Spin-Dependent Bohmian Electronic Trajectories for Helium

We examine “de Broglie-Bohm” causal trajectories for the two electrons in a nonrelativistic helium atom, taking into account the spin-dependent momentum terms that arise from the Pauli current. Given that this many-body problem is not exactly solvable, we examine approximations to various helium eigenstates provided by a low-dimensional basis comprised of tensor products of one-particle hydrogenic eigenstates. First to be considered are the simplest approximations to the ground and first-excited electronic states found in every introductory quantum mechanics textbook. For example, the trajectories associated with the simple 1s(1)1s(2) approximation to the ground state are, to say the least, nontrivial and nonclassical. We then examine higher-dimensional approximations, i.e., eigenstates Ψ _α of the Hamiltonian in this truncated basis, and show that ∇ _i S _α =0 for both particles, implying that only the spin-dependent momentum term contributes to electronic motion. This result is independent of the size of the truncated basis set, implying that the qualitative features of the trajectories will be the same, regardless of the accuracy of the eigenfunction approximation. The electronic motion associated with these eigenstates is quite specialized due to the condition that the spins of the two electrons comprise a two-spin eigenfunction of the total spin operator. The electrons either (i) remain stationary or (ii) execute circular orbits around the z-axis with constant velocity.     

Dynamic topological solitons in a two-dimensional ferromagnet

It is shown that stable, skyrmion-type, dynamic solitons can be constructed for a wide class of two-dimensional models of anisotropic ferromagnets. These solitons are stabilized as a result of the conservation of various integrals of motion: the z projection of the total spin S _z or the orbital angular momentum L _z of the magnetization field. A class of two-parameter solitons with quite complicated (almost periodic) magnetization-field dynamics exists for a purely uniaxial model (in the sense of both spin and spatial rotations) with maximum symmetry. Stable solitons with periodic magnetization dynamics exist for ferromagnets with lower symmetry (only S _z or L _z or the total angular momentum J _z =L _z +S _z is conserved). Zh. éksp. Teor. Fiz. 115, 1511–1530 (April 1999)     

Angular Momentum and Positive Mass Theorem

Total angular momentum for asymptotically flat manifolds is defined. Positive mass theorem for initial (spin) data set (M, g _ij , p _ij ) with nonsymmetric p _ij is proved. As an application, we establish positive mass theorems involving total linear momentum and total angular momentum. This gives an answer to a problem of S. T. Yau in his Problem Section [Ya2] and a partial answer to his recent conjecture on the relationship among total energy, total linear momentum, total angular momentum and entropy of black hole. Received: 26 October 1998 / Accepted: 10 March 1999     

On long-time limit behavior of the solution of atom's master equation

We study long-time limit behavior of the solution of atom's master equation, for the first time we derive that the probability of the atom being in the α-th (α =j+1-j_z, j is the angular momentum quantum number, j_z is the z-component of angular momentum) state is {(1-K/G)/[1-(K/G)^2j+1]}(K/G)^α-1 as t→+∞, which coincides with the fact that when K/G > 1, the larger the α is, the larger probability of the atom being in the α-th state (the lower excited state). We also consider the case for some possible generalizations of the atomic master equation.     

Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Gravitational energy-momentum: The Einstein pseudotensor reexamined

By using a suitable two-point scalar field, a covariant formulation of the Einstein pseudotensor is given. A unique choice of scalar field is made possible by examining the role of linear and angular momentum in their correct geometric context. It is shown that, contrary to many text-book statements, linear momentum is not generated by infinitesimal coordinate transformations on space-time. Use is made of the nonintersecting lifted geodesies on the tangent bundle,T M, to space-time, to define a globally regular three-dimensional Lagrangian submanifold ofT M, relative to an observer at some pointz in space-time. By integrating over this submanifold rather than a necessarily singular spacelike hypersurface, gravitational linear and angular momentum, relative toz, are defined, and shown to have sensible physical properties.This essay received an honorable mention from the Gravity Research Foundation for the year 1979-Ed.     

