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A quadratic time-dependent quantum harmonic oscillator.

F E Onah1,2, E García Herrera3, J A Ruelas-Galván3

  • 1Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2501, 64849, Monterrey, NL, Mexico. a00834081@tec.mx.

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Summary
This summary is machine-generated.

We developed a Lie algebraic method for solving complex quantum harmonic oscillator models with time-varying parameters. This approach offers analytic solutions for driven and parametric oscillators, even in unstable numerical regimes.

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Area of Science:

  • Quantum mechanics
  • Mathematical physics
  • Quantum optics

Background:

  • Quantum harmonic oscillators are fundamental in quantum mechanics.
  • Time-dependent parameters in quantum systems present significant theoretical challenges.
  • Existing methods often struggle with numerical stability for complex models.

Purpose of the Study:

  • To present a novel Lie algebraic approach for solving a general class of time-dependent quantum harmonic oscillators.
  • To provide analytic solutions for driven and parametric quantum harmonic oscillators.
  • To demonstrate the framework's applicability to numerically unstable models.

Main Methods:

  • Utilized a Lie algebraic framework.
  • Employed unitary transformations to solve the time-dependent Schrödinger equation.
  • Developed a general quadratic time-dependent quantum harmonic model.

Main Results:

  • Obtained analytic solutions for driven and parametric quantum harmonic oscillators with time-varying parameters.
  • Provided an analytic solution for the periodically driven quantum harmonic oscillator without the rotating wave approximation.
  • Showcased a unitary transformation connecting the Caldirola-Kanai oscillator to the Paul trap Hamiltonian.
  • Demonstrated the method's ability to handle models with numerically unstable dynamics.

Conclusions:

  • The Lie algebraic approach offers a powerful and versatile tool for analyzing complex quantum harmonic oscillator systems.
  • This method provides exact analytic solutions where approximations or numerical methods fail.
  • The framework unifies solutions for various historically significant quantum oscillator models.