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Physical Review A

Wigner-Kirkwood expansion of N -body Green's function: The case with magnetic field

E. Tirapegui, Florentin Langouche, D. Roekaerts

Published May 1, 1983• 9 citations
Publisher

Abstract

A systematic method is given to compute the semiclassical Wigner-Kirkwood expansion of the off-diagonal thermal kernel of an N-body quantum system. © 1983 The American Physical Society.
DOI
10.1103/physreva.27.2649

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Equation for the WKB approximation and Wigner-Kirkwood expansions for systems with magnetic field

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We discuss semiclassical expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent, magnetic field. In the process, we derive a new equation for the WKB approximation for such systems. In particular, we provide recursion relations and a diagrammatic method for calculating explicitly the coefficients in these expansions. As an illustration we present the off-diagonal coefficients up to order ${\ensuremath{\hbar}}^{3}$, and the on-diagonal coefficients up to ${\ensuremath{\hbar}}^{6}$ for a uniform magnetic field. We also give the on-diagonal coefficients up to order ${\ensuremath{\hbar}}^{4}$ for a nonuniform field.

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As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].

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We discuss semiclassical expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent, magnetic field. In the process, we derive a new equation for the WKB approximation for such systems. In particular, we provide recursion relations and a diagrammatic method for calculating explicitly the coefficients in these expansions. As an illustration we present the off-diagonal coefficients up to order ${\ensuremath{\hbar}}^{3}$, and the on-diagonal coefficients up to ${\ensuremath{\hbar}}^{6}$ for a uniform magnetic field. We also give the on-diagonal coefficients up to order ${\ensuremath{\hbar}}^{4}$ for a nonuniform field.
This paper constructs a general method for obtaining series expansions of the logarithm of the configuration function FFF. For an NNN-body quantum system with Hamiltonian HHH and inverse temperature $\ensuremath{\beta}$, the configuration function is the ratio of the exact to the free coordinate-space heat kernels: $F=\frac{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}H}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}{〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}|{e}^{\ensuremath{-}\ensuremath{\beta}{H}_{0}}|{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}}〉}$. Expansion of lnF\mathrm{ln}FlnF in Planck's constant hhh leads to the semiclassical expansion of Wigner and Kirkwood, whereas the series in the variable $\ensuremath{\beta}$ provides the high-temperature expansion. By the introduction of an appropriate linked-graph method, it is shown how to obtain explicit formulas for the coefficient functions that enter either of these two series. Further, it is established that the same results can be derived by using the Feynman-Kac path-integral description of the partition function.
The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.
We study the expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent magnetic field in physically relevant parameters. In particular, the high-temperature expansion and a new heavy-mass expansion is discussed. We provide recursion relations and a diagrammatic method to calculate the coefficients in these expansions. We compare these expansions in detail with the semiclassical expansion derived previously [Phys. Rev. A 30, 2024 (1984)]. As an illustration we present the first few coefficients explicitly.
As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].