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.
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.
Quantum mechanical streamlines and probability density contours help in the understanding of collision dynamics by showing what happens during the collision. This is illustrated by considering the elastic scattering of two particles which interact with a spherically symmetric square potential, V (r) =0 when r≳a and V (r) =C when r<a, where the constant C is negative for a potential well and positive for a barrier. The streamlines and density contours completely determine the wavefunction (which is separable in r and ϑ and independent of φ) since the pre-exponential factor is the square root of the density and the phase is everywhere perpendicular to the streamlines. The streamlines often form quantized vortices surrounding wavefunction nodes. Since the nodal regions are circular rings, the vortices are toroidal (like smoke rings). If one or both of the particles is charged, the vortex ring corresponds to a collisional magnetic moment. The distortion of the streamlines provides a visual explanation of the large collision cross sections. In S-wave resonances 4πr2=σ0, where r is the distance from the origin to the center of the outermost vortex and σ0 is the cross section of the S-wave component. Thus, r is essentially the ’’scattering length.’’ In P-wave resonances (where again r is the radial distance to the outermost vortex), 1.85πr2?σ1 and in D-wave resonances, 1.45πr2?σ2. In the square well resonances, the density inside of the interaction sphere has the shape of the dominant wave and often has maxima hundreds to thousands of times greater than the density of the incident wave. By happenstance, whenever the potential well has the proper depth to form a newly bound D state, there is a Ramsauer–Townsend effect so that (for small energy of the incident wave) the total scattering cross section is very small;outside the interaction zone, the streamlines are only slightly deflected by the potential well;inside, the streamlines are deflected towards the symmetry axis and the probability density is concentrated near this axis
An atom–diatomic molecule collision is simulated by considering an idealized potential energy surface which is a two-dimensional duct with an adjustable potential in the corner region. This potential is symmetric with respect to an interchange of the x and y Cartesian coordinates. Explicit expressions for the wavefunctions are obtained which make use of this symmetry. Also analytical relations are obtained between the transmission and reflection coefficients and their phases. Quantum mechanical streamlines are computer graphed for a large number of energies and positive, negative, and zero values of the potential energy in the corner region. Special attention is given to the quantized vortices (surrounding wavefunction nodes) which appear in the streamlines. When only one energy channel is open, the streamlines are symmetric and the flux is antisymmetric. This occurs because the wavefunction is a linear combination (with complex coefficients) of two real solutions, one symmetric, the other antisymmetric. When two energy channels are open, the streamlines are no longer symmetric. This is because the two wavefunctions are now linear combinations (with complex coefficients) of four real solutions. Although the second wavefunction is expressed in terms of the first, their streamlines have quite different patterns. From the Argand diagrams, it is clear that the reflection, transmission, and Feshbach-type resonances which occur correspond to the Breit–Wigner type. At a reflection resonance energy all of the streamlines disappear; at slightly higher energy, the streamlines reappear with the same vortices as before but rotating in the opposite direction. The probability density is only symmetric at transmission resonance energies. At energies higher than the Feshbach-type resonance energy (where a second energy channel becomes open), the streamlines pattern ceases to be symmetric and the flux antisymmetric. The exact quantum mechanical wavefunction for this idealized model is the same as the exact semiclassical wavefunction. These wavefunctions are infinite series. It is found that for the low energies which we consider, an excellent approximation is obtained by truncating the series after the first two terms. This corresponds to a semiclassical wavefunction formed from two manifolds of classical trajectories.
The quantum hydrodynamic approach is based on the de Broglie-Bohm formulation of quantum mechanics. The resulting fluid-like equations of motion describe the flow of probability and an accurate solution to these equations is equivalent to solving the time-dependent Schroedinger equation. Furthermore, the hydrodynamic approach provides new insight into the mechanisms as well as an alternative computational approach for treating tunneling phenomena. New concepts include well-defined 'quantum trajectories', 'quantum potential', and 'quantum force' all of which have classical analogues. The quantum potential and its associated force give rise to all quantum mechanical effects such as zero point energy, tunneling, and interference. A new numerical approach called the Iterative Finite Difference Method (IFDM) will be discussed. The IFDM is used to solve the set of non-linear coupled hydrodynamic equations. It is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a 'smart' Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is also capable of treating anharmonic potentials. Example calculations using the IFDM will be presented which include: a one-dimensional Gaussian wave packet tunneling through an Eckart barrier, a one-dimensional bound-state Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. Approximate treatments of the quantum hydrodynamic equations will also be discussed which could allow scaling of the calculations to hundreds of degrees of freedom which is important for treating tunneling phenomena in condensed phase systems.
In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, psi = psi1 + psi2, was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper].
Abstract We consider reflection and transmission of 2D quantum wavepackets with phase vortices (also known in optics as spatiotemporal vortex pulses) at potential step-like, delta-function, and rectangular barriers. The presence of a vortex significantly modifies the Wigner time delays and Goos–Hänchen shifts, previously studied for Gaussian-like wavepackets. In particular, the scattered wavepackets undergo non-zero time delays and lateral shifts even for purely real scattering coefficients, when the standard Wigner and Artmann formulae vanish. We derive analytical expressions for the vortex-induced times delays and spatial shifts and verify these with numerical calculations of the Schrödinger equation. The time delays and shifts are resonantly enhanced in the vicinity of the critical-angle incidence for a step-like potential and near transmission resonances for a rectangular barrier.
In this paper we compute quantum trajectories arising from Bohm’s causal description of quantum mechanics. Our computational methodology is based upon a finite-element moving least-squares method (MWLS) presented recently by Wyatt and co-workers [Lopreore and Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. This method treats the “particles” in the quantum Hamilton–Jacobi equation as Lagrangian fluid elements that carry the phase, S, and density, ρ, required to reconstruct the quantum wave function. Here, we compare results obtained via the MWLS procedure to exact results obtained either analytically or by numerical solution of the time-dependent Schrödinger equation. Two systems are considered: first, dynamics in a harmonic well and second, tunneling dynamics in a double well potential. In the case of tunneling in the double well potential, the quantum potential acts to lower the barrier, separating the right- and left-hand sides of the well, permitting trajectories to pass from one side to another. However, as probability density passes from one side to the other, the effective barrier begins to rise and eventually will segregate trajectories in one side from the other. We note that the MWLS trajectories exhibited long time stability in the purely harmonic cases. However, this stability was not evident in the barrier crossing dynamics. Comparisons to exact trajectories obtained via wave packet calculations indicate that the MWLS trajectories tend to underestimate the effects of constructive and destructive interference effects.