Quantized vortices can occur around nodal points in wavefunctions. The derivation depends only on the wavefunction being single valued, continuous, and having continuous first derivatives. Since the derivation does not depend upon the dynamical equations, the quantized vortices are expected to occur for many types of waves such as electromagnetic and acoustic. Such vortices have appeared in the calculations of the H + H2 molecular collisions and play a role in the chemical kinetics. In a companion paper, it is shown that quantized vortices occur when optical waves are internally reflected from the face of a prism or particle beams are reflected from potential energy barriers.
Quantized vortices can occur around nodal points in wavefunctions. The derivation depends only on the wavefunction being single valued, continuous, and having continuous first derivatives. Since the derivation does not depend upon the dynamical equations, the quantized vortices are expected to occur for many types of waves such as electromagnetic and acoustic. Such vortices have appeared in the calculations of the H + H2 molecular collisions and play a role in the chemical kinetics. In a companion paper, it is shown that quantized vortices occur when optical waves are internally reflected from the face of a prism or particle beams are reflected from potential energy barriers.
Quantized vortices can occur around nodes of wavefunctions. This fact, discovered by Dirac (1931) but little noted since, is rederived here and examples are discussed. The derivation depends only on the wavefunction being single valued and continuous. Since the derivation does not depend upon the dynamical equations, the quantized vortices are expected to occur for many types of waves (i.e., electromagnetic, acoustic, etc.). Such vortices have appeared in the calculations (McCullough and Wyatt, Kuppermann) of the H + H2 molecular collisions and play a role in the chemical kinetics. In a companion paper, it is shown that quantized vortices occur when optical waves are internally reflected from the face of a prism or particle beams are reflected from potential energy barriers.
The flux of probability density corresponding to a complex wavefunction, ψ=ψr+iψi, can form vortices which rotate about a nodal region of ψ, and have integer values of the circulation numbers. For one particle wavefunctions (or for approximate natural spin molecular orbitals), the vortices are either axial or toroidal. The axial vortices have angular momentum dipole moments which are usually quantized only if the wave function is an eigenfunction of Lz. The axial vortices interact with homogeneous magnetic fields whereas the toroidal interact with inhomogeneous magnetic fields. Toroidal vortices are easily created or annihilated by a perturbation which moves the nodal surface of ψi relative to the nodal surface of ψr so that these surfaces intersect or become separated. Thus, toroidal vortices are unstable and may be important only near resonance. However, axial vortices are stable since their creation requires (a) photon absorption or emission, (b) a perturbation (such as a magnetic field) which converts a real into a complex wavefunction, or (c) conversion of a toroidal into an axial vortex by expanding the nodal loop until a part of it reaches the boundary of configuration space. Thus, axial vortices should play an important role in energy transfer, photochemical processes, etc., and their circulation numbers should be good quantum numbers. A method for determining generalized first-order density matrices and natural spin–orbitals for use in scattering problems is proposed.
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.
The time-dependent Schrödinger equation describes dynamical processes of one-electron species in terms of complex wave functions. The functions are inherently complex; therefore zeros occur only when both the real and imaginary parts of the wave functions vanish. If this happens at isolated points rather than on nodal surfaces one can show that the zeros must correspond to vortices. An imaging theorem is given which shows how such vortices can be seen experimentally. Since the theorem requires time propagation from microscopic to macroscopic scales, a method is developed that does just the same. Examples of vortices that emerge in dynamical processes are given. The vortices that we find are linked to the hydrodynamic interpretation of Schrödinger's time-dependent equation.
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
Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.