Generalized angular momentum

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

The operator J is any hermitian operator which obeys the relation 

We implicitly assume that these eigenfunctions are uniquely determined by only the two parameters X and m. 

Since Jx and Jy are hermitian, the expectation values of J and are real and positive, so that 

We have already introduced the use of ladder operators in Chapter 4 to find the eigenvalues for the harmonic oscillator. We employ the same technique here to obtain the eigenvalues of and Jz. The requisite ladder operators and J-are defined by the relations 

Neither 7+ nor J is hermitian. Application of equation (3.33) shows that they are adjoints of each other. Using the definitions (5.18) and (5.14) and the commutation relations (5.13) and (5.15), we can readily prove the following relationships 

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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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