Quantum mechanical streamlines. I. Square potential barrier

Abstract

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.

Nonclassical polarization dynamics in classical-like states

“…Or, in topological terms, it accounts for the number of jumps between different equivalent points of the Riemann surface described by the logarithm of the wave function. In all cases examined in this work, the trajectories around the zeros will be nearly circular in a neighborhood of the node, giving rise to a vortical dynamics [20][21][22][23][24]. It is worth noting that in quantum mechanics, it was Dirac who first noticed this effect [28], suggesting the existence of magnetic monopoles.…”

Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems

“…There has been similar efforts for visualizing quantum phenomena in terms of graphical representations of the flows and wave structures in scattering theory. 25,26 Finally, several wave packet calculations, 27,28 semiclassical methods, 29 the WKBJ method for both the Schrödinger and Dirac equations, 30 and discrete variable representation ͑DVR͒ scheme 31 as well as general integration schemes of the timedependent Schrödinger equation 32,33 share features resembling those addressed in the present formulation.…”

Manifestations of vortices during ultracold-atom propagation through waveguides

“…Hirschfelder and collaborators, for example, examined vortex creation around wave-function holes for plane-wave scattering from 2-D potentials such as partially and totally reflecting walls [11,12]. In the present paper, the fundamental mechanism for generating what we call interference vortices (also known as optical vortices [13]) is examined using two simple geometries.…”