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 $\ensuremath{\hbar}$ 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.
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 \ensuremathℏ 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.
In a recent paper [B. Poirier, Chem. Phys. 370, 4 (2010)], a formulation of quantum mechanics was presented for which the usual wavefunction and Schrödinger equation are replaced with an ensemble of real-valued trajectories satisfying a principle of least action. It was found that the resultant quantum trajectories are those of Bohmian mechanics. In this paper, analogous ideas are applied to Bohmian Mechanics with Complex Action (BOMCA). The standard BOMCA trajectories as previously defined are found not to satisfy an action principle. However, an alternate set of complex equations of motion is derived that does exhibit this desirable property, and an approximate numerical implementation is presented. Exact analytical results are also presented, for Gaussian wavepacket propagation under quadratic potentials.
In a recent publication, Goldfarb et al. have proposed a new trajectory scheme to solve quantum problems: Bohmian mechanics with complex action BOMCA . According to these authors, their contribution represents “a novel formulation of Bohmian mechanics.” In our opinion, rather than a new formulation of Bohmian mechanics, the proposal of Goldfarb et al. consists of a novel quantum computational scheme, which is based on solving the complex trajectory equations resulting from an already known complex version of Bohmian dynamics. On the other hand, Goldfarb et al. also claimed that their numerical scheme describes “fully local complex quantum trajectories,” with the reward of a significantly higher degree of localization than in the conventional Bohmian mechanics by lowering the magnitude of the quantum potential. At least from a theoretical and interpretative point of view, we consider that it is very important to clarify what “fully local quantum trajectories” means. Bohmian mechanics is a nonlocal theory because standard quantum mechanics is a nonlocal theory. This nonlocality manifests via the presence of a quantum potential, and therefore the corresponding quantum trajectories will display features that reflect such nonlocality. Something similar should also occur for any trajectory-based approach that intends to be equivalent to standard quantum mechanics. In this Comment we would like to provide a general discussion on these two aspects arising from the work of Goldfarb et al., which can be of much interest to the Chemical Physics community. In particular, we would like to stress the special interest relying on how the term locality should be employed. The approach of Goldfarb et al. starts with the timedependent quantum Hamilton-Jacobi equation. This equation was considered by Leacock and Padgett in 1983 within the context of the quantum transformation theory as a way to obtain bound-state energy levels of quantum systems with no need to calculate the corresponding eigenfunctions. According to Leacock and Padgett, the quantum Hamilton-Jacobi equation can be either stated as a postulate as done by these authors or derived from the Schrodinger equation through a simple connection formula. In the second case, the quantum action corresponds to the complex phase of the wave function when the latter is expressed as a pure complex exponential function. This relationship allows one to pass straightforwardly from the quantum Hamilton-Jacobi formalism to the standard one in terms of the time-dependent Schrodinger and vice versa . Although both the Ansatz and the formulation used by Leacock and Padgett are similar to those considered in the standard quantum WKB semiclassical approximation, as pointed out by these authors, their approach conceptually differs from the WKB one. Within Bohmian mechanics, energy eigenstates present an inconvenience: they give rise to zero velocity fields. Hence, when particles are described by this type of states, they will remain steady motionless . To overcome this problem, Floyd and Faraggi and Matone developed timeindependent quantum Hamilton-Jacobi-like formulations starting from real bipolar Ansatze, though they did not claim full equivalence with standard quantum mechanics regarding their predictions. Later, John proposed a timedependent complex quantum trajectory formalism based on the same connection formula mentioned by Leacock and Padgett to study the dynamics associated with some simple analytical cases, such as the harmonic oscillator or the step barrier. This “modified de Broglie–Bohm approach to quantum mechanics,” as termed by John, is formally equivalent to the BOMCA formulation of Goldfarb et al. Nevertheless, we would like to specify that, due to the fundamental theoretical nature of John’s contribution, a separate calculation of the wave function is required in order to obtain the corresponding trajectories John makes use of the “guiding wave” idea from Bohmian mechanics . Meanwhile, in the version of Goldfarb et al. the modified de Broglie–Bohm approach is formulated in terms of a closed set of equations in the spirit of quantum hydrodynamical formalisms , which avoids the calculation of the wave functions, thus providing a new, practical use to such an approach. Furthermore, within the approach of Goldfarb et al., the expression for the quantum force happens to be typically much smaller than in standard Bohmian mechanics. On the other hand, we would also like to note that a related formulation has also been used by Chou and Wyatt, though they developed a different, novel numerical scheme to tackle the problem of bound-energy levels and scattering in one dimension. When trying to understand the quantum-classical correspondence from the perspective of a WKB-like formulation of Bohmian mechanics WKB-BM , Sanz and co-workers reached the complex quantum HamiltonJacobi equation the same year as John, but independently. The WKB-BM formulation constitutes an excellent, compreTHE JOURNAL OF CHEMICAL PHYSICS 127, 197101 2007
We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.
We present a unified derivation of Bohmian methods that serves as a common starting point for the derivative propagation method (DPM), Bohmian mechanics with complex action (BOMCA), and the zero-velocity complex action method (ZEVCA). The unified derivation begins with the ansatz psi = eiS/Planck's where the action (S) is taken to be complex, and the quantum force is obtained by writing a hierarchy of equations of motion for the phase partial derivatives. We demonstrate how different choices of the trajectory velocity field yield different formulations such as DPM, BOMCA, and ZEVCA. The new derivation is used for two purposes. First, it serves as a common basis for comparing the role of the quantum force in the DPM and BOMCA formulations. Second, we use the new derivation to show that superposing the contributions of real, crossing trajectories yields a nodal pattern essentially identical to that of the exact quantum wavefunction. The latter result suggests a promising new approach to deal with the challenging problem of nodes in Bohmian mechanics.
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.
The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V ( x ) = 1/ x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.