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This study introduces a new Hamiltonian wave equation for classical-quantum coupling, enabling bidirectional influence and preserving system properties. It resolves issues in prior models, demonstrating robust classical-quantum dynamics.

Keywords:
Koopman–von Neumann theoryclassical–quantumdynamicsquantum density matrix

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Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Theoretical physics

Background:

  • Koopman-von Neumann theory offers a framework for classical-quantum dynamics.
  • Existing models struggle with bidirectional classical-quantum coupling and property preservation.
  • Formulating a consistent dynamical theory for classical-quantum interactions remains a challenge.

Purpose of the Study:

  • To develop a novel Hamiltonian wave equation for classical-quantum coupling.
  • To accurately describe both classical influence on quantum systems and quantum backreaction.
  • To address and overcome limitations of current classical-quantum interaction models.

Main Methods:

  • Revisiting the Hamiltonian structure of classical wavefunctions within Koopman-von Neumann theory.
  • Proposing a new Hamiltonian wave equation to model hybrid classical-quantum systems.
  • Utilizing an exactly solvable model of a two-level quantum system coupled to a classical harmonic oscillator for illustration and comparison.

Main Results:

  • The new model successfully describes bidirectional classical-quantum influence, including quantum backreaction.
  • The quantum subsystem's density matrix remains positive definite, unlike in some previous models.
  • The Liouville density of the classical subsystem is shown to preserve its sign over time for a specific class of systems.

Conclusions:

  • The proposed Hamiltonian wave equation provides a more robust and consistent framework for classical-quantum dynamics.
  • This model overcomes key shortcomings of existing theories, particularly regarding property preservation.
  • The findings offer a significant advancement in understanding and modeling hybrid classical-quantum systems.