Hadron properties and Dyson-Schwinger equations

An overview of the theory and phenomenology of hadrons and QCD is provided from a Dyson-Schwinger equation viewpoint. Following a discussion of the definition and realization of light-quark confinement, the nonperturbative nature of the running mass in QCD and inferences from the gap equation relating to the radius of convergence for expansions of observables in the current-quark mass are described. Some exact results for pseudoscalar mesons are also highlighted, with details relating to the U_A(1) problem, and calculated masses of the lightest J=0,1 states are discussed. Studies of nucleon properties are recapitulated upon and illustrated: through a comparison of the ln-weighted ratios of Pauli and Dirac form factors for the neutron and proton; and a perspective on the contribution of quark orbital angular momentum to the spin of a nucleon at rest. Comments on prospects for the future of the study of quarks in hadrons and nuclei round out the contribution.     

High excitation energy structures in208Pb

Level-crossing signals of impact radiation of He I have been analyzed with regard to the determination of excitation matrices. The signals have been registered at various electricfield strengths by sweeping a magnetic field strength ?_z through the positions ofΔm=2 crossings of Zeeman substates ofn ³ D ₂ andn ³ D ₃ levels (n=4 and 5) of He I excited by 35keV-He⁺-He collisions. We analyzed the signal amplitudes and signal shapes with regard to a spin-dependence of the excitation matrix usually discarded by theoretical arguments. To this end a parametrization of spin-dependent excitation matrices has been performed, and the dependence of the level-crossing amplitudes on the 19 parameters obtained have been studied. It turned out that the observed relative signal amplitudes can satisfactorily be explained by only two parameters, the two multipole components of orderk=2 andk=4 of the alignment of the orbital angular momentum. Thus in agreement with theoretical estimations no spin-dependence must be taken into account.     

On the Dirac equation with anomalous magnetic moment term in a beam of electromagnetic wave

The object of the paper is to obtain the solution of the Dirac equation with the Pauli-term in an electromagnetic field depending on the single variable (ct -nr) along the directionn.     

Dirac Particle in an Aharonov-Bohm Potential: The Structure of the First Order S-matrix

The structure of the interaction Hamiltonian in the first order S-matrix element of a Dirac particle in an Aharonov-Bohm (AB) field is analyzed and shown to have interesting interesting algebraic properties. It is demonstrated that as a consequence of these properties, this interaction Hamiltonian splits both the incident and outgoing waves in the the first order components (eigenstates of the third component of the spin). The matrix element can then be viewed as the sum of two transitions taking place in these two channels of the spin. At the level of partial waves, each partial wave of the conserved total angular momentum is split into two partial waves of the orbital angular momentum in a manner consistent with the conservation of the total angular momentum quantum number.     

Dual electromagnetism and magnetic monopoles

Dual electromagnetism (proposed some time ago) allows the fractional electric charges and the magnetic monopoles to exist simultaneously. In fact, the Dirac quantization condition can be numerically reduced (with some plausible assumptions) to the third component of the particle total weak isospin, which by definition is always quantized. The field angular momentum,L, of a static particle-magnetic monopole configuration is evaluated exactly; it is found that because the dual photon has a mass,M _c ,L generally depends onr, the separation between a particle and a monopole. However, sinceM _c - 130 GeV, atr > M _c ^–1 ,L is basically dominated by ordinary electromagnetism and as such very weakly dependent onr.     

Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

In this paper we show how the dynamics of the Schr?dinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, C_1,3, C_3,0{\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}, and C_0,1{\mathcal{C}}_{0,1}. Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.     

Rotating QCD string and the meson spectrum

The spectra of light–light and heavy–light mesons are described by spinless Salpeter equation and Dirac equation respectively, which predict linear dependence of the meson mass squared M² on angular momentum J and number of radial nodes n. Both spectra are computed by the WKB method and shown to agree with exact numerical data within few percent even for the lowest levels. The drawback of Salpeter and Dirac equation is that (inverse) Regge slopes do not coincide with the string ones, 2πσ and πσ respectively, because the string dynamics is not taken into account properly. The lacking string rotation is introduced via effective Hamiltonian derived from QCD which generates linear Regge trajectories for light mesons with the correct string slope.     

