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# Probability current tornado loops in three-dimensional scattering

**Article**

*in*Physics Letters A 245(s 1–2):35–39 · August 1998

*with*16 Reads

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DOI: 10.1016/S0375-9601(98)00366-1 · Source: arXiv

Cite this publicationAbstract

We consider scattering of a three-dimensional particle on a finite family of δ potentials. For some parameter values the scattering wavefunctions exhibit nodal lines in the form of closed loops, which may touch but do not entangle. The corresponding probability current forms vortical singularities around these lines; if the scattered particle is charged, this gives rise to magnetic flux loops in the vicinity of the nodal lines. The conclusions extend to scattering on hard obstacles or smooth potentials.

- ... As a first example the field of an tornado loop [11] is shown. It shows the probability current of a quantum mechanical plane wave scattered on a d-function potential in three dimensions. ...Article
- Jan 2004

Lines are widely use in three-dimensional graphics. The most graphics systems render lines with a color independent from the lighting conditions in the scene. Correct lighting of lines can improve the visual appearance of scenes with many lines or long lines significant. The paper will explain the lighting calculations used in MathGL3d and show some applications for the visualization of dynamical systems and field lines. MathGL3d uses free programmable 3d graphics hardware to modify the lightning calculation for lines and can display scenes with 3·10 6 vertices in the lines in real-time. - Article
- Jan 2006

Mathematica is today's most advanced technical computing system. It features a rich programming environment, two-and three-dimensional graphics capabilities and hundreds of sophisticated, powerful programming and mathematical functions using state-of-the-art algorithms. Combined with a user-friendly interface, and a complete mathematical typesetting system, Mathematica offers an intuitive easy-to-handle environment of great power and utility. "The Mathematica GuideBook for Symbolics" (code and text fully tailored for Mathematica 5.1) deals with Mathematica's symbolic mathematical capabilities. Structural and mathematical operations on single and systems of polynomials are fundamental to many symbolic calculations and they are covered in considerable detail. The solution of equations and differential equations, as well as the classical calculus operations (differentiation, integration, summation, series expansion, limits) are exhaustively treated. Generalized functions and their uses are discussed. In addition, this volume discusses and employs the classical orthogonal polynomials and special functions of mathematical physics. To demonstrate the symbolic mathematics power, a large variety of problems from mathematics and phyics are discussed. © 2006 Springer Science+Business Media, Inc. All rights reserved. - Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.

- Article
- Mar 1979
- J CHEM PHYS

We show how the presence of a conical intersection in the adiabatic potential energy hypersurface can be handled by including a new vector potential in the nuclear‐motion Schrödinger equation. We show how permutational symmetry of the total wave function with respect to interchange of nuclei can be enforced in the Born–Oppenheimer approximation both in the absence and the presence of conical intersections. The treatment of nuclear‐motion wave functions in the presence of conical intersections and the treatment of nuclear‐interchange symmetry in general both require careful consideration of the phases of the electronic and nuclear‐motion wave functions, and this is discussed in detail. - Hermann: Paris 1967, Sec. III.8; V. Jarník, Differential Calculus II, Academia: Prague
- Jan 1956

- L Schwartz
- Analyse Mathématique

L. Schwartz, Analyse Mathématique, Hermann: Paris 1967, Sec. III.8; V. Jarník, Differential Calculus II, Academia: Prague 1956, Sec.VIII.1. - Article
- Oct 1996
- PHYS LETT A

In addition to the conventional renormalized-coupling-constant picture, point interactions in two and three dimensions are shown to model within a suitable energy range scattering on localized potentials, both attractive and repulsive. - Article
- Jan 1992
- REV MOD PHYS

The fundamental theory of the geometric phase is summarized in a way suitable for use in molecular systems treated by the Born-Oppenheimer approach. Both Abelian and non-Abelian cases are considered. Applications discussd include the Abelian geometric phase associated with an intersection of two electronic potential-energy surfaces; screening of nuclei by the electrons from an external magnetic field; non-Abelian gauge potentials in molecular systems with Kramers degeneracy; and the coupling between different electronic levels (Born-Oppenheimer breakdown) represented as a gauge potential. Experimental tests for these systems are discussed, as well as a number of experiments on spin systems. - We consider resonant vortices around nodal points of the wave function describing electron transport through a mesoscopic device. With a suitable choice of the device geometry, the dominating role is played by single vortices with a preferred orientation. To characterize the strength of the resulting magnetic moment, we have introduced a ``magnetance,'' the quantity defined in analogy with the device conductance. Its basic properties and possible experimental detection are discussed.
- In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.
- Article
- Dec 1977
- J CHEM PHYS

