External Scalar Potential Research Articles

HIGHER ORDER FIRST INTEGRALS OF MOTION IN A GAUGE COVARIANT HAMILTONIAN FRAMEWORK

The higher order symmetries are investigated in a covariant Hamiltonian formulation. The covariant phase-space approach is extended to include the presence of external gauge fields and scalar potentials. The special role of the Killing–Yano tensors is pointed out. Some nontrivial examples involving Runge–Lenz type conserved quantities are explicitly worked out.

Connection between Coulomb and harmonic oscillator potentials in relativistic quantum mechanics

The Levi-Civita transformation is applied in the two-dimensional (2D) Dirac and Klein–Gordon (KG) equations with equal external scalar and vector potentials. The Coulomb and harmonic oscillator problems are connected via the Levi-Civita transformation. These connections lead to an approach to solve the Coulomb problems using the results of the harmonic oscillator potential in the relativistic systems mentioned above.

Dirac Field on Moyal-Minkowski Spacetime and Non-commutative Potential Scattering

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov's formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.

Influence of Auxiliary Equation on Wave Functions for Time-Dependent Pauli Equation in Presence of Aharonov–Bohm Effect

Invariant operator method for discrete or continuous spectrum eigenvalue and unitary transformation approach are employed to study the two-dimensional time-dependent Pauli equation in presence of the Aharonov–Bohm effect (AB) and external scalar potential. For the spin particles the problem with the magnetic field is that it introduces a singularity into wave equation at the origin. A physical motivation is to replace the zero radius flux tube by one of radius R, with the additional condition that the magnetic field be confined to the surface of the tube, and then taking the limit R → 0 at the end of the computations. We point that the invariant operator must contain the step function θ(r – R). Consequently, the problem becomes more complicated. In order to avoid this difficulty, we replace the radius R by ρ(t)R, where ρ(t) is a positive time-dependent function. Then at the end of calculations we take the limit R → 0. The qualitative properties for the invariant operator spectrum are described separately for the different values of the parameter C appearing in the nonlinear auxiliary equation satisfied by ρ(t), i.e., C > 0, C = 0, and C < 0. Following the C's values the spectrum of quantum states is discrete (C > 0) or continuous (C ≤ 0).

Dirac equation for electrodynamic model particles

We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.

Timelike trajectories with fixed energy under a potential in static spacetimes

By variational methods, we study the motions of a relativistic particle under the action of an external scalar potential. We consider standard static spacetimes whose metric coefficients grow at most quadratically at infinity. Fixing suitable values for the energy, we obtain timelike trajectories.

An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation

Presented is quantum lattice-gas model for simulating the time-dependent evolution of a many-body quantum mechanical system of particles governed by the non-relativistic Schrödinger wave equation with an external scalar potential. A variety of computational demonstrations are given where the numerical predictions are compared with exact analytical solutions. In all cases, the model results accurately agree with the analytical predictions and we show that the model's error is second order in the temporal discretization and fourth order in the spatial discretization. The difficult problem of simulating a system of fermionic particles is also treated and a general computational formulation of this problem is given. For pedagogical purposes, the two-particle case is presented and the numerical dispersion of the simulated wave packets is compared with the analytical solutions.

Synoptic models of high latitude magnetic activity and equivalent ionospheric and induced currents

Hourly models of the northern high latitude magnetic activity and the equivalent ionospheric and induced currents have been made for the entire 1980 period using data from 55 magnetic observatories. Vector observations and the method of spherical cap harmonic analysis are used to simultaneously model the external and internal scalar potentials from which the disturbance vector field and the corresponding equivalent currents can be determined anywhere and at anytime on, over and under the cap. This modelling technique, when used with observatory mean hourly values to reduce spatial aliasing of irregular disturbances and with the quiet nighttime (undisturbed) reference level, provides reasonable synoptic (basic) models (errors 5–80 nT). Only the statistically significant coefficients are retained in the hourly models so those for a full year (8784 models) can be readily stored and used on small computer systems. However, there are numerous regions that are in need of observatories and such additional data could significantly improve the delineation of large irregular disturbances. These synoptic models of the high latitude disturbance fields and the equivalent currents have a variety of applications. The models of the disturbance field might be used for correcting some magnetic surveys for the temporal variations, for the analyses of the large scale equivalent ionospheric and induced current systems and for determining parameters of the activity. Maps of the external currents provide an indication of their spatial extent and, with other observations, can be used to study the electrodynamics and the heating of the upper atmosphere and the magnetosphere-ionosphere coupling. Similarly, maps of the induced currents show their spatial extent, variability and the influences of the conducting oceans and other crustal features. Timely data and models of the activity could be used to help mitigate the effects of active conditions on surveys and also the hazardous effects of storms on some technological facilities.

Semiclassical expansions up to -order in relativistic nuclear physics

We present the first calculation of the Wigner - Kirkwood corrections to a relativistic system of fermions in the presence of external scalar and vector potentials. The method we propose allows the efficient computation of semiclassical corrections to one-body operators such as mean energies and local densities. It also preserves gauge invariance and produces explicitly convergent results despite some apparent divergences at the classical turning points. As a byproduct we obtain the corrections stemming from the polarization of the Dirac sea. We apply our results to the relativistic - Lagrangian in the Hartree valence approximation. We compare the semiclassical expansion with the exact result and with the Strutinsky average whenever it can be obtained. We find that the corrections are much smaller than typical shell effects. Our results provide convincing arguments for neglecting higher than second order effects in the Wigner - Kirkwood scheme on relativistic nuclear physics in the mean field approximation.

