Commutation relations orbital angular-momentum operators

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... 

We postulate that the spin angular-momentum operators obey the same commutation relations as the orbital angular-momentum operators. Analogous to [Lj, Ly = ihL [Ly, LJ = ihL [Lj, Lj,] = ihLy [Eqs. (5.46) and (5.48)], we have... 

The observed spin components behave like components of a vector, on rotating the coordinate axes, and from the general postulates of quantum mechanics it is inferred that the spin operators satisfy commutation relations exactly like the orbital angular-momentum operators (see... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

Conserved operators commute with the Hamiltonian. We want to know if the orbital angular momentum L (3.59) is conserved. Consider Lx, using the commutation relations (3.6) and equn. (3.161). 

Comparing with the commutation relations above, we see that for r and p at least, K has the effect of an antiunitary operator. Expressing orbital angular momentum as f = r X p, we see that = —1. For spin we can draw on the analogies between the transformation of commutation relations for spin and orbital angular momentum. From these we see that the transformed commutation relations are consistent with Cs0 = -s, and the spin has the same transformation properties as orbital angular momentum. 

L, S, J AAA L, S, J L, S, J A J j =jl +j2 orbital, spin, and total angular momenta quantum mechanical operators corresponding to L, S, and J quantum numbers that quantize L2, S2, and J2 operator that obeys the angular momentum commutation relations total (j) and individual (ji, j2, ) angular momenta, when angular momenta are coupled... 

Tensors (15.39)—(15.41) meet commutation relations (14.2) for irreducible components of the momentum operator, and, in addition, they commute with the operators of orbital (14.15) and spin (14.16) angular momenta for the lN configuration, since they are scalars in their respective spaces. Accordingly, the states of the lN configuration can be characterized by the eigenvalues of operators L2, Lz, S2, Sz, Q2, Qz. 

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