Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Correlation functions of the nonstationary quantum singular oscillator

S. M. Chumakov, V. V. Dodonov, Vladimir I. Man’ko

Published Nov 11, 1986• 23 citations
Publisher

Abstract

The exact density matrix elements and the correlation functions of the one-dimensional non-stationary singular oscillator (the potential U(x,t)= omega 2 (t)x 2 /2+g/x 2 ) excited from the thermodynamic equilibrium states are obtained. Generating functions for rho nm and also for transition probabilities are constructed.
DOI
10.1088/0305-4470/19/16/021

Related papers

Nonstationary Deformed Singular Oscillator: Quantum Invariants and the Factorization Method

Kevin Zelaya

Abstract New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a nonstationary quantum invariant can be constructed for the singular oscillator. Such a quantum invariant depends on coefficients related to solutions of the Ermakov equation, where the latter guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant rather than to the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators that become the constants of motion for the new time-dependent Hamiltonians. At the appropriate limit, the initial quantum invariant reproduces the stationary singular oscillator Hamiltonian. Some families of stationary potentials already reported by other authors are also recovered as particular cases. A striking feature of the method is that the singular barrier of the potential can be managed to vanish, which leads to non-singular time-dependent potentials.

arXiv:2001.06764•Journal of Physics Conference Series•Apr 1, 2020•5 citations

Exact solutions for the general nonstationary oscillator with a singular perturbation

D. A. Trifonov

Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrödinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU (1,1) group-related coherent states for the SO minimize the Robertson and Schrödinger uncertainty relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.

arXiv:quant-ph/9811081•Journal of Physics A Mathematical and General•Jan 1, 1999•20 citations

Quantum transitions between highly excited atomic levels induced by external time-dependent forces

Леонид Петрович Пресняков, A. M. Urnov

A model of an excited quantum system is considered. An exact solution is obtained for a non-stationary problem in which the interaction potential is an arbitrary function of time and coordinates. The obtained solution reduces in particular cases to well-known results for transition probabilities between highly excited levels of a quantum oscillator.

Journal of Physics B Atomic and Molecular Physics•Oct 1, 1970•24 citations

Exactly Solvable Models of Stochastic Quantum Mechanics within the Framework of Langevin-Schroedinger Type Equations

Ashot S. Gevorkyan

A closed, uncountably infinite dimensional system of “quantum object + thermostat” is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrödinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in potential determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial differential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase transitions of second kind may occur in the expressions for microscopic quantum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscillator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermodynamics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.

Kluwer Academic Publishers eBooks•Feb 4, 2006•0 citations

Feynman method for disentangling operators and a singular oscillator

Vladimir S Popov, M. A. Trusov

The problem of the evolution of a singular quantum oscillator with a frequency exhibiting an arbitrary time dependence has been solved. The probabilities w mn of transitions in the oscillator spectrum and generating functions have been calculated, and the sum rules for w mn have been derived. The calculations are based on the Feynman disentangling method and the theory of representations of the SU(1, 1) group.

Journal of Experimental and Theoretical Physics•Feb 1, 2012•1 citation

Statistical mechanics of quantum one-dimensional damped harmonic oscillator

E. N. M. Borges, O. N. Borges, L. A. Amarante Ribeiro

We calculate the thermal correlation functions of the one-dimensional damped harmonic oscillator in contact with a reservoir, in an exact form by applying Green's function method. In this way the thermal fluctuations are incorporated in the Caldirola–Kanai Hamiltonian.

Canadian Journal of Physics•May 1, 1985•6 citations
Abstract New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a nonstationary quantum invariant can be constructed for the singular oscillator. Such a quantum invariant depends on coefficients related to solutions of the Ermakov equation, where the latter guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant rather than to the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators that become the constants of motion for the new time-dependent Hamiltonians. At the appropriate limit, the initial quantum invariant reproduces the stationary singular oscillator Hamiltonian. Some families of stationary potentials already reported by other authors are also recovered as particular cases. A striking feature of the method is that the singular barrier of the potential can be managed to vanish, which leads to non-singular time-dependent potentials.
Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrödinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU (1,1) group-related coherent states for the SO minimize the Robertson and Schrödinger uncertainty relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.
A closed, uncountably infinite dimensional system of “quantum object + thermostat” is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrödinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in potential determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial differential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase transitions of second kind may occur in the expressions for microscopic quantum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscillator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermodynamics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.