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

Physica Scripta

Quantum and Classical Exact Solutions of the Time-Dependent Driven Generalized Harmonic Oscillator

Mai-Lin Liang, WU Hai-bo

Published Jan 1, 2003• 11 citations
Publisher

Abstract

Using the trial function method, exact wave functions of the time-dependent driven generalized harmonic oscillator (TDGHO) were derived. The wave functions are wave packets with the center moving according to the classical equation of motion. Correspondingly, the geometrical phase is divided into two parts with one part generated by the shape change of the wave packet and the other part by the motion of the wave packet center. The exact classical solution as well as the classical phase was obtained from a quantum description by the Heisenberg correspondence principle. Finally, the relationship between the quantum and classical phases was achieved.

DOI
10.1238/physica.regular.068a00041

Related papers

The exact evolution of the time-dependent driven damped harmonic oscillator with time-dependent mass and frequency

M Liang, B Yuan, Kai Zhong

Under a new quantization scheme, the exact wave functions of the time-dependent driven damped harmonic oscillator with time-dependent mass and frequency are obtained. The wave functions are shape-unchanging wave packet with the center moving along the classical trajectory. The total phase of the wave function is explicitly expressed as the sum of the dynamical phase and the geometrical phase. PACS Nos.: 03.65-w, 05.40-a

Canadian Journal of Physics•Dec 1, 2002•3 citations

Quantum and Classical Exact Solutions of Harmonic Oscillator in a Time-Dependent Magnetic Field

Mai-Lin Liang, Liyan Liu

In cylindrical coordinate, exact wave functions of the two-dimensional time-dependent harmonic oscillator in a time-dependent magnetic field are derived by using the trial function method. Meanwhile, the exact classical solution as well as the classical phase is obtained too. Through the Heisenberg correspondence principle, the quantum solution and the classical solution are connected together.

Communications in Theoretical Physics•Jun 1, 2005•2 citations

Quantum and Classical Exact Solutions of Harmonic Oscillator in a Time-Dependent Magnetic Field

LIANGMai-Lin, LIULi-Yan

In cylindrical coordinate, exact wave functions of the two-dimensional time-dependent harmonic oscillator in a time-dependent magnetic field are derived by using the trial function method. Meanwhile, the exact classical solution as well as the classicalphase is obtained too. Through the Heisenberg correspondence principle, the quantum solution and the classical solution are connected together.

理论物理通讯：英文版•Jan 1, 2005•0 citations

Exact quantum states of a general time-dependent quadratic system from classical action

Dae-Yup Song

A generalization of a driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered. The generalization, which gives a general quadratic Hamiltonian system, does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous solutions and one particular solution. The kernels are then used to find wave functions that satisfy the Schr\"odinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization, which gives a rather complicated quadratic Hamiltonian, is simply interpreted as applying unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.

arXiv:quant-ph/9810034•Physical Review A•Apr 1, 1999•40 citations

Exact wave functions and coherent states of a damped driven harmonic oscillator

H. G. Oh, H. R. Lee, Thomas F. George +1 more

For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.

Physical Review A•Jun 1, 1989•40 citations

Evolution of Gaussian wave packet and nonadiabatic geometrical phase for the time-dependent singular oscillator

Mustapha Maamache, Hacene Bekkar

We investigate the dynamics and geometric phases of a time-dependent singular oscillator. We construct certain Gaussian wave packet solutions of the corresponding Schrödinger equation, relate the latter with the classical equation of motion and explore the relationship between the associated quantum and phase angles. It is shown by a simple geometrical approach that the geometrical phase is connected with the classical nonadiabatic Hannay angle of the generalized harmonic oscillator. Our geometric approach is based on a rule for a 'natural transport' of the complex two-dimensional vector in the phase space and the results obtained are quite suggestive of similarities to the quantum mechanical two-state evolution.

arXiv:quant-ph/0204132•Journal of Physics A Mathematical and General•May 28, 2003•17 citations
Using the trial function method, exact wave functions of the time-dependent driven generalized harmonic oscillator (TDGHO) were derived. The wave functions are wave packets with the center moving according to the classical equation of motion. Correspondingly, the geometrical phase is divided into two parts with one part generated by the shape change of the wave packet and the other part by the motion of the wave packet center. The exact classical solution as well as the classical phase was obtained from a quantum description by the Heisenberg correspondence principle. Finally, the relationship between the quantum and classical phases was achieved.
A generalization of a driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered. The generalization, which gives a general quadratic Hamiltonian system, does not change the classical equation of motion. Based on the observation by Feynman and Hibbs, the propagators (kernels) of the systems are calculated from the classical action, in terms of solutions of the classical equation of motion: two homogeneous solutions and one particular solution. The kernels are then used to find wave functions that satisfy the Schr\"odinger equation. One of the wave functions is shown to be that of a Gaussian pure state. In every case considered, we prove that the kernel does not depend on the way of choosing the classical solutions, while the wave functions depend on the choice. The generalization, which gives a rather complicated quadratic Hamiltonian, is simply interpreted as applying unitary transformation to the driven harmonic oscillator system in the Hamiltonian formulation.
For a damped harmonic oscillator forced by a time-dependent field, the exact wave function is obtained by three different methods: (i) path integral, (ii) second quantization, and (iii) dynamical invariant. The explicit form of the dynamical invariant involves a solution to a corresponding auxiliary equation. The coherent states, defined as eigenstates of a new destruction operator, form a nonorthogonal, overcomplete set and correspond to the minimum uncertainty states. These coherent states give the exact classical motion of the damped driven harmonic oscillator.
We investigate the dynamics and geometric phases of a time-dependent singular oscillator. We construct certain Gaussian wave packet solutions of the corresponding Schrödinger equation, relate the latter with the classical equation of motion and explore the relationship between the associated quantum and phase angles. It is shown by a simple geometrical approach that the geometrical phase is connected with the classical nonadiabatic Hannay angle of the generalized harmonic oscillator. Our geometric approach is based on a rule for a 'natural transport' of the complex two-dimensional vector in the phase space and the results obtained are quite suggestive of similarities to the quantum mechanical two-state evolution.