Oscillator Wave Functions Research Articles

Bose-Einstein condensation in an optical lattice: A perturbation approach

We derive closed analytical expressions for the order parameter $\Phi (x)$ and for the chemical potential $\mu $ of a Bose-Einstein Condensate loaded into a harmonically confined, one dimensional optical lattice, for sufficiently weak, repulsive or attractive interaction, and not too strong laser intensities. Our results are compared with exact numerical calculations in order to map out the range of validity of the perturbative analytical approach. We identify parameter values where the optical lattice compensates the interaction-induced nonlinearity, such that the condensate ground state coincides with a simple, single particle harmonic oscillator wave function.

Generating Spectra from Ground-State Wave Functions: Unraveling Anharmonic Effects in the OH−·H2O Vibrational Predissociation Spectrum

An approach is described for calculating anharmonic spectra for polyatomic molecules using only the ground-state probability amplitude. The underlying theory is based on properties of harmonic oscillator wave functions and is tested for Morse oscillators with a range of anharmonicities. More extensive tests are performed with H(3)O(2)(-), using the potential and dipole surfaces of Bowman and co-workers [J. Am. Chem. Soc. 2004, 126, 5042]. The resulting energies are compared to earlier studies that employed the same potential surface, and the agreement is shown to be very good. The vibrational spectra are calculated for both H(3)O(2)(-) and D(3)O(2)(-). In the case of H(3)O(2)(-), comparisons are made with a previously reported experimental spectrum below 2000 cm(-1). We also report the spectrum of H(3)O(2)(-) from 2400-4500 cm(-1), which extends 500 cm(-1) above the region reported earlier, revealing several new bands. As the only fundamentals in this spectral region involve the OH stretches, the spectrum is surprisingly rich. On the basis of comparisons of the experimental and calculated spectra, assignments are proposed for several of the features in this spectral region.

Unscreened Coulomb Interactions and the Quantum Spin Hall Phase in Neutral Zigzag Graphene Ribbons

A study of the effect of unscreened Coulomb interactions on the quantum spin Hall phase of finite-width neutral zigzag graphene ribbons is presented. By solving a tight-binding Hamiltonian that includes the intrinsic spin-orbit (ISO) interaction, exact expressions for band structures and edge-state wave functions are obtained. These analytic results, supported by tight-binding calculations, show that chiral spin-filtered edge states are composed of localized and damped oscillatory wave functions, reminiscent of the ones obtained in armchair ribbons. The addition of long-range Coulomb interactions opens a gap in the charge sector with a gapless spin sector. In contrast to armchair terminations, the charge gap vanishes exponentially with the ribbon width and its amplitude and decay length are strongly dependent on the ISO coupling. Comparison with reported ab initio calculations are presented.

Density functional theory and Møller-Plesset studies of hindered rotations of acetone

The hindered rotations of acetone were studied density functional theory (B3LYP) and second order Møller-Plesset approaches using 6-31G** and 6-311G** basis sets. One of the CH(3) groups of acetone with fixed heavy atoms was rotated from 0.0 to 120 degrees, and CCH angles were scanned from 90.3 to 130.3 degrees to cover the potential energy surface of interest; a circular valley was obtained with the deepest potential value at a CCH angle equal to 109.3 degrees. Potential energy profiles were then calculated by assuming that the molecular geometry could relax during rotation (i.e., each value of the torsion angle of the molecular geometry was optimized). Next, the two methyl groups were both rotated clockwise, and then one was rotated clockwise and the other counterclockwise. Using the variation method, and utilizing the first 20 harmonic oscillator wave functions, the energy levels, relative transition moment and relative transition intensities of the component of the hindered rotation nu(2) (125.16 cm(-1)) were computed in a one-dimensional Schrodinger equation. The first three energy levels were almost degenerate; the next three were opened up, and the seventh energy level appeared above the level where tunneling can occur.

Solution of the Cauchy problem for a time-dependent Schrödinger equation

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products.

The electron g factorof cylindrical GaAs–Ga1−xAlxAs quantum well wires under magnetic fields applied along the wire axis

The effects of confinement and magnetic fields on the effective electron Landég factorof GaAs–Ga1−xAlxAs cylindrical quantum well wires are studied. Calculations were carried out via theOgg–McCombe effective Hamiltonian which is used to describe the non-parabolicity andanisotropy effects on the electron states in the conduction band. The applied magneticfield is taken along the wire axis, and the Schrödinger equation corresponding toelectron spin projections parallel and antiparallel to the magnetic field is solved byusing an expansion of the electron wavefunctions in terms of two-dimensionalharmonic oscillator wavefunctions. Calculations for the electron factor in GaAs–Ga1−xAlxAs cylindrical quantum well wires are compared with results from previoustheoretical work. Moreover, the present results clearly indicate the importanceof taking into account the non-parabolicity/anisotropy of the conductionband if one is interested in a quantitative understanding of the electrong factorin GaAs–Ga1−xAlxAs quantum well wires.

