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The Journal of Chemical Physics

Complex-valued classical trajectories for tunneling and well-defined paths

Thomas F. George

Published Jan 15, 1975• 2 citations
Publisher

Abstract

A paper by Hirschfelder, Christoph, and Palke has suggested analogies between quantum streamlines and complex−valued classical trajectories in a semiclassical treatment of tunneling in reactive molecular collisions. This Comment points out that there are basic differences between trajectories and the streamlines and that care must be taken ataching any physical significance to complex−valued trajectories or in relating them to quantum mechanical quantities such as streamlines. (AIP)

DOI
10.1063/1.430451

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A paper by Hirschfelder, Christoph, and Palke has suggested analogies between quantum streamlines and complex−valued classical trajectories in a semiclassical treatment of tunneling in reactive molecular collisions. This Comment points out that there are basic differences between trajectories and the streamlines and that care must be taken ataching any physical significance to complex−valued trajectories or in relating them to quantum mechanical quantities such as streamlines. (AIP)
In the case of tunneling in chemical reactions [see figure1.1(c)], the total energy of reagents cannot be generally considered to be small; treating it as the first-order term of h¯ as was successfully done in Chapters 6 and 7 does not lead to any sensible semiclassical theory. The semiclassical mechanics requires (1) construction of Lagrange manifolds, (2) determination of amultidimensional caustic surface, and (3) connection of wave functions between classically allowed and forbidden regions [59]. Unfortunately, these problems have not yet been solved mathematically. As was discussed in Section 3.1, in the framework of the ordinary WKB approach we can construct Lagrange manifolds and caustics in the case of two dimensions and the connection problem can be treated relatively well by assuming the local separability. Extension of this method to higher dimensions, however, cannot be practical. The complexvalued (complex coordinate and/or momentum) WKB theory has been devised, as briefly explained there, but cannot be simply extended to multidimensions (see, for instance, [15]). This formidable task can hardly be accomplished mainly because of the complexity of the Lagrange manifold and the fact that analytical properties of any realistic potential energy surfaces extended into the complex-valued coordinate space cannot be guaranteed. The use of a complex trajectory was nicely materialized by the double-ended complex trajectory in the classical S-matrix theory [206], in which the quantized boundary conditions are satisfied at both (initial and final) ends. Unfortunately, this method turned out not to be practically useful, because finding such trajectories is also a formidable task in a multidimensional complex coordinate (and/or momentum) space. We can, however, make a break through the difficulty of detecting caustics in multidimensional space. An efficient way of doing that has been devised along classical trajectory [47]. This is not a trivial task at all, since it is not possible to find caustics in high-dimensional space by just shooting classical trajectories. Since the caustics can be found along each classical trajectory, chemical dynamics with tunneling effects incorporated can now be treated by the on-the-fly method. In this subsection, this method is explained.
A model for describing barrier tunneling (or other classically forbidden processes) using purely real-valued classical trajectories is presented. The basic idea is to introduce an auxiliary degree of freedom that allows for fluctuations in the potential-energy surface for the original classical degrees of freedom. The model can be applied purely classically or better semiclassically (e . g ., via the initial value representation). Numerical results for 1D barrier tunneling are presented to illustrate the model.
Exact numerical calculations are made for the scattering of quantum mechanical particles from a square two-dimensional potential barrier. This treatment is an exact analog of both frustrated total reflection of perpendicularly polarized light and the longitudinal Goos-Hänchen shift. Quantum mechanical streamlines (which are analogous to either classical mechanical trajectories or optical rays) are plotted. These streamlines are smooth, continuous, and have continuous first derivatives even through the classically forbidden region. The streamlines form quantized vortices surrounding each of the nodal points (which result from interference between the incident and reflected waves). Similar vortices occur in reactive collisions of H + H2 (McCullough and Wyatt; Kuppermann, Adams, and Truhlar) and undoubtedly play an important role in molecular collision dynamics. The theory for these vortices is given in a companion paper. A comparison is given between our numerical calculations and the stationary phase approximation (SPA). Although our incident wave packet has a half-width of more than one de Broglie wavelength in contrast to the SPA which replaces the diffuse beams by rays with delta function profiles, the agreement was surprisingly good for both the Goos-Hänchen shifts and for the reflection coefficients. However, we found that the Goos-Hänchen shift for the transmitted beam is significantly smaller than for the reflected beam, although in the stationary phase approximation these two shifts are equal. Furthermore, we found that the scattering from the potential barrier has very little effect on the shape of the wave packets. The power series expansion of the incident Debye-Picht wave packet ψI has an extremely small radius of convergence, whereas the power series for ψI*ψI has a radius of convergence of more than two de Broglie wavelengths. The imaginary velocity viis introduced into the Madelung-Landau-London hydrodynamical formulation of quantum mechanics. The corresponding imaginary streamlines will be considered in a forthcoming paper. The time-independent Schrödinger equation for real wavefunctions is reduced to solving the nonlinear first order partial differential equation: ℏ▿·vi = 2(E − V) + vi2. Here, vi is irrotational. This equation may lead to interesting new methods of solving the Schrödinger equation. It does lead to a generalization of the Prager-Hirschfelder perturbation scheme which invokes an electrostatic analogy. The philosophical implications of the hydrodynamical formulation of quantum mechanics are discussed. Penetration of a one-dimensional square potential barrier is used to demonstrate exactly how tunneling occurs by particles ``riding over the barrier'' à la Bohm. Cases are cited where quantum and classical mechanical motions are identical.
It is a strict requirement that physical observables should be real-valued; the position and momentum of a particle obeying the Newtonian equations of motion are both real, and any dynamical variables derived from them are also real. The wavefunction in quantum mechanics may be complex-valued, but observables obtained averaged over the square modulus of the wavefunction turn out to be real. Occasionally, one examines the nature of singularities in the complex plane in order to judge whether the system under consideration is completely integrable or not, but in almost all cases complex variables appear only implicitly in the description of physical phenomena. Indeed, there is no serious demand or even motivation to investigate dynamics whose variables are complex.