TY - JOUR
T1 - Path-integral derivations of complex trajectory methods
AU - Schiff, Jeremy
AU - Goldfarb, Yair
AU - Tannor, David J.
N1 - Israel Science Foundation [807/08]J.S. wishes to thank David Kessler and Harry Dym for many useful discussions. Much of this work was done while J.S. was on sabbatical leave in the Departments of Mathematics and Chemical Physics at the Weizmann Institute. This work was supported by the Israel Science Foundation (807/08).
PY - 2011/1/18
Y1 - 2011/1/18
N2 - Path-integral derivations are presented for two recently developed complex trajectory techniques for the propagation of wave packets: complex WKB and Bohmian mechanics with complex action (BOMCA). The complex WKB technique is derived using a standard saddle-point approximation of the path integral, but taking into account the dependence of both the amplitude and the phase of the initial wave function, thus giving rise to the need for complex classical trajectories. The BOMCA technique is derived using a modification of the saddle-point technique, in which the path integral is approximated by expanding around a near-classical path, chosen so that up to some predetermined order there is no need to add any correction terms to the leading-order approximation. Both complex WKB and BOMCA techniques give the same leading-order approximation; in the complex WKB technique higher accuracy is achieved by adding correction terms, while in the BOMCA technique no additional terms are ever added: higher accuracy is achieved by changing the path along which the original approximation is computed. The path-integral derivation of the methods explains the need to incorporate contributions from more than one trajectory, as observed in previous numerical work. On the other hand, it emerges that the methods provide efficient schemes for computing the higher-order terms in the asymptotic evaluation of path integrals. The understanding we develop of the BOMCA technique suggests that there should exist near-classical trajectories that give exact quantum dynamical results when used in the computation of the path integral keeping just the leading-order term. We also apply our path-integral techniques to give a compact derivation of the semiclassical approximation to the coherent-state propagator.
AB - Path-integral derivations are presented for two recently developed complex trajectory techniques for the propagation of wave packets: complex WKB and Bohmian mechanics with complex action (BOMCA). The complex WKB technique is derived using a standard saddle-point approximation of the path integral, but taking into account the dependence of both the amplitude and the phase of the initial wave function, thus giving rise to the need for complex classical trajectories. The BOMCA technique is derived using a modification of the saddle-point technique, in which the path integral is approximated by expanding around a near-classical path, chosen so that up to some predetermined order there is no need to add any correction terms to the leading-order approximation. Both complex WKB and BOMCA techniques give the same leading-order approximation; in the complex WKB technique higher accuracy is achieved by adding correction terms, while in the BOMCA technique no additional terms are ever added: higher accuracy is achieved by changing the path along which the original approximation is computed. The path-integral derivation of the methods explains the need to incorporate contributions from more than one trajectory, as observed in previous numerical work. On the other hand, it emerges that the methods provide efficient schemes for computing the higher-order terms in the asymptotic evaluation of path integrals. The understanding we develop of the BOMCA technique suggests that there should exist near-classical trajectories that give exact quantum dynamical results when used in the computation of the path integral keeping just the leading-order term. We also apply our path-integral techniques to give a compact derivation of the semiclassical approximation to the coherent-state propagator.
UR - http://www.scopus.com/inward/record.url?scp=78751497830&partnerID=8YFLogxK
U2 - https://doi.org/10.1103/PhysRevA.83.012104
DO - https://doi.org/10.1103/PhysRevA.83.012104
M3 - مقالة
SN - 1050-2947
VL - 83
JO - Physical Review A - Atomic, Molecular, and Optical Physics
JF - Physical Review A - Atomic, Molecular, and Optical Physics
IS - 1
M1 - 012104
ER -