Terms Of Wave Functions Research Articles

Hilbert space theory of classical electrodynamics

Classical electrodynamics is reformulated in terms of wave functions in the classical phase space of electrodynamics, following the Koopman–von Neumann–Sudarshan prescription for classical mechanics on Hilbert spaces sans the superselection rule which prohibits interference effects in classical mechanics. This is accomplished by transforming from a set of commuting observables in one Hilbert space to another set of commuting observables in a larger Hilbert space. This is necessary to clarify the theoretical basis of the much recent work on quantum-like features exhibited by classical optics. Furthermore, following Bondar et al, Phys. Rev. A 88, 052108 (2013), it is pointed out that quantum processes that preserve the positivity or nonpositivity of the Wigner function can be implemented by classical optics. This may be useful in interpreting quantum information processing in terms of classical optics.

Two-species Bose–Einstein condensate in an optical lattice: analytical approximate formulae

Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.

Simple, accurate, and efficient implementation of 1-electron atomic time-dependent Schrödinger equation in spherical coordinates

Modelling atomic processes in intense laser fields often relies on solving the time-dependent Schrödinger equation (TDSE). For processes involving ionisation, such as above-threshold ionisation (ATI) and high-harmonic generation (HHG), this is a formidable task even if only one electron is active. Several powerful ideas for efficient implementation of atomic TDSE were introduced by H.G. Muller some time ago (Muller, 1999), including: separation of Hamiltonian terms into tri-diagonal parts; implicit representation of the spatial derivatives; and use of a rotating reference frame. Here, we extend these techniques to allow for non-uniform radial grids, arbitrary laser field polarisation, and non-Hermitian terms in the Hamiltonian due to the implicit form of the derivatives (previously neglected). We implement the resulting propagator in a parallel Fortran program, adapted for multi-core execution. Cost of TDSE propagation scales linearly with the problem size, enabling full-dimensional calculations of strong-field ATI and HHG spectra for arbitrary field polarisations on a standard desktop PC. Program summaryProgram title: SCID-TDSE: Time-dependent solution of 1-electron atomic Schrödinger equation in strong laser fields.Catalogue identifier: AEYM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 334254No. of bytes in distributed program, including test data, etc.: 20005596Distribution format: tar.gzProgramming language: Fortran-2003 with OpenMP extensions.Computer: Portable code. Tested on x86_64 Linux.Operating system: Portable code. Tested on x86_64 Linux.RAM: Memory requirements depend on the input parameters. Time propagation of a given initial state requires O(32∗NR∗(LMAX+1)∗(MMAX−MMIN+1)) bytes of memory (double precision), where NR is the number of the radial grid points; LMAX is the maximum desired angular momentum; MMIN and MMAX are respectively the smallest and largest desired angular momentum projections. Preparation of the initial atomic states and analysis of the final wavefunction in terms of the field-free atomic states may require O(64∗NR2∗NCPU) bytes of RAM (double precision), where NCPU is the number of processing threads used.Classification: 2.5, 4.3.External routines: BLAS and LAPACK (required); libhugetlbfs (optional), DGEFA and DGEDI (LINPACK); routines included with the code.Nature of problem: Time propagation of non-relativistic 1-electron Schrödinger equation for a central potential, under the influence of a long-wavelength laser field treated in the velocity-gauge dipole approximation.Solution method: The propagator is constructed by separating the Hamiltonian into a large number of non-commuting terms, where each term can be handled simply and computationally efficiently (linear scaling). Time-reversibility of the propagator is ensured by combining the individual terms symmetrically around time midpoint (See ref. [1] and the text). The numerical accuracy is achieved through implicit representation of derivatives (accurate to O(d4) for a uniform grid), combined with variable grid spacing.Restrictions: Ill-conditioned Hamiltonians can occur for some choices or radial grids. The propagator is only approximately norm-conserving; small time steps may be necessary to achieve stable propagation.Unusual features: Due to the implicit representation of the spatial derivative operators, the overall Hamiltonian is not Hermitian. As the result, the left wavefunction is no longer given by a complex conjugate of the right wavefunction, and must be propagated explicitly.The code makes no assumptions on the accuracy of numerical types, and can be built for any real or integer kinds supported by the compiler. Detailed instructions for building the code in single, double, and quadruple precision are included.Running time: Running time is input dependent. Time propagation of the Schrödinger equation scales as O(NR∗(LMAX+1)∗(MMAX−MMIN+1)) per time step. Preparation of the initial atomic state and analysis of the results in terms of the field-free atomic states may require O(NR3) and O(NR3∗(LMAX+1)) work, respectively. On a 3.6 GHz core i7 desktop with 4 CPU cores available, run times are from 2 s (probability of a perturbative bound- to-bound transition in hydrogen atom) to 1 h (high-harmonic spectrum of a hydrogen atom driven by intense elliptically-polarised IR field).

Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model

We obtain the exceptional part of the eigenspectrum of the generalized Rabi model, also known as the driven Rabi model, in terms of the roots of a set of algebraic equations. This approach provides a product form for the wavefunction components and allows an explicit connection with recent results obtained for the wavefunction in terms of truncated confluent Heun functions. Other approaches are also compared. For particular parameter values the exceptional part of the eigenspectrum consists of doubly degenerate crossing points. We give a proof for the number of roots of the constraint polynomials and discuss the number of crossing points.

Construction of exchange repulsion in terms of the wave functions at QM/MM boundary region.

We developed a simple method to calculate exchange repulsion between a quantum mechanical (QM) solute and a molecular mechanical (MM) molecule in the QM/MM approach. In our method, the size parameter in the Buckingham type potential for the QM solute is directly determined in terms of the one-electron wave functions of the solute. The point of the method lies in the introduction of the exchange core function (ECF) defined as a Slater function which mimics the behavior of the exterior electron density at the QM/MM boundary region. In the present paper, the ECF was constructed in terms of the Becke-Roussel (BR) exchange hole function. It was demonstrated that the ECF yielded by the BR procedure can faithfully reproduce the radial behavior of the electron density of a QM solute. The size parameter of the solute as well as the exchange repulsion are, then, obtained using the overlap model without any fitting procedure. To examine the efficiency of the method, it was applied to calculation of the exchange repulsions for minimal QM/MM systems, hydrogen-bonded water dimer, and H3O(+)-H2O. We found that our approach is able to reproduce the potential energy curves for these systems showing reasonable agreements with those given by accurate full quantum chemical calculations.

Intermediate coupling for core-level excited states: Consequences for X-Ray absorption spectroscopy

The origin of the complex NEXAFS features of X-Ray Absorption, XAS, spectra in transition metal complexes is analyzed and interpreted in terms of the angular momentum coupling of the open shell electrons. Especially for excited configurations where a core-electron is promoted to an open valence shell, the angular momentum coupling is intermediate between the two limits of Russell-Saunders, RS, coupling where spin-orbit splitting of the electron shells is neglected and j–j coupling, where this splitting is taken as dominant. The XAS intensities can be understood in terms of two factors: (1) The dipole selection rules that give the allowed excited RS multiplets and (2) the contributions of these allowed multiplets to the wavefunctions of the intermediate coupled levels. It is shown that the origin of the complex XAS spectra is due to the distribution of the RS allowed multiplets over several different intermediate coupled excited levels. The specific case that is analyzed is the L2,3 edge XAS of an Fe3+ cation, because this cation allows a focus on the angular momentum coupling to the exclusion of other effects; e.g., chemical bonding. Arguments are made that the properties identified for this atomic case are relevant for more complex materials. The analysis is based on the properties of fully relativistic, ab initio, many-body wavefunctions for the initial and final states of the XAS process. The wavefunction properties considered include the composition of the wavefunctions in terms of RS multiplets and the occupations of the spin–orbit split open shells; the latter vividly show whether the coupling is j–j or not.

Electron–phonon coupling and associated scattering rates in diamond

We calculate the variation of electron–phonon (e–ph) coupling as a function of phonon wavevector and the initial electron energy over the entire Brillouin zone (BZ) in diamond. We consider three cases of the initial electron energy, the first corresponds to an electron at the top of the valence band, the second where the electron is at the conduction band minimum at ∆, and the third is for electrons at the secondary conduction band minimum at Γ. The e–ph interaction exhibits a strong phonon wavevector dependence which indicates a need to go beyond the long-wavelength approximation typically used for treating electron–phonon scattering in transport simulations, especially for capturing high field effects. Within an ab initio electronic structure framework, we describe electron and phonon wavefunctions in terms of localized Wannier functions which allow the evaluation of e–ph coupling over a dense grid in the BZ with tractable computational effort. The full description of the e–ph coupling over the BZ is used to compute the electron linewidth which is applied to predict the associated total phonon scattering rates. Calculations at 0K and 300K are presented and show the corresponding linewidth increases with temperature and that the method is suitable for producing inputs to transport calculations.

