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arXiv:2110.00248 · Physical Review Research

Conditional wave theory of environmental interactions with a quantum particle

Rory van Geleuken, Andrew V. Martin

Published Aug 4, 2020• 2 citations
arXivHTMLPDF

Abstract

This paper presents a Conditional Wave Theory of collisional quantum decoherence, which reproduces the behavior of the Joos-Zeh master equation via a Schr"odinger-like equation with a time-dependent logarithmic non-linearity.
DOI
10.1103/physrevresearch.2.033189

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A logarithmic Schr\"odinger equation with time-dependent coupling to the nonlinearity is presented as a model of collisional decoherence of the wave function of a quantum particle in position space. The particular mathematical form of the logarithmic Schr\"odinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schr\"odinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the nonlinear equation of motion for the wave function numerically and compare it to the solution of the standard Joos-Zeh master equation for the density matrix. We find good agreement for the time-dependent behavior of the ensemble widths between the two approaches, but note curious zero-pinning behavior of the logarithmic Schr\"odinger equation, whereby the zeros of the wave function are not erased by continued propagation. By examining the derivation of the logarithmic Schr\"odinger equation from conditional wave theory, we indicate possible avenues of resolution to this zero-pinning problem.
We discuss a recent proposal by Albert (1994a,b; 2000, Chapter 7) to recover thermodynamics on a purely dynamical basis, using the quantum theory of the collapse of the wave function of Ghirardi, Rimini and Weber (1986). We propose an alternative way to explain thermodynamics within no-collapse interpretations of quantum mechanics. Our approach relies on the standard quantum mechanical models of environmental decoherence of open systems, \eg Joos and Zeh (1985) and Zurek and Paz (1994). This paper presents the two approaches and discusses their advantages. The problems they face will be discussed in a sequel (Hemmo and Shenker 2002b).
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