Racah algebra of two-body double-tensor operators

1985 ◽  
Vol 63 (9) ◽  
pp. 1220-1227 ◽  
Author(s):  
J. A. Tuszyński ◽  
R. Chatterjee ◽  
J. M. Dixon
Keyword(s):  
Crystal Field ◽  
Time Reversal ◽  
Numerical Evaluation ◽  
Charge Conjugation ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Selection Rules ◽  
Definition Of ◽  
Racah Algebra ◽  
Tensor Operators

Two-body double-tensor (TBDT) operators are introduced in the LS- and jj-coupling schemes following Racah's definition of irreducible tensor operators. These operators are shown to contain well-known one-body double tensors as a subset. The result of applying parity, charge conjugation, and time reversal to the TBDT operators is investigated so as to obtain selection rules for their matrix elements. On the basis of their rotation properties, these Hermitian tensor operators are shown to satisfy an SU(2) algebra, and consequently the Wigner–Eckart theorem is applied to calculate their matrix elements. The derivation of the reduced matrix elements is provided. We show some examples of interactions investigated in the literature that can be represented as TBDT operators. For one such operator, the spin-correlated crystal field, a detailed numerical evaluation of its matrix elements is given.

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 .

Parity, time, charge and Hermitian conjugations of Racah tensor operators

1983 ◽  
Vol 61 (12) ◽  
pp. 1613-1617 ◽  
Author(s):  
R. Chatterjee ◽  
J. A. Tuszyński ◽  
H. A. Buckmaster
Keyword(s):  
Magnetic Properties ◽  
Electric Properties ◽  
Tensor Operator ◽  
Matrix Elements ◽  
Selection Rules ◽  
Angular Momenta ◽  
The Matrix ◽  
Polar Tensor ◽  
The Relationship ◽  
Tensor Operators

The relationship between the parity P, time θ, charge C, and Hermitian h conjugation operators and the irreducible Racah tensor operators is reexamined. Polar tensor operators (describing electric properties) are distinguished from axial tensor operators (describing magnetic properties and angular momenta) on the basis of their individual parity and time conjugation properties. However, the effect of the Pθ product conjugation is identical for both classes and for even rank is equivalent to the Racah definition for the Hermitian conjugation of a tensor operator. It is shown that this property separates the Racah tensor operators from other vector quantities like linear momentum which cannot be represented by such operators. The selection rules due to parity and time conjugation and Hermitian conjugation that arise in the calculation of the matrix elements of the tensor operators and their products are then obtained self-consistently using the Wigner–Eckart theorem.

scholarly journals WIGNER–ECKART THEOREM FOR TENSOR OPERATORS OF HOPF ALGEBRAS

2005 ◽  
Vol 02 (03) ◽  
pp. 393-408 ◽  
Author(s):  
MAREK MOZRZYMAS
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Hopf Algebras ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Schur Lemma ◽  
Completely Reducible ◽  
Representations Of Hopf Algebras ◽  
Tensor Operators

We prove Wigner–Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representations is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of Ut(su(2)) is given.

scholarly journals Time reversal and charge conjugation in an embedding quantum simulator

2015 ◽  
Vol 6 (1) ◽  
Author(s):  
Xiang Zhang ◽  
Yangchao Shen ◽  
Junhua Zhang ◽  
Jorge Casanova ◽  
Lucas Lamata ◽  
...  
Keyword(s):  
Time Reversal ◽  
Charge Conjugation ◽  
Quantum Simulator
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