function collapse'' of a two-state quantum system. The collapse to a classical state is driven by a nonlinear evolution equation with an extreme sensitivity to absolute phase. It is hypothesized that the phase, or part of it, is displaying chaotic behaviour. This chaotic behaviour can then be responsible for the apparent indeterminacy we are experiencing for a single quantum system. Through this randomness, the statistical ``ensemble'' behaviour, due to Born, to describe a single quantum system, is no longer needed.
Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector σ in the Bloch sphere.
The measurement problem in quantum mechanics represents a disconnect between the Schr\"{o}dinger equation's unitary evolution and the non-unitary collapse of the wave function. This work resolves this issue by modeling measurement as a deterministic, thermodynamically driven process. We modify the Liouville-von Neumann equation by introducing a dual mechanism: a dephasing term that eliminates quantum coherence and a state-resolution dynamic that amplifies one outcome to certainty. An infinitesimal environmental perturbation breaks the initial symmetry, determining which outcome emerges. Numerical simulations of a qubit in superposition demonstrate that (1) coherence is conserved during unitary evolution, (2) a rapid phase transition occurs upon measurement, and (3) Born Rule statistics emerge from ensemble averaging over random environmental fluctuations. We conclude that quantum collapse need not be a separate postulate but can be modeled as a unified, deterministic dynamical law with testable experimental predictions.
We propose that the mechanism responsible for the ``collapse of the wave function" (or "decoherence" in its broadest meaning) in quantum mechanics is the nonlinearities already present in the theory via nonabelian gauge interactions. Unlike all other models of spontaneous collapse, our proposal is, to the best of our knowledge, the only one which does not introduce any new elements into the theory. Indeed, unless the gauge interaction nonlinearities are not used for exactly this purpose, one must then explain why the violation of the superposition principle which they introduce does not destroy quantum mechanics. A possible experimental test of the model would be to compare the coherence lengths for, e.g., electrons and photons in a double-slit experiment. The electrons should have a finite coherence length, while photons should have a much longer (in principle infinite) coherence length.
This work is an attempt to recreate Quantum Mechanics conceptual foundations. First, we claim that the wave function in quantum mechanics is a definition of random discontinuous particle motion, and that the wave function's modulus square gives the particle probability density at some space locations. First, we demonstrate that the linear, non-relativistic evolution of an isolated system's wave function obeys the free Schr ̈odinger equation due to the spacetime translation invariance and relativistic invariance requirements. Thirdly, we argue that the discontinuous random motion of the particles will lead to a stochastic, nonlinear collapse of the wave function. A discrete model of the energy-conserved collapse of the wave function is proposed and demonstrated to be consistent with current experiments and our macroscopic experience. We also offer a crucial review of the de Broglie-Bohm theory, the explanation of several worlds and other theories of dynamic collapse, and briefly discuss the question of the incompatibility between quantum mechanics and particular relativity.
Several of the imponderables associated with the Copenhagen interpretation of quantum mechanics have alternative, parallel explanations in terms of nonlinear dynamics and chaos. For example, an exponential law for processes such as radioactive decay can be obtained by iterating unimodal maps operating in chaotic domains, utilizing their extreme sensitivity to initial conditions. Such alternative explanations raise the question of the possibility of underlying nonlinearities in quantum mechanics. This is a ‘from the bottom up’, empirical approach to the question and will be compared with other, more theoretical, ‘from the top down’ investigations into the possibility of nonlinear extensions to quantum mechanics. Nonlinear dynamics, although deterministic, in its chaotic extreme provides only statistical information about the outcomes of state-space orbits. Thus in principle it provides the determinism so dear to Einstein; yet in practice it exhibits the probabilistic behaviour of Bohr. Perhaps Einstein and Bohr could both have been right.