50 algebra Casimir operators

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

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... 

Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term E(). The algebra U(l) has a linear invariant... 

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... 

For any Lie algebra, one can construct a set of operators, called invariant or Casimir operators, C, such that... 

A Casimir operator containing p operators X, is called of order p. Only unitary algebras U(n) have linear Casimir operators. All other algebras have 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. 

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

This appendix provides a summary of the functional form of the algebraic Hamiltonian used in the text for tri- and tetratomic molecules. Values of the parameters are reported both for a low-order realistic representation of the spectrum and for accurate fits using terms quadratic in the Casimir operators. 

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

It can be easily checked by direct computation that we have really obtained a realization of the Lie algebra g in a Hilbert (Fock) space, [T a, T fc] = ifabc fc, in accordance with (11), where Ta = T f/aL . For an irreducible representation R, the second-order Casimir operator C2 is proportional to the identity operator I, which, in turn, is equal to the number operator N in our Fock representation, that is, if T" —> Ta, then I /V 5/,/a . Thus we obtain an important for our further considerations constant of motion N ... 

The Casimir operator is a useful tool for identifying a representation of the Lie algebra 5m(2). In this section we investigate Casimir operators and apply them to the classification of the finite-dimensional irreducible representations of the Lie algebra so 4 ). 

Like the raising and lowering operators, the Casimir operator does not correspond to any particular element of the Lie algebra 5m(2). However, for any vector space V, both squaring and addition are well defined in the algebra gt (V) of linear transformations. Given a representation, we can define the Casimir element of that representation. ... 

Proposition 8.11 Suppose (su(2 ), V,p) is a finite-dimensional irreducible Lie algebra representation. Then the Casimir operator is a scalar multiple of the identity on V. 

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

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. 

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... 

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) ... 

Finally, we would like to mention that the set of operators which commute with all elements of the Lie algebra is very useful in the study of Lie algebras (cf. Section III). In general these operators do not belong to the Lie algebra and are called Casimir operators. 

The Casimir operators of a Lie algebra are important since they commute with all generators and can therefore be simultaneously diagonalized. The Casimir operators for so(4) or so(3,1) are given by... 

It is well known that the angular momentum of a quantum mechanical system is specified by a representation of the SU(2) algebra. If the corresponding enveloping algebra contains a uniquely defined scalar (the Casimir operator), the polar decomposition of the angular momentum can be obtained [51]. This polar decomposition determines a dual representation of the SU(2) algebra expressed in terms of so-called phase states [51], In particular, the Hermitian operator of the SU(2) quantum phase can be constructed [51],... 

The enveloping algebra of (37)-(38) contains the uniquely defined Casimir operator... 

By analogy, in the su,(2)-algebra one can find an operator (the g-deformed Casimir operator or g-deformed angular momentum length ) with the form ... 

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). 

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