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 the Korean Physical Society

Exact Quantum Theory of the Harmonic Oscillator with the Classical Solution in the Form of Mathieu Functions

Chung‐In Um, Jeong Ryeol Choi, Kyu Hwang Yeon, Thomas F. George

Published Jun 1, 2002• 13 citations
OpenAlex

Related papers

Quantum Theory of Harmonic Oscillator

Carlo Jacoboni

Springer series in solid-state sciences•Jan 1, 2010•0 citations

Geometric phases in the simple harmonic and perturbative Mathieu's oscillator systems

JeongHyeong Park, Dae-Yup Song

Geometric phases of simple harmonic oscillator system are studied. Complete sets of "eigenfunctions" are constructed, which depend on the way of choosing classical solutions. For an eigenfunction, two different motions of the probability distribution function (pulsation of the width and oscillation of the center) contribute to the geometric phase which can be given in terms of the parameters of classical solutions. The geometric phase for a general wave function is also given. If a wave function has a parity under the inversion of space coordinate, then the geometric phase can be defined under the evolution of half of the period of classical motions. For the driven case, geometric phases are given in terms of Fourier coefficients of the external force. The oscillator systems whose classical equation of motion is Mathieu's equation are perturbatively studied, and the first term of nonvanishing geometric phase is calculated.

arXiv:quant-ph/9911030•Nov 8, 1999•0 citations

Exact solution of a time-dependent quantal harmonic oscillator with damping and a perturbative force

Dinkar C. Khandekar, S. V. Lawande

The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.

Journal of Mathematical Physics•Sep 1, 1979•85 citations

Exact wave functions for generalized harmonic oscillators

Nathan A. Lanfear, Raquel M. López, Sergeĭ K. Suslov

Journal of Russian Laser Research•Jul 1, 2011•39 citations

Quantum Mechanics of the Harmonic Oscillator

WORLD SCIENTIFIC eBooks•Sep 1, 2012•1 citation

Quantum Parametric Resonance

Weigert, S

The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically unstable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.

•Jun 25, 2001•10 citations
Geometric phases of simple harmonic oscillator system are studied. Complete sets of "eigenfunctions" are constructed, which depend on the way of choosing classical solutions. For an eigenfunction, two different motions of the probability distribution function (pulsation of the width and oscillation of the center) contribute to the geometric phase which can be given in terms of the parameters of classical solutions. The geometric phase for a general wave function is also given. If a wave function has a parity under the inversion of space coordinate, then the geometric phase can be defined under the evolution of half of the period of classical motions. For the driven case, geometric phases are given in terms of Fourier coefficients of the external force. The oscillator systems whose classical equation of motion is Mathieu's equation are perturbatively studied, and the first term of nonvanishing geometric phase is calculated.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically unstable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation.