Quantum mechanical version of classical Liouville theorem
Xie Chuan-Mei (谢传梅)a, Fan Hong-Yi (范洪义)b
a College of Physics & Material Science, Anhui University, Hefei 230039, China; b Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China
Abstract In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of coherent state from |z〉to |sz-rz*〉angle corresponds to the motion from a point z(q,p) to another point sz-rz* with |s|2-|r|2=1. The evolution is governed by the so-called Fresnel operator U(s,r) recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation and obeys the group product rules. In another word, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.
Fund: Project supported by the Doctoral Scientific Research Startup Fund of Anhui University, China (Grant No. 33190059), the National Natural Science Foundation of China (Grant No. 10874174), the Research Fund for the Doctoral Program of Higher Education of China (New Teacher) (Grant No. 20113401120004), and the Open Funds from the National Laboratory for Infrared Physics, Chinese Academy of Sciences (Grant No. 201117).
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