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. - Jan 1957
- 485

- Jetp Abrikosov

Abrikosov, JETP 32, 123 (1957); Y. Aharonov, D. Bohm, Phys. Rev. 56, 485 (1959).- Article
- Jan 1994
- MATH RES LETT

A b s t r a c t . The Ginzburg-Landau energy minimization problem for a vec-tor field on a two dimensional disc is analyzed. This is the simplest non-trivial example of a vector field minimization problem and the goal is to show that the energy minimizer has the full geometric symmetry of the problem. The standard methods that are useful for similar problems in-volving real valued functions cannot be applied to this situation. Our main result is that the minimizer in the class of symmetric fields is stable, i.e., the eigenvalues of the second variation operator are all nonnegative. - Article
- Dec 1993
- PHYS LETT A

Viewing electron transmission as a probability flow problem, we study the stream lines of the flow. Velocity nodes are found at the points of wavefunction stationary phase, while vortices occur at wavefunction nodes. Analytical results are given for stream lines around such points. The sum of the action along a closed path and the charge times the enclosed magnetic flux is quantized in terms of the number of “quantum vortices”. - Article
- Aug 1997
- PHYS LETT A

We examine the quantum motion of two particles interacting through a contact force which are confined in a rectangular domain in two and three dimensions. When there is a difference in the mass scale of two particles, adiabatic separation of the fast and slow variables can be performed. Appearance of the Berry phase and magnetic flux is pointed out. The system is reduced to a one-particle Aharonov-Bohm billiard in two-dimensional case. In three dimension, the problem effectively becomes the motion of a particle in the presence of closed flux string in a box billiard. - Article
- Oct 1995
- Phys Rev B

The Aharonov-Bohm effect has been measured in nanostructures such as those obtained with two quantum point contacts in series. The measured conductance agrees with a transmission given by the Airy formula. We perform a full quantum mechanical calculation in terms of scattering of the edge states using a generalization of the Airy formula for the nondiagonal transmitted amplitude matrix of one of the quantum point contacts. Our model consists of two δ barriers interrupting a parabolic confining potential acting on a two-dimensional electron gas under a perpendicular magnetic field. We perform a critical analysis of the use of the standard Airy formula within this model. Our main result is the good agreement, due to a strong backscattering suppression, of the full calculation with the semiclassical one that uses the phase acquired by the wave function along the classical path contained in the cavity. - The two-dimensional spatial distribution of the current and Fermi carriers around localized elastic scatterers in phase-coherent electron transport has been calculated using a generalized scattering-matrix approach. The distributions show dramatic differences depending on whether the scatterers are attractive (donorlike) or repulsive (acceptorlike). We find that attractive scatterers can produce strong vortices in the current, resulting in localized magnetic moments, while repulsive scatterers produce much weaker vortices and may not produce any at all in quasiballistic transport (few impurities). This is a significant difference between majority-carrier transport (when the scatterers are attractive) and minority-carrier transport (scatterers are repulsive). The vortices are caused by quantum-mechanical interference between scatterers and are accentuated by evanescent modes which have a stronger effect in the case of attractive scatterers owing to the formation of quasidonor states. We also examine the influence of the impurity configuration (positions of the scatterers) on the nature of the vortices.
- Article
- Apr 1993
- Phys Rev B

We have analyzed the spatial flow pattern in two-dimensional ballistic electron waveguides with circular bends. For a double bend the conductance may still be perfectly quantized in spite of the strong mixing of modes. In narrow energy regions just below the subband thresholds there is strong interference between localized and propagating solutions, causing an interference blockade. Within these regions the current flow becomes vortical. Minor changes in, e.g., energy cause drastic changes in the flow pattern. - Article
- Aug 1993
- Phys Rev B

We study a microjunction in a two-dimensional electron gas under the action of a uniform magnetic field, orthogonal to the gas plane. The microjunction is modeled as an infinite rectilinear waveguide with parabolic walls interrupted by a barrier. Low magnetic fields suppress the backscattering and improve the conductance quantization with sharper transmission steps. At higher field values an unexpected feature emerges: the resonant tunneling through the barrier. Just near the threshold, at energies for which the traversal of the barrier is classically forbidden, well-defined transmission peaks appear which correspond to quasibound states trapped inside the barrier. - We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width $ \ell $ in the common boundary. We show that such a system has at least one bound state for any $ \ell>0 $. We find the corresponding eigenvalues and eigenfunctions numerically using the mode--matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase--shift spacing statistics shows that in distinction to closed pseudo--integrable billiards, the present system is essentially non--chaotic. Finally, we illustrate time evolution of wave packets in the present model. Comment: LaTeX text file with 12 ps figures