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arXiv:1304.7368

Solution to the measurement problem within linear, unitary, no collapse, no particle quantum mechanics

Casey Blood

Published Apr 27, 2013• 1 citation
arXivHTMLPDF

Abstract

The measurement problem is to explain why a system which is in a linear combination of states appears, upon measurement, to be in just one of those states. The solution given here is to first show that if one assumes linear, unitary, no collapse, no particle quantum mechanics, the different versions of reality are completely isolated from each other. This then implies only the eigenstates appropriate to the measurement setup will be perceived. In a Stern-Gerlach experiment, for example, only spin up or spin down will be perceived, but never a combination of the two.

DOI
10.48550/arxiv.1304.7368

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The measurement problem is to explain why a system which is in a linear combination of states appears, upon measurement, to be in just one of those states. The solution given here is to first show that if one assumes linear, unitary, no collapse, no particle quantum mechanics, the different versions of reality are completely isolated from each other. This then implies only the eigenstates appropriate to the measurement setup will be perceived. In a Stern-Gerlach experiment, for example, only spin up or spin down will be perceived, but never a combination of the two.
The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.
A discussion of quantum measurement theory is presented. It is argued that the quantum measurement problem consists in the failure of any formalism based on unitary evolution to describe fully the observed behaviour in measurement-like processes. It is shown that the problem is essentially concerned with the phase of the coherence between orthogonal eigenstates of an object/measuring device system. It is then proposed that the observed behaviour is fully described if the value of this phase is formally undecidable.
Paper I of this series introduced a nonlinear version of quantum mechanics that blocks cats, and paper II postulated a random part of the wavefunction to explain outcomes in experiments such as Stern-Gerlach or EPRB. However, an ad hoc extra parameter was assumed for the randomness. Here I provide some analytic and simulation evidence that the nonlinear theory exhibits sensitive dependence on initial conditions in measurement scenarios, perhaps implying that the magnitude of randomness required is determined by structural features of the model, and does not require a free parameter.
We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation, the position of a quantum particle exists and is piloted by the phase of the wave function. We show how this position explains determinism and realism in the three most important experiments of quantum measurement: double-slit, Stern-Gerlach, and EPR-B. First, we demonstrate the conditions in which the de Broglie-Bohm interpretation can be assumed to be valid through continuity with classical mechanics. Second, we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961 with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics. Third, we present an analytic expression of the wave function in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinor with a spatial extension. This solution enables us to demonstrate the decoherence of the wave function and the three postulates of quantum measurement. Finally, we study the Bohm version of the Einstein-Podolsky-Rosen experiment. Its theoretical resolution in space and time shows that a causal interpretation exists where each atom has a position and a spin.
Although the solution, within standard quantum physics, of the problem of outcomes has been published several times, many authors continue to treat measurement as an unsolved fundamental dilemma. The solution lies in the formation of entangled subsystems, the non-local nature of the measurement state, and the resulting distinction between mixed-state local outcomes and the pure-state global outcome. Upon "measurement" (i.e. entanglement), the quantum system and its measurement apparatus both decohere and collapse into local mixed states while the unitarily-evolving global state remains coherent and un-collapsed. The states we observe are the local, collapsed states. Considerable experimental evidence supports this conclusion. Theoretical objections to this conclusion are rebutted, and a new perspective on measurement and entanglement is noted.