Isospin basis and its properties

Let us take a closer look at the concept of isospin referring to two shells of equivalent electrons with the same orbital quantum numbers. Our reasoning will partially follow the work [123]. 

The concept of isospin was introduced by Heisenberg who suggested that the proton and neutron can be treated as two states of the nucleon 

Let the quantities ajfy and amSf = (— l)l+s+m+fld fl be creation and annihilation operators for the electrons in the nilNl shell and and bmp = (—l)(+s+m+ ih-m- be creation and annihilation operators for the electrons in the n2lNl shell. We shall now introduce the isospin angular momentum operator Tj1 1 given by 

Consequently, in the n lNln2lNl configuration the creation and annihilation operators for electrons 

Out of the operators (18.22) we can compose the following triple irreducible tensors  

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