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arXiv:hep-th/9303121 · Physical Review A

Quantum dynamical model for wave-function reduction in classical and macroscopic limits

Chang-Pu Sun

Published Aug 1, 1993• 94 citations
arXivHTMLPDF

Abstract

In this paper, a quantum dynamical model describing the quantum-measurement process is presented as an extensive generalization of the Coleman-Hepp model. In both the classical limit with very large quantum number and the macroscopic limit with very large particle number in the measuring instrument, this model generally realizes the wave-packet collapse in quantum measurement as a consequence of the Schr\"odinger time evolution in either the exactly solvable case or the non-exactly-solvable case. For the latter, the quasiadiabatic case is explicitly analyzed by making use of the high-order adiabatic-approximation method, which manifests the wave-packet collapse as well as in the exactly solvable case. By highlighting these analyses, it is finally found that an essence of the dynamical model of wave-packet collapse is the factorization of the Schr\"odinger evolution rather than the exact solvability. Therefore many dynamical models including the previous well-known ones, whether they are exactly solvable or not, can be shown to be only the concrete realizations of this factorizability.

DOI
10.1103/physreva.48.898

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In this paper, a quantum dynamical model describing the quantum-measurement process is presented as an extensive generalization of the Coleman-Hepp model. In both the classical limit with very large quantum number and the macroscopic limit with very large particle number in the measuring instrument, this model generally realizes the wave-packet collapse in quantum measurement as a consequence of the Schr"odinger time evolution in either the exactly solvable case or the non-exactly-solvable case. For the latter, the quasiadiabatic case is explicitly analyzed by making use of the high-order adiabatic-approximation method, which manifests the wave-packet collapse as well as in the exactly solvable case. By highlighting these analyses, it is finally found that an essence of the dynamical model of wave-packet collapse is the factorization of the Schr"odinger evolution rather than the exact solvability. Therefore many dynamical models including the previous well-known ones, whether they are exactly solvable or not, can be shown to be only the concrete realizations of this factorizability.
By building a general dynamical model for quantum measurement process,it is shown that the factorization of reduced evolution operator sufficiently results in the quantum mechanical realization of the wave packet collapse and the state correlation between the measured system and the measuring instrument-detector.This realizability is largely independent of the details of both the interaction and Hamiltonian of detector. The Coleman-Hepp model and all its generalizations are only the special cases of the more universal model given in this letter.An explicit example of this model is finally given in connection with coherent state.
Introduction In a very elegant and rigorous paper, K. Hepp has discussed quantum measurement theory. He uses the C * algebra description of infinite quantum systems. Here an attempt is made to give a more popular account of some of his reasoning. Such an attempt seems worthwhile because many people not familiar with the C * algebra approach, and even somewhat intimidated by it, have been intrigued by the following statement in Hepp's abstract: In several explicitly soluble models, the measurement leads to macroscopically different ‘pointer positions’ and to a rigorous ‘reduction of the wave packet’ with respect to all local observables. This could look like a clean solution at last to the infamous measurement problem. But it is not so, nor thought by Hepp to be so. Here we will take one of his models and analyse it in elementary text-book terms. It will be insisted that the ‘rigorous reduction’ does not occur in physical time but only in an unattainable mathematical limit. It will be argued that the distinction is an important one. We will work at first in the Schrödinger picture, but later, with the extension to relativistic systems in mind, it will be argued that such considerations become particularly clear in the Heisenberg picture.
The systematical studies on the dynamical approach of wavefunction collapse in quantum measurement are reported in this paper based on the Hepp-Coleman’s model and its generalizations. Under certain physically reasonable conditions, which are easily satisfied by the practical problems, it is shown that the off-diagonal elements of the reduced density matrix vanish in quantum mechanical evolution process in the macroscopic limit with a very large particle number N. Various examples with detector made up of oscillators of different spectrum distribution are used to illustrate this observations. With the two-level system as an explicit illustration, the quantum information entropy is exactly obtained to quantitatively describe the degree of decoherence for the so-called partial coherence caused by detector. The entropy for the case with many levels is computed based on perturbation method in the limits with very large and very small N. As an application of this general approach for quantum measurement, a dynamical realization of the quantum Zeno effect are present to analyse its recent testing experiment in connection with a description of transition in quantum information entropy. Finally, the Cini’s model for the correlation between the states of the measured system and the detector is generalized for the case with many energy-level. It is shown that this generalization can also be invoked to give the dynamical realization of wavefunction collapse. PACS number(s): 0365-w, 0380+r.
The systematical studies on the dynamical approach of wavefunction collapse in quantum measurement are report in this paper based on the Hepp-Coleman's model and its generalizations. Under certain physically reasonable conditions, which are easily satisfied by the practical problems, it is shown that the off-diagonal elements of the reduced density matrix for the measured system vanish in quantum mechanical evolution process of the universe formed by the measured system plus the measuring instrument-detector at the macroscopic limit with a very large particle number N. Various examples with detector made up of oscillators of different spectrum distribution are used to illustrate this observations. With the two-level system as an explicit illustration, the quantum information entropy is exactly obtained to quantitatively describe the degree of decoherence for the so-called partial coherence caused by detector. The entropy for the case with many levels is computed based on perturbation method in the limits with very large and very small N. As an application of this general approach for quantum measurement, a dynamical realization of the quantum Zeno effect are present to analyse its recent testing experiment in connection with a description of transition in quantum information entropy. Finally, the Cini's model for the correlation between the states of the measured system and the detector is generalized for the case with many energy-level. It is shown that this generalization can also be invoked to give the dynamical realization of wavefunction collapse.
We formulate the dynamics of the generic quantum system S_{c} comprising a microsystem S and a macroscopic measuring instrument I, whose pointer positions are represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces are simultaneous eigenspaces of a set of coarse grained intercommuting macroscopic observables and, most crucially, their dimensionalities are astronomically large, increasing exponentially with the number, N, of particles comprising I. We formulate conditions under which the conservative dynamics of S_{c} yields both a reduction of the wave packet describing the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant eigenstate of S; and we show that these conditions are fulfilled, up to utterly negligible corrections that decrease exponentially with N, by the finite version of the Coleman-Hepp model.
The Born-Oppenheimer adiabatic approximation is used to describe the dynamic realization of wave-function collapse in quantum measurement. In the adiabatic limit, it is shown that the wave function of the total system formed by the measured quantum system plus the measuring apparatus can be factorized as an entangled state with correlation between adiabatic quantum states and quasiclassical motion configurations of the large system. When the apparatus effectively behaves as a classical object, this adiabatic entanglement leads to the wave-function collapse, which creates an ideal quantum measurement process.