Abstract:
We show that the generalized Schrödinger uncertainty relations have the meaning of fundamental restrictions on the characteristics of the state space in any theory of a probabilistic type. Both quantum mechanics and the theory of Brownian motion for arbitrary time intervals are among these theories. We compare the “position-momentum” uncertainty relation in the theory of Brownian motion and a similar uncertainty relation for a microparticle in the Gaussian wave-packet state. We establish that the two theories are conceptually similar despite a serious distinction between their mathematical apparatus. This similarity manifests itself in alternative regimes such that small times in one theory correspond to large times in the other theory, and vice versa. In each of the theories, an uncontrollable effect of either quantum or thermal type is of crucial importance.
Citation:
A. D. Sukhanov, “The Generalized “Position–Momentum” Uncertainty Relation in Quantum Mechanics and in the Theory of Brownian Motion”, TMF, 139:1 (2004), 129–144; Theoret. and Math. Phys., 139:1 (2004), 557–570
\Bibitem{Suk04}
\by A.~D.~Sukhanov
\paper The Generalized ``Position--Momentum'' Uncertainty Relation in Quantum Mechanics and in the Theory of Brownian Motion
\jour TMF
\yr 2004
\vol 139
\issue 1
\pages 129--144
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\crossref{https://doi.org/10.4213/tmf47}
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\zmath{https://zbmath.org/?q=an:1178.81013}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2004TMP...139..557S}
\transl
\jour Theoret. and Math. Phys.
\yr 2004
\vol 139
\issue 1
\pages 557--570
\crossref{https://doi.org/10.1023/B:TAMP.0000022747.19258.45}
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Linking options:
https://www.mathnet.ru/eng/tmf47
https://doi.org/10.4213/tmf47
https://www.mathnet.ru/eng/tmf/v139/i1/p129
This publication is cited in the following 6 articles:
Rudoy Yu.G., Chekmareva I O., “Geometrical Aspects of the Equilibrium Statistical Thermodynamics”, J. Mech. Contin. Math. Sci., 2019, no. 1, 537–547
Golubjeva O., Sidorov S., “Quantum-thermal self-diffusion as a hydrodynamic mechanism for the fluctuations? relaxation”, Can. J. Phys., 94:3 (2016), 310–319
Kazakov K.A., Nikitin V.V., “On the infrared singularity of the effective electromagnetic field of free electrons”, J. Phys. A: Math. Theor., 44:31 (2011), 315402
O. N. Golubjeva, A. D. Sukhanov, “Elements of nonequilibrium (ћ, k) dynamics at zero and finite temperatures”, Phys. Part. Nuclei Lett., 8:1 (2011), 1
A. D. Sukhanov, “Schrödinger uncertainty relation for a quantum oscillator in a thermostat”, Theoret. and Math. Phys., 148:2 (2006), 1123–1134
Sukhanov AD, “Einstein's statistical thermodynamic ideas in a modern physical picture of the world - (To the 100th anniversary of Einstein's early works)”, Physics of Particles and Nuclei, 36:6 (2005), 667–698
Citation in format AMSBIB
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