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Zeitschrift für Naturforschung A

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

W. Ochs

Published Feb 1, 1971• 0 citations
Publisher

Abstract

Abstract In part I of this paper we constructed a model representing quantum states by ensembles of dispersionsfree states. In part II we study further properties of this model and analyse their logical relations. It is also shown that the main results do not depend on the assumption that each self-adjoint operator corresponds to an observable.

DOI
10.1515/zna-1971-0204

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Abstract In part I of this paper we constructed a model representing quantum states by ensembles of dispersionsfree states. In part II we study further properties of this model and analyse their logical relations. It is also shown that the main results do not depend on the assumption that each self-adjoint operator corresponds to an observable.
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
The purpose of the paper is to study the foundations of the main axioms of Quantum Mechanics. From a general study of the mathematical properties of the models used in Physics to represent systems, we prove that the states of a system can be represented in a Hilbert space, that a self-adjoint operator is associated to any observable, that the result of a measure must be the eigen value of the operator and appear with the usual probability. Furthermore an equivalent of the Wigner's theorem holds, which leads to the Schr{ö}dinger equation. These results are based on well known mathematics, and do not involve any specific hypothesis in Physics. They validate and explain the methods currently used, which are made simpler and safer, and open new developments. In the third edition of this paper developments have been added about the estimation of physical anomalies.
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
In quantum mechanics, many observables for a quantum system are represented as self-adjoint linear operators (possibly unbounded) on a complex Hilbert space H\mathcal{H}H. In this paper, we introduce a binary relation ⊥\perp⊥ on S(H)S(\mathcal{H})S(H) of all self-adjoint linear operators (including unbounded) on H\mathcal{H}H. We obtain that A⊥BA\perp BA⊥B implies that A,BA,BA,B are affiliated with some abelian von Neumann algebra at the same time. Depending on ⊥\perp⊥,S(H), S(\mathcal{H}),S(H) can be constructed to be a generalized orthoalgebra. Then we present the orthoalgebra order ⪯\preceq⪯ and the traditional order ≤\leq≤ on S(H)S(\mathcal{H})S(H) and give some properties about these two orders. We prove that for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H), A⪯BA\preceq BA⪯B if and only if AAA has a value in Δ\DeltaΔ implies that BBB has a value in Δ\DeltaΔ for every Borel set Δ\DeltaΔ not containing 000. Moreover, we study the existence of the infimum A∧BA\wedge BA∧B and supremum A∨BA\vee BA∨B for A,B∈S(H)A,B\in S(\mathcal{H})A,B∈S(H) with respect to ⪯\preceq⪯. Finally, we prove that Q∧P=0Q\wedge P=0Q∧P=0 where QQQ and PPP are the significant quantum observables corresponding to the position and momentum.}