Non-Hamiltonian commutators in quantum mechanics.

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Theoretical physics

Background:

• Non-Hamiltonian dynamics are crucial for describing open quantum systems and complex classical systems.
• Existing formalisms for quantum and classical non-Hamiltonian dynamics are often disparate.
• The symplectic structure of quantum commutators has not been fully explored for unifying these dynamics.

Purpose of the Study:

• To unveil the symplectic structure of quantum commutators.
• To develop a unified approach to classical and quantum-classical non-Hamiltonian dynamics.
• To illustrate the application of non-Hamiltonian commutators in defining thermodynamic constraints.

Main Methods:

• Exploiting the symplectic structure of quantum commutators to define generalized non-Hamiltonian brackets.
• Applying the formalism to quantum-classical systems.
• Deriving specific equations of motion and stationary density matrices for thermodynamic ensembles.

Main Results:

• The symplectic structure of quantum commutators is elucidated.
• A unified framework for classical and quantum-classical non-Hamiltonian dynamics is presented.
• Quantum-classical Nosé-Hoover equations and stationary density matrices are derived.
• Non-Hamiltonian commutators for Nosé-Hoover chains and Nosé-Andersen dynamics are provided.

Conclusions:

• The developed formalism offers a unified perspective on non-Hamiltonian dynamics across classical and quantum regimes.
• Non-Hamiltonian commutators provide a powerful tool for incorporating thermodynamic constraints into quantum-classical systems.
• The work opens avenues for further exploration of quantum-classical thermodynamics and dynamics.