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

A class of exactly solved time-dependent quantum harmonic oscillators

Sang Pyo Kim

Published Jun 7, 1994• 37 citations
Publisher

Abstract

We consider a class of time-dependent harmonic oscillators, H(t)=p 2 /2mt alpha + m omega 2 t b q 2 /2, whose mass and frequency vary as non-negative powers of time. Classically they describe damping oscillators slowly decaying as negative powers of time. Using the connection between classical and quantum harmonic oscillators we find analytically the Lewis-Riesenfeld invariants, obtain the exact quantum states, and compare these with the Caldirola-Kanai oscillator.

DOI
10.1088/0305-4470/27/11/039

Related papers

Time-dependent coupled harmonic oscillators: Classical and quantum solutions

D. X. Macedo, I. Guedes

In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.

International Journal of Modern Physics E•Aug 25, 2014•4 citations

Wave functions of a time-dependent harmonic oscillator with and without a singular perturbation

I. A. Pedrosa, G. P. Serra, I. Guedes

We use the Lewis and Riesenfeld invariant method to obtain the exact Schr\"odinger wave functions for a time-dependent harmonic oscillator with and without an inverse quadratic potential. As a particular case we also obtain the wave functions for the Caldirola-Kanai oscillator.

Physical Review A•Nov 1, 1997•120 citations

Harmonic oscillators coupled time-dependent

Diego Ximenes Macedo

In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.

•Jan 1, 2012•0 citations

Exact wave functions of a harmonic oscillator with time-dependent mass and frequency

I. A. Pedrosa

We use the Lewis and Riesenfeld invariant method [J. Math. Phys. 10, 1458 (1969)] to obtain the exact Schr\"odinger wave functions for a harmonic oscillator with time-dependent mass and frequency. Exact coherent states for such system are also constructed.

Physical Review A•Apr 1, 1997•189 citations

Osciladores log-periódicos e tipo Caldirola-Kanai

Vagner Henrique Loiola Bessa

In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ and (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. In all three cases $omega(t)=omega_0frac{t_0}{t}$. For class (b) we study the Caldirola-Kanai oscillator (IV)where $omega(t)=omega_0$ and $m(t)=m_0 ext{exp}ajust{gamma t}$ and the oscillator with $omega(t)=omega_0$ and $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, for $alpha=2$ (V) and $alpha=4$ (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of $q(t)$, $p(t)$ as well as the phase diagram $q(t)$ vs $p(t)$. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.

•Jan 1, 2012•0 citations

Time-dependent diffeomorphisms as quantum canonical transformations and the time-dependent harmonic oscillator

Ali Mostafazadeh

Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this result leads to the reduction of the Schrödinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.

arXiv:quant-ph/9807002•Journal of Physics A Mathematical and General•Jul 31, 1998•37 citations
We consider a class of time-dependent harmonic oscillators, H(t)=p 2 /2mt alpha + m omega 2 t b q 2 /2, whose mass and frequency vary as non-negative powers of time. Classically they describe damping oscillators slowly decaying as negative powers of time. Using the connection between classical and quantum harmonic oscillators we find analytically the Lewis-Riesenfeld invariants, obtain the exact quantum states, and compare these with the Caldirola-Kanai oscillator.
In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.
In this work we present the classical and quantum solutions of time-dependent coupled harmonic oscillators. In these systems the masses, frequencies and coupling parameter (k) are functions of time. Four systems are investigated. To obtain the classical solutions we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. We find the analytical expression for position and velocity of each oscillator of the systems. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld invariant method. The wave functions obtained are written in terms of a c-number quantity () which is solution of the Milne-Pinney equation. For each system we solve the respective Milne-Pinney equation and discuss how the quantum fluctuations and the uncertainty product evolve with time.
In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) m(t)=m0fractt0m(t)=m_0frac{t}{t_0}m(t)=m0​fractt0​, (II) m(t)=m0m(t)=m_0m(t)=m0​ and (III) m(t)=m0ajustfractt02m(t)=m_0ajust{frac{t}{t_0}}^2m(t)=m0​ajustfractt0​2. In all three cases omega(t)=omega0fract0tomega(t)=omega_0frac{t_0}{t}omega(t)=omega0​fract0​t. For class (b) we study the Caldirola-Kanai oscillator (IV)where omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0extexpajustgammatm(t)=m_0 ext{exp}ajust{gamma t}m(t)=m0​extexpajustgammat and the oscillator with omega(t)=omega0omega(t)=omega_0omega(t)=omega0​ and m(t)=m0ajust1+fractt0alpham(t)=m_0ajust{1+frac{t}{t_0}}^alpham(t)=m0​ajust1+fractt0​alpha, for alpha=2alpha=2alpha=2 (V) and alpha=4alpha=4alpha=4 (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of q(t)q(t)q(t), p(t)p(t)p(t) as well as the phase diagram q(t)q(t)q(t) vs p(t)p(t)p(t). To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.
Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this result leads to the reduction of the Schrödinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.