Hamiltonian Representation Research Articles

Three-dimensional statistical reduction of theN-body Schrödinger equation for electrons with pairwise Coulomb interactions

Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of $N$ electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the $N$-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.

Modelling of piezoelectric structures–a Hamiltonian approach

This contribution is dedicated to the geometric description of infinite-dimensional port Hamiltonian systems with in- and output operators. Several approaches exist, which deal with the extension of the well-known lumped parameter case to the distributed one. In this article a description has been chosen, which preserves useful properties known from the class of port controlled Hamiltonian systems with dissipation in the lumped scenario. Furthermore, the introduced in- and output maps are defined by linear differential operators. The derived theory is applied to the piezoelectric field equations to obtain their port Hamiltonian representation. In this example, the electrical field strength is assumed to act as distributed input. Finally it is shown, that distributed inputs, that are in the kernel of the input map act similarly on the system as certain boundary inputs.

A new integrable hierarchy and constrained flows, its expanding integrable system

A new subalgebra of loop algebra A ∼ 2 is first constructed. It follows that an isospectral problem is established. Using Tu-pattern gives rise to a new integrable hierarchy, which possesses bi-Hamiltonian structure. Furthermore, by making use of bi-symmetry constraints, the generalized Hamiltonian regular representations for the hierarchy are obtained. Finally, we obtain an expanding integrable system of this hierarchy by applying a scalar transformation between two isospectral problems and constructing a five-dimensional loop algebra G ∼ .

On the vierbein formalism of general relativity

Both the Einstein–Hilbert action and the Einstein equations are discussed under the absolute vierbein formalism. Taking advantage of this form, we prove that the “kinetic energy” term, i.e., the quadratic term of time derivative term, in the Lagrangian of the Einstein–Hilbert action is non-positive definitive. And then, we present two groups of coordinate conditions that lead to positive definitive kinetic energy term in the Lagrangian, as well as the corresponding actions with positive definitive kinetic energy term, respectively. Based on the ADM decomposition, the Hamiltonian representation and canonical quantization of general relativity taking advantage of the actions with positive definitive kinetic energy term are discussed; especially, the Hamiltonian constraints with positive definitive kinetic energy term are given, respectively. Finally, we present a group of gauge conditions such that there is not any second time derivative term in the ten Einstein equations.

Enigma of probability amplitudes in Hamiltonian formulation of integrable semidiscrete nonlinear Schrödinger systems

An attempt to find a probability-amplitude-based Hamiltonian representation of the symmetrical version of integrable semidiscrete multicomponent nonlinear Schrödinger systems is made. Thus the on-cell locality of the general point transformation in combination with the model multicomponentness is shown to contradict the concept of canonical Hamiltonian representation in terms of probability amplitudes. Nevertheless, the above concept can be realized in a slightly adjusted semidiscrete multicomponent nonlinear Schrödinger system that preserves some physically valuable solutions of the original (either symmetric or asymmetric) integrable model. The advantages of the adjusted Hamiltonian model for the analysis of real physical systems are formulated. Examples of longitudinal and lateral soliton dynamics on multichain tubular lattices subjected to uniform electric and magnetic fields are given.

Законы сохранения, иерархия динамики и явное интегрирование неголономных систем

This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

Alternative Hamiltonian representation for gravity

By using a Hamiltonian formalism for fields wider than the canonical one, we write the Einstein vacuum field equations in terms of alternative variables. This variables emerge from the Ashtekar's formalism for gravity.

Floquet perturbation theory: Applicability of the finite level approximation in different gauges

The Floquet perturbation theory represents an important technique for studying the response of atoms or molecules exposed to weak monochromatic fields. In this paper, we discuss an explicit implementation of the Floquet perturbation theory starting from different gauge representations of the matter-light Hamiltonian, namely, from the velocity gauge, the length gauge, and the acceleration gauge. Interestingly, the development of the second-order Floquet perturbation theory in different gauges gives rise to formally different quasienergy corrections, whose mutual equivalence is not self-evident. Nevertheless, our derivation shows how the perturbation formulas associated with different gauges can be converted to one another, provided that a complete basis set of the atomic-molecular electronic states has been used. On the other hand, it turns out that an inappropriate basis set truncation employed for a particular gauge (such as, e.g., the two-level approximation applied within the velocity gauge) may sometimes lead toward completely wrong results. This fact should serve as a warning against an uncautious use of various gauge transformations in practical calculations with a finite basis set.

The rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation: I. Bound states

This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound-state energy spectrum by diagonalizing the finite-dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is handy, very efficient, and it enhances accuracy by combining analytic power with a convergent and stable numerical technique.

Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation

An effective and general algorithm is suggested for variational vibrational calculations of N-atomic molecules using orthogonal, rectilinear internal coordinates. The protocol has three essential parts. First, it advocates the use of the Eckart-Watson Hamiltonians of nonlinear or linear reference configuration. Second, with the help of an exact expression of curvilinear internal coordinates (e.g., valence coordinates) in terms of orthogonal, rectilinear internal coordinates (e.g., normal coordinates), any high-accuracy potential or force field expressed in curvilinear internal coordinates can be used in the calculations. Third, the matrix representation of the appropriate Eckart-Watson Hamiltonian is constructed in a discrete variable representation, in which the matrix of the potential energy operator is always diagonal, whatever complicated form the potential function assumes, and the matrix of the kinetic energy operator is a sparse matrix of special structure. Details of the suggested algorithm as well as results obtained for linear and nonlinear test cases including H(2)O, H(3) (+), CO(2), HCNHNC, and CH(4) are presented.

Frequency-dependent stabilization ofHe−by a superintense laser field

Using the alternative representation of time-dependent Hamiltonians that describe laser-driven systems [I. Gilary and N. Moiseyev, Phys. Rev. A 66, 063415 (2002)], we have performed calculations for the stability of He{sup -} in a superintense linearly polarized laser field. We estimated the laser parameters, amplitude, and frequency needed to stabilize He{sup -}. If we choose the laser frequency {omega}=5 eV, the laser intensity needed for stabilization is I{sup critical}=9.0x10{sup 15} W/cm{sup 2} and the maximal detachment energy is 1.0 eV. Our results show that high frequencies and large intensities are preferred for stabilizing multiply negative atomic ions in superintense laser fields.

Cluster structure of EU-15 countries derived from the correlation matrix analysis of macroeconomic index fluctuations

The statistical distances between countries, calculated for various moving average time windows, are mapped into the ultrametric subdominant space as in classical Minimal Spanning Tree methods. The Moving Average Minimal Length Path (MAMLP) algorithm allows a decoupling of fluctuations with respect to the mass center of the system from the movement of the mass center itself. A Hamiltonian representation given by a factor graph is used and plays the role of cost function. The present analysis pertains to 11 macroeconomic (ME) indicators, namely the GDP (x1), Final Consumption Expenditure (x2), Gross Capital Formation (x3), Net Exports (x4), Consumer Price Index (y1), Rates of Interest of the Central Banks (y2), Labour Force (z1), Unemployment (z2), GDP/hour worked (z3), GDP/capita (w1) and Gini coefficient (w2). The target group of countries is composed of 15 EU countries, data taken between 1995 and 2004. By two different methods (the Bipartite Factor Graph Analysis and the Correlation Matrix Eigensystem Analysis) it is found that the strongly correlated countries with respect to the macroeconomic indicators fluctuations can be partitioned into stable clusters.

Relativistic extension of the complex scaling method

We construct a tridiagonal matrix representation for the three-dimensional Dirac-Coulomb Hamiltonian that provides for a simple and straightforward relativistic extension of the complex scaling method. Besides the Coulomb interaction, additional vector, scalar, and pseudoscalar coupling to short-range potentials are also included in the same representation. Using that, we are able to obtain highly accurate values for the relativistic bound states and resonance energies. A simple program code is developed to perform the calculation for a given charge, angular momentum, and potential configuration. The resonance structure in the complex relativistic energy plane is also shown graphically. Illustrative examples are given and we verify that in the nonrelativistic limit one obtains known results. As an additional advantage of this tridiagonal representation, we use it to obtain highly accurate evaluation of the relativistic bound state energies for the Woods-Saxon potential (as a model of nuclear interaction) with the nucleus treated as a solid sphere of uniform charge distribution.

