Use Of First-order Perturbation Theory Research Articles

Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions

We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent two-qubit ZZ and XX+YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space–time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.

The Rotating Morse–Pekeris Oscillator Revisited

The Morse–Pekeris oscillator model for the calculation of the vibration–rotation energy levels of diatomic molecules is revisited. This model is based on the realization of a second-order exponential expansion of the centrifugal term about the minimum of the vibrational Morse oscillator and the subsequent analytical resolution of the resulting approximate radial eigenvalue equation for the rotating Morse oscillator. It is, however, possible to develop the Morse–Pekeris model by transforming the second-order exponential expansion of the rotating Morse oscillator into an effective Morse oscillator and by then taking advantage of the well-known solutions for the Morse oscillator. Such a transformation can be carried out in a straightforward and instructive way merely by making a suitable change of variable of the internuclear coordinate. In addition, this alternative method allows us to correct the Morse–Pekeris model by simple use of first-order perturbation theory. The efficiency of this treatment is demonstrated by carrying out a numerical application to the H2 molecule.

The equilibrium structure of thiophene by the combined use of electron diffraction, vibrational spectroscopy and microwave spectroscopy guided by theoretical calculations

The equilibrium molecular geometry of thiophene has been determined from a combination of gas-phase electron diffraction, vibrational and microwave data and ab initio and DFT calculations. The quadratic and cubic force constants of thiophene calculated theoretically and empirically improved by harmonic scale factors were incorporated in the analysis in which equilibrium distances and harmonic scale factors were refined simultaneously. The diffraction intensities were calculated by the use of first-order perturbation theory. The commonly used ra distances and amplitudes of vibration were also estimated and found to agree reasonably well with those from an earlier investigation. Anharmonic phase shift parameters for all atom pairs and the various distance correction terms are presented.

Time-evolution operator for a forced parametric oscillator

We apply an algebraic technique to describe the evolution of a parametric harmonic oscillator forced by a constant quartic potential. As the first step, we make use of iterative Bogolubov transformations (IBT) to incorporate information from the anharmonic part of the interaction in a nonperturbative form, yielding a unitary time-evolution operator. Later on, we make use of first-order perturbation theory to deal with that part of the interaction which was not incorporated previously. We show numerically that the resulting time-evolution operator is closer to unitarity than is the one obtained if no IBT is applied. The quantum fluctuations of position and momentum are evaluated for “the ground” state. Squeezing and correlation effects are observed. © 1997 John Wiley & Sons, Inc.

Annihilation shifts and widths of the p-bar-d atomic levels.

The strong interaction shifts and widths of the antiproton-deuteron atom are calculated within a simple three-body model. The nuclear p\ifmmode\bar\else\textasciimacron\fi{}-d optical potential is obtained from a projected form of the Faddeev equations with rank one, separable, S-wave two-body potentials. The N-N\ifmmode\bar\else\textasciimacron\fi{} potential parameters are fit to the zero-energy scattering results of the Graz potential. The Coulomb potential is combined with the nuclear optical potential to obtain the atomic levels from the Schr\"odinger equation. The truncation of the Faddeev model optical potential to the level of the standard t\ensuremath{\rho} impulse approximation overestimates the shifts and widths by up to 10% and 30%, respectively. The use of first-order perturbation theory for the atomic eigenvalue overestimates the widths by one order of magnitude.

A Hele-Shaw Cell Study Of A New Approach To Instability Theory In Porous Media

Abstract Whether a displacement is stable or unstable has a profound effect on how efficiently one fluid immiscibly displaces another within a porous medium. As a consequence, it is of interest to be able to predict the boundary which separates stable displacements from those which are unstable. Recently, a new approach to stability theory has been developed which is based on the assumption that the immiscible displacement of one fluid by another can be treated as (J moving boundary problem. This paper describes a number of experiments undertaken to validate the newly developed theory. Because the assumption of a sharp front between the two fluids is satisfied exactly in a Hele-Shaw cell, these preliminary experiments were undertaken in such a model. Sixty-three experiments, using fluids having different viscosity ratios, interfacial tensions and wetting characteristics, have been performed. In general, these experiments have confirmed the validity of the new theory. In particular, the stability boundary and subsequent mode boundaries are predicted correctly. Moreover, the time constant vs wavelength behaviour predicted by the theory has been verified experimentally. Also, the relative widths of oppositely directed viscous fingers are predicted correctly. In addition, the distance travelled by the tip o/the oil and the water fingers, with respect to the initially plane interface separating two fluids, was found to be a linear function of time. Finally, the shape of the interface obtained in the experiments W(2) in good agreement with the postulated functional farm. Introduction Early work on the stability analysis of immiscible fluid displacement is normally based on first-order perturbation theory and the concept of a velocity potential(3–10). Also, it is usual to use a sharp interface approximation. While such an approach can deal with an incipient finger, it cannot deal with the subsequent growth of the viscous finger. Moreover, there are problems with the use of first-order perturbation theory and with the concept of a velocity potential. A velocity potential exists only for flow fields involving a fluid of constant density and viscosity and a porous medium which is homogeneous and isotropic throughout(11). Therefore, it cannot, in principle, deal with a real porous medium. Moreover, it is the force potential and not the velocity potential which governs the flow of fluids through porous media(12). When the velocity potential approach is taken, the postulated velocity potential will satisfy Laplace's equation only if the divergence of the perturbation velocity is zero. But the divergence of the perturbation velocity is zero only when a plane interface separates the displacing fluid from the displaced one. As soon as the interface is perturbed, two fluids will occupy a region where there was previously only one, and the divergence of the perturbation velocity will no longer be zero. Rather, the divergence of the sum of the perturbation velocities, one for each fluid, will be zero(22). First-order perturbation theory is not rigorous and the conclusions obtained from it are not always correct, as linearization of the equations is valid only if the deviations are small.

