Wigner matrix elements

The 1/j.p terms in the above equations contain products of the Wigner matrix elements and thus may be expressed as a hnear arrangement (up to the fourth order) of Wigner matrices [5]. The frequency sphtting due to the second-order quadrupolar interaction may be written as a sum of three terms ... 

Table 1. The values of the reduced Wigner matrix elements dmn(l3 Definition D (a, p, y) =...

Table 1. Second-rank reduced Wigner matrix elements...

Since F (cos )=Doo(/ ), eqn (4.32) can be rewritten in terms of Wigner matrix elements introduced earlier ... 

In the latter expression the matrix element of operator dq> is transformed according to the Wigner-Eckart theorem and the definition used is... 

The matrix element of operator is written in terms of the Wigner-Eckart theorem, and the integral part is denoted as... 

Owing to liquid isotropy, the averaged matrix elements of X are expressed in the Wigner-Eckart form (omitting further on the overbar denoting averaging)... 

In lack of analytical or numerical methods to obtain the spectra of complicated Hamiltonians, Wigner and Dyson analyzed ensembles of random matrices and were able to derive mathematical expressions. A Gaussian random matrix ensemble consists of square matrices with their matrix elements drawn from a Gaussian distribution... 

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,... 

The reduced matrix elements of the spherical harmonics can be written in terms of a Wigner 3 - j symbol... 

By Wigner-Eckart s theorem [6] Eq. (2) can be expressed in terms of a reduced matrix element that is independent of M and M, ... 

We now use Ae fact that the space-fixed functions (R, f) have parity (—1) = (—1). We can therefore multiply the second Wigner D matrix element in the curly bracket in Eq. (A.ll) by 1 = (—1) - This yields... 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

The last two matrix elements in Eq. (8.45) can be evaluated using the Wigner-Eckart theorem [5] ... 

These expansions can be used to evaluate the matrix elements of the interaction potential in the basis (8.43) by the direct application of the Wigner-Eckart... 

From the Wigner-Eckart theorem, these matrix elements may be written in terms of 3-j symbols, as [350]... 

A general development of matrix elements of an irreducible tensor operator leads to the Wigner-Eckart theorem (see, for example, Tinkham [2] or Chisholm [7]), which relates matrix elements between specific symmetry species to a single reduced matrix element that depends only on the irrep labels, but this is beyond the scope of the present course. 

A fundamental role is played in theoretical atomic spectroscopy by the Wigner-Eckart theorem, the utilization of which allows one to find the dependence of any matrix element of an arbitrary irreducible tensorial operator on projection parameters,... 

Usually the Wigner-Eckart theorem (5.15) is utilized to find the dependence of the matrix elements on the projections of angular and spin momenta. Its use in the quasispin space... 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

---