Terms Of Harmonic Oscillators Research Articles

Static and dynamic density functional investigation of hydrated beryllium dications

Using density functional theory with Becke's gradient correction for the exchange part and Vanderbilt's ultrasoft pseudopotentials, we investigated [Be(H2O)n]2+ clusters for n = 1 and 3. A new scheme implemented with the Car-Parrinello method is used, upon which no periodic boundary conditions are imposed so that isolated and possibly charged molecules can also be treated dynamically in a plane-wave basis. Harmonic vibrational frequencies are obtained via a fit of the molecular dynamics trajectory in terms of harmonic oscillators, for which we use a fragmentation scheme to analyze complex spectra. Based on this combination of techniques, we find good agreement with data from SCF calculations and correlated methods. © 1996 John Wiley & Sons, Inc.

Empirical dielectric function of amorphous materials for spectroscopic ellipsometry

An empirical dielectric function (EDF) is proposed for optical characterization of amorphous materials. The EDF consists of the sum of damped harmonic oscillator terms whose square root amplitudes are distributed according to a hyperbolic function of photon energy connected to an exponential function. The usefulness of EDF is demonstrated by fitting it to the table values of dielectric constants for a-Si and a-Si3N4 and by applying it to the spectro-ellipsometric analysis of a-Si films deposited by vacuum evaporation and rf glow discharge methods.

Most general and simplest algebraic relationship between energy eigenstates of a hydrogen atom and a harmonic oscillator of arbitrary dimensions.

In this paper, we work out a most general and the simplest relationship between the energy eigenstates of a d-dimensional hydrogen atom and those of a D-dimensional harmonic oscillator in terms of the su(1,1) algebra.

Calculation of baryon spectra based on an improved quark potential

An improved QCD-inspired quark potential is introduced and employed to calculate baryon spectra. This potential has some novel features: (i) the one-gluon exchange potential in it is derived from the exact three-dimensional relativistic equation and contains a retardation term which is logarithmic; (ii) the confining part of the potential is generated from a linear potential Bethe-Salpeter (BS) kernel and a general Lorentz structure of the confinement and includes therefore various spin-dependent terms added to the linear potential; (iii) the off-shell correction part of the potential which is related to the negative energy component of the BS amplitude automatically gives rise to a harmonic oscillator term. We analyse the contributions of all parts of the potential to the baryon spectrum and find that the potential we use makes some improvements to theoretical baryon masses, especially for the Roper resonance.

Interacting protons in two double minimum potentials: Quantum mechanical calculations

Abstract A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.

A new ring-shaped potential and its dynamical invariance algebra

A new ring-shaped potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, is investigated. The Schrodinger equation is solved in spherical, circular cylindrical, prolate and oblate spheroidal coordinates. As in the case of the Hartmann potential, the 'accidental' degeneracies occurring in the spectrum are shown to be due to an su(2) dynamical invariance algebra. This establishes a close connection between both ring-shaped potentials.

Stochastic mechanics and the Kepler problem.

The stochastic mechanics of Nelson and Guerra is formulated for the hydrogen atom. We demonstrate that this simple quantum system can be described in terms of three independent Gaussian Markov processes which are driven (controlled) by the classical Kepler problem. It reveals a manifest connection between the classical and quantized versions of the Kepler problem. I. MOTIVATION The idea of stochastic quantization, as developed in Refs. 1 and 2, amounts to associating the stochastic processes to quantum states of the dynamical system. The procedure works quite successfully as long as ground states of the simplest models are considered; the determination of the Madelung fiuid representation for higher excited states is much more involved. In fact, the stochastic strategy works in full generality for an example of the harmonic oscillator and its most straightforward generalizations (see also the studies of its two-level, Fermi version ' ). However, the formulation of the stochastics mechanics for another simple quantum system, that of the hydrogen atom, except for the ground state, is yet to be accomplished. This fact is a bit puzzling since, like many other simple quantum systems, the quantized Kepler problem admits a realization in terms of (a quartet of) harmonic oscillators, and should in principle allow for the generalization of the arguments of Refs. 2 and 12. Moreover the concept of related coherent states was introduced in Ref. 11, and the construction of oscillator stochastic processes is most transparent with respect to the coherent basis. It is our aim to take advantage of the oscillator reconstruction of the Kepler problem, to formulate the stochastic mechanics of the latter. While working with the fouroscillator system, the functions of Madelung densityphase variables p;(x), S;(x), i=1,2,3,4, arise through computing the coherent-state expectation values (a ~ A ~ a) =Z(p, S) of operator-valued quantities. To recover the hydrogen problem, the constraints must be accounted for. As we demonstrate in the course of the paper, the (analytic) stochastic mechanics of the problem, if formulated in the Madelung [p(x),S(x)] parametrization, is in all respects equivalent to the standard classical mechanics of the singular (constrained) Hamiltonian system, whose phase manifold is parametrized by holom orphic coherent-state labels (a,a): 4 ~ a)=exp g (a;tt — a;tt;) ~ 0) . consequence, we identify the coherent-state domain for the Kepler problem, whose a~, az, a3, a4 parameters are completely determined in terms of the canonical variables of the standard classical Kepler problem. It allows for the final conclusion that the three independent GaussianMarkov processes can be associated with the hydrogen atom. Moreover, these processes are driven (controlled in the language of Ref. 12) by the classical Kepler motion. It establishes an apparent link between the quantized and classical versions of the Kepler problem, the connection which could hardly have been seen from the path-integral computation presented in Ref. 6.