The higher derivative expansion of the effective action by the string-inspired method. I

The higher derivative expansion of the one-loop effective action for an external scalar potential is calculated to order O(T**7), using the string-inspired Bern-Kosower method in the first quantized path integral formulation. Comparisons are made with standard heat kernel calculations and with the corresponding Feynman diagrammatic calculation in order to show the efficiency of the present method.

Aharonov-Bohm effects of the photon

Although completely neutral systems are not expected to manifest the Aharonov-Bohm effect, it is shown that photons propagating through the vector potential field of a string of magnetic flux can, in principle, give rise to flux-dependent phenomena by means of nonlinear quantum electrodynamic processes involving virtual electtron-positron pairs. The magnetic flux dependence is determined for lowest order contributions to (i) vacuum polarisation, (ii) light scattering by an external scalar potential, and (iii) light-light scattering.

Classical path approximation for one-electron Green’s function in a magnetic field and the density functional model

This paper is an extension of our previous work on density functional models of electrons in magnetic fields. A full ‘‘classical path’’ (CP) one-body Green’s function is used in the calculation of the kinetic functional and the one particle density. This treatment leads to a more precise expression of the density functional theory in the strong B case, in which magnetic fields are much stronger than the external scalar potential. In the semiclassical limit t→0, the CP Green’s function is checked by an exact solvable system, interactionless electrons in a central harmonic potential and a constant magnetic field. It turns out that the Green’s function is correct through the order O(t2). Furthermore, in the high B limit (but still t→0), the leading terms in B of the CP Green’s function are the same as those of the exact Green’s function in the entire t-expansion series.

A time-dependent spin density functional theory for the dynamical spin susceptibility

Within the framework of the time-dependent spin density functional formalism, we study the dynamical spin susceptibility [Formula: see text] of a general system of interacting electrons in the presence of a static external scalar potential. Based on the stationary property of the action functional, a variational expression for the real part of [Formula: see text] is derived: [Formula: see text]. The result is valid for a general exchange-correlation (XC) action functional. In particular, the XC factor 1(q,ω) is manifestly dependent on q and ω. The imaginary part Im [Formula: see text] is then determined through the Kramers–Kronig relation. Practical approximations for the required XC kernel are also discussed

A class of solvable Pauli–Schrödinger Hamiltonians

The Pauli–Schrödinger equation for a spin 1/2, neutral particle with a nonvanishing magnetic moment μ0, interacting with an external scalar potential V and a static magnetic field B, both functions of only one of the coordinates, is solved exactly for four different choices of the potential and the field. By choosing in the examples the coordinate y, we present these solutions in the following cases: (i) V( y)=0, B( y)=(B0sin κy, 0, B0cos κy) where B0 and κ are two arbitrary constants. (ii) V( y)=λ′y2, B( y)=‖αy‖(cos 2κy, 0, sin 2κy) where 6* λ=ℏ2/2m and α,κ arbitrary constants and λ′=λ(μ0/2κλ)2. (iii) V( y)=λ′(c̄1tan αy+c̄2cot αy)2, B( y)=B( y) (sin (2κy+2δ(y)), 0, cos (2κy+2δ( y)) where B2( y)=(c̄1tan αy+c̄2cot αy)2 (+α2/4κ2)(c̄1sec2αy−c̄2csc2αy)2 and tan 2δ( y)=(2κ/λ)((c̄1tan αy+c̄2cot αy) /(c̄1sec2αy−c̄2csc2αy)); c̄1, c̄2, α, and κ are arbitrary constants. (iv) V( y)=λ′(ā tanh z+b̄)2 where z =[(y−cd)/d], B2( y)=(ā tanh z+b̄)2+(ā2/4κ2d2)sech2z, tan 2δ( y)=(2κd/ā)((ā tanh z+b̄)/sech2z); ā, b̄, c, d, and κ are arbitrary constants restricted only by (μ0/2κλ)ā2+ā/d&amp;gt;0. The functions B( y) and δ( y) define the vector B( y) as in (iii). Our method of solution is based on the familiar factorization technique for solving the Schrödinger eigenvalue equation. Several interesting physical results of these solutions are discussed.

Dynamic Response of Inhomogeneous Fermi Systems

This paper presents a theory of the linear density response to an external time-dependent scalar potential for a Fermi system whose unperturbed density varies slowly. Simple local-density-functional response theory is valid except in a region of small $q$ and $\ensuremath{\omega}$ (wave number and frequency of the perturbation). For this region we have worked out a generalization, to the case of inhomogeneous systems of Landau Fermi-liquid theory.

Infrared divergence cancellation in pure Yang-Mills theory

Virtual and real corrections to massless external lines in pure Yang-Mills theory are considered in order to look for general features of the infrared divergence cancellation. Use of the Ward identities and sums over transverse polarization states give rise to terms formally corresponding to real ghost emission, cancelling ghost loop singularities, and to a factorisation of the hard narrow single gauge boson emission. Other virtual corrections are examined in the soft region and a graph cancellation is also found. An illustrative explicit calculation of scattering of a gauge particle in an external scalar potential, including hard narrow angle emission is presented.

The ideal Boson gas in an external scalar potential

We consider a new way of going to the infinite volume thermodynamic limit for a finite density quantum system and apply it to the case of an ideal Boson gas. We describe two procedures for calculating the particle density in the thermodynamic limit, one local and one global, and show that they give different values for the density. Further calculations show that this discrepancy is caused by lack of macroscopic translation invariance of the system, which is not apparent at the microscopic level. We calculate the limiting value of the expectation function of the Weyl operators both above and below the critical density for Bose-Einstein condensation, and show that the condensate has paradoxical properties of a similar type to those recently discovered for the rotating Boson gas.

Nonlinear conductivity in non-relativistic classical electron plasmas

A non-relativistic magnetic field-free classical electron plasma is perturbed from equilibrium by a small external scalar potential and the second order current response and conductivity are calculated.