Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system

V. V. Belov, M. F. Kondratieva, A. Yu Trifonov

Research output: Contribution to journalArticle

Abstract

Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton-Ehrenfest M-systems (M ≤ 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schrödinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton-Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.

Original languageEnglish
Article number015
Pages (from-to)10821-10847
Number of pages27
JournalJournal of Physics A: Mathematical and General
Volume39
Issue number34
DOIs
Publication statusPublished - 25 Aug 2006

Fingerprint

trajectories
linear equations
differential equations
derivation
Trajectory
Semiclassical Approximation
Quantum State
approximation
System of Ordinary Differential Equations
Superposition
Linear equation
Correspondence
Valid
kernel
Series
Class
Family

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system. / Belov, V. V.; Kondratieva, M. F.; Trifonov, A. Yu.

In: Journal of Physics A: Mathematical and General, Vol. 39, No. 34, 015, 25.08.2006, p. 10821-10847.

Research output: Contribution to journalArticle

@article{a6b8fbddf67f410ea774b3edd9793a2c,
title = "Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system",
abstract = "Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton-Ehrenfest M-systems (M ≤ 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schr{\"o}dinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton-Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.",
author = "Belov, {V. V.} and Kondratieva, {M. F.} and Trifonov, {A. Yu}",
year = "2006",
month = "8",
day = "25",
doi = "10.1088/0305-4470/39/34/015",
language = "English",
volume = "39",
pages = "10821--10847",
journal = "Journal of Physics A: Mathematical and General",
issn = "0305-4470",
publisher = "IOP Publishing Ltd.",
number = "34",

}

TY - JOUR

T1 - Semiclassical spectrum for a Hartree-type equation corresponding to a rest point of the Hamilton-Ehrenfest system

AU - Belov, V. V.

AU - Kondratieva, M. F.

AU - Trifonov, A. Yu

PY - 2006/8/25

Y1 - 2006/8/25

N2 - Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton-Ehrenfest M-systems (M ≤ 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schrödinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton-Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.

AB - Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate a quantum-mechanical state as → 0. This idea leads to a family of systems of ordinary differential equations, called Hamilton-Ehrenfest M-systems (M ≤ 0, 1, 2, ...). As noted in the authors' previous works, every M-system is formally equivalent to the semiclassical approximation of order M for the linear Schrödinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Hamilton-Ehrenfest systems, without solving the quantum equation: the semiclassical spectral series are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.

UR - http://www.scopus.com/inward/record.url?scp=33747175767&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747175767&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/39/34/015

DO - 10.1088/0305-4470/39/34/015

M3 - Article

VL - 39

SP - 10821

EP - 10847

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

SN - 0305-4470

IS - 34

M1 - 015

ER -