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arXiv:quant-ph/0410095 · Optics and Spectroscopy

Spontaneous collapse of an unstable quantum superposition state: A single-particle model of modified quantum dynamics

Takuya Okabe

Published Sep 1, 2005• 0 citations
arXivHTMLPDF

Abstract

A modified dynamics of quantum mechanics is proposed, in which the classical mechanics of a point mass is derived intrinsically in the massive limit of a single-particle model. On the premise that the position basis plays a special role in wave function collapse, we deduce a formalization of the spontaneous localization of a wave function with an analogy drawn from thermodynamics, in which a characteristic energy scale and a time scale are introduced to separate quantum and classical regimes.

DOI
10.1134/1.2055939

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A modified dynamics of quantum mechanics is proposed, in which the classical mechanics of a point mass is derived intrinsically in the massive limit of a single-particle model. On the premise that the position basis plays a special role in wave function collapse, we deduce a formalization of the spontaneous localization of a wave function with an analogy drawn from thermodynamics, in which a characteristic energy scale and a time scale are introduced to separate quantum and classical regimes.
On the basis of a proposed model of wave function collapse, we investigate spontaneous localization of a quantum state. The model is similar to the Ghirardi-Rimini-Weber model, while we postulate the localization functions to depend on the quantum state to suffer collapse. According to the model, dual dynamics in quantum mechanics, deterministic and stochastic time evolution, are algorithmically implemented in tandem. After discussing the physical implications of the model qualitatively, we present numerical results for one-dimensional systems by way of example.
We discuss a model of spontaneous collapse of the quantum state that does not require adding any stochastic processes to the standard dynamics. The additional ingredient with respect to the wave function is a position in the configuration space which drives the collapse in a completely deterministic way. This new variable is equivalent to a set of positions of all the particles, i.e., a set of Bohmian positions, which obey the usual guiding equation of Bohmian theory. Any superposition of quantum states of a macroscopic object occupying different regions of space is projected by a localization process onto the region occupied by the positions. Since the Bohmian positions are well defined in a single realization of the experiment, a space localization into one region is produced. The mechanism is based on the correlations between these positions arising from the cohesive forces inside macroscopic objects. The model introduces two collapse parameters which play a role very similar to those of the Ghirardi-Rimini-Weber (GRW) [Ghirardi et al., Phys. Rev. D 34, 470 (1986)] and continuous spontaneous localization (CSL) [Pearle, Phys. Rev. A 39, 2277 (1989); Ghirardi et al., Phys. Rev. A 42, 78 (1990)] theories. With appropriate values of these parameters, we check that the corresponding dynamics rapidly projects superpositions of macroscopic states localized in different regions of space into a single region, while it maintains a negligible effect in all situations where the predictions of standard quantum dynamics are known to be correct. We briefly speculate about the possible relations with gravity. We then study the evolution of the density operator and a mean-field approximation of the dynamical equations of this model, as well as the change of the evolution of the momentum introduced by the localization process. Possible theoretical interpretations are discussed. Generally speaking, this model introduces a sharper border between the quantum and classical worlds than the GRW and CSL theories and leaves a broader range of acceptable values for the parameters.
The basic strategy underlying models of spontaneous wave function collapse (collapse models) is to modify the Schrödinger equation by including nonlinear stochastic terms, which tend to localize wave functions in space in a dynamical manner. These terms have negligible effects on microscopic systems—therefore their quantum behaviour is practically preserved. On the other end, since the strength of these new terms scales with the mass of the system, they become dominant at the macroscopic level, making sure that wave functions of macro-objects are always well-localized in space. We will review these basic features. By changing the dynamics of quantum systems, collapse models make predictions, which are different from standard quantum mechanical predictions. Although they are difficult to detect, we discuss the most relevant scenarios, where such deviations can be observed.
We study a model of spontaneous wavefunction collapse for a free quantum particle. We analyze in detail the time evolution of the single-Gaussian solution and the double-Gaussian solution, showing how the reduction mechanism induces the localization of the wavefunction in space; we also study the asymptotic behavior of the general solution. With an appropriate choice for the parameter λλλ which sets the strength of the collapse mechanism, we prove that: i) the effects of the reducing terms on the dynamics of microscopic systems are negligible, the physical predictions of the model being very close to those of standard quantum mechanics; ii) at the macroscopic scale, the model reproduces classical mechanics: the wavefunction of the center of mass of a macro-object behaves, with high accuracy, like a point moving in space according to Newton's laws.
Physical systems to exist in multiple states simultaneously. While standard quantumtheory permits superposition to persist indefinitely in the absence of mea-surement or environmental decoherence, this work proposes that superpositionstates possess an intrinsic temporal constraint. We argue that quantum super-position cannot remain stable forever and that every superposed system ischaracterized by a finite lifetime, beyond which the superposition inevitablybreaks down and the wave function collapses into a single definite state. Thisproposed time-limited superposition introduces an objective mechanism for wave-function collapse that does not rely on observation or classical interaction.Conceptual arguments, mathematical considerations, and potential experimentalimplications are discussed, including connections to macroscopic quantum lim-its, decoherence timescales, and fundamental temporal structure. If validated,this framework would offer a new perspective on the measurement problem andsuggest that time itself plays an active role in determining quantum outcomes.This perspective is conceptually related to objective collapse models such as theGhirardi–Rimini–Weber (GRW) theory and Penrose’s gravitationally inducedcollapse, while differing in that the collapse mechanism proposed here arisesfrom an intrinsic temporal instability of superposition rather than stochastic orgravity-driven effects.
Quantum mechanics is grounded on the superposition principle, which is the source both of its tremendous success and technological power, as well as of the problems in understanding it. The reason why superpositions do not propagate from the microscopic to the macroscopic world are subject to debate. Spontaneous wave function collapse models have been formulated to take into account a progressive breakdown of quantum superpositions when systems are large enough; they do so by modifying the Schrödinger dynamics, and therefore they are empirically testable. Deviations are tiny, and require precision measurements. I will review collapse models, and present the most recent experimental tests.