Quasirelativistic theory. II. Theory at matrix level

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to the matrix representation of a quasirelativistic Hamiltonian that has the same electronic eigenstates as the original Dirac matrix (but no positronic eigenstates). This transformation involves a matrix X, for which an exact identity is derived and which can be constructed either in a noniterative way or by various iteration schemes, not requiring an expansion parameter. Both linearly convergent and quadratically convergent iteration schemes are discussed and compared numerically. The authors present three rather different schemes, for each of which even in unfavorable cases convergence is reached within three or four iterations, for all electronic eigenstates of the Dirac operator. The authors present the theory both in terms of a non-Hermitian and a Hermitian quasirelativistic Hamiltonian. Quasirelativistic approaches at the matrix level known from the literature are critically analyzed in the frame of the general theory.

Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization

Abstract The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.

ON SOME DISSIPATIVITY PROPERTIES OF A CLASS OF POWER SYSTEM MODELS

This paper studies the dissipativity properties of a class of power system models, characterized by the absence of resistive loads and leaky lines. A Port-Controlled Hamiltonian (PCH) representation is given for each component of the network and its dissipativity properties are shown. The linear model around the equilibrium is shown to meet a convex condition in the frequency domain, able to be exploited in the stability analysis of interconnected systems. The application of this property to a classical example shows that it can be computationally exploited even in the case of non-idealized models.

Regularization in the J-matrix method of scattering revisited

Abstract We present an alternative, but equivalent, approach to the regularization of the reference problem in the J-matrix method of scattering. After identifying the regular solution of the reference wave equation with the “sine-like” solution in the J-matrix approach we proceed by direct integration to find the expansion coefficients in an L 2 basis set that ensures a tridiagonal representation of the reference Hamiltonian. A differential equation in the energy is then deduced for these coefficients. The second independent solution of this equation, called the “cosine-like” solution, is derived by requiring it to pertain to the L 2 space. These requirements lead to solutions that are exactly identical to those obtained in the classical J-matrix approach. We find the present approach to be more direct and transparent than the classical differential approach of the J-matrix method.

Relativistic interactions for the meson-two-nucleon system in the clothed-particle unitary representation

The method of unitary clothing transformations put forward in relativistic quantum field theory (QFT) by Greenberg and Schweber and developed by Shirokov is applied to construct a new family of interactions in the meson-two-nucleon system. Along with a brief exposition of its basic elements we show a specific transition from the initial “bare” one-meson and one-nucleon operators and states to their physical “clothed” counterparts. We emphasize that the clothing transformations in question do not alter the original total Hamiltonian, but provides a conceptually more transparent representation of the same Hamiltonian in terms of a new set of operators for particles with physical properties and their relativistic interactions. The Hermitian and energy-independent interaction operators for the processes π N → π N, NN → NN and NN ↔ π NN are derived starting from the Yukawa-type couplings between fermions (nucleons and antinucleons) and bosons (π−, η−, ρ−, ω− mesons, etc.). These types of interaction have a distinctive off-energy-shell structure which is naturally generated by the unitary transformation that removes from the Hamiltonian the (three-leg) π NN vertex coupling.

Force constants and transition intensities in the U ( ν + 1) model for molecular vibrational excitations

A connection between the unitary group approach U ( ν + 1) and the traditional description in configuration space of vibrational excitations is proposed. Local operators b ˆ i † ( b ˆ i ) satisfying the su (2) commutation relations are used to establish approximate algebraic expansions of the local coordinates and momenta. The use of the proposed relations allows to obtain an algebraic representation of traditional Hamiltonians in terms of the U ( ν + 1) model. This approach provides in natural form the connection between the spectroscopic parameters and force constants. Using the linear expansion of the coordinates in terms of the b ˆ i † ( b ˆ i ) operators, an approach to study local dipole transition intensities based on traditional descriptions is proposed. A closed general analytical expression for the local dipole operators is obtained. The analysis of the stretching vibrational excitations of arsine is taken as an example for the determination of both force constants and the description of dipole transition intensities.

Energy Spectrum of a Bipartite Complicated-Coupled-Oscillator System

We present a general discussion on the eigenstate problem of abipartite complicated-coupled-oscillator system. By use of ageneralized intermediate entangled state representation, theeigenvalue and eigenfunction of Hamiltonian are analyticallyobtained. As its application, we obtain the energy spectrum for twospecial situations of Hamiltonian.