LANDAU LEVELS AND GEOMETRIC QUANTIZATION

1989 ◽  
Vol 04 (15) ◽  
pp. 3939-3949 ◽  
Author(s):  
J. R. KLAUDER ◽  
E. ONOFRI
Keyword(s):  
Magnetic Field ◽  
Landau Level ◽  
Free Particle ◽  
Geometric Quantization ◽  
Point Of View ◽  
Geometrical Approach ◽  
Higher Dimensional ◽  
Definition Of ◽  
The Magnetic Field ◽  
Quantum Counterpart

The geometrical approach to phase-space quantization introduced by Klauder [KQ] is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted to a specific polarization; in this way the first Landau level acquires the expected degeneracy. This unification with GQ makes it clear how algebraic relations between classical observables are or are not preserved under quantization. From this point of view all quantum systems appear as the low energy sector of a generalized theory in which all classical observables have a uniquely assigned quantum counterpart such that Poisson bracket relations are isomorphic to the commutation relations.

scholarly journals No-z Model for Magnetic Fields of Different Astrophysical Objects and Stability of the Solutions

Data ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 4
Author(s):  
Evgeny Mikhailov ◽  
Daniela Boneva ◽  
Maria Pashentseva
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Large Scale ◽  
Point Of View ◽  
Accretion Discs ◽  
Wide Range ◽  
Astrophysical Objects ◽  
Alpha Effect ◽  
The Stability ◽  
The Magnetic Field

A wide range of astrophysical objects, such as the Sun, galaxies, stars, planets, accretion discs etc., have large-scale magnetic fields. Their generation is often based on the dynamo mechanism, which is connected with joint action of the alpha-effect and differential rotation. They compete with the turbulent diffusion. If the dynamo is intensive enough, the magnetic field grows, else it decays. The magnetic field evolution is described by Steenbeck—Krause—Raedler equations, which are quite difficult to be solved. So, for different objects, specific two-dimensional models are used. As for thin discs (this shape corresponds to galaxies and accretion discs), usually, no-z approximation is used. Some of the partial derivatives are changed by the algebraic expressions, and the solenoidality condition is taken into account as well. The field generation is restricted by the equipartition value and saturates if the field becomes comparable with it. From the point of view of mathematical physics, they can be characterized as stable points of the equations. The field can come to these values monotonously or have oscillations. It depends on the type of the stability of these points, whether it is a node or focus. Here, we study the stability of such points and give examples for astrophysical applications.

scholarly journals Topology of Coronal Magnetic Fields: Extending the Magnetic Skeleton Using Null-like Points

Solar Physics ◽  
2020 ◽  
Vol 295 (12) ◽  
Author(s):  
Daniel T. Lee ◽  
Daniel S. Brown
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Free Field ◽  
Point Sources ◽  
Topological Features ◽  
Perpendicular Component ◽  
Definition Of ◽  
Magnetic Null ◽  
The Magnetic Field ◽  
Continuous Source

AbstractMany phenomena in the Sun’s atmosphere are magnetic in nature and study of the atmospheric magnetic field plays an important part in understanding these phenomena. Tools to study solar magnetic fields include magnetic topology and features such as magnetic null points, separatrix surfaces, and separators. The theory of these has most robustly been developed under magnetic charge topology, where the sources of the magnetic field are taken to be discrete, but observed magnetic fields are continuously distributed, and reconstructions and numerical simulations typically use continuously distributed magnetic boundary conditions. This article investigates the pitfalls in using continuous-source descriptions, particularly when null points on the $z=0$ z = 0 plane are obscured by the continuous flux distribution through, e.g., the overlap of non-point sources. The idea of null-like points on the boundary is introduced where the parallel requirement on the field $B_{\parallel }=0$ B ∥ = 0 is retained but the requirement on the perpendicular component is relaxed, i.e. $B_{\perp }\ne 0$ B ⊥ ≠ 0 . These allow the definition of separatrix-like surfaces which are shown (through use of a squashing factor) to be a class of quasi-separatrix layer, and separator-like lines which retain the x-line structure of separators. Examples are given that demonstrate that the use of null-like points can reinstate topological features that are eliminated in the transition from discrete to continuous sources, and that their inclusion in more involved cases can enhance understanding of the magnetic structure and even change the resulting conclusions. While the examples in this article use the potential approximation, the definition of null-like points is more general and may be employed in other cases such as force-free field extrapolations and MHD simulations.

scholarly journals Landau problem on the ellipsoid, hyperboloid and paraboloid of revolution

