Vol. 19, No. 1, 1966

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Uniqueness and existence properties of bounded observables

Stanley P. Gudder

Vol. 19 (1966), No. 1, 81–93

Until recently observables have been nothing more than self-adjoint operators. However, due to axiomatic formulations of quantum mechanics, observables have now been placed in a more abstract setting. With the advent of this abstract concept comes the natural questions concerning uniqueness and existence. The uniqueness problem considered here seeks to answer the question: if two bounded observables have the same expectations in every state, are the observables equal? We say that an observable z is the sum of two bounded observables x and y if the expectation of z is the sum of the expectations of x and y for every state. The existence problem would pose the question: does the sum of two bounded observables exist? The author has found only partial answers to these questions. It is shown that the uniqueness property holds for simultaneous observables and certain classes of nonsimultaneous or complementary observables. The existence property holds for simultaneous observables, and a counterexample is given to show that this property does not hold in general. The last section of this paper considers systems in which the existence and uniqueness properties are known to hold.

Mathematical Subject Classification
Primary: 81.02
Received: 27 July 1965
Revised: 5 January 1966
Published: 1 October 1966
Stanley P. Gudder