The orthogonality postulate in axiomatic quantum mechanics |
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Authors: | M J Maczyński |
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Institution: | (1) Institute of Mathematics, Technical University, 00-661 Warsaw, Poland |
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Abstract: | Letp(A,,E) be the probability that a measurement of an observableA for the system in a state will lead to a value in a Borel setE. An experimental function is a function f from the set of all statesI into 0,1] for which there are an observableA and a Borel setE such thatf()=p(A, , E) for all I. A sequencef
1,f
2,... of experimental functions is said to be orthogonal if there is an experimental functiong such thatg+f
1+f
2+...=1, and it is said to be pairwise orthogonal iff
i+f
j 1 forij. It is shown that if we assume both notions to be equivalent then the setL of all experimental functions is an orthocomplemented partially ordered set with respect to the natural order of real functions with the complementationf=1–f, each observableA can be identified with anL-valued measure
A, each state can be identified with a probability measurem
onL and we havep(A,,E)=m
oA(E). Thus we obtain the abstract setting of axiomatic quantum mechanics as a consequence of a single postulate. |
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Keywords: | |
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