Ukrainian Mathematical Journal

, Volume 57, Issue 6, pp 891–899 | Cite as

Quantum-Classical Wigner-Liouville Equation

  • R. Kapral
  • A. Sergi


We consider a quantum system that is partitioned into a subsystem and a bath. Starting from the Wigner transform of the von Neumann equation for the quantum-mechanical density matrix of the entire system, the quantum-classical Wigner-Liouville equation is obtained in the limit where the masses M of the bath particles are large as compared with the masses m of the subsystem particles. The structure of this equation is discussed and it is shown how the abstract operator form of the quantum-classical Liouville equation is obtained by taking the inverse Wigner transform on the subsystem. Solutions in terms of classical trajectory segments and quantum transition or momentum jumps are described.


Density Matrix Quantum System Entire System Operator Form Liouville Equation 
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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • R. Kapral
    • 1
  • A. Sergi
    • 1
  1. 1.University of TorontoTorontoCanada

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