Reply to ‘On the Use of First-Order Perturbation Theory in Interface Shift Problems,’

"Reply to ‘On the Use of First-Order Perturbation Theory in Interface Shift Problems,’." Nuclear Science and Engineering, 85(1), p. 81This article refers to:Use of Boundary-Fitted Coordinate Transformation in Neutron Diffusion Calculations for Arbitrary Two-Dimensional Geometries

On the Use of First-Order Perturbation Theory in Interface Shift Problems

"On the Use of First-Order Perturbation Theory in Interface Shift Problems." Nuclear Science and Engineering, 84(1), pp. 72–73

Reply to “On the Use of First-Order Perturbation Theory in Interface Shift Problems”

This letter points out that the classical first-order perturbation formula will not correctly treat the case of a slight internal interface shift. Indicates that the traditional derivation, equation, and interpretation of classic first-order perturbation theory are all entirely correct if the perturbed problem differs from the unperturbed one by a material property perturbation.

Microscopic calculation of mass transport coefficients in heavy ion collisions

We present results of microscopic calculations of the drift and diffusion coefficients in deep inelastic heavy ion collisions. Assuming the two nuclei to interact weakly, we make use of first-order perturbation theory (Golden Rule) for nucleon transfer and take into account the Pauli blocking effect. The nucleon transfer probability from one nucleus to the other is obtained using single particle wave functions. The results obtained are consistent with empirical estimates.

Excited levels 1s n, n−1 of the two-electron atom

Closed formulas for the Coulomb and exchange integrals in the excited states 1snl, l=n−1, of the two-electron atom are applied to the computation of the ionization energy of the singlet and tripet 1s2p states from He i up to O vii and to singlet and triplet 1s3d and 1s4f states of He i and Li ii. Two methods are compared: (a) the use of first-order perturbation theory considering the mutual electrrostatic repulsion e2/r12 as a small perturbation and (b) imposing as solution of the Schrödinger equation a symmetrized product of a Z and a screened Z−1 hydrogenic wave functions. Computed and experimental values of the ionization energy and of the singlet-triplet interval are compared and the evolution of agreement with increasing Z and n is discussed.

Inversion of first-order perturbation theory and its application to structural design

Present methods of structural design, such as the finite-element method, NASTRAN, are capable of accurately determining the shape and frequencies of the normal modes of complicated structures, if the stiffness and mass matrices are specified. The inverse problem, that of determining the stiffness and mass matrices of a structure in order to arrive at a desired set of normal mode shapes and frequencies, in general is unsolved. A mathematical method is presented here which addresses the inverse problem by the use of first-order perturbation theory. In this approach, the usual perturbation calculation is inverted. That is, the structural changes necessary to effect a given change in vibration modes are computed, rather than computing the effect on the vibration modes resulting from a small change in the design of that structure. The application of this technique requires a specification of a number of the normal modes of vibration of a base line structural design. This information can be obtained by a conventional finite-element calculation or by measurement of the normal modes of an existing prototype. In the former case, the inverse perturbation calculation can be used to proceed from the baseline design to an optimum design by an iterative procedure. In the latter case, the inverse perturbation calculation employs normal mode data from an existing structure to determine the changes in that structure necessary to improve its vibratory response. The basic mathematical structure of the inverse perturbation technique is presented herein, together with two illustrative numerical examples, the first employing four springs and three masses, and the second treating the modification of the modes of a cantilever beam.

