Time-dependent WKB expansion for the non-relativistic coulomb propagator

V. V. Dodonov, Vladimir I. Man’ko, Dmitri E. Nikonov

Domenico P. L. Castrigiano, F. Staerk

Yair Goldfarb, Jeremy Schiff, David J. Tannor

We present a significant improvement to a complex time-dependent WKB (CWKB) formulation developed by Boiron and Lombardi [J. Chem. Phys. 108, 3431 (1998)] in which the time-dependent WKB equations are solved along classical trajectories that propagate in complex space. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper, we show that this limitation can be overcome by superposing the contributions of crossing trajectories. Secondly, we demonstrate that the approximation improves when incorporating higher order terms in the expansion. Thirdly, equations of motion for caustics and Stokes lines are implemented to help overcome Stokes discontinuities. These improvements could make the CWKB formulation a competitive alternative to current time-dependent semiclassical methods.

F. H. Molzahn, T. A. Osborn

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 ).

Lochlainn O'Raifeartaigh, Andreas Wipf

T. A. Osborn, F. H. Molzahn

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.