Quantum-Classical Wigner-Liouville Equation
- 136 Downloads
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.
KeywordsDensity Matrix Quantum System Entire System Operator Form Liouville Equation
Unable to display preview. Download preview PDF.
- 1.J. C. Tully, Modern Methods for Multidimensional Dynamics Computations in Chemistry, World Scientific, New York (1998).Google Scholar
- 2.V. I. Gerasimenko, “Uncorrelated equations of motion of the quantum-classical systems,” Rept. Acad. Sci. Ukr. SSR, No. 10, 65–68 (1981).Google Scholar
- 5.I. V. Aleksandrov, Z. Naturforsch, 36a, 902 (1981).Google Scholar
- 7.R. Kapral and G. Ciccotti, “A statistical mechanical theory of quantum dynamics in classical environments,” in: P. Nielaba, M. Mareschal, and G. Ciccotti (editors), Bridging Time Scales: Molecular Simulations for the Next Decade, Springer, Berlin (2003).Google Scholar
- 8.D. Mac Kernan, G. Ciccotti, and R. Kapral, J. Phys.: Condens. Matt., 14, 9069 (2002).Google Scholar
- 10.V. S. Filinov, Y. V. Medvedev, and V. L. Kamskyi, Mol. Phys., 85, 711 (1995).Google Scholar
- 11.V. S. Filinov, Mol. Phys., 88, 1517–1529 (1996).Google Scholar
- 12.V. S. Filinov, S. Bonella, Y. E. Lozovik, A. Filinov, and I. Zacharov, Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore (1998).Google Scholar
- 13.D. Mac Kernan, G. Ciccott, and R. Kapral, “Surface-hopping dynamics of a spin-boson system,” J. Chem. Phys., 116, 2346–2356 (2002).Google Scholar
- 14.A. Sergi and R. Kapral, “Quantum-classical dynamics of nonadiabatic chemical reactions,” J. Chem. Phys., 118, No.19, 8566–8575.Google Scholar