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 TMF, 2017, Volume 192, Number 1, Pages 164–184 (Mi tmf9232)

Scale transformations in phase space and stretched states of a harmonic oscillator

V. A. Andreeva, D. M. Davidovićb, L. D. Davidovićc, Milena D. Davidovićd, Miloš D. Davidovićb

a Lebedev Physical Institute, Russian Academy of Sciences, Moscow, Russia

Abstract: We consider scale transformations $(q,p)\to(\lambda q,\lambda p)$ in phase space. They induce transformations of the Husimi functions $H(q,p)$ defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of $n$-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-of-states operator. We give the form of the Heisenberg and Robertson–Schrödinger uncertainty relations for stretched states.

Keywords: phase space, Husimi function, scale transformation, harmonic oscillator, stretched state, uncertainty relation.

 Funding Agency Grant Number Ministry of Education, Science and Technical Development of Serbia OI 171031OI 171028 This research is performed within the scientific collaboration between the Russian Academy of Sciences and the Serbian Academy of Sciences and Arts on the subject "Fundamental study in the field of quantum information theory and quantum calculations and their application" and is supported by the Serbian Ministry of Education, Science, and Technological Development. The research of L. D. Davidović is supported in part by the Serbian Ministry of Education, Science, and Technological Development (Project No. OI 171031). The research of Milena D. Davidović and Miloš D. Davidović is supported in part by the Serbian Ministry of Education, Science, and Technological Development (Project No. OI 171028).

DOI: https://doi.org/10.4213/tmf9232

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English version:
Theoretical and Mathematical Physics, 2017, 192:1, 1080–1096

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Revised: 11.07.2016

Citation: V. A. Andreev, D. M. Davidović, L. D. Davidović, Milena D. Davidović, Miloš D. Davidović, “Scale transformations in phase space and stretched states of a harmonic oscillator”, TMF, 192:1 (2017), 164–184; Theoret. and Math. Phys., 192:1 (2017), 1080–1096

Citation in format AMSBIB
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\by V.~A.~Andreev, D.~M.~Davidovi\'c, L.~D.~Davidovi\'c, Milena~D.~Davidovi\'c, Milo{\v s}~D.~Davidovi\'c
\paper Scale transformations in phase space and stretched states of a~harmonic oscillator
\jour TMF
\yr 2017
\vol 192
\issue 1
\pages 164--184
\mathnet{http://mi.mathnet.ru/tmf9232}
\crossref{https://doi.org/10.4213/tmf9232}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3670252}
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\transl
\jour Theoret. and Math. Phys.
\yr 2017
\vol 192
\issue 1
\pages 1080--1096
\crossref{https://doi.org/10.1134/S0040577917070091}
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