Arbitrary External Field Research Articles

Magnetic field dependence of hydrogenic energy levels

The energy levels of hydrogen-like systems in arbitrary external magnetic fields are calculated using hybrid variational wave functions. Such functions enable the calculation energies for levels corresponding to n = 1 through n = 5 levels of the free hydrogen atom. We use much smaller determinants than in previously reported variational calculations. In many cases there is noticeable improvement on previously reported energies in low, intermediate and high field regimes.

Thermodynamics of interfacial oscillations in two-dimensional systems

The quiescent two-phase interface density profile and its capillary wave broadening are obtained heuristically. In two-dimensional space, the drumhead model in an arbitrary external field is transformed into a pair of Bloch equations and solved in special cases. It is extended to encompass a fixed liquid volume change. The original model is then solved completely and expressed as a free energy density functional, which is likewise extended. Finally, the genesis of the singular structure of the free energy is discussed. 14 references, 1 figure.

Final-state interferences in the radiative decay of hydrogen (n=2)

The angular and spectral distributions of the radiation emitted by metastable $n=2$ hydrogenic systems in arbitrary external electric and magnetic fields are analyzed. Special attention is given to level crossings as a means to enhance some of the final-state interactions which are possible only through the coupling with the radiative sector and the radiative reaction. A complete classification of the final-state interferences is given in terms of the possible scalar invariants and their symmetry with respect to time reversal. It is shown that the final-state interferences can dominate the radiative decay at one of the level crossings where the decoupling of the electronic and nuclear spin is important. The evaluation of the line shapes of these final-state interferences through the relevant level crossings allows one to consider their applications to lifetime measurements. Finally the presence of very small radiative corrections and recoil effects are analyzed and their physical consequences are discussed.

Local momentum-space representation of graviton propagators in an external gravitational field and one-loop counterterms in quantum gravity

Local momentum-space representation of graviton and ghost propagators in an arbitrary external gravitational field is constructed. It is shown that this representation allows to easily obtain one-loop counterterms in quantum gravity.

Nonlinear response theory—I

A general nonlinear response theory for the case of linear coupling of physical systems to arbitrary external fields is formulated for applications in different branches of physics. This is done within the framework of non-relativistic density matrix approach of quantum mechanics. Some simple properties of response functions and other related functions, which are introduced here for convenience, are studied to obtain suitable representations of the nonlinear response functions, including important sum-rules. As an example, the sum rule for the second-order response function is applied to electronic dipole nonlinearity at optical frequencies which includes both the Raman nonlinearity arising from perturbation to the electronic motion from external ionic displacement field and the usual optical sum, difference and harmonic generations. This immediately allows us to visualize a rigorous connection between these two types of non-linearities.

Mayer expansions for gas of contours at low temperatures and in arbitrary external fields for the multicomponent ising model

Mayer expansions for gas of contours at low temperatures and in arbitrary external fields for the multicomponent ising model

Quasiclassical trajectory-coherent states of a particle in an arbitrary electromagnetic field

In this paper we show that for a nonrelativistic charged particle moving in an arbitrary external electromagnetic field there exist approximate solutions of the Schrödinger equation, such that the quantum-mechanical averages of the coordinates and the momenta with respect to these states are general exact solutions of the classical Hamiltonian equations. Such states are called trajectory-coherent states. The wave functions of the trajectory-coherent states are obtained by the complex germ method by V. P. Maslov. The simplest properties of these states are studied.

Dipole moment fluctuation formulas in computer simulations of polar systems

The fundamental equation of electrostatics, i.e. the integral equation for the polarization of a macroscopic dielectric in an arbitrary external field, is written in a form that allows explicit inclusion of the toroidal boundary conditions as well as the cutoff of dipolar interactions often used in the computer simulation of polar systems. The toroidal boundary conditions are accounted for in a natural way if the integral equation is formulated (and solved) in Fourier space. Rigorous expressions for the polarization induced by a homogeneous field as well as for the equilibrium dipole moment fluctuations, both as a function of the dielectric constant, are then easily derived for general systems in two and three dimensions. The equations obtained for spherical cutoff geometry with reaction field are identical to those valid for an infinite periodic system (Ewald sum plus reaction field). When applied to the case of a highly polar Stockmayer system (μ*2=3·0, θ*=0·822, T*=1·15) the formulas lead to a consiste...

Some results of high-temperature expansions for the ising model in an arbitrary magnetic field

The high-temperature expansions of the function ϕ(χ,α), viz., the Legendre transform of the specific logarithm of the statsum of the Ising model in an arbitrary external field, are investigated. The computed surface of the thermodynamic potential ϕ is well described by the scaling form of the similarity hypothesis. For lattices with second-neighbor interaction the calculations clearly indicate the weak nonuniversality of the critical behavior (the index γ for the fcc-2 and sc-2 lattices with coordination number q=18 is closer to the value 1.21 than to the standard value 1.25). It is noted that the bcc-2 lattice has a universal behavior, i.e., nonuniversality is observed for lattices with coordination number q⩾18. High-temperature series in temperature-magnetization variables are presented in the Appendix for three-space lattices with both first-neighbor as well as second-neighbor interactions, to within the ninth and the eighth orders, respectively.