On the inclusion of collisional correlations in quantum dynamics

We present a formalism to describe collisional correlations responsible for thermalization effects in finite quantum systems. The approach consists in a stochastic extension of time dependent mean field theory. Correlations are treated in time dependent perturbation theory and loss of coherence is assumed at some time intervals allowing a stochastic reduction of the correlated dynamics in terms of a stochastic ensemble of time dependent mean-fields. This theory was formulated long ago in terms of density matrices but never applied in practical cases because of its complexity. We propose here a reformulation of the theory in terms of wave functions and use a simplified 1D model of cluster and molecules allowing to test the theory in a schematic but realistic manner. We illustrate the performance in terms of several observables, in particular global moments of the density matrix and single particle entropy built on occupation numbers. The occupation numbers remain fixed in time dependent mean-field propagation and change when evaluating the correlations, then taking fractional values. They converge asymptotically towards Fermi distributions which is a clear indication of thermalization.

Escort distribution function of work done and diagonal entropies in quenched Luttinger liquids

We study the escort probability distribution function of work done during an interaction quantum quench of Luttinger liquids. It crosses over from the thermodynamic to the small system limit with increasing $a$, the order of the escort distribution, and depends on the universal combination $(|K_i-K_f|/(K_i+K_F))^a$ with $K_i$, $K_f$ the initial and final Luttinger liquid parameters, respectively. From its characteristic function, the diagonal R\'enyi entropies and the many body inverse participation ratio (IPR) are determined to evaluate the information content of the time evolved wavefunction in terms of the eigenstates of the final Hamiltonian. The hierarchy of overlaps is dominated by that of the ground states. The IPR exhibits a crossover from Gaussian to power law decay with increasing interaction quench parameter.

Rapidly converging bound state eigenenergies for the two dimensional quantum dipole

We examine the effectiveness of a new spectral method in solving the two dimensional dipole problem (DP), as originally formulated by Dasbiswas et al (2010 Phys. Rev. B: At. Mol. Opt. Phys. 81 064516), and recently analysed by Amore and Fernandez (AF, 2012 Phys. Rev. B: At. Mol. Opt. Phys. 45 235004), through a large, non-orthogonal basis, Rayleigh–Ritz (RR) analysis. This deceptively simple problem has a long history of poorly approximated energy values, particularly for the ground state, until the recent work by AF. In contrast to their approach, we implement an orthogonal polynomial projection quantization (OPPQ) analysis (Handy and Vrinceanu 2013 J. Phys. A: Math. Theor. 46 135202), involving expanding the wavefunction in terms of a complete basis, , where are the orthogonal polynomials relative to the weight . For systems transformable into a moment equation, such as DP, the projection coefficients are determinable in closed form, yielding an efficient quantization procedure, particularly when the weight assumes the asymptotic form of the physical solutions. There are several theoretical reasons why the OPPQ should be more effective than the above RR approach. Indeed, comparable results are achieved with significantly fewer OPPQ variational parameters as compared to RR-variational parameters. For instance, with regards to the delicate ground state energy, 130 OPPQ variables are required to achieve Egr = −0.137 7614 (Egr = −0.137 7514 after a Shanks transform) as opposed to the 821 required within the RR formulation: Egr = −0.137 7478. Despite this, the relative slow convergence for low lying even parity states, within both the OPPQ and RR formulations, suggests that significant logarithmic contributions to the wavefunction, at the origin, have been ignored by all previous investigators. Modifying the RR variational analysis to include log-dependent basis, affirms this through an accelerated RR convergence for the ground state: Egr = −0.137 748 utilizing 68 variational parameters. This result suggests further improvements within either the OPPQ or RR formulation.

Tomographic probability representation in the problem of transitions between Landau levels

The problem of quantum motion of a charged particle in a magnetic field is considered in the framework of the tomographic probability representation. The coherent and Fock states of a charge moving in a time-dependent homogeneous magnetic field are studied in the tomographic probability representation. These states are expressed in terms of the quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. These results are then generalized and applied to calculate the transition probabilities between the Landau levels for the case of transitions between stationary states of different magnetic fields, when the initially constant field becomes time dependent and then stays at some constant value again. The problem is studied in terms of wave functions and symplectic tomograms as well.

Hermitian and non-Hermitian description of quantum wave propagation

Two different approaches that provide solutions of the time-dependent Schrödinger equation for the quantum shutter initial condition, are used and compared in the description of transient quantum wave propagation. One of them is a (non-Hermitian) resonance formalism that involves a discrete resonant state expansion of the wavefunction in terms of the resonances of the system, and the other one is a (Hermitian) approach based on a continuum wavefunction expansion of the solution. We investigate the equivalence of these different methods in resonant structures finding that the two approaches lead to results that are numerically indistinguishable from each other. In particular, we verify that the continuum wavefunction expansion predicts the existence of resonance forerunners in the probability density, which are identical to those obtained with the resonance state formalism. We also provide with useful criteria for the numerical evaluation of the solutions obtained with both formulations based on the knowledge of the relevant contributions of the spectrum in momentum k space.