The geometry of ethylene in the ground ionic state

Franck—Condon analyses were carried out on the first photoelectron band of ethylene to deduce the CC bond length, the CH bond length and the HCH bond angle in the ground ionic state 1 B 3u|to be 1.42 ± 0.01 Å, 1.09 ± 0.01 Å and 120 ± 1° respectively. These structructural parameters have values close to those previously obtained in the 1 B 3u Rydberg state (see ref. 10). With regard to the vibrational energies of the torsional mode of ethylene and its deuterated species in the 2 B 3u and 1 B 3u Rydberg states, they were analysed by using a one-deminsional potential function constructed from the harmonic oscillator term perturbed by a Gaussian barrier. The equilibrium torsional angles of ethylene thus obtained are found to be 28° and 25°, respectively, in the 2 B 3u and 1 B 3u states. In addition, the observed vibrational intensity distributions of the torsional mode accompanying electronic transitions from the ground state to the two 2 B 3u and 1 B 3u states are well reproduced by the proposed potential. Moreover, it is found that these vibrational transition probabilities are sensitive to the magnitude of the barrier heights, the shape of the potentials and the frequency changes upon electronic transitions and thus should be useful in providing structural information about the molecules under study.

Realisation of Lie algebras and of representations of Lie groups in terms of harmonic oscillators

The algebras and irreducible representations (irreps) of compact Lie groups, including the exceptional groups, are realised in terms of sets of Bose oscillator (SHO) creation and annihilation operators. In particular, not only the tensor irreps but also the spinor irreps of orthogonal groups of all ranks can be constructed using Bose (rather than the more customary Fermi) oscillators.

New raising and lowering operators for highly anharmonic coupled oscillators

New raising and lowering operators A+, A are proposed for vibrations in the highly anharmonic regime (roughly corresponding to the zone of overlapping nonlinear resonances and the transition to classical chaos). All matrix elements 〈ν+k‖0‖ν〉 of p and powers xλ of x=2Ce−aq between bound states ‖ν〉 of the Morse oscillator can be written exactly in terms of powers of the new operators, (A+)k and (A)k, and a supplementary operator α̂. In terms of α̂, A+, A, the momentum and coordinate operators take a form similar to that of the harmonic oscillator in terms of a+, a. It is shown that it is the operator α̂ that is crucial for representing the novel qualitative behavior of dynamical operators in the highly anharmonic regime. It is therefore suggested that the operators α̂(A+)k and α̂(A)k be used in place of the conventional (a+)k and (a)k in fits and dynamical models for systems of coupled anharmonic vibrations.

Domain walls and the Holstein-Primakoff transformations

The problem of determining the interaction of spin waves in the presence of a domain wall is examined. The Hamiltonian for a ferromagnet with a uniaxial anisotropy is rewritten in terms of harmonic oscillators with the use of the Holstein-Primakoff transformations. In the continuum limit a nonrelativistic scalar field theory is obtained. The equation of motion in three spatial dimensions admits domain-wall solutions. The Hamiltonian is expanded about the classical solution and the quadratic terms are used to generate the spin-wave spectrum. The results are shown to be consistent with other methods. The space of states and propagators are exhibited, and the extension to the higher-order terms governing spin-wave interactions is discussed.

Time symmetries involving magnetic monopoles

Time symmetries of a single-particle system in the presence of a U1 magnetic monopole are further discussed especially in connection with its quantization and related spectrum. Emphasis is directly given to the class of Lagrangians containig isotropic harmonic-oscillator terms.