Pseudoclassical description of relativistic spinning particles with anomalous magnetic moment

We present a completely consistent lagrangian formulation for a supersymmetric test particle with an anomalous magnetic moment in the presence of an arbitrary external electromagnetic field. We show that such a description is also possible in a supersymmetrical approach without breaking supersymmetry.

Gravitational perturbation of the hydrogen spectrum

The perturbations of the energy levels of a freely falling one-electron atom in an arbitrary external gravitational field are considered. The energy-level shifts are calculated to first order in the Riemann tensor for the relativistic $2{P}_{\frac{3}{2}}$ levels and the nonrelativistic $3S$, $3P$, and $3D$ levels. These and earlier results are evaluated explicitly for atoms in radial orbits of the Schwarzschild and parametrized post-Newtonian metrics and for atoms in circular orbits of the Schwarzschild metric. Highly excited Rydberg atoms are also discussed.

Higher spin one-dimensional Ising lattice in arbitrary external field

We consider an integer lattice in one dimension whose site variables take on the values ν = 0,1,...,D with a fixed nearest neighbor interaction but an arbitrary site-dependent external potential. By first eliminating the external potential in favor of the site probability density, an expression is found in principle for the potential as a functional of the density. This relation is worked out in detail for basic spin 1/2 model, Z3 lattice, random walk ensemble, and a special continuous spin model. The direct correlation function in all cases has only nearest neighbor support, and the thermodynamic potential as a functional of the density couples only nearest neighbor sites.

One-dimensional classical fluid with nearest-neighbor interaction in arbitrary external field

We consider the equilibrium statistical mechanics of a classical one-dimensional simple fluid, with nearest-neighbor interactions, and in an arbitrary external potential. The external potential is eliminated to yield relations between the truncated partition functions and the one-body density. These relations are solved for pure cores and for sticky cores, resulting in each case in both an explicit potential density relation and grand potential density functional. Both models maintain finite-range direct correlations and have grand potentials expressible in terms of simple linear density transforms.

Renormalization-group calculation of superfluid density and helicity modulus

The field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous, $d$-dimensional systems, which (close to their critical temperature) can be described in terms of the Landau-Ginzburg-Wilson functional integral with a two-component order parameter, and applied to obtain the scaling form of the helicity modulus, defined as a measure of the free-energy increment associated with a metastable, helical twisting of the order parameter. The general scaling form of this quantity is determined even in the presence of an external helical field, for arbitrary helix pitch and temperature. In the case of vanishing external field, this quantity represents the superfluid density and is a function of the temperature and the superfluid velocity (which corresponds to the inverse of the helix pitch). In the limit of small velocities this expression exhibits the well-known Josephson asymptotic form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{\nu}(d\ensuremath{-}2)}={({T}_{c}\ensuremath{-}T)}^{2\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$. An explicit $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion for the helicity modulus is presented in the case of arbitrary external field to the one-loop approximation.

Model grand potential for a nonuniform classical fluid

Classical equilibrium statistical mechanics of one-dimensional hard cores in an arbitrary external field is reviewed. The grand canonical potential is shown to be an extension of local thermodynamics in which suitable averages of density and pressure are used. A general three-dimensional system with arbitrary internal and external potentials is modeled in corresponding form, and the external potential represented as a functional of the density. Comparison with assumed bulk data permits evaluation of the required averages. It is shown that the wall equation of state for a wall-bounded fluid is satisfied exactly. Application is also made to the self-maintained density profile of the two-phase fluid at a first order phase transition.

Analytical properties of the quark polarization operator in an external self-dual field

It is shown that in the lowest orders in α s, light quark polarization operators in an arbitrary multi-instanton external field have the asymptotic form Π( q 2) ∼ Π (0) + C/ q 2 + exponential terms and the terms ∼1/ q 4, ∼1/ q 6, are absent. We use the operator expansion language that corresponds to the vanishing of the contributions of higher-dimension operators to the polarization operator in the lowest order in α s in a self-dual external field. It is shown that in a self-dual field quark and gluon vacuum condensate densities are strictly connected: m〈 ψψ〉 0=− 1 8π 〈α sG a μνG a μν〉 0. This relation is strongly violated in experiment, displaying the inapplicability of the saddle approximation in calculating continual integrals. The abelian case is also discussed. The physical pictures in the abelian and non-abelian cases are very similar. It is shown, in particular, that in the constant electromagnetic self-dual field, zero-mode solutions are present.

