Casimir’s operators for

Exercise 8.22 Is there anything in group representations ofSU (2) cir (S) analogous to the Casimir operator for Lie algebra sulL) ... 

This is called a chain. Each subalgebra has one (or more) Casimir operator(s) C(G,) which commute with all the operators of that subalgebra. The Casimir operator is usually bilinear in the generators and the number of linearly independent Casimir operators is the rank of the algebra. In (59) the Casimir operator of the last subalgebra necessarily commutes with all the Casimir operators of the earlier subalgebras. The Hamiltonian is given as a linear combination of the Casimir operators for the chain of Eq. (59). 

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

Here wi, W2, W3 are parameters characterizing the representations of group Rj u, U2 stand for the corresponding quantities of group G2 v is the seniority quantum number, defined in a simpler way in Chapter 9. On the other hand, the eigenvalues of the Casimir operator of group i 2(+i may be expressed in the following way by v and S quantum numbers... 

In [90] the relationship between eigenvalues of the Casimir operators of higher-rank groups and quantum numbers v, N, L, S is taken into account to work out algebraic expressions for some of the reduced matrix elements of operators (Uk Uk) and (Vkl Vkl). However, the above formulas directly relate the operators concerned, and some of these formulas are not defined by the Casimir operators of respective groups. 

Again, using the properties of the Casimir operators, we can establish the following simple algebraic expressions for these corrections (0 <, N < 41 + 2, for s-electrons the orbit-orbit interaction vanishes) ... 

By the mid-1960 s it was recognized that this simple picture was not adequate. Sandars and Beck (1965) showed how relativistic effects of the type first described by Casimir (1963) could be accommodated by generalizing the non-relativistic Hamiltonian to the form given by (108). A rather profound mental adjustment was required instead of setting the relativistic Hamiltonian between products of four-component Dirac eigenfunctions, they asked for the effective operator that accomplishes the same result when set between non-relativistic states. The coefficients ujf now involve sums over integrals of the type dr, where Fj and Gj,... 

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