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Communications in Theoretical Physics

Application of Weyl-Wigner Method in Calculating Thermal Averages

Fan Hong-Yi

Published Jul 1, 1991• 5 citations
Publisher

Abstract

The Weyl-Wigner correspondence between quantum operators and their classic analogues is employed in calculating thermal averages, which enables one to treat the Wigner function of mixed states on the same footing as that of the thermal vacuum state which plays an essential role in thermodynamics. Both the bosonic and fermionic cases are considered in this work by virtue of the coherent state representation of the Wigner operators.

DOI
10.1088/0253-6102/16/1/123

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The Weyl-Wigner correspondence between quantum operators and their classic analogues is employed in calculating thermal averages, which enables one to treat the Wigner function of mixed states on the same footing as that of the thermal vacuum state which plays an essential role in thermodynamics. Both the bosonic and fermionic cases are considered in this work by virtue of the coherent state representation of the Wigner operators.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
We point out that there is a close connection between the Weyl correspondence and coherent state correspondence. By virtue of differential and integral method within normal ordered products we demonstrate that the essence of the Weyl correspondence is just coherent state correspondence in both boson and fermion cases. Furthermore, coherent state form and explicit operator form for the Wigner function are given. With these forms we can derive some important operator formulas from the Weyl correspondence. The Wigner function's zero-point value is also discussed.
On the basis of the preceding paper we present some new applications of both the normal product form and the coherent state form of the Wigner operator, which involve deriving some new quantum operator formulas, giving the coherent state generalization of the Moyal theorem, evaluating some quantum operators which correspond to the given classical functions in the weyl manner and vice versa. Were it not for the Wigner operator's coherent state formulation given by us the above-mentioned calculations would be hard to perform.