Summary
In this paper we examine, in the framework of Hilbert-space quantum mechanics, the mathematical problem of the existence of the mean value and variance of unbounded observables. We consider both pure states and quantum mixtures, with particular care for the problems that arise in the latter case.
Riassunto
In questo lavoro si esamina, nell’ambito della formulazione della meccanica quantistica in spazi di Hilbert, il problema matematico dell’esistenza del valor medio e della varianza di osservabili non limitate. Si considera sia il caso di stati puri che di misture, con particolare cura ai problemi che sorgono in quest’ultimo caso.
Резюме
В этой работе в квантовомеханическом формализме в Гильбертовом пространстве исследуется математическая проблема существования среднего значения и изменения неограниченных наблюдаемых. Мы рассматриваем чистые состояния и кванторые смеси. Особое внимание уделяется проблемам, возникающим в случае квантовых смесей.
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References
M. Reed andB. Simon:Methods of Modern Mathematical Physics, Vol.1 (New York, N.Y., 1972).
E. G. Beltrametti andG. Cassinelli:The Logic of Quantum Mechanics (Reading, Mass., 1981).
V. S. Varadarajan:Geometry of Quantum Theory, Vol.1 (Princeton, N.J., 1970),
K. Krans andJ. Schoter:Int. J. Theor. Phys.,8, 431 (1973).
J. Dieudonné:Treatise on Analysis, Vol.2 (New York, N.Y., London, 1970), p. 377.
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In this paper we use freely some simple facts and results from the theory of operators in Hilbert spaces. We refer for them to (1).
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Cassinelli, G., Olivieri, G. The statistics of unbounded observables in Hilbert-space quantum mechanics. Nuov Cim B 84, 43–52 (1984). https://doi.org/10.1007/BF02721647
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DOI: https://doi.org/10.1007/BF02721647