Interaction of molecules with electromagnetic fields. I. Classical particles and fields

Classical models of electrons and nuclei are developed which behave surprisingly like their quantum mechanical counterparts. The particles are finite in size and have both electrical charge density and current density. Classical spin results from the angular momentum of the particles rotating about their ’’center of mass’’. Internal electromagnetic fields are produced by the motion of the charged particles. These internal fields have dynamical properties of their own and produce the interaction between the particles. In this paper, the internal fields are derived through the second order of retardation or terms in 1/c2. Since higher order terms in the internal fields are not included, no difficulties are encountered with respect to the classical particles radiating. This same result could also be obtained by using equal parts of the retarded and advanced potentials. The Lagrangian, the (canonical, mechanical, and angular) momenta, the energy, and the Hamiltonian are derived in a gauge invariant manner for a ’’molecular system’’ of particles in the presence of an arbitrary (time and space varying) external electromagnetic field. The classical Hamiltonian corresponds term-by-term with the comparable quantum mechanical Breit–Pauli Hamiltonian. The correspondence includes the spin–orbit interaction terms, the Darwin terms (usually attributed to zitterbewegung), and the nuclear quadrupole terms (usually omitted from molecular Hamiltonians). A detailed comparison of classical and quantum mechanical dynamical properties will be given in a subsequent paper. Delta functions occur in both the classical and the quantum mechanical Hamiltonians to approximate the effect of overlapping when particles come close together. Otherwise, overlapping is generally not considered in either the classical or quantum treatments. In deriving the kinetic energy of a particle rotating about its center of mass, there was discovered a new relation which the rate of change of intrinsic spin (in the laboratory reference frame) must satisfy in order that the Lagrangian should completely describe the motion of the particle. This led to the discovery of a condition of compatibility between the dynamical properties of classical and quantum mechanical systems. This condition then led to a novel derivation of the relationship between the explicit expressions for the multipole moments of a particle in its laboratory reference frame (the observed moments) and the multipole moments in the particle’s rest frame (the intrinsic moments). It is assumed that an electron has only an intrinsic magnetic dipole moment whereas a nucleus has both an intrinsic magnetic dipole moment and an intrinsic electric quadrupole moment. The condition of compatibility also led to the derivation of canonical equations of change and generalized Poisson brackets, in both of which the spins are included as basic dynamical variables. The generalized Poisson brackets of the positions, the canonical momenta, and the spins exhibit the same forms as the commutation relations of their quantum mechanical counterparts. The (time and space varying) functions of the fields are expanded in Taylor series which have many of the features of Cartesian tensorial representations. If these functions were expanded in terms of spherical harmonics, the formulation would have been much more complicated.

Cylindrical prominences and the magnetic influence of the photospheric boundary

We construct exact, non-linear, solutions for an horizontal, cylindrical, current-carrying, prominence supported against solar gravity by the action of a Lorentz force. The solutions incorporate the photosphere boundary condition, proposed by van Tend and Kuperus (1978), and analyzed by them for line filaments. Our solutions have finite radius for the prominence material and, as well as satisfying the equations of magnetostatic equilibrium, they allow for the continuity of gas pressure, and of the normal and tangential components of magnetic field across the circular prominence boundary. We show that an infinity of solutions is possible and we illustrate the basic behavior by investigation of a special case. We also give a prescription for constructing equilibrium fields for any horizontal prominence with arbitrary cross-section and with an arbitrary external magnetic field. The prescription is ideally suited for numerical codes and we suggest that both the equilibrium of such shapes can easily be accomplished numerically together with their evolutionary history.

?Time inversion? and mobility of many particle systems

The unitary operations which can be generated on many particle states in non-relativistic quantum mechanics are discussed. These operations depend on an arbitrary external field which is in the experimenter's control, whereas the pairwise potential of interaction between the particles is fixed. The various kinds of systems ofN identical particles interacting via the potentials\(V_I = \sum\limits_{k,j} {r_{kj}^p w(r_{kj} )} \) are studied. For every system in question, the semigroup spanned by evolution transformations is proved to contain all the unitary operators in the Hilbert space of states. In particular, it is shown that the natural evolution operation can be reversed by a certain prescribed sequence of maneouvres involving only external fields.

Excitation of a spherical resonant medium by an external field

The coherent radiation of N atoms that are excited by an external field is studied close to a resonant transition. The resonant atoms are uniformly distributed over a sphere and the external field is a pulse that propagates in a fixed direction. The linear classical continuum theory is used for the above configuration. Exact expressions are obtained for the polarization, for the fields everywhere in space, and for the energy radiated and absorbed by the resonant atoms. Two equivalent methods for the derivation of the exact expression of the polarization are given, i.e., either by solving Maxwell’s equations and boundary conditions, or by solving an integral equation inside the sample. The equivalence of the two methods, in the general case of an arbitrary external field, has been proved elsewhere. Approximate expressions are also obtained for the physical quantities mentioned above in the extreme cases of a dilute and a dense resonant sample with radius much smaller than the radiation wavelength. Inside the dense small sample the polarization depends on position, and this space dependence is expressed in terms of spherical Bessel functions and vector spherical harmonics. It is shown that outside the dense small sample, a broadening of the original resonant line and a shift of the resonance frequency occur in the radiation spectrum of the atoms.