Operators eigenvalues of Casimir

Another ingredient one needs in the application of algebraic methods to problems in physics and chemistry is the eigenvalues of Casimir operators in the representations of Section A.8. The known solution is given in Table A.5. 

A. 11 Example of representations of Lie algebras, 204 A. 12 Eigenvalues of Casimir operators, 204... 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

The eigenvalues / have been evaluated for any Casimir operator of any Lie algebra, and a summary of the results is given in Appendix A. Using the expressions of the appendix, we find, for example, that the eigenvalues of the Casimir operator of SO(3), J2, in the representation 1/ > is... 

It is convenient to take the U(2) z> 0(2) symmetry of the preceding section as the starting point for approximations. Since it is unnecessary to carry the index z or x, the wave functions can be written simply as IN, m >. Denoting by C2 the Casimir operator of 0(2) with eigenvalues... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

We have stated several times that whenever the Hamiltonian can be written in terms of invariant (Casimir) operators of a chain, its eigenvalue problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of Eq. (4.43). 

Table A.5 gives the eigenvalues of the Casimir operator of SO(3) in the representation J as...

Table A.5 Eigenvalues of some Casimir operators of Lie algebras...

The expressions for eigenvalues of the Casimir operators are presented below. They are completely defined by the transformation properties of the wave functions with regard to the corresponding groups and depend only on indices characterizing the representations of these groups. For the cases discussed above they are, respectively... 

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. 

The method of CFP is an elegant tool for the construction of wave functions of many-electron systems and the establishment of expressions for matrix elements of operators corresponding to physical quantities. Its major drawback is the need for numerical tables of CFP, normally computed by the recurrence method, and the presence in the matrix elements of multiple sums with respect to quantum numbers of states that are not involved directly in the physical problem under consideration. An essential breakthrough in this respect may be finding algebraic expressions for the CFP and for the matrix elements of the operators of physical quantities. For the latter, in a number of special cases, this can be done using the eigenvalues of the Casimir operators [90], however, it would be better to have sufficiently simple but universal formulas for the CFP themselves. 

As in a non-relativistic case, making use of the eigenvalues of the corresponding Casimir operators, we find an interesting relationship... 

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