The simplest way to say is:

a operator can be projected on another one, for example, The orbital angular momentum cab be projected on the total angular momentum.

L = L\cdot J \frac {J}{j(j+1)}

a simple application is on the Zeeman effect on spin-orbital coupling. the Hamiltonian is:

H_B = - \mu \cdot B = - ( \mu_l L + \mu_s S ) \cdot B

by the Wigner-Eckart theorem:

L = L\cdot J \frac {J}{j(j+1)}

S = S\cdot J \frac {J}{j(j+1)}

then the Hamiltonian becomes:

H_B = - \frac{1}{j(j+1)} ( \mu_l (L \cdot J) + \mu_s (S \cdot J) ) J\cdot B

and introduce the Bohr Magneton and g-factor:

H_B = - g \mu_B J \cdot B

g = - \frac{1}{j(j+1)} ( g_l (L \cdot J) + g_s (S \cdot J) )

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