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Lecture notes in physics

Structural connections between the WKB and Wigner-Kirkwood semiclassical approximations

T. A. Osborn, F. H. Molzahn

Published Jan 1, 1986• 1 citation
Publisher
DOI
10.1007/3-540-17894-5_370

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The time evolution operator U(t, s) of a spinless nonrelativistic N-body quantum system in Euclidean space with real analytic time-dependent scalar interaction v(x, t) is studied. A complete formal asymptotic expansion involving simple connected graphs is derived for the full coordinate, and mixed coordinate-momentum representation propagators. The derivation is based on Dyson's series for U(t, s), and the combinatorics involved in the cluster expansion of the classical grand partition function. These results provide an efficient means of generating nonperturbative propagator expansions in the physical variables: mass m, Planck's constant h, time displacement t-s. The structural bridge between the WKB and Wigner-Kirkwood expansions is sketched for mixed representations of U(t, s). In the heat equation context the graphical expansions are found for mixed representations of the density operator e - beta H . Finally, an explicit differential formula is obtained for Wigner's distribution function f(x, p; beta ).

Journal of Physics A Mathematical and General•Aug 1, 1987•14 citations
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 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.
The time evolution operator U(t, s) of a spinless nonrelativistic N-body quantum system in Euclidean space with real analytic time-dependent scalar interaction v(x, t) is studied. A complete formal asymptotic expansion involving simple connected graphs is derived for the full coordinate, and mixed coordinate-momentum representation propagators. The derivation is based on Dyson's series for U(t, s), and the combinatorics involved in the cluster expansion of the classical grand partition function. These results provide an efficient means of generating nonperturbative propagator expansions in the physical variables: mass m, Planck's constant h, time displacement t-s. The structural bridge between the WKB and Wigner-Kirkwood expansions is sketched for mixed representations of U(t, s). In the heat equation context the graphical expansions are found for mixed representations of the density operator e - beta H . Finally, an explicit differential formula is obtained for Wigner's distribution function f(x, p; beta ).