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Physical Review

Measurement of Quantum Mechanical Operators

Huzihiro Araki, Mutsuo M. Yanase

Published Oct 15, 1960• 353 citations
Publisher

Abstract

The limitation on the measurement of an operator imposed by the presence of a conservation law is studied. It is shown that an operator which does not commute with a conserved (additive) quantity cannot be measured exactly (in the sense of von Neumann). It is also shown for a simple case that an approximate measurement of such an operator is possible to any desired accuracy.

DOI
10.1103/physrev.120.622

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The limitation on the measurement of an operator imposed by the presence of a conservation law is studied. It is shown that an operator which does not commute with a conserved (additive) quantity cannot be measured exactly (in the sense of von Neumann). It is also shown for a simple case that an approximate measurement of such an operator is possible to any desired accuracy.
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The demonstrations of Wigner and others, that observables which do not commute with additive conserved quantities cannot be measured precisely, are reexamined. A proposed new formulation of the claim is shown to be valid for observables with a continuous spectrum whenever the conserved quantities are bounded. However, a countermodel is constructed, and it suggests that the position can be measured as precisely as the momentum even though the measuring interaction conserves the total linear momentum.
A new classification of quantum mechanical quantities according to their measurability is discussed. We have three different types of quantities: first, exactly measurable quantities which commute with all the conserved quantities; second, approximately measurable quantities which do not commute with all additive conserved quantities; and finally nonmeasurable quantities which are conjugate to the absolute invariants. The epistemological implications of the measurement process are discussed and it is pointed out that the quantum mechanical description of this process does not furnish an adequate explanation of the acquisition of knowledge.