Semiclassical Perturbation Scattering by a Rigid Dipole

A uniform semiclassical S matrix has been developed for collisions of charged particles with rotating rigid dipoles, with use of first-order perturbation theory. The resulting expression is analytical, depending on tabulated functions, and trivial to calculate; it allows evaluations of quantum transitions in classically forbidden regions, and of quantum interference effects. (AIP)

Ro-vibrational energy Jacobians with respect to Coriolis zeta coefficients

Algebraic Jacobian expressions are given describing the sensitivity of upper-state ro-vibrational energy levels to changes in Coriolis zeta coefficients. For two and three interacting states closed formulae are derived. For non-degenerate, multiple interactions two alternative approaches are used. One method is novel and consists of the application of the “mixing matrices”, the other is the use of first-order perturbation theory.1

Seismic Wave Scattering by Obstacles on Interfaces

Summary A unified treatment of the scattering of seismic waves by small obstacles is presented. The method is based on an integral equation formulation and the use of first-order perturbation theory. This approach is used to consider elastic wave scattering by interface irregularities and obstacles at the interface of two elastic media, whose elastic properties are different from those of the bounding media. The results obtained by this method are shown to reduce to those of previous authors when applied to their more specialized problems. It is well known that in regions of irregular topography much noise is present in both reflection and refraction seismic records; this is attributed to scattering by surface and sub-surface topography and inclusions with elastic properties which differ from those of the surrounding matrix. Also much of the signal-related noise at seismic recording stations appears to be due to scattering by obstacles at or below the surface in the vicinity of the station. Gilbert & Knopoff (1960) considered the scattering produced by a surface irregularity on a uniform half space. Their treatment was subsequently extended by Hudson (1967) to deal with a three-dimensional surface obstacle whose elastic properties differ from those of the half space. This method is suitable for small obstacles whose bounding surfaces have only small slopes. Recently Aki & Larner (1970) and Slavin & Wolf (1970) have considered larger interface irregularities by constructing solutions in which the boundary condition mismatch is minimized by a least squares technique. In this paper we will consider only the first order perturbations for small obstacles, andpresentaunified treatment of the scattering of elastic wavesby interface irregularities and obstacles at the interface of two elastic media, whose elastic properties differ from those of the bounding media. The method is based on an integral equation approach and has the advantage that it is an extension of the normal methods for layered media. The results obtained reduce to those of Gilbert & Knopoff and of Hudson when applied to their more specialized problems. 2. Basic equations for an irregular interface We consider isotropic media, and for simplicity only two-dimensional problems where all stresses and displacements are independent of the coordinate y. A harmonic time dependence of the form exp (- iwt) will be assumed throughout, though it will not be explicitly represented. Away from the neighbourhood of irregular interfaces we shall consider media

The dashen-frautschi method and parity-conserving baryon-meson couplings

The Dashen-Frautschi method (1), which was successfully applied in particular to the mass differences among the baryon octet and decuplet (2), is here used to treat parity-conservingSU3-noninvariant baryon-meson couplings (the method being « universal », the same arguments hold for medium-strongSU3 breaking and for parity-conserving weak interactions). The results are not very gratifying: 1) Both octet and 27-like coupling shifts are enhanced by comparable factors, which means in particular that the ΔI=1/2 rule forP-wave nonleptonic hyperon decays does not come out as a consequence of a dynamical enhancement, as it did in theS-wave (parity-violating) case. 2) Furthermore, if one nevertheless tries the hypothesis of octet dominance, the model investigated is incapable of fitting the experimental results concerning the relevant hyperon decays. As for medium-strong interactions, the situation is better: the calculated widths for the 3/2+ resonances are reasonable; however, the over-all magnitude of theSU3-breaking constants needed is so large as to make the use of first-order perturbation theory quite questionable.

Vibrational Energy Transfer in Molecular Collisions Involving Large Transition Probabilities

The usual calculations of vibrational energy exchange in a diatomic gas make use of first-order perturbation theory and are therefore limited to low transition probabilities. In actual practice, the thermally averaged transition probabilities in most gases are indeed small. However, this is partly because it is the relatively few molecules in the high-velocity ``tail'' of the velocity distribution which account for most of the energy exchange. The actual microscopic transition probabilities in these collisions may be too great to justify the perturbation method for many gases at high temperatures. We have therefore solved the time-dependent Schrödinger equation based on a semiclassical collision, subject to an assumed form of potential, to get exact transition probabilities in a molecular collision. At low velocities, our result reduces to the perturbation result. A thermal average of our transition probabilities should eliminate errors due to use of the perturbation solution in previous calculations. For N2, these errors only become important above 5000°K. For gases with lower vibrational frequencies such as O2, or strong attractive forces such as NO, these effects become important at much lower temperatures.