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Open quantum system, namely system-reservoir model, is described by a master equation of density operator. For example, the Caldeira-Leggett eqaution describes dissipative phenomenon of solid physics. Although some efforts have been made to derive the exact expression of this master equation, so far as we know, it has not been reported in the literature. The purpose of this paper is to provide a new approach to solving the Caldeira-Leggett equation, via this approach the explicit integral-form expression of ρ(t) can be obtained. The main point of this approach is to convert equation of density operator into an equation of density state vector, and then project density state vector into thermo entangled state representation and convert it into wave function by using the technique of integration within an ordered product of operators. Thus the master equation for Caldeira-Leggett model is converted into an differential equation of wave function. Wave function is also a function. The wave function can be obtained via the approach to solving the differential equation in mathematics. It can be converted into a density state vector and density operator. Using the technique of integration within an ordered product of operators again, the integra-form solution of the Caldeira-Leggett equation is obtained.
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Keywords:
- thermoentangled state representation /
- the master equation for Caldeira-Leggett model /
- density operator /
- the technique of integration within an ordered product of operators
[1] Feynman R P, Vernon F L 1963 Ann. Phys. 24 118
[2] Caldeira A O, Leggett A J 1983 Physics 121A 587
[3] Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (New York: Oxford University Press Inc.)
[4] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics Progress of Dirac's Symbolic Method (Shanghai: Shanghai Scientific and Technical Publishers)
[5] Gradshteyn I S, Ryzhik I M 1980 Tables of Integrals, Series, and Products (London: Fourth Edition Academic Press)
[6] Fan H Y 2005 From Quantum Mechanics to Quantum Optics-Development of the Mathematical Physics (Shanghai: Shanghai Jiao Tong University Press)
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[1] Feynman R P, Vernon F L 1963 Ann. Phys. 24 118
[2] Caldeira A O, Leggett A J 1983 Physics 121A 587
[3] Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (New York: Oxford University Press Inc.)
[4] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics Progress of Dirac's Symbolic Method (Shanghai: Shanghai Scientific and Technical Publishers)
[5] Gradshteyn I S, Ryzhik I M 1980 Tables of Integrals, Series, and Products (London: Fourth Edition Academic Press)
[6] Fan H Y 2005 From Quantum Mechanics to Quantum Optics-Development of the Mathematical Physics (Shanghai: Shanghai Jiao Tong University Press)
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