General angular momenta

For the determination of matrix elements, it is often more convenient to use linear combinations of the Cartesian components of the angular momentum operator instead of the Cartesian components themselves. In the literature, two different kinds of operators are employed. The first type is defined by 

Each of these forms will be used later. 

A second set of operators, the so-called tensor operators, differ only slightly from the ladder operators. They are introduced here without further 

Phase conventions have been chosen to be consistent with those of Condon and Shortley.13 In terms of tensor operators, the square modulus of f becomes 

We shall come back to these operators after learning what a tensor is. Commutation Relations 

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We now prove several identities that are needed to diseover the information about the eigenvalues and eigenfunetions of general angular momenta that we are after. Later in this Appendix, the essential results are summarized. 

Based on an analogy with orbital angular momentum, we define a generalized angular-momentum operator J with components Jx, Jy, Jz... 

We now apply the results of the quantum-mechanical treatment of generalized angular momentum to the case of orbital angular momentum. The orbital angular momentum operator L, defined in Section 5.1, is identified with the operator J of Section 5.2. Likewise, the operators I , L, Ly, and are identified with J, Jx, Jy, and Jz, respectively. The parameter j of Section 5.2 is denoted by I when applied to orbital angular momentum. The simultaneous eigenfunctions of P and are denoted by Im), so that we have... 

Thus, the quantum-mechanical treatment of generalized angular momentum presented in Section 5.2 may be applied to spin angular momentum. The spin operator S is identified with the operator J and its components Sx, Sy, Sz with Jx, Jy, Jz- Equations (5.26) when applied to spin angular momentum are... 

It was shown in Section 1.7 that when the operators Px, PY, Pz °t>ey general angular-momentum commutation relations, as in (5.41), then the eigenvalues of P2 and Pz are J(J+ )h2 and Mh, respectively, where M ranges from — J to J, and J is integral or half-integral. However, we exclude the half-integral values of the rotational quantum number, since these occur only when spin is involved. 

F.T. Smith, Generalized angular momentum in many-body collisions, Phys. Rev. 120 (1960) 1058. 

In this section we extend the theory of photodissociation and rotational excitation outlined in Section 3.2 for J = 0 to general angular momentum states J 7 0 of a triatomic system ABC. We will closely follow the detailed presentation of Balint-Kurti and Shapiro (1981) [see also Hutson (1991), Glass-Maujean and Beswick (1989), Beswick (1991), and Roncero et al. (1990)]. The discussion in this section is not meant to be a substitute for reading the original literature we merely want to outline the general methodology and underline the complexity of the theory. 

As for the orbital angular momentum, the commutation relations between the Cartesian components of a general angular momentum / and its square modulus A read... 

These components are now independent of the orbital and spin variables and so commute with L and S. They also obey the commutation relations for a general angular momentum, provided only that the anomalous sign of i is used. 

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