Multi-Quark Hadrons in the Joined-Spring Quark Model

The level structures of typical multi-quark exotic hadrons, (q2 q2 )-meson, (q' q )-baryon and (q6)-di-baryon with only nuclear quarks, are investigated systematically in the joined spring quark model, where hadron masses squared are described in terms of harmonic oscillators with various quanta. This model was proposed by us as a simple idealization of the string-junction model so as to make it possible a quantum mechanical treatment of constituent quarks. An interesting prediction of our model is that there exist orbital Regge trajectories with the steeper slope than the usual one for non-exotic hadrons_ Somewhat strong experimental suggestions for existence of these trajectories are seen for all respective cases of exotic hadrons.

Existence of a complex structure for quadratic Hamiltonians?

We consider a classical system with finite degrees of freedom and arbitrary quadratic Hamiltonian. We prove that Segal's Hilbertization procedure is possible if and only if the system is stable and can be analysed in terms of independent harmonic oscillators. In this case, the complex structure J on phase space is unique for each Hamiltonian and we give its explicit form. Applications to the theory of infinite degree systems, including fields with external potential, the BCS model and the Schwinger model, are suggested.

Quantum mechanical model for conformons

The purpose of this paper is the investigation of the role of entropy in the conformon, based on a model Hamiltonian approach. There are three stages of our considerations. (1) Quadratic terms are introduced into the potential energy terms of harmonic oscillators in the presence of an electron, leading to an increase in entropy and thus to an increased number of thermally activated electrons. (2) The number of activated electrons is not just increased, but they are also mobile if electronphonon bound states are formed. (3) A compensation rule cannot be obtained in this harmonic approximation, and therefore anharmonic coupling is introduced. This leads to a mobility edge and the analogy to amorphous semiconductors is pointed out.

THE THEORY OF THE POLARIZATION OF CORONAL FORBIDDEN LINES

ABSTRACT A brief review, presented in simplified terms, is given for the theory of the origin of coronal emission-line polarization. A classical view of the scattering problem in terms of harmonic oscillators is first presented where the influence of the magnetic field is demonstrated. The Van Vleck depolarization phenomena is described in these terms. Next, a more precise physical picture of the emission-line polarization is established through a discussion of the Zeeman effect in scattering. Sample results are discussed for three-dimensional coronal emission4ine models based upon the solution of the statistical equilibrium equations for magnetic sublevels of the Fe xiv ion. Finally, comments are directed toward the determination of magnetic fields based upon emission-line polarization observations where both medeling and deconvolution procedures are mentioned. Key words: corona - polarization - magnetic fields - forbidden lines

String-plucking model for vibrational excitation of molecules

A ``string-plucking'' model has been developed to describe the translational to vibrational energy transfer between an atom and a harmonic oscillator in terms of the time-dependent force constant and equilibrium distance of the molecule. The model is compared with exact classical trajectory calculations of collinear and ``breathing sphere'' collisions involving repulsive exponential or Lennard-Jones interaction potentials. In the adiabatic limit for simple collisions (i.e., in the absence of multiple collisions produced by a light middle atom or a deep potential well), the energy transfer predicted by the model is in good agreement with the exact trajectory results. In the case of the Lennard-Jones potential, both model and exact calculations predict that the effect of decreasing the initial kinetics energy is to increase the fractional energy transfer up to 100%. For breathing sphere collisions with a Lennard-Jones potential, the model correctly predicts a sharp decline in energy transfer at a critical impact parameter which is found in the exact trajectory calculations to equal the rainbow impact parameter. For simple collisions with either potential both model and exact calculations show that the effect of increasing the vibrational energy is equivalent to diminishing the initial radial kinetic energy by an equal amount. The model is also compared with the Landau-Teller, Nikitin, and Attermeyer-Marcus theories, and their regions of validity are explored.

Antiferromagnetism by the Spin Wave Method I: The Energy Levels

The Hamiltonian operator for an exchange antiferromagnet is expressed in terms of the deviations of the spins from the basic ordered antiparallel arrangement By a Fourier transformation the Hamiltonian may be approximately represented as a sum of harmonic oscillator terms, corresponding to spin waves. The role of anisotropy energy in fixing the domain axis is discussed, and the energy levels are calculated for the case of an external field parallel to this axis The ground-state energy includes the zero-point energy of the spin waves and is in good agreement with other theories Formulae are quoted for a Hamiltonian including dipole-dipole interaction between non-isotropic spins