Matrix Elements of Irreducible Representations of $SU(2) \times SU(2)$ and Vector-Valued Orthogonal Polynomials

1985 ◽  
Vol 16 (3) ◽  
pp. 602-613 ◽  
Author(s):  
Tom H. Koornwinder
Keyword(s):  
Orthogonal Polynomials ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Vector Valued

Matrix factorizations for the Coulomb interaction between electrons in atoms

1969 ◽  
Vol 309 (1497) ◽  
pp. 185-194 ◽  
Keyword(s):  
Coulomb Interaction ◽  
Irreducible Representations ◽  
Matrix Factorizations ◽  
Matrix Elements ◽  
Spin Orientation ◽  
Straightforward Application ◽  
Basic States

In contrast to what would be expected from a straightforward application of the Wigner-Eckart theorem, a simple factorization can often be made of matrix elements of operators that represent parts of the Coulomb interaction between f electrons in atoms. This property, which was discovered by Racah, is explained by using as basic states those in which the electrons are divided into two classes according to their spin orientation. The two collections of electrons are coupled through the irreducible representations of the group G 2 to which they belong, and this permits many of Racah’s proportionality constants to be related to isoscalar factors. The analogous factorizations that Feneuille found for the configurations (s + d) N are also discussed.

scholarly journals Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

2021 ◽  
Author(s):  
Mohammad Walid AlMasri
Keyword(s):  
Energy Spectrum ◽  
Orthogonal Polynomials ◽  
Complex Potentials ◽  
Complex Conjugate ◽  
Matrix Elements ◽  
Morse Oscillator ◽  
Supersymmetric Partner ◽  
Natural Home ◽  
Symmetric Complex ◽  
Physical Quantities

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

Theoretical studies in nuclear structure. II. Nuclear d 2 , d 3 and d 4 configurations. Fractional parentage coefficients and central force matrix elements

1951 ◽  
Vol 205 (1081) ◽  
pp. 192-237 ◽  
Keyword(s):  
Group Theory ◽  
Nuclear Structure ◽  
Central Force ◽  
Irreducible Representations ◽  
Theoretical Studies ◽  
Matrix Elements ◽  
Energy Matrix ◽  
Body Interaction ◽  
Fractional Parentage

The orbital and charge-spin fractional parentage coefficients for the nuclear d 3 and d 4 con­figurations are derived using group theory. The orbital coefficients are given in a form appropriate to the new subclassification of the states according to irreducible representations of R 5 discussed in part I (Jahn 1950). Using these coefficients the complete energy matrices for the d 3 and d 4 configurations are derived, for a general charge-symmetric central two-body interaction, from the known energy matrix for the d 2 configuration.

scholarly journals Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1120
Author(s):  
Carlo Heissenberg ◽  
Franco Strocchi
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Goldstone Bosons ◽  
The Matrix ◽  
Symmetry Breaking Term ◽  
Breaking Term

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group G are governed by the well-known Wigner–Eckart relations. In the case of infinite-dimensional systems, with G spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases.

scholarly journals Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(m|n)

2014 ◽  
Vol 55 (1) ◽  
pp. 011703 ◽  
Author(s):  
Mark D. Gould ◽  
Phillip S. Isaac ◽  
Jason L. Werry
Keyword(s):  
Lie Superalgebra ◽  
Irreducible Representations ◽  
Matrix Elements

The matrix elements of tensor operators for configurations of three equivalent electrons

1959 ◽  
Vol 250 (1263) ◽  
pp. 562-574 ◽  
Keyword(s):  
Irreducible Representations ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Quantum Numbers ◽  
Fractional Parentage ◽  
The Matrix ◽  
Tensor Operators

The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .

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