Generalized Spherical Harmonics as Representation Matrix Elements of Rotation Group

1952 ◽  
Vol 7 (3) ◽  
pp. 307-312 ◽  
Author(s):  
Takehito Takahashi
Keyword(s):  
Spherical Harmonics ◽  
Rotation Group ◽  
Matrix Elements ◽  
Generalized Spherical Harmonics

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths

1964 ◽  
Vol 54 (2) ◽  
pp. 571-610
Author(s):  
George E. Backus
Keyword(s):  
Spherical Harmonics ◽  
Rayleigh Waves ◽  
Seismic Station ◽  
Great Circle ◽  
Explicit Formulas ◽  
Phase Velocities ◽  
Rapid Accumulation ◽  
Generalized Spherical Harmonics ◽  
Geographical Position ◽  
The Difference

ABSTRACT If the averages of the reciprocal phase velocity c−1 of a given Rayleigh or Love mode over various great circular or great semicircular paths are known, information can be extracted about how c−1 varies with geographical position. Assuming that geometrical optics is applicable, it is shown that if c−1 is isotropic its great circular averages determine only the sum of the values of c−1 at antipodal points and not their difference. The great semicircular averages determine the difference as well. If c−1 is anisotropic through any cause other than the earth's rotation, even great semicircular averages do not determine c−1 completely. Rotation has negligible effect on Love waves, and if it is the only anisotropy present its effect on Rayleigh waves can be measured and removed by comparing the averages of c−1 for the two directions of travel around any great circle not intersecting the poles of rotation. Only great circular and great semicircular paths are considered because every earthquake produces two averages of c−1 over such paths for each seismic station. No other paths permit such rapid accumulation of data when the azimuthal variations of the earthquakes' radiation patterns are unknown. Expansion of the data in generalized spherical harmonics circumvents the fact that the explicit formulas for c−1 in terms of its great circular or great semicircular integrals require differentiation of the data. Formulas are given for calculating the generalized spherical harmonics numerically.

scholarly journals Addition Formula and Related Integral Equations for Heine—Stieltjes Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1231
Author(s):  
Hans Volkmer
Keyword(s):  
Integral Equations ◽  
Explicit Form ◽  
Spherical Harmonics ◽  
Reproducing Kernel ◽  
Addition Formula ◽  
Symmetric Products ◽  
Special Cases ◽  
Related Integral ◽  
Generalized Spherical Harmonics ◽  
The Relationship

It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found.

A New Method for Orientation Distribution Function Analysis

1985 ◽  
Vol 29 ◽  
pp. 443-449
Author(s):  
Munetsugu Matsuo ◽  
Koichi Kawasaki ◽  
Tetsuya Sugai
Keyword(s):  
Distribution Function ◽  
Spherical Harmonics ◽  
Orientation Distribution Function ◽  
Function Analysis ◽  
Orientation Distribution ◽  
Symmetry Operation ◽  
Composite Function ◽  
Odd Order ◽  
Pole Figures ◽  
Generalized Spherical Harmonics

AbstractAs a means for quantitative texture analysis, the crystallite orientation distribution function analysis has an important drawback: to bring ghosts as a consequence of the presence of a non-trivial kernel which consists of the spherical harmonics of odd order terms. In the spherical hamonic analysis, ghosts occur in the particular orientations by symmetry operation from the real orientation in accordance with the symmetry of the harmonics of even orders. For recovery of the odd order harmonics, the 9th-order generalized spherical harmonics are linearly combined and added to the orientation distribution function reconstructed from pole figures to a composite function. The coefficients of the linear combination are optimized to minimize the sum of negative values in the composite function. Reproducibility was simulated by using artificial pole figures of single or multiple component textures. Elimination of the ghosts is accompanied by increase in the height of real peak in the composite function of a single preferred orientation. Relative fractions of both major and minor textural components are reproduced with satisfactory fidelity In the simulation for analysis of multi-component textures.

scholarly journals Spin lattices, state transfer, and bivariate Krawtchouk polynomials

2015 ◽  
Vol 93 (9) ◽  
pp. 979-984 ◽  
Author(s):  
Vincent X. Genest ◽  
Hiroshi Miki ◽  
Luc Vinet ◽  
Alexei Zhedanov
Keyword(s):  
Three Dimensional ◽  
Rotation Group ◽  
Matrix Elements ◽  
Quantum State Transfer ◽  
Krawtchouk Polynomials ◽  
Triangular Domain ◽  
State Transfer ◽  
Exactly Solvable ◽  
Spin Lattices ◽  
Transfer Properties

The quantum state transfer properties of a class of two-dimensional spin lattices on a triangular domain are investigated. Systems for which the 1-excitation dynamics is exactly solvable are identified. The exact solutions are expressed in terms of the bivariate Krawtchouk polynomials that arise as matrix elements of the unitary representations of the rotation group on the states of the three-dimensional harmonic oscillator.

Quasi-particles in atomic shell theory

1970 ◽  
Vol 315 (1520) ◽  
pp. 27-37 ◽  
Keyword(s):  
Shell Theory ◽  
Particle Creation ◽  
Rotation Group ◽  
Coupling Scheme ◽  
Matrix Elements ◽  
Spin Orientation ◽  
Quasi Particle ◽  
Quantum Numbers ◽  
Fractional Parentage ◽  
Electronic Configurations

A new scheme is described for defining and classifying the states of the electronic configurations l N . The spaces for which the spin orientation is either up or down are both factored into two parts. Each of these parts (distinguished by a symbol Ɵ) corresponds to the irreducible representatio n (½ ½ ... ½ ) of the rotation group R Ɵ (2 l +1). The generators for this group are constructed from quasi-particle creation and annihilation operators. The angular momentum quantum numbers l Ɵ arising from the decomposition of (½ ½ ... ½) into representations of R Ɵ (3) can be used to couple the four parts together. No ambiguities arise when l < 9, thereby giving a very satisfactory coupling scheme. No coefficients of fractional parentage (c. f. p.) are required in the calculation of matrix elements. Simple explanations are given for some null c. f. p. and for some repeated eigenvalues of an operator that had previously been used to classify the state s of g N .

Still another way to calculate representations of the three-dimensional pure rotation group

1967 ◽  
Vol 63 (2) ◽  
pp. 273-275 ◽  
Author(s):  
R. H. Albert
Keyword(s):  
Explicit Formula ◽  
Three Dimensional ◽  
Rotation Group ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Pure Rotation ◽  
The Matrix ◽  
Finite Sum

AbstractAn explicit formula is derived for exp (iβJz) as a finite sum of irreducible tensor components. With this formula, a technique is developed to obtain the matrix elements of exp (iβJy).

The vector-ladder operators for the rotation group O3

1993 ◽  
Vol 71 (3-4) ◽  
pp. 152-154
Author(s):  
J. E. Hardy
Keyword(s):  
Rotation Group ◽  
Matrix Elements ◽  
Ladder Operators ◽  
Straightforward Calculation ◽  
Angular Momenta ◽  
State Formula

The two vector-ladder operators, which step from any state [Formula: see text] in an irreducible multiplet in the space of product states of two commuting angular momenta, are defined and all their nonvanishing matrix elements are given, facilitating direct, straightforward calculation of the six nearby nonsibling states [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].

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