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.
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.
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
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.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.
A new facet was added to the study of dilute gases by the production and subsequent Bose–Einstein condensation of diatomic molecules from a gas of fermionic atoms. Feshbach resonances which, as we have seen in Sec. 5.4, make it possible to tune the atom–atom interaction, play a crucial role in the experiments. At the magnetic field strength for which the binding energy of the molecule vanishes, the inverse of the scattering length, which determines the low-energy effective interaction between atoms, passes through zero. In the experiments to produce molecules, one starts with a mixture of two species of fermion, most commonly different hyperfine states of the same isotope, in a magnetic field of such a strength that the molecular state has an energy higher than that of two zero-momentum atoms in the open channel. The magnetic field is then altered to a value at which the molecular state is bound with respect to two atoms in the open channel, and in this process, many of the atoms combine to form molecules. These molecules have binding energies in the 10 −9 eV range, and are thus extremely weakly bound by the standards of conventional molecular physics. In addition, they are very extended, with atomic separations as large as one micron. These molecules, being bosons, can undergo Bose–Einstein condensation, just as bosonic atoms do.