2014 ◽  
Vol 29 (29) ◽  
pp. 1450148
Author(s):  
Eva Gevorgyan ◽  
Armen Nersessian ◽  
Vadim Ohanyan ◽  
Evgeny Tolkachev
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Free Particle ◽  
Two Dimensional ◽  
Qualitative Character ◽  
Surface Of Revolution ◽  
Paraboloid Of Revolution ◽  
Kepler System ◽  
The Magnetic Field ◽  
Dimensional Surface

We define the Landau problem on two-dimensional ellipsoid, hyperboloid and paraboloid of revolution. Starting from the two-center McIntosh–Cisneros–Zwanziger (MICZ)–Kepler system and making the reduction into these two-dimensional surfaces, we obtain the Hamiltonians of the charged particle moving on the corresponding surface of revolution in the magnetic field conserving the symmetry of the two-dimensional surface (Landau problem). For each case we figure out the values of parameter for which the qualitative character of the motion coincides with that of a free particle moving on the same two-dimensional surface. For the case of finite trajectories we construct the action-angle variables.

scholarly journals Computation of the magnetic potential induced by a collection of spherical particles using series expansions

2020 ◽  
Vol 54 (4) ◽  
pp. 1073-1109
Author(s):  
Stéphane Balac ◽  
Laurent Chupin ◽  
Sébastien Martin
Keyword(s):  
Magnetic Field ◽  
Magnetic Potential ◽  
Point Of View ◽  
Spherical Particles ◽  
Main Magnetic Field ◽  
Minimal Distance ◽  
Metallic Particles ◽  
Scalar Magnetic Potential ◽  
The Magnetic Field ◽  
Made In

In Magnetic Resonance Imaging there are several situations where, for simulation purposes, one wants to compute the magnetic field induced by a cluster of small metallic particles. Given the difficulty of the problem from a numerical point of view, the simplifying assumption that the field due to each particle interacts only with the main magnetic field but does not interact with the fields due to the other particles is usually made. In this paper we investigate from a mathematical point of view the relevancy of this assumption and provide error estimates for the scalar magnetic potential in terms of the key parameter that is the minimal distance between the particles. A special attention is paid to obtain explicit and relevant constants in the estimates. When the “non-interacting assumption” is deficient, we propose to compute a better approximation of the magnetic potential by taking into account pairwise magnetic field interactions between particles that enters in a general framework for computing the scalar magnetic potential as a series expansion.

scholarly journals Time development of electric fields and currents in space plasmas

2006 ◽  
Vol 24 (3) ◽  
pp. 1137-1143
Author(s):  
A. T. Y. Lui
Keyword(s):  
Magnetic Field ◽  
Electric Current ◽  
Electric Fields ◽  
Time Development ◽  
Space Plasmas ◽  
Simplifying Assumptions ◽  
Electric Fields And Currents ◽  
Definition Of ◽  
The Magnetic Field ◽  
Made In

Abstract. Two different approaches, referred to as Bu and Ej, can be used to examine the time development of electric fields and currents in space plasmas based on the fundamental laws of physics. From the Bu approach, the required equation involves the generalized Ohm's law with some simplifying assumptions. From the Ej approach, the required equation can be derived from the equation of particle motion, coupled self-consistently with Maxwell's equation, and the definition of electric current density. Recently, some strong statements against the Ej approach have been made. In this paper, we evaluate these statements by discussing (1) some limitations of the Bu approach in solving the time development of electric fields and currents, (2) the procedure in calculating self-consistently the time development of the electric current in space plasmas without taking the curl of the magnetic field in some cases, and (3) the dependency of the time development of magnetic field on electric current. It is concluded that the Ej approach can be useful to understand some magnetospheric problems. In particular, statements about the change of electric current are valid theoretical explanations of change in magnetic field during substorms.

scholarly journals On the structure of the magnetic field of sunspots

1968 ◽  
Vol 35 ◽  
pp. 215-229 ◽  
Author(s):  
E. I. Mogilevsky ◽  
L. B. Demkina ◽  
B. A. Ioshpa ◽  
V. N. Obridko
Keyword(s):  
Magnetic Field ◽  
Free Field ◽  
Large Field ◽  
Doppler Velocity ◽  
Small Scale ◽  
High Excitation ◽  
Proposed Model ◽  
Background Magnetic Field ◽  
Definition Of ◽  
The Magnetic Field

The model of the magnetic field of sunspots, taking account of fine structure of magnetic field in solar plasma, is considered. Small-scale subgranules with their own field form magnetic filaments in the external current-free field. The filaments are vertical in the umbra, while in the penumbra they run along the surface with sharp bends. In a number of spot umbra the relation between Doppler velocity and the field is established on polarized spectrograms. The π-component splitting in umbra is interpreted as a result of a weak background magnetic-field existence together with a large field of magnetic filaments. Spectrographic definition of the magnetic field in spot umbra is accomplished on the effect of magnetic-lines intensification and directly on spectrograms of low-excitation (Fe I, Ti I) and high-excitation (Fe II) lines. Magnetic field measured in low-excitation lines exceeds twice the field value obtained in high-excitation lines. This result has been considered in the light of the proposed model of sunspot field.