The Harmonic Oscillator with a Gaussian Perturbation: Evaluation of the Integrals and Example Applications

A general result for the integrals of the Gaussian function over the harmonic oscillator wavefunctions is derived using generating functions. Using this result, an example problem of a harmonic oscillator with various Gaussian perturbations is explored in order to compare the results of precise numerical solution, the variational method, and perturbation theory. This model problem will provide instructors of quantum chemistry with an additional example for instructional use. An additional example in which a potential of this form is used to model the ammonia vibrational spectrum is included in summary form.

A review of <sup>20</sup>Ne structure in a full microscopic self-consistent shell–model calculation with tensor correlations

A set of single-particle energies together with a set of two-body matrix- elements derived in a self –consistent manner from the Reid soft–core potential are used to calculate the energy levels of 20 Ne. We used a harmonic oscillator wave function folded with two-body correlation functions in our calculation. It is found that the calculated spectra agree very well with experiment and the best available shell-model calculations by other workers. As a result we have demonstrated that it is possible to calculate the spectroscopy of nuclei microscopically and self-consistently in such a way that both the single –particle energies and the effective two-body interactions are derived from the same procedure. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 63-68

Efficient configuration selection scheme for vibrational second-order perturbation theory

A fast algorithm of vibrational second-order Moller-Plesset perturbation theory is proposed, enabling a substantial reduction in the number of vibrational self-consistent-field (VSCF) configurations that need to be summed in the calculations. Important configurations are identified a priori by assuming that a reference VSCF wave function is approximated well by harmonic oscillator wave functions and that fifth- and higher-order anharmonicities are negligible. The proposed scheme has reduced the number of VSCF configurations by more than 100 times for formaldehyde, ethylene, and furazan with an error in computed frequencies being not more than a few cm(-1).

Two-Variable Hermite Function as Quantum Entanglement of HarmonicOscillator's Wave Functions

We reveal that the two-variable Hermite function hm,n, whichis the generalized Bargmann representation of the two-mode Fock state, involvesquantum entanglement of harmonic oscillator's wave functions. The Schmidtdecomposition of hm,n is derived. It also turns out that hm,n canbe generated by windowed Fourier transform of the single-variable Hermitefunctions. As an application, the wave function of the two-variable Hermitepolynomial state S(r)Hm,n(μa1†,μa2†)|00⟩, which is the minimum uncertaintystate for sum squeezing, in ⟨η| representationis calculated.

Nucleon Meson Form Factors and Coupling Constants

The nucleon-nucleon (NN) interaction has been an object of active study for over fifty years. In the past decade high precision potentials based on one boson exchange (OBE) were developed: Argonne 18, Nijmegen '93, CD Bonn, etc. Unfortunately they are highly parametrized (≈ 40 parameters) and less theoretically sound than their predecessors: Bonn B, Nijmegen '78, Paris, etc. Historically OBE potentials parametrize the coupling constants/form factors of an effective Lagrangian. We have calculated these form factors using quark model simple harmonic oscillator wave functions and the 3P0 model of hadronic decays. We present preliminary results for phase shifts of a toy nuclear interaction model, pion (π) and scalar/iso- scalar (σ) exchange. Prospects for future work, including self-consistently extending this work to non-zero strangeness hadrons, are discussed.

Mass Parameter Quantization in the FRW Cosmological Model

Using the canonical quantum theory apply to spherically symmetric pure gravitational systems, we present the study of the closed Friedmann-Robertson-Walker (FRW) cosmological model filled with pressureless matter (dust) content as a toy model. The Wheeler-DeWitt equation is view as the Schrodinger equation for the linear harmonic oscillator with energy E. We show that such type of universe has a quantized masses of the order of the Planck mass and harmonic oscillator wave functions, where a dual symmetry emerge among the quantum parameters.

Improvement of the Heisenberg and Fisher-information-based uncertainty relations for D-dimensional central potentials

The Heisenberg and Fisher-information-based uncertainty relations are improved for stationary states of single-particle systems in a D-dimensional central potential. The improvement increases with the squared orbital hyperangular quantum number. The new uncertainty relations saturate for the isotropic harmonic oscillator wavefunction.

Gamow shell model and realistic nucleon-nucleon interactions

We present a new and efficient method to obtain a Gamow shell-model basis and matrix elements generated by realistic nucleon-nucleon interactions. We derive a self-consistent Hartree-Fock potential from the renormalized N3LO interaction model. The corresponding Gamow one-body eigenstates are generated in a plane wave basis in order to build a Gamow shell-model set of basis states for the closed shell nuclei 4He and 16O. We address also the problem of representing a realistic nucleon-nucleon interaction in a two-particle Berggren basis in the laboratory frame. To achieve this, an expansion of matrix elements of the residual nucleon-nucleon interaction in a finite set of harmonic oscillator wave functions is used. We show that all loosely bound and narrow resonant states converge surprisingly fast. Even broad resonances in these two-particle valence systems converge within a reasonable number of harmonic oscillator functions. Examples of 6He and 18O Gamow shell-model calculations using 4He and 16}O as closed shell cores are presented. This procedure allows Gamow shell-model calculations to be performed with all realistic nucleon-nucleon interactions and with either momentum or position space representations for the Gamow basis. The possibility to remove the center of mass spuriosity of Gamow shell-model nuclear states with this method is also discussed. Perspectives for nuclear structure calculations of dripline nuclei are outlined.

