ar X iv : q ua nt - p h / 95 07 00 4 v 1 7 Ju l 9 5 Generalized uncertainty relations : Theory , examples , and Lorentz invariance ∗ Samuel
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چکیده
The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter—e.g., elapsed time—may be determined via arbitrary data analysis of arbitrary measurements on N identically prepared quantum systems. The limits are expressed as generalized Mandelstam-Tamm uncertainty relations, which involve the operator that generates displacements of the parameter—e.g., the Hamiltonian operator in the case of elapsed time. This approach avoids entirely the problem of associating a Hermitian operator with the parameter. We illustrate the general formalism, first, with nonrelativistic uncertainty relations for spatial displacement and momentum, harmonic-oscillator phase and number of quanta, and time and energy and, second, with Lorentzinvariant uncertainty relations involving the displacement and Lorentz-rotation parameters of the Poincaré group. ∗Supported in part by the Office of Naval Research (Grant No. N00014-93-1-0116) †Supported by a Feinberg Fellowship at the Weizmann Institute of Science. Present address: Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel ‡Permanent address: Department of Physics, University of Queensland, St. Lucia 4072, Australia
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تاریخ انتشار 2002