aqora

Position and momentum observables on R and on R3

Claudio Carmeli, Teiko Heinosaari, Alessandro Toigo

We characterize all position and momentum observables on ↛ and on R3. We study some of their operational properties and discuss their covariant joint observables.

arXiv:quant-ph/0403108Journal of Mathematical PhysicsMay 14, 200422 citations

POSITION AND MOMENTUM OBSERVABLES ON R AND ON R 3

Claudio Carmeli, Teiko Heinonen, Alessandro Toigo

Abstract. We characterize all position and momentum observables on R and on R 3. We study some of their operational properties and discuss their covariant joint observables. 1.

Jan 1, 200419 citations

Structures in the Set of Quantum Observables: Coexistence, Unsharpness, Approximation

Paul Busch, Garrett Birkhoff, John Von Neumann

refer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will berefer to the joint measurement problem for noncommuting quantum observables: “One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations).” The object of the present paper is to consider what approximate solution one may hope to find to the joint measurement problem. Our main conclusion, based on a rigorous notion of unsharp observable, is that one can reasonably expect to find approximate joint measurements for any pair of noncommuting sharp observables, provided a trade-off in the sense of a Heisenberg uncertainty relation for measurement inaccuracies is taken into account. In addition, it will be seen that as a consequence of the noncom-mutativity of the given observables, the approximating observables must have degrees of unsharpness that obey yet another trade-off relation in the spirit of Heisenberg’s principle. These connections will be

Dec 26, 20140 citations

On joint lower bounds of position and momentum observables in quantum mechanics

Paul Busch

A new theorem on joint lower bounds of unsharp position and momentum observables is formulated and proven. It is developed as a straightforward formal generalization of a similar theorem for the spectral projections of the usual observables which has been employed to express the strict incompatibility of position and momentum. The relevance of our theorem to the problem of joint measurements in quantum mechanics has been discussed elsewhere.

Journal of Mathematical PhysicsJun 1, 198419 citations

On the localization of momentum observables in quantum mechanics

K. Kong Wan, Tony Jackson, I. H. McKenna

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BJun 1, 198411 citations

The commutation relations

Keith Hannabuss

Abstract Our results so far, in particular Section 3.4, strongly suggest that although it is possible to find formalisms in which either momentum or position are easy to handle, one cannot simultaneously manage both. The underlying reason for this is that the operators P and X do not commute.

Mar 20, 19971 citation

On the coexistence of position and momentum observables

Abstract

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