A uniform WKB approximation for spheroidal wave functions

The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.

Power series expansions for spheroidal wave functions with small arguments

An approximate light-cone wave function for the pion effective quark-antiquark Fock sector corresponding to a small value of the renormalization group parameter is presented. The approximate wave function is motivated by the LF-holography and the quadratic confinement potential in the front form of Hamiltonian dynamics, which is in harmony with the linear confining potential in the instant form. The pion radius, decay constant and form-factor are also presented.

Supersymmetry quantum mechanics method had been used to solve the time independent cosmic string equation for trigonometric non-central potential system. Variable separation method was applied to the time independent cosmic string equation, and three one-dimensional Schrodinger equation had been obtained. Radial wave functions, angular wave functions, azimuthal wave functions, and energy levels were analyzed using supersymmetry quantum mechanics method. The effect of the trigonometric non-central potential caused energy levels to increase. Energy levels for ≠ λ0 ≠ κ,0 ≠ ϖ,0, a ≠0, and b ≠0 had a higher value than energy levels for = λO = κ,0 = ϖ,0, a =0, and b = 0. The increased value of parameter λ, parameter κ, parameter ϖ, parameter a , parameter b , and the decreased of the cosmic string parameter a, caused an increase in energy levels.Supersymmetry quantum mechanics method had been used to solve the time independent cosmic string equation for trigonometric non-central potential system. Variable separation method was applied to the time independent cosmic string equation, and three one-dimensional Schrodinger equation had been obtained. Radial wave functions, angular wave functions, azimuthal wave functions, and energy levels were analyzed using supersymmetry quantum mechanics method. The effect of the trigonometric non-central potential caused energy levels to increase. Energy levels for ≠ λ0 ≠ κ,0 ≠ ϖ,0, a ≠0, and b ≠0 had a higher value than energy levels for = λO = κ,0 = ϖ,0, a =0, and b = 0. The increased value of parameter λ, parameter κ, parameter ϖ, parameter a , parameter b , and the decreased of the cosmic string parameter a, caused an increase in energy levels.

FORTRAN computer programs have been developed which accurately and rapidly evaluate for a wide range of parameters both prolate and oblate spheroidal radial wave functions of the first and second kind and their first derivatives, and both prolate and oblate spheroidal angle wave functions of the first kind and their first and second derivatives. This letter briefly discusses available documentation on these programs and describes some of the computational procedures necessary for their success. In addition, recently published tables of spheroidal radial wave functions calculated using these programs are described.

Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force, magnetic pressure, or anisotropic equations of state. The stellar surface therefore slightly or significantly deviates from a sphere depending on the strength of these anisotropic perturbations. Aims. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. Methods. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. The solutions are expanded in series of radial and angular spheroidal wave functions. Particular emphasis is put on the magnetic dipole radiation. Second, we compute approximate solutions by integrating the time-dependent Maxwell equations in spheroidal coordinates numerically. Results. We show that the spin-down luminosity corrections compared to a perfect sphere are, to leading order, given by terms involving (a/rL)2 and (a/R)2 where a is the stellar oblateness or prolateness, R the smallest star radius, and rL the light-cylinder radius. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. At large distances r ≫ a, the solution tends asymptotically to the perfect spherical case of a rotating dipole.

In quantum mechanics, 3d orbital can be expressed using quantum wave function. Though five 3d orbitals have the same radial quantum wave function, they have different angular quantum wave functions. In molecular orbital (MO) calculation, Gaussian-type quantum wave function is applied, instead of Slater-type quantum wave function. $$3{d}_{{x}^{2}-{y}^{2}}$$ and $$3{d}_{{z}^{2}}$$ orbitals are expressed by the superposition of $${x}^{2}$$ , $${y}^{2}$$ and $${z}^{2}$$ components. Ligand field theory is constructed in semi-quantum manner, based on point charge. Though one specific spin configuration can be predicted, the correct orbital energy diagram cannot be predicted. For example, in OCuO cluster, ligand field theory predicts that highest orbital energy is given in $$3{d}_{{3x}^{2}-{r}^{2}}$$ orbital. However, in fact, the orbital has lower orbital energy than other 3d orbitals. Two spin configurations: (1) $$3{d}_{{3x}^{2}-{r}^{2}}$$ type and (2) $$3{d}_{{z}^{2}-{x}^{2}}$$ type are here given. Next, we consider realistic model, inverse Cu4O4 cluster, where oxygen atoms are allocated at edges of square, instead of copper atom. As same as OCuO cluster, two spin configurations are given. In $$3{d}_{{z}^{2}-{x}^{2}}$$ type spin configuration, there is no orbital overlap between copper and oxygen atoms. In other words, $$3{d}_{{z}^{2}-{x}^{2}}$$ type electron works as barrier to oxygen electron. The effect is called “Barrier Effect”. Further, in $$3{d}_{{z}^{2}-{x}^{2}}$$ type spin configuration, quintet and singlet spin states are almost degenerated. When controlling the total energy difference, inverse Cu4O4 cluster can be applied for quantum spin memory.

