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CRC Press eBooks

Proper and Improper Mixtures

Bernard d’Espagnat

Published May 4, 2018• 0 citations
Publisher

Abstract

Observable dynamical quantities are described by Hermitean operators, and in the early days of quantum mechanics the correspondence was believed to be one-to-one. It was assumed that (i) every observable can be related to some Hermitean operator and (ii) every Hermitean operator corresponds to one observable. The arguments concerning the difference between proper and improper mixtures are conceived of for the benefit of the physicists with a "realist" turn of mind who, prima facie, would tend to consider that, quite on the contrary, the statement in question is meaningful and true. Two ensembles are said to be different with respect to some observable A if the statistical frequencies of the prospective measurement outcomes on observable A are different on the two ensembles. Also, two ensembles are said to be different if there exists at least one observable A with respect to which they are different.

DOI
10.1201/9780429503283-7

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Observable dynamical quantities are described by Hermitean operators, and in the early days of quantum mechanics the correspondence was believed to be one-to-one. It was assumed that (i) every observable can be related to some Hermitean operator and (ii) every Hermitean operator corresponds to one observable. The arguments concerning the difference between proper and improper mixtures are conceived of for the benefit of the physicists with a "realist" turn of mind who, prima facie, would tend to consider that, quite on the contrary, the statement in question is meaningful and true. Two ensembles are said to be different with respect to some observable A if the statistical frequencies of the prospective measurement outcomes on observable A are different on the two ensembles. Also, two ensembles are said to be different if there exists at least one observable A with respect to which they are different.
The relevant mathematical theory was developed with regard to mixtures and their description. That theory could of course be expanded to a considerable degree. However, an abundant literature exists already on that subject. As a first, elementary way of clarifying them this chapter introduces an unconventional but useful distinction between improper mixtures and proper mixtures. It then investigates the question of the homogeneity or heterogeneity of ensembles, as well as two related questions, which bear on the measurability of the density matrix and on the relation between Hermitean operators and observables, respectively. More precisely, the assumptions were made that every observable can be related to some Hermitean operator, and every Hermitean operator corresponds to one observable. The chapter includes a discussion of the meaning that can be given to this sentence: "Such and such an observable has such and such a value on a system.".
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.
The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state ). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question. A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme. It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.
All quantum mixtures are what d'Espagnat has termed "improper." His "proper" mixture cannot be created -- if welcher weg, or distinguishing, information exists, an improper mixture results, while in the absence of such information, the resulting "mixture" is a pure state. D'Espagnat has claimed that an interpretation of the improper mixture in terms of subensembles leads to logical inconsistency; this claim is shown to be incorrect, as d'Espagnat's argument fails to account for the indistinguishability of the pure-state subensembles.