The measurements of the z-components of the spins of the individual particles are made most easily when these energy eigenvectors are transformed to the basis 
. Doing this involves the use of transformation coefficients called Clebsch-Gordon coefficients, which are real, are the overlaps in the expansion of an energy eigenvector in terms of 
eigenvectors
Defining 
as this coefficient, Racah demonstrated
where
The proedure used to transform the energy eigenbasis vectors to the z-component eigenbasis vactors is given in appendix 13, section 13.2. In this routine, each energy eigenvector is expanded recursively in terms of the 
eigenvectors, using equation 12.2 repeatedly.