Cavity polaritons: Classical behavior of a quantum parametric oscillator

We address theoretically the optical parametric oscillator based on semiconductor cavity exciton polaritons under a pulsed excitation. A ``hyperspin'' formalism is developed which allows, in the case of large number of polaritons, to reduce quantum dynamics of the parametric oscillator wave function to the Liouville equation for the classical probability distribution. Implications for the statistics of polariton ensembles are analyzed.

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.

Applying reproducing kernels to the evaluation and approximation of the simple and time-dependent imaginary time harmonic oscillator path integrals

Reproduction of kernel Hilbert spaces offers an attractive setting for imaginary time path integrals, since they allow to naturally define a probability on the space of paths, which is equal to the probability associated with the paths in Feynman's path integral formulation. This study shows that if the propagator is Gaussian, its variance equals the squared norm of a linear functional on the space of paths. This can be used to rederive the harmonic oscillator propagator, as well as to offer a finite-dimensional perturbative approximation scheme for the time-dependent oscillator wave function and its ground state energy, and its bound error. The error is related to the rate of decay of the Fourier coefficients of the time-dependent part of the potential. When the rate of decay increases beyond a certain threshold, the error in the approximation over a subspace of dimension n is of order (1/n 3).

Exact analytical expressions and numerical analysis of two-center Franck–Condon factors and matrix elements over displaced harmonic oscillator wave functions

A detailed study is undertaken, using various techniques, in deriving analytical formula of Franck–Condon overlap integrals and matrix elements of various functions of power ( x l ) , exponential ( exp ( − 2 c x ) ) and Gaussian ( exp ( − c x 2 ) ) over displaced harmonic oscillator wave functions with arbitrary frequencies. The results suggested by previous experience with various algorithms are presented in mathematically compact form and consist of generalization. The relationships obtained are valid for the arbitrary values of parameters and the computation results are in good agreement with the literature. The numerical results illustrate clearly a further reduction in calculation times. Program summary Program name:FRANCK Catalogue identifier:ADXX_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXX_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Programming language:Mathematica 5.0 Computer:Pentium M 1.4 GHz Operating system:Mathematica 5.0 RAM:512 MB No. of lines in distributed program, including test data, etc.:825 No. of bytes in distributed program, including test data, etc.:16 344 Distribution format:tar.gz Nature of problem:The programs calculate the Franck–Condon factors and matrix elements over displaced harmonic oscillator wave functions with arbitrary quantum numbers ( n , n 1 ) , frequencies ( a , a 1 ) and displacement ( d ) for the various functions of power ( x l ) , exponential ( exp ( − 2 c x ) ) and Gaussian ( exp ( − c x 2 ) ) . Solution method:The Franck–Condon factors and matrix elements are evaluated using binomial coefficients and basic integrals. Restrictions:The results obtained by the present programs show great numerical stability for arbitrary quantum numbers ( n , n 1 ) , frequencies ( a , a 1 ) and displacement ( d ) . Unusual features:None Running time:As an example, for the value of Franck–Condon Overlap Integral I n n ′ ( d ; α , α ′ ) = 0.004405001887372332 with n = 3 , n 1 = 2 , a = 4 , a 1 = 3 , d = 2 , the compilation time in a Pentium M 1.4 GHz computer is 0.18 s. Execution time depends on the values of integral parameters n, n ′ , d, α, α ′ .

Coulomb breakup in a transformed harmonic oscillator basis

The problem of Coulomb breakup in the scattering of a two-body loosely bound projectile by a heavy target is addressed. A basis of transformed harmonic oscillator (THO) wave functions is used to discretize the projectile continuum and to diagonalize the Hamiltonian of the two-body system. Results for the reaction $^{8}$B+$^{58}$Ni at subcoulomb energies are presented. Comparison of different observables with those obtained with the standard Continuum Discretized Coupled-Channels (CDCC) method shows good agreement between both approaches.

Analytical Evaluation of Two-Center Franck-Condon Overlap Integrals over Harmonic Oscillator Wave Function

A unified treatment of Franck-Condon (FC) overlap integrals with arbitrary values of parameters is described. These integrals are represented in terms of binomial coefficients. For quick calculations, the binomial coefficients are stored in the memory of the computer. Therefore, the CPU time has been greatly reduced. Numerical results presented agree excellently with those obtained in the literature