Theoretical and Mathematical Physics, 192(1): 1080–1096 (2017)
SCALE TRANSFORMATIONS IN PHASE SPACE AND STRETCHED
STATES OF A HARMONIC OSCILLATOR
V. A. Andreev,
D. M. Davidovi´c,
L. D. Davidovi´c,
Milena D. Davidovi´c,
We consider scale transformations (q, p) → (λq, λp) 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 arbi-
trary 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–Schr¨odinger uncertainty relations for
Keywords: phase space, Husimi function, scale transformation, harmonic oscillator, stretched state,
We propose a new method for constructing the quantum states arising in certain physical processes.
This method is based on using quasiprobability distributions to describe quantum states. These distribu-
tions are deﬁned on the phase space, and physical processes are associated with some transformations of this
space. Phase-space transformations induce transformations of the functions deﬁned on them. Determining
the physical states corresponding to the transformed quasiprobability distributions, we can ﬁnd the result
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia, e-mail: firstname.lastname@example.org.
Vinˇca Institute for Nuclear Sciences, University of Belgrade, Belgrade, Serbia, e-mail: email@example.com,
Institute of Physics, University of Belgrade, Belgrade, Serbia, e-mail: firstname.lastname@example.org.
Faculty of Civil Engineering, University of Belgrade, Belgrade, Serbia, e-mail: email@example.com.
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. Davidovi´c is supported in part by the Serbian Ministry of Education, Science, and
Technological Development (Project No. OI 171031).
The research of Milena D. Davidovi´c and Miloˇs D. Davidovi´c is supported in part by the Serbian Ministry of
Education, Science, and Technological Development (Project No. OI 171028).
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 192, No. 1, pp. 164–184, July, 2017. Original
article submitted May 18, 2016; revised July 11, 2016.
2017 Pleiades Publishing, Ltd.