Operator orbital angular momentum

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... 

The eigenvalues and eigenfimctions of the orbital angular momentum operator may also be obtained by solving the differential equation I ip = Xh ip using the Frobenius or series solution method. The application of this method is presented in Appendix G and, of course, gives the same results... 

The orbital angular momentum operations needed to calculate integrals for other orbitals are summarized in Table 4.2. 

Apart of historical reasons, there are several features of the Dirac-Pauli representation which make its choice rather natural. In particular, it is the only representation in which, in a spherically-symmetric case, large and small components of the wavefunction are eigenfunctions of the orbital angular momentum operator. However, this advantage of the Dirac-Pauli representation is irrelevant if we study non-spherical systems. It appears that the representation of Weyl has several very interesting properties which make attractive its use in variational calculations. Also several other representations seem to be worth of attention. Usefulness of these ideas is illustrated by an example. 

Spherical harmonics are the eigenfunctions of the orbital-angular-momentum operators /. We shall give them the symbol Yim... 

Because V depends only on r, one finds that this Hamiltonian commutes with the orbital angular-momentum operators L2 and Lz. Hence the... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

For the components of the orbital angular momentum operator we have... 

Formally, orbital angular momentum operator L of a particle moving with linear momentum p = —itiV at a position r with respect to some... 

To take advantage from the pseudo-angular momentum representation we shall employ the technique of the irreducible tensor operators as suggested in Ref. [10]. One can easily establish the following interrelations between the matrices Orr and the orbital angular momentum operators ... 

The same relations can be applied to the spin operators. Then, using the Clebsch-Gordan decomposition [11] one can express the bilinear forms of the orbital angular momentum operators in equation (6) in terms of the irreducible tensorial products ... 

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

In principle, knowledge of Eqs. [18]—[22] is sufficient to set up differential equations for the orbital angular momentum operators and to solve for eigenvalues and eigenfunctions. The solutions are most easily obtained employing spherical coordinates r, 0,< ) (see Figure 7). The solutions, called spherical harmonics, can be found in any introductory textbook of quantum chemistry and shall be given here only for the sake of clarity. 

The eigenvalues and eigenfunctions of the orbital angular momentum operators can also be derived solely on the basis their commutation relations. This derivability is particularly attractive because the spin operators and the total angular momentum obey the same commutation relations. 

The commutation relations of the orbital angular momentum operators can be derived from those between the components of r and p. If we denote the Cartesian components by the subindices i, k, and /, we can use the short-hand notation... 

Now the orbital angular momentum operator hL has components in the space-fixed axis system which are defined as... 

Each term on the right-hand side of this equation consists of the product of a direction cosine and an orbital angular momentum operator. Of these two factors, only the first depends on the Euler angles which define the instantaneous orientation of the molecule. In the corresponding equation (5.151) for./,, however, both factors depend on the Euler angles. 

In our discussion of the far-infrared laser magnetic resonance spectrum of NiH in chapter 9, a fairly general effective Hamiltonian was presented. This Hamiltonian included terms which would produce A-doubling in a A state, an unusual situation because one requires electronic orbital angular momentum operators to connect A = + 2 and A = - 2 components [77], The effective Hamiltonian used to analyse the mi-crowave/optical double resonance spectrum of NiH was as follows ... 

Kronecker s delta X is the spin-orbit coupling constant and Ag j is defined in terms of the matrix elements of the orbital angular momentum operator L by... 

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