Computation of resonant states using explicitly correlated coordinates in Be-like B, C and N

We apply an efficient computational method to obtain accurate values for energy positions and widths of autoionizing states in Be-like atomic systems. The method is based on the configurration interaction (CI) expansion of the two-active (outer) electron wavefunction in terms of Hylleraas-type correlated configurations. A model potential is used to represent the outer electron interaction with the frozen 1s2 core. The series of unphysical virtual 1snℓ states are removed by using a Phyllips-Keniman pseudopotential projector. All matrix elements can be written in closed form. The resonance parameters are computed within the stabilization method.

Incorporation of quantum effects for selected degrees of freedom into the trajectory-based dynamics using spatial domains

The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the "quantum" degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom.

An efficient method to evaluate energy variances for extrapolation methods

The energy variance extrapolation method consists of relating the approximate energies in many-body calculations to the corresponding energy variances and inferring eigenvalues by extrapolating to zero variance. The method needs a fast evaluation of the energy variances. For many-body methods that expand the nuclear wavefunctions in terms of deformed Slater determinants, the best available method for the evaluation of energy variances scales with the sixth power of the number of single-particle states. We propose a new method which depends on the number of single-particle orbits and the number of particles rather than the number of single-particle states. We discuss as an example the case of 4He using the chiral N3LO interaction in a basis consisting up to 184 single-particle states.

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Natural bond orbital methods

AbstractNatural bond orbital (NBO) methods encompass a suite of algorithms that enable fundamental bonding concepts to be extracted from Hartree‐Fock (HF), Density Functional Theory (DFT), and post‐HF computations. NBO terminology and general mathematical formulations for atoms and polyatomic species are presented. NBO analyses of selected molecules that span the periodic table illustrate the deciphering of the molecular wavefunction in terms commonly understood by chemists: Lewis structures, charge, bond order, bond type, hybridization, resonance, donor–acceptor interactions, etc. Upcoming features in the NBO program address ongoing advances inab initiocomputing technology and burgeoning demands of its user community by introducing major new methods, keywords, and electronic structure system/NBO communication enhancements. © 2011 John Wiley & Sons, Ltd.This article is categorized under:Structure and Mechanism > Molecular Structures

Dynamic responses of a sediment-filled valley with a fluid layer subject to incident waves

The diffraction of elastic waves by a sedimentary valley in a homogeneous elastic half-space is studied in this paper. The sediment-filled valley is composed of a fluid layer over a soft soil deposit whose characteristics may be significant and should be carefully considered when designing long span bridges with high piers. The method of analysis adopted in the paper is to decompose the problem into an interior region and an exterior region. In the exterior region, the scattered wave fields are constructed with the linear combinations of two independent sets of Lamb’s singular solutions, i.e., the integral solutions for two concentrated surface loads in two directions; and their derivatives are used to represent the scattered wave fields. A technique is proposed to calculate the integrals in the wave-number domain based on the method of steepest descent. For the interior region, the wave fields for the fluid layer and soft soil deposit are expressed in terms of wave functions which satisfy the equation of motion. The continuity condition at the interface of the media is satisfied in the least square sense. The effects of geometric topography, soil amplification and fluid layer subject to different types of incident harmonic plane waves are analyzed and discussed.

Dynamic Responses of a Sedimentary Valley to Incident Waves

This paper presents the analysis of the dynamic response of a sedimentary valley subjected to incident plane waves. The sediment-filled valley is composed by a fluid layer over a soft soil deposit whose effect is of importance in the design of long span bridges with high piers. The method of analysis makes use of decomposing the problem into interior and exterior regions. In the exterior region, the scattered wave fields are constructed with two linearly independent sequences of Lamb’s singular solutions which satisfy the traction free conditions at ground surface and radiation conditions at infinity. For the interior regions, the wave fields for the fluid layer and soft soil deposit are expressed in terms of wave functions which satisfy the equation of motion. The continuity conditions at the interface of the media are satisfied in the least square sense. The effects of geometric topography, soil amplification and fluid layer under different types of incident harmonic plane waves are analyzed and discussed in detail.

Dynamics of a truncated elastic cone

The problem of determining the nonstationary wave field of an elastic truncated cone with nonzero dead weight is formulated in terms of wave functions. The Laplace transform with respect to time and an integral transform with respect to time polar angle are used to reduce the problem to a one-dimensional vector problem in the transform domain. The transforms of the wave functions are expanded into series in inverse powers of the Laplace transform parameter, which makes it possible to study the wave process at the initial instants of interaction. A method is proposed to solve the problem for an elastic cone doubly truncated by spherical surfaces