On various joint measurements of position and momentum observables in quantum theory

Abstract

In this paper the problem of simultaneous measurability of complementary physical quantities, and the relevance of the uncertainty relations to this question, is analyzed. Formalizing carefully the crucial notations involved in the problem it is shown that the well-known apparently contradictory viewpoints on the subject matter are all true in pointing to a different aspect of one and the same fundamental fact expressed here as complementarity.1634

On the coexistence of position and momentum observables

“…[1]). According to a common view sharp position and momentum observables are complementary quantities and therefore are not jointly measurable.…”

Prospects of ultra-relativistic heavy-ion collisions

“…, which is again excluded by (13). It follows that I cannot be repeatable, and we have the following result.…”

“…The above considerations also show that for unsharp observables condition (14) always implies (13), but the reverse implication need not hold. We take condition (14) as the formal definition of the complementarity of observables E 1 and E 2 .…”

“…If these effects are projection operators, then E 1 (X)ϕ = ϕ and E 2 (Y )ϕ = ϕ holds exactly when ϕ ∈ E 1 (X)(H) ∩ E 2 (Y )(H). Treating similarly the two other cases excluded by (13), [i.e., p E 1 T (X) = 1 and p E 2 T (Y ) = 0, or p E 1 T (X) = 0 and p E 2 T (Y ) = 1] one observes, first of all, that in the case of sharp observables the probabilistic complementarity of a given pair of observables is equivalent to the disjointness of their spectral projections:…”

“…It would allow one to construct a measurement (state transformer) which provides some probabilistic information on both E 1 (X) and E 2 (Y ). 13 We say that two state transformers are mutually exclusive if no such test exists, that is, if…”