David Bohm had shown that the Schrödinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions — action and probability density. The first equation is the Hamilton–Jacobi (HJ) equation, a "visiting card" of classical mechanics, is modified by the Bohmian quantum potential. This potential is a nonlinear function of the probability density. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed into two Bohmian quantum correctors. The first corrector modifies the kinetic energy term of the HJ equation, and the second one modifies the potential energy term. The unification of the quantum HJ equation and the entropy balance equation gives a complexified HJ equation containing complex kinetic and potential terms. The imaginary parts of these terms have an order of smallness about the Planck constant. The Bohmian quantum corrector is an indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to an imaginary sector. The difference between the Bohmian and Feynman's trajectories is that the former satisfies the principle of least action and they bifurcate on interfaces. The latter covers all possible paths from a source to a detector. They can split and annihilate.
David Bohm had shown that the Schrödinger equation, that is a "visiting card" of quantum mechanics, can be decomposed onto two equations for real functions — action and probability density. The first equation is the Hamilton–Jacobi (HJ) equation, a "visiting card" of classical mechanics, is modified by the Bohmian quantum potential. This potential is a nonlinear function of the probability density. And the second is the continuity equation. The latter can be transformed to the entropy balance equation. The Bohmian quantum potential is transformed into two Bohmian quantum correctors. The first corrector modifies the kinetic energy term of the HJ equation, and the second one modifies the potential energy term. The unification of the quantum HJ equation and the entropy balance equation gives a complexified HJ equation containing complex kinetic and potential terms. The imaginary parts of these terms have an order of smallness about the Planck constant. The Bohmian quantum corrector is an indispensable term modifying the Feynman's path integral by expanding coordinates and momenta to an imaginary sector. The difference between the Bohmian and Feynman's trajectories is that the former satisfies the principle of least action and they bifurcate on interfaces. The latter covers all possible paths from a source to a detector. They can split and annihilate.
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
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpretation to nonrelativistic quantum mechanics [1-3]. In the hydrodynamic formulation of quantum mechanics, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE) are obtained by substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation. Bohm’s analytical approach has been used to compute and interpret realvalued quantum trajectories from a precomputed wave function for a diverse range of physical processes [3-8]. In the synthetic approach, the quantum trajectory method has been developed as a computational tool to generate the wave function by evolving ensembles of real-valued quantum trajectories through the integration of the hydrodynamic equations on the fly [9]. Remarkable progress has been made in the use of T&F Cat # K11224, Chapter 18, Page 284, 15-9-2010 real-valued quantum trajectories for solving a wide range of quantum mechanical problems [10].
It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton–Jacobi equation, namely, it is the quantum Hamilton's principal function (or quantum action). Therefore, the Feynman propagator can be computed either by means of the path integration, or by the way of the Hamilton–Jacobi equation. This is analogous to what happens in classical mechanics, where the Hamilton's principal function can be computed either by integrating the Lagrangian along the extremal paths, or as a solution of partial differential equation, namely the classical Hamilton–Jacobi equation. If the path is decomposed in the classical one and quantum fluctuations, the contribution of these quantum fluctuations satisfies a non-linear partial differential equation, whose coefficients depend on the classical action. When the contribution of the quantum fluctuations depend only on the time, it can be computed by means of a simple integration. The final results for the propagators in this case are equal to the Van Vleck–Pauli–Morette expressions, even though the two derivations are quite different. Quanta 2023; 12: 22–26.
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.
Short introduction to Bohmian mechanics Sheldon Goldstein in a Foundations of Physics paper [1] indicates (p. 341): “Since macroscopic objects are normally regarded as built out of microscopic constituents …there can be no problem of macroscopic reality per se in Bohmian mechanics.” Goldstein in the same paper remarks (p. 342) that this | ψ | 2 has “a status very much the same as that of a thermodynamic equilibrium.” Basil Hiley the closest collaborator of David Bohm, says the following (Hiley [2] (p. 2)): “What Bohm (1952a; 1952b) did was to show how to retain a description of all the usual properties of a classical world and yet remain completely within the quantum formalism.” The Bohm (1952) [3] [4] references in that quote refer to the original work of Bohm in which he sets out the basics of the Bohmian mechanics. As we will see later in the next section of this chapter, the appearance of an “additional term” in the Hamilton-Jacobi equation, is the hallmark of Bohmian mechanics and it is very often termed the “quantum potential.” Hiley [2] (p. 2) remarks that since the Bohmian momentum is a well-defined function of position and time, an ensemble of trajectories can be found when the quantum potential is non-zero. A unique, classical path is found when the quantum potential is zero. From the outset, we hope the reader can savor the beauty of this simple but very powerful result. Bohmian mechanics indeed shows this very gentle transition from quantum mechanics to classical mechanics via the value of the quantum potential.