Mixing Quantum and Classical Mechanics

Using a group theoretical approach we derive an equation of motion for a mixed quantum-classical system. The quantum-classical bracket entering the equation preserves the Lie algebra structure of quantum and classical mechanics: The bracket is antisymmetric and satisfies the Jacobi identity, and, therefore, leads to a natural description of interaction between quantum and classical degrees of freedom. We apply the formalism to coupled quantum and classical oscillators and show how various approximations, such as the mean-field and the multiconfiguration mean-field approaches, can be obtained from the quantum-classical equation of motion. PACS numbers: 03.65.Sq, 03.65.Db, 03.65.Fd E-mail: [email protected] Current address: Vakgroep Wiskundige Analyse, Universiteit Gent, Galglaan 2, B-9000, Gent, BELGIE. E-mail: [email protected]