Angular momentum commutators involving

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. 

The proof of the theorem affirming that J8 is a proper quantum mechanical angular momentum involves only an expansion of (Ji + J2) x (Ji + J2) with subsequent use of the commutation rules for Jj and J2, and the fact that Jj and J2 commute because they act in... 

Since Hj does not have spherical symmetry like the hydrogen atom the angular momentum operator L2 does not commute with the Hamiltonian, [L2,H] 7 0. However, Hj does have axial symmetry and therefore Lz commutes with H. The operator Lz = —ih(d/d) involves only the 0 coordinate and hence, in order to calculate the commutator, only that part of H that involves need be considered, i.e. 

Finally, some important commutators involving the angular momentum are... 

The exact solution of the Schrbdinger equation for the rotations of asymmetric molecules is possible because of the existence of complete sets of commuting operators, namely the respective Hamiltonians H and H, and the square of angular momentum H. The practical advantage of using the Hamiltonian H over H is that the latter involves three independent parameters a,b,c), whereas H involves only one independent parameter. 

The application of angular-momentum theory to atomic spectroscopy is not limited to bringing eqs. (3)-(7) into play. In their book, Condon and Shortley (1935, ch. 3) developed the theory with particular attention to operators T that are vectors. They did this by specifying the commutation relations of Twith respect to J rather than by stating the transformation properties of the components of T under rotations. They considered angular momenta J built from two parts S and L) and obtained formulas for the matrix elements of operators that behave as a vector with respect to one part (say L) and a scalar with respect to the other (S, in this case). These formulas involve proportionality constants that would be called reduced matrix elements today. Condon and Shortley systematized the methods that had come into current use but which were often only hinted at, if that, by many theorists. For example. Van Vleck (1932) gave the formula... 

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