ADVERTISEMENT RETURN TO ISSUEPREVTech ReportNEXTConstruction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen AtomMark Ellison , John Tippin , and Thomas S. Kuntzleman View Author InformationDepartment of Chemistry, Ursinus College, Collegeville, PA 19426Department of Mathematics, Spring Arbor University, Spring Arbor, MI 49283Department of Chemistry, Spring Arbor University, Spring Arbor, MI 49283Cite this: J. Chem. Educ. 2007, 84, 11, 1886Publication Date (Web):November 1, 2007Publication History Received3 August 2009Published online1 November 2007Published inissue 1 November 2007https://doi.org/10.1021/ed084p1886RIGHTS & PERMISSIONSArticle Views305Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (104 KB) Get e-AlertsSupporting Info (3)»Supporting Information Supporting Information SUBJECTS:Hydrogen,Wave function,Students,Mathematical methods Get e-Alerts

The time-dependent variational principle in the stationary action principle form is formulated with constraint conditions for parametrized wave functions. The constraint conditions are classified into the first and the second classes as in Dirac's constrained classical mechanics based on the commutability of operators. If the local bases in the parametrized wave functions construct a complete basis set, the classification of constraints with the commutator becomes equivalent to that with the complex generalized Poisson brackets (CGPB). However, in approximate wave functions with limited variational parameters, the equivalence between the CGPB and the commutator of operators is lost. The first-class constraints and the constants of motion classified with the commutator should be reconsidered as the pseudo-second-class constraints in the approximate wave functions.

Quantum-selected configuration interaction (QSCI) is an approach for quantum chemical calculations using current quantum computers. In conventional QSCI, Slater determinants used for the wave function expansion are sampled by iteratively performing approximate wave function preparation and subsequent measurement in the computational basis, and then the subspace Hamiltonian matrix is diagonalized on a classical computer. In this approach, preparation of a high-quality approximate wave function is necessary to accurately compute total energies. Here we propose a Hamiltonian simulation-based QSCI (HSB-QSCI) to avoid this difficulty, by sampling the Slater determinants from quantum states generated by the real-time evolution of approximate wave functions. We provide numerical simulations for the lowest spin-singlet and triplet states of oligoacenes (benzene, naphthalene, and anthracene), phenylene-1,4-dinitrene, and hexa-1,2,3,4,5-pentaene. We found that the HSB-QSCI is applicable not only to molecules where the Hartree-Fock provides a good approximation of the ground state, but also to strongly correlated systems where preparing a high-quality approximate wave function is hard. Hardware demonstrations of the HSB-QSCI are also reported for carbyne molecules expressed by up to 36 qubits, using an IBM quantum processor. The HSB-QSCI captures more than 99.18% of the correlation energies in the active space by considering about 1% of all the Slater determinants in 36 qubit systems, illustrating the ability of the proposed method to efficiently consider important electronic configurations.

Taking the ground state of helium as an example, evidence is presented in support of the theory that when an approximate wave function satisfies a hypervirial theorem, then the expectation values of certain operators calculated using that wave function will be quite accurate. Firstly, it is discovered which hypervirial theorems are satisfied by various simple approximate wave functions when their disposable parameters are chosen to optimize the energy. Appropriate expectation values are tested, and found to be reasonably accurate. Secondly, the optimum energy criterion is abandoned in certain cases, and parameters are chosen to satisfy alternative hypervirial theorems. This leads to improvement in expectation values related to the hypervirial operators. Some advantages and drawbacks of the hypervirial method of fixing parameters in approximate wave functions are briefly discussed.

In this paper we report on a package, written in the Mathematica computer algebra system, which has been developed to compute the spheroidal wave functions of Meixner [J. Meixner and R.W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen, 1954] and is available online (www.physics.uwa.edu.au/~falloon/spheroidal/spheroidal.html). This package represents a substantial contribution to the existing software, since it computes the spheroidal wave functions to arbitrary precision for general complex parameters mu, nu, gamma and argument z; existing software can only handle integer mu, nu and does not give arbitrary precision. The package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter gamma. The spheroidal wave functions of Flammer [C. Flammer, Spheroidal Wave Functions, 1957] are included as a special case of Meixner's more general functions. This paper presents a concise review of the general theory of spheroidal wave functions and a description of the formulas and algorithms used in their computation, and gives high-precision numerical examples.

Ground-state energy for periodic Anderson model

Approximate wave functions for the hydrogen molecule ground state previously available in the literature are analyzed quantitatively into approximate natural spin orbital functions with particular attention to the corresponding occupation numbers. The analysis demonstrates the very great similarity of all such trial wave functions, and especially the largely molecular orbital nature of the wave function. In addition it shows a close relationship between the molecular orbital and valence bond functions, and the importance for allowing for angular correlation of the electrons by including terms dependent upon the azimuthal coordinate. The analysis particularly demonstrates that approximate natural spin orbital occupation numbers are nearly invariant under a wide variety of choices of basis functions, and therefore are particularly suitable for comparison of different approximate functions and for discussion of their respective properties.

The energy-variance-based optimization procedures have proven to be useful tools to describe N-electron spectra. However, the resulting wave functions usually present spin-contaminant contributions. The goal of this work is to reduce the spin contamination of the results arising from the unrestricted doubly occupied configuration interaction method in its energy variance minimization version [Alcoba et al., J. Chem. Phys. 160, 164107 (2024)]. We propose to incorporate the half-projection technique, which allows removing the spin components with even or odd spin quantum number of an approximate N-electron wave function, into the framework of the unrestricted doubly occupied configuration interaction treatment. This implementation can be carried out following several possible ways, whose results are analyzed in detail, in order to show the behavior of each procedure. Numerical determinations performed on selected strongly correlated N-electron systems, in ground and excited states, allow us to assess the most suitable procedure.