中国物理B ›› 2013, Vol. 22 ›› Issue (3): 30501-030501.doi: 10.1088/1674-1056/22/3/030501

• GENERAL • 上一篇    下一篇

Quantum mechanical version of classical Liouville theorem

谢传梅a, 范洪义b   

  1. 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
  • 收稿日期:2012-06-07 修回日期:2012-08-29 出版日期:2013-02-01 发布日期:2013-02-01
  • 基金资助:
    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).

Quantum mechanical version of classical Liouville theorem

Xie Chuan-Mei (谢传梅)a, Fan Hong-Yi (范洪义)b   

  1. 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
  • Received:2012-06-07 Revised:2012-08-29 Online:2013-02-01 Published:2013-02-01
  • Contact: Xie Chuan-Mei E-mail:xiecmei@mail.ustc.edu.cn
  • Supported by:
    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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摘要： 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.

关键词: Fresnel operator, Liouville theorem, quantum mechanical version

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.

Key words: Fresnel operator, Liouville theorem, quantum mechanical version

中图分类号:  (Classical statistical mechanics)

  • 05.20.-y
05.30.-d (Quantum statistical mechanics)