Orbital angular momentum eigenvalues

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... 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

The relationship between different components of orbital angular momentum such as Lz and Lx can be investigated by multiple SG experiments as discussed for electron spin and photon polarization before. The results are in fact no different. This is a consequence of the noncommutativity of the operators Lx and Lz. The two observables cannot be measured simultaneously. While total angular momentum is conserved, the components vary as the applied analyzing field changes. As in the case of spin or polarization, measurement of Lx, for instance, disturbs any previously known value of Lz. The structure of the wave function does not allow Lx to be made definite when Lz has an eigenvalue, and vice versa. 

Hence, = I + 1 if k > 0 and = I — 1 if k < 0. Consequently, in the Dirac-Pauli representation and have definite parity, (—1) and (—1) respectively. It is customary in atomic physics to assign the orbital angular momentum label I to the state fnkm.j- Then, we have states lsi/2, 2si/2) 2ri/2, 2p3/2, , if the large component orbital angular momentum quantum numbers are, respectively, 0,0,1, ,... while the corresponding small components are eigenfunctions of to the eigenvalues 1,1,0,2,. 

The molecular electronic wave functions ipe] are classified using the operators that commute with Hei. For diatomic (and linear polyatomic) molecules, the operator Lz for the component of the total electronic orbital angular momentum along the internuclear axis commutes with Hel (although L2 does not commute with tfel). The Lz eigenvalues are MLh,... 

It has the same form as the cartesian components and the solution, = ke tmf describes rotation about the polar axis in terms of the orbital angular momentum vector LZ1 specified by the eigenvalue equation... 

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 values e(t+ l)/i are found to be the eigenvalues of the operator for the square of the orbital angular momentum P. Thus, is an orbital angular momentum quantum number. The z component of the angular momentum is given by mfi, the quantum number varying between — and + , in unit jumps. For plots of the wavefunctions, see, for example, ref. 22. 

Here, the familiar eigenvalues of the orbital angular momentum operators have already been incorporated, reminding the reader that their form and numerical values are determined by the periodicity condition + 2n) = (0) for the first one, and by guaranteeing good behavior for the second at its regular singularity points, 9 = 0 and n. 

The works in [36,37] have recently revisited the evaluation of the rotational spectra of asymmetric molecules. Such an evaluation requires the simultaneous solution of the eigenvalue equations for the square of the orbital angular momentum and for the purely asymmetric part of the rotational Hamiltonian,... 

Stone applied similar reasoning to the problem of a three-dimensional cluster. Here, the solutions of the corresponding free-particle problem for an electron-on-a-sphere are spherical harmonics. These functions should be familiar because they also describe the angular properties of atomic orbitals.Two quantum numbers, L and M, are associated with the spherical harmonics, Yim 0,total orbital angular momentum and its projection on the z-axis, respectively. It is more convenient to use the real linear combinations of Yim 9,(p)dinA (except when M = 0), and... 

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