scholarly journals Equatorial Geodesics Around the Magnetars

2017 ◽  
Vol 45 ◽  
pp. 1760050
Author(s):  
Viviane A. P. Alfradique ◽  
Orlenys N. Troconis ◽  
Rodrigo P. Negreiros
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Neutron Stars ◽  
Complete Solution ◽  
Compact Object ◽  
Relativistic Effects ◽  
Schwarzschild Solution ◽  
Soft Gamma Repeaters ◽  
Definition Of ◽  
The Magnetic Field

Neutron stars manifest themselves as different classes of astrophysical sources that are associated to distinct phenomenology. Here we focus our attention on magnetars (or strongly magnetized neutron stars) that are associated to Soft Gamma Repeaters and Anomalous X-ray Pulsars. The magnetic field on surface of these objects, reaches values greater than [Formula: see text] G. Under intense magnetic fields, relativistic effects begin to be decisive for the definition of the structure and evolution of these objects. We are tempted to question ourselves to how strengths fields affect the structure of neutron star. In this work, our objective is study and compare two solutions of Einstein-Maxwell equations: the Bonnor solution, which is an analytical solution that describe the exterior spacetime for a massive compact object which has a magnetic field that is characterize as a dipole field and a complete solution that describe the interior and exterior spacetime for the same source found by numerical methods). For this, we describe the geodesic equations generated by such solutions. Our results show that the orbits generated by the Bonnor solution are the same as described by numerical solution. Also, show that the inclusion of magnetic fields with values up to [Formula: see text]G in the center of the star does not modify sharply the particle orbits described around this star, so the use of Schwarzschild solution for the description of these orbits is a reasonable approximation.

scholarly journals The secondary corpuscular rays produced by homogeneous X-rays

1923 ◽  
Vol 104 (727) ◽  
pp. 455-479 ◽  
Keyword(s):  
Magnetic Field ◽  
Photographic Plate ◽  
High Vacuum ◽  
Radius Of Curvature ◽  
X Rays ◽  
Metallic Surfaces ◽  
Philosophical Magazine ◽  
Narrow Strip ◽  
Definition Of ◽  
The Magnetic Field

In the summer of 1914 I began, with the assistance of Mr. W. F. Rawlinson, an investigation of the velocities of the secondary cathode particles ejected from metallic surfaces by X-rays of known frequency. The source of X-rays then available was not very powerful, and little progress had been made when the work was interrupted by the war. A preliminary account of the experimental method, and of the few results which had been obtained, was communicated to the ‘ Philosophical Magazine,' and published in August, 1914. Briefly, the method consisted in exposing a narrow strip of metal to the beam of X-rays, and using a uniform magnetic field to bend the stream of ejected electrons onto a photographic plate, the experiment being performed in a high vacuum. The product r H, where H is the strength of the magnetic field and r the radius of curvature of the path of an electron moving normally to the field, is characteristic of that electron : from it the kinetic energy of the electron may readily be calculated. A special focussing device was used, and described in the paper referred to, by means of which quite a wide beam of electrons could be employed without impairing the definition of the resulting image on the photographic plate: this device not only appreciably shortens the exposures required, but also enhances the effect of the rays under investigation, relatively to that of the general scattered radiations. These latter are always troublesome, and cannot be entirely eliminated.

scholarly journals Cyclotron Line Emission from Accretion onto a Magnetized Neutron Star

1981 ◽  
Vol 93 ◽  
pp. 233-233
Author(s):  
E. E. Salpeter
Keyword(s):  
Magnetic Field ◽  
Neutron Star ◽  
Landau Level ◽  
Optical Depth ◽  
Radiative Decay ◽  
Line Emission ◽  
Collisional Excitation ◽  
Magnetic Field Axis ◽  
Cyclotron Line ◽  
The Magnetic Field

For material accreting along the magnetic field axis of a neutron star, electrons are quantized into Landau orbits. Collisional excitation of the first excited Landau level, followed by radiative decay, leads to the emission of a cyclotron line. The expected line is broad, because the optical depth is large, and its shape is difficult to calculate. Redshifts due to the recoil of a scattering electron and blueshifts due to scattering from the infalling accretion column are being calculated by I. Wasserman, as well as the proton stopping length in the presence of a magnetic field.

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