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Zeitschrift für Naturforschung A

Lassen sich Quantenzustände als Ensembles streufreier Zustände darstellen? I/Can quantum-states be considered as ensembles of dispersionfree states? I

Wilhelm Ochs

Published Nov 1, 1970• 1 citation
Publisher

Abstract

Abstract We consider the problem, whether quantum-states can be represented by Gibbsian ensembles of dispersionfree microstates having well defined values for all observables. Such a statistical representation is shown to be possible if and only if the concept and mathematical representation of observables is modified by abandoning the 1-1-correspondance between observables and operators. A particular representation of quantum-states by Gibbsian ensembles is constructed.

DOI
10.1515/zna-1970-1103

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Abstract We consider the problem, whether quantum-states can be represented by Gibbsian ensembles of dispersionfree microstates having well defined values for all observables. Such a statistical representation is shown to be possible if and only if the concept and mathematical representation of observables is modified by abandoning the 1-1-correspondance between observables and operators. A particular representation of quantum-states by Gibbsian ensembles is constructed.
Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific categories. The corresponding Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. It is shown that commutation relations corresponding to a system with generalized statistics can be constructed from relations corresponding to Bolzman statistics.
In this paper I propose an interpretation of classical statistical mechanics that centers on taking seriously the idea that probability measures represent complete states of statistical mechanical systems. I show how this leads naturally to the idea that the stochasticity of statistical mechanics is associated directly with the observables of the theory rather than with the microstates (as traditional accounts would have it). The usual assumption that microstates are representationally significant in the theory is therefore dispensable, a consequence which suggests interesting possibilities for developing non-equilibrium statistical mechanics and investigating inter-theoretic answers to the foundational questions of statistical mechanics.
General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.
Abstract We examine the possibility of reformulating quantum theory (QT) as a deterministic ensemble theory which (a) interprets observables as objective properties of physical systems and (b) coincides with QT in all quantitative statements. As will be demonstrated, such an Ensemble-Quantum-Theory (EQT) can only be constructed if (1) one accepts a modified observable-concept, and (2) as long as the theory of measurement is left out of account. A correct treatment of the measuring process is impossible within such an EQT. Consequently, there exist no Hidden-Variable Theories with the properties (a) and (b).
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.