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

International Journal of Quantum Chemistry

On the linearly driven parametric oscillator

Francisco Marcelo Fernández, Eduardo Alberto Castro

Published Nov 1, 1987• 3 citations
Publisher

Abstract

Abstract The time‐dependent Schrouml;dinger equation for the oscillator H = α(t)p2 + β(t)x2 + γ(t)x is exactly solved. The time evolution operator is easily obtained by means of an operator algebra and the quantum‐mechanical equations of motion. The problem is reduced to solving the classical equations of motion. The method is shown to apply to multidimensional oscillators.

DOI
10.1002/qua.560320505

Related papers

Exponential time-evolution operator for the time-dependent harmonic oscillator

Francisco Marcelo Fernández

The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.

Journal of Mathematical Physics•Dec 1, 1987•16 citations

Propagator of the general driven time-dependent oscillator

Chi‐Fai Lo

In this paper we generalize Wang's approach [J. Phys. A 20, 5041 (1987)] and investigate the algebraic structure of the Schr\"odinger equation associated with a general driven time-dependent oscillator. Using the Lie-algebraic technique we obtain an exact form of the time-evolution operator which, in turn, enables us to derive the propagator of the system readily. Since the propagator is for the most general time-dependent oscillator, results for any special case can be easily deduced from it. These results will be useful for future studies in quantum optics as well as in atomic and molecular physics.

Physical Review A•Jan 1, 1993•39 citations

Some remarks on analytical solutions for a damped quantum parametric oscillator

Dieter Schuch

Abstract The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined by solving a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.

Journal of Physics Conference Series•Sep 1, 2019•2 citations

Operator Methods for the Time Evolution of Wave Functions

K. P. Michnicki

A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.

arXiv:0811.3431•Nov 20, 2008•0 citations

Bilinear oscillators

Francisco Marcelo Fernández, Eduardo A. Castro

The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.

CRC Press eBooks•Jan 16, 2020•0 citations

Exact quantum solution for the general time-dependent linear potential and related problems

Mai-Lin Liang, Sun Yu-Jing, (1)天津大学理学院物理系,天津　300072; (2)天津大学理学院物理系,天津　300072；南开大学天津大学刘徽应用数学中心,天津　300071

For the general time-dependent linear potential, the exact and complete solution of the Schr?dinger equation was obtained by assuming a form of wave function. Meanwhile, it was pointed out that the matrices φ(t)〉和ψ(t)〉 defined by the two wave functions 〈φ(t)xψ(t)〉和〈φ(t)pψ(t)〉 satisfy classical equations of motion. According to the ensemble theory of quantum mechanics, such classical equations describe the motion of fluid. A further research shows that similar conclusions apply to any linear system.

Acta Physica Sinica•Jan 1, 2004•1 citation
Abstract The time‐dependent Schrouml;dinger equation for the oscillator H = α(t)p2 + β(t)x2 + γ(t)x is exactly solved. The time evolution operator is easily obtained by means of an operator algebra and the quantum‐mechanical equations of motion. The problem is reduced to solving the classical equations of motion. The method is shown to apply to multidimensional oscillators.
In this paper we generalize Wang's approach [J. Phys. A 20, 5041 (1987)] and investigate the algebraic structure of the Schr\"odinger equation associated with a general driven time-dependent oscillator. Using the Lie-algebraic technique we obtain an exact form of the time-evolution operator which, in turn, enables us to derive the propagator of the system readily. Since the propagator is for the most general time-dependent oscillator, results for any special case can be easily deduced from it. These results will be useful for future studies in quantum optics as well as in atomic and molecular physics.
Abstract The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined by solving a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied by linearizing the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. The effect of a dissipative, linear velocity dependent friction force on these systems is discussed.
A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.
The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.
For the general time-dependent linear potential, the exact and complete solution of the Schr?dinger equation was obtained by assuming a form of wave function. Meanwhile, it was pointed out that the matrices φ(t)〉和ψ(t)〉 defined by the two wave functions 〈φ(t)xψ(t)〉和〈φ(t)pψ(t)〉 satisfy classical equations of motion. According to the ensemble theory of quantum mechanics, such classical equations describe the motion of fluid. A further research shows that similar conclusions apply to any linear system.