ABSTRACT
In Section 3 we saw how to combine the intrinsic spin s and the orbital angular momentum L to form eigenstates of J 2 where J = L + s. The coefficients in (3.30) and (3.31) are special cases of a general class of transformation coefficients. The states on the r.h.s. of (3.30) and (3.31) are eigenstates of s 2, sz , L 2, Lz . On the left hand side there are eigenstates of s 2, L 2, J 2 and Jz . In general, we may start with any two sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq354.tif"/> of two different systems. The spins j 1 and j 2 may be either orbital angular momenta or total spins of two particles, two intrinsic spins or one orbital angular momentum and one intrinsic spin. The latter case was considered in Section 3. In other cases j 1 may be the rank of an irreducible tensor operator and j 2 the angular momentum of a wave function or both j 1 and j 2 may be the ranks of two irreducible tensor operators. To simplify the notation we will use, for the transformation of the sets ψ j 1 m 1 , ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq355.tif"/> under infinitesimal rotations, the expressions (6.20). This will not detract from the applicability of the following results to other tensorial sets. As pointed out several times above, the following mathematical derivations have general validity and are independent of the physical nature of ψ j 1 m 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq356.tif"/> and ψ j 2 m 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq357.tif"/> . They are due only to the transformation properties of the two sets under rotations.
