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This study compares classical and quantum mechanics for nanoscopic systems using an anharmonic oscillator. It reveals periodic variations in energy transition probabilities, offering insights into the classical-quantum correspondence.

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Area of Science:

  • Thermodynamics of nanoscopic systems
  • Quantum mechanics and classical mechanics correspondence

Background:

  • The relationship between classical and quantum descriptions is crucial for nanoscopic systems.
  • Anharmonic oscillators driven by external forces are key models for studying thermodynamic properties.

Purpose of the Study:

  • To scrutinize the correspondence between classical and quantum mechanical descriptions.
  • To investigate the energy transition probabilities in a driven anharmonic oscillator.

Main Methods:

  • Classical and quantum mechanical analysis of a driven anharmonic oscillator.
  • Numerical simulations and approximate analytical results (pendulum approximation).
  • Floquet theory for quantum mechanical analysis.

Main Results:

  • Nondiagonal energy transitions in the classical case are primarily due to separatrix crossing.
  • Approximate analytical results align with numerical simulations for the classical oscillator.
  • Numerically exact quantum results, supported by Floquet theory, show periodic variations in energy transition probabilities with driving amplitude.

Conclusions:

  • Provides an intuitive explanation for the periodic variation of energy transition probabilities in both classical and quantum regimes.
  • Highlights the importance of separatrix crossing in classical nondiagonal transitions.
  • Demonstrates the utility of Floquet theory in analyzing quantum transitions.