Complex Parameters in Quantum Mechanics

The Schrödinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes, it is shown that in such equation the coefficient of r ^–2 is an even function of a parameter, say , depending on a linear combination of q and of the angular momentum quantum number, say l. Thus, the case of complex values of , which is useful in scattering theory, involves, in general, both a complex value of the parameter originally viewed as the spatial dimension and complex values of the angular momentum quantum number. The paper ends with a proof of the Levinson theorem in an arbitrary number of spatial dimensions, when the potential includes a non-local term which might be useful to understand the interaction between two nucleons.     

Hawking Radiation of Charged Particles from a Rotating Black String

Using Damour-Ruffini method, Hawking radiation of rotating black strings is studied. Under the condition that the total energy, total angular momentum and total charge are conservative, the transition probability from initial state (energy M+ω, charge Q+e and angular momentum J+m) to final state (energy M, charge Q and angular momentum J) for black strings is derived considering the reaction of radiation particles to spacetime. That is, the probability that black strings radiate particles with energy ω, charge e and angular momentum m is obtained. The real spectrum is not a strictly pure thermal spectrum. Our result is consistent with Parikh and Wilczek’s result. It satisfies the unitary principle of quantum mechanics. However, in our result there are not only the term that denotes effect of energy and charge of radiation particles but also the term that denotes effect of radiation particles angular momentum on rotating black strings angular momentum. We provide a new way for investigating radiation of black strings.     

Three-Qubit Entangled Embeddings of <Emphasis Type="Italic">CPT</Emphasis> and Dirac Groups within <Emphasis Type="Italic">E</Emphasis><Subscript>8</Subscript> Weyl Group

In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group ℘, and two-qubit Pauli group ℘₂, respectively. It has been found (Socolovsky, Int. J. Theor. Phys. 43: 1941, 2004) that the CPT group of the Dirac equation is isomorphic to ℘. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E ₈) is generated (Planat, Preprint , 2009). In this paper, one derives three-qubit entangling groups [(P)\tilde]\tilde{\mathcal{P}} and [(P)\tilde]₂\tilde{\mathcal{P}}_{2}, isomorphic to the CPT group ℘ and to the Dirac group ℘₂, that are embedded into W(E ₈). One discovers a new class of pure three-qubit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families, and also chain-type states, encode the new representation of the Dirac group and its CPT subgroup.     

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.     

EXACT SOLUTION OF THE PAULI EQUATION IN THE FIELD OF A 'T HOOFT-POLYAKOV MONOPOLE

In this paper, the motion for the spin one half particles with arbitrary isospin in the field of a't Hooft-Polyakov magnetic monopole (in the Prasad-Sommerfield limit) is studied by the method of the moving frame. An exact solution of the Pauli equation for the total angular momentum J=0 and isospin I=1/2 is presented. The question of Proton decay catalysis is briefly discussed.     

Precession of angular momentum vector in a slowly rotating Kerr metric

Specializing Penrose and Floyd's resultF _ab;c =F _[ab;c] to the Kerr metric, we explicitly construct the skew symmetric tensorF _ab and Carter's quadratic integral of geodesic motion.F _ab is then shown to be closely related to the orbital angular momentum encountered in Newtonian mechanics. Furthermore, F_ab can be decomposed additively intoL _ab andM _ab , whereL _ab has the character of angular momentum, andM _ab exists only for a nonzero rotation parameter,a, of the Kerr metric. It turns out that the equation of precession = _a ^b L _b has a nontrivial solution only for the case of a slowly rotating Kerr metric valid to first order in rotation parameter. In this case, Carter's integral can be interpreted as the squared length of the precessing angular momentum vectorL _a =L _a ^b P _b . The equation of precession is solved, and a vector ^a describing angular velocity of precession is derived.     

Calculating coefficients for a system of moment equations used to describe the magnetization kinetics of a superparamagnetic particle in a fluctuating field

A system of equations is derived for moments [averages of spherical harmonics 〈Y _l,m 〉(t)] that determine the dynamics of the magnetization M of a superparamagnetic particle in a fluctuating field. The system is derived by representing the Gilbert equation in a fluctuating field, and the corresponding Fokker-Planck equation for the distribution function of M, in terms of angular momentum operators, which in turn makes it possible to express the coefficients of the system of moment equations in terms of Clebsch-Gordan coefficients. Fiz. Tverd. Tela (St. Petersburg) 41, 2020–2027 (November 1999)