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Progress of Theoretical Physics

Theory of Measurement in Quantum Mechanics: Mechanism of Reduction of Wave Packet. II

S. Machida, Mikio Namiki

Published Jun 1, 1980• 67 citations
Publisher

Abstract

General scope and general formulation of the theory of quantum-mechanical measurement are given. Essential points of our theory are that macroscopic apparatus is associated with many Hilbert spaces and the dynamical scattering theory is applied to microscopic interactions between object and apparatus. It is shown that our theory gets over the Furry-Wigner-Fine criticisms.

DOI
10.1143/ptp.63.1833

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Progress of Theoretical Physics•Feb 1, 1992•2 citations
General scope and general formulation of the theory of quantum-mechanical measurement are given. Essential points of our theory are that macroscopic apparatus is associated with many Hilbert spaces and the dynamical scattering theory is applied to microscopic interactions between object and apparatus. It is shown that our theory gets over the Furry-Wigner-Fine criticisms.
New theory of measurement is presented to show that quantum mechanics can describe the whole processes of quantum-mechanical measurement leading to the so-called reduction of wave packet if applied properly to the total system of object and apparatus. We introduce a simple model, where momenta of particles are measured using a perfect rigid mirror, to yield the reduction of wave packet in a clear-cut way. Microscopic uncertainties of macroscopic quantities and the dynamical scattering theory play the essential roles. Before entering into the main part, we briefly summarize the controversial points in the theory of measurement.
The question as to whether quantum mechanics can be applied to a macroscopic system, which has recently been raised through discussions of the problem of how to interpret the so-called “reduction of a wave packet”, is discussed. It is shown by a detailed analysis of the destruction of interference which brings about the reduction, that a supplementary assumption on the preciseness of measurements must at least be introduced in order to interpret the reduction process as an objective process, and it does not, however, necessarily imply inapplicability of quantum mechanics to a macroscopic system.
The "reduction of the wave packet" is an essential feature of the quantum description of measurement. Since it means an abrupt change in the wave function, it therefore also implies an abrupt change in the statistical description of the system. If the differential-space theory of quantum systems rests on a description of reality more detailed than quantum mechanics, it must contain an account of this change which is consistent with the strictly probabilistic point of view of the theory. It is here shown that the theory does satisfy this necessary condition. The need of a special postulate for the measurement process is eliminated in this formulation, and a new explicitness is introduced into the description.
Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising an object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann. Time needed for measurement is also discussed. The Einstein-Podolsky-Rosen state is described as an eigenstate of correlation. The argument lends support to the standpoint that the physical state of a system is to be represented by the statistical operator, not by the wave function, in the conceptual foundation of quantum mechanics.