The Total Angular Momentum Operator

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

This new operator is referred to as the square of the total angular momentum operator. 

The total momentum operator P, as well as the total angular momentum operator M, commute with H and hence are constants of the motion. However, they do not commute with another, their commutator being equal to... 

We first inquire as to the constants of the motion in this situation. Since h is invariant under the group of spatial rotations, and under spatial inversions, the total angular momentum and the parity operator are constants of the motion. The total angular momentum operator is... 

From (27) and (29) it follows that every component of the total angular momentum operator J = L + S and J2 commute with the Dirac Hamiltonian. The eigenvalues of J2 and Jz are j(j + 1 )h2 and rrijh respectively and they can be defined simultaneously with the energy eigenvalues E. 

Since the total angular momentum operator L and spin-momentum operator S commute with Eq. (18), the solution of the Hamiltonian containing the first two terms in Eq. (5) can be an eigenfunction of L2, Lz, 52, and Sz ... 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

A detailed analysis (Chapter 11) shows that this result depends upon the commutation relations for the L operators, and, since the spin and the total angular momentum operators obey the same commutation relations (CRs), this formula holds also for S and for J ... 

In this expression, Pq = — hi jZQ, V(Q) is the potential energy of equation (8) re-expressed in terms of Q by using equation (10), 11 is the total angular momentum operator, and / is the instantaneous moment of inertia given by... 

Here, the operator J is the total angular momentum operator in the space-fixed frame, and Tx, X=A and B, is defined by Eq. (1-262). Note, that the present coordinate system corresponds to the so-called two-thirds body-fixed system of Refs. (7-334). Therefore, the internal angular momentum operators jA and jB, and the pseudo angular momentum operator J do not commute, so the second term in Eq. (1-265) cannot be factorized. 

To construct the Dirac-Fock equations, it is assumed that the wave function for an atom having N electrons may be expressed as an antisymmetrized product of four-component Dirac spinors of the form shown in Eq. (9). For cases where a single antisymmetrized product is an eigenfunction of the total angular momentum operator J2, the JV-electron atomic wave function may be written... 

The invariance of H under other transformations of the coordinate system [5] imposes certain symmetries upon the elements of the set f (r,R)>. In particular, rotational invariance implies that these states must be eigenfunctions of the total angular momentum operators,... 

In Eq. (3.20), (a =x, y, z) are the components of the total angular momentum operator with respect to the molecule-fixed axes. The quantity is classically/p = (bT/bojp) and quantum mechanically... 

The norm of the density operator o- = (Tr o-V ) is always conserved under the unitary transformation of Eq. (71). However, in general, additional constants of motion exist. If the effective mixing Hamiltonian is composed exclusively of zero-quantum operators, it commutes with the z component of the total angular momentum operator ... 

The operator f a g contains the cross-terms that give rise to the Coriolis coupling that mixes states with different which is the quantum number of the projection of the total angular momentum operator J on the intermolecular axis. This term contains first derivative operators in y, and its matrix elements change on application of (18) according to... 

Simplifications. In the form we give to tp, the use of half integers, which is a complication, is avoided. Only the integers m G Z that appear in the associated Legendre polynomials and P 1 1 are employed. Half integers m = m + 1/2 appear, for example, in the formula implying the total angular momentum operator of the electron (see Appendix C) and will be introduced in the Zeeman-effect (Chap. 14). 

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