Wigner—Eckart theory

Weisskopf generalization, adiabatic broadening theory 132, 136 Wigner D-fiinotions 86, 275 Wigner-Eckart theory 232, 244, 253... 

As was mentioned in the previous paragraph, the Wigner-Eckart theorem (5.15) is fairly general, it is equally applicable for both approaches considered, for tensorial operators, acting in various spaces (see, for example, Chapters 15,17 and 18, concerning quasispin and isospin in the theory of an atom). 

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. 

The use of quarks in atomic shell theory provides an alternative basis to the traditional one. The transformations between these bases can be complicated, but there are many special cases where our quarks can account for unusual selection rules and proportionalities between sets of matrix elements that, when calculated by traditional methods, go beyond what would be predicted from the Wigner-Eckart theorem [4,5], This is particularly true of the atomic f shell. An additional advantage is that fewer phase choices have to be made if the quarks are coupled by the standard methods of angular-momentum theory, for which the phase convention is well established. This is a strong point in favor of quark models when icosahedral systems are considered. A number of different sets of icosahedral Clebsch-Gordan (CG) coefficients have been introduced [6,7], and the implications of the different phases have to be assessed when the CG coefficients are put to use. 

This is the Wigner-Eckart theorem, a very important result which underpins most applications of angular momentum theory to quantum mechanics. It states that the required matrix element can be written as the product of a 3- j symbol and a phase factor, which expresses all the angular dependence, and the reduced matrix element (rj, j T/ (d) if. j ) which is independent of component quantum numbers and hence of orientation. Thus one quantity is sufficient to determine all (2j + 1) x (2k + 1) x (2/ + 1) possible matrix elements (rj, j, mfIkq(A) rj, jf m ). The phase factor arises because the bra (rj, j, m transforms in the same way as the ket (— y m rj, j, —m). The definition of the reduced matrix element in equation (5.123), which is due to Edmonds [1] and also favoured by Zare [4], is the one we shall use throughout this book. The alternative definition, promoted by Brink and Satchler [3],... 

One of our main motivations for pursuing the development of a density functional response theory for open-shell systems has been to calculate spln-Hamiltonian parameters which are fundamental to experimental magnetic resonance spectroscopy. It is only within the context of a state with well-defined spin we can speak of effective spin Hamiltonians. The relationship between microscopic and effective Hamiltonians rely on the Wigner-Eckart theorem for tensor operators of a specific rank and states which transform according to their irreducible representations [45]. 

One of the most remarkable features of the analysis of Racah (1949) for the Coulomb energies in the f shell is that the theory works much better and shows more simpliiications than could possibly have been anticipated in 1949. As was explained in section 4.3.6, Racah s use of the lie groups SO(7) and Gi provides explanations for the vanishing of many matrix elements and for the proportionalities that some matrices bear to others but it also goes beyond a straightforward application of the Wigner-Eckart theorem that such examples represent. The principal surprises are as follows ... 

There is a general result from angular momentum theory in quantum mechanics, known as the Wigner-Eckart theorem [4], which allows the magnetic dipole moment to be directly related to the nuclear spin, according to ... 

The Wigner-Eckart theorem. In the theory of the interaction of radiation with atoms we are interested in the matrix elements of tensor operators taken between states of different quantum numbers. The very powerful Wigner-Eckart theorem states that these matrix elements are given by ... 

If information on the reaction path is available, as, for instance, in variational transition state theory, this can be used to calculate k [69,70]. In transition state theory, only the knowledge of the energy and its first and second derivatives at the reactant and transition state locations is needed and the barrier is typically approximated by a simple functional form. One possibility is to describe the reaction barrier by an Eckart potential [75] (also called a sech potential, depending on the literature source), k in Eq. (7.19) is defined as the ratio of transmitted quantum particles to classical particles and the resulting integral for the Eckart potential can be solved numerically. An approximate solution is the Wigner tunneling correction ... 

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