Casimir operators examples

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... 

In the local basis, all terms involving Casimir operators are diagonal. For example, the term XuCf has an expectation value of... 

Situations in which the Hamiltonian does not have a dynamic symmetry (i.e., it contains Casimir operators of both chains), as, for example,... 

In order to obtain spectroscopic accuracy it was found necessary, as discussed in Chapter 4, to introduce higher-order terms, for example, products and powers of Casimir operators. These higher-order terms can be dealt with easily within the approach discussed here. For example, a Hamiltonian of the type... 

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

For example, consider the representation of 5m(2) on polynomials in two variables defined by Equation 8.8. The Casimir operator for this representation is... 

Exercise 8.11 Find an example of a reducible (i. e., not irreducible) representation (su (2), V, p) such that the Casimir operator is a scalar multiple of the identity on V. (This implies that the converse of Proposition 8.11 is false.)... 

They are constructed from powers of the operators Xs and can be linear, quadratic, cubic,. Quite often a subscript is attached to C in order to indicate the order. For example, C2 denotes a quadratic invariant. The number of independent Casimir invariants of an algebra is called the rank of the algebra. It is easy to see, by using the commutation relation (2.3) that the operator... 

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