Theory Wigner matrices

Around a fixed energy E, the average reaction rate is given by the famous RRKM formula, which can be derived from both quasiclassical and quantal considerations [71, 72]. In the context of the Wigner matrix theory [133], the rate is given by the sum of the half-widths of all the open channels. The rate is thus the product of the number v(E) of open channels and the rate per channel fcchannei(E) = l/hnav(E), where h is the Planck constant. The average reaction rate is obtained as [134, 135]... 

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],... 

Wigner was the first to suggest the application of random matrix theory to complicated quantum mechanical systems, such as atoms and nuclei. 

From a computational standpoint, the usefulness of the method relies on the simplicity of the calculations needed for the determination of the three equivalent crystals associated with each atom i. This is accomplished by building on the simple concepts of Equivalent Crystal Theory (ECT) [25,26], as will be discussed in detail below. The procedure involves the solution of one simple transcendental equation for the determination of the equilibrium Wigner-Seitz radius i WSE) of ch equivalent crystal. These equations are written in terms of a small number of parameters describing each element in its reference state, and a matrix of perturbative parameters Ay , which describe the changes in the electron density in the vicinity of atom / due to the presence of an atom j (of a different chemical species), in a neighboring site. The determination of parameters for each atom in... 

The first part of the chapter contains a brief summary of Wigner s scattering theory, presented so as to emphasise the underlying similarity with the closely related approach of MQDT (chapter 3). This is followed by a discussion of the properties of S-, R- and K-matrices, in which we give the motivation for choosing one or the other, depending on the application in hand. Finally, we turn to some explicit applications of K-matrix theory to cases of interacting resonances in atomic physics. 

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