Seniority in ln Configurations With T = n/2. The O(2l + 1) Group

The classification of states according to the irreducible representations of the unitary group and its sub-groups may be applied to the case of LS-coupling. For this case, in atomic spectroscopy, it was introduced by Racah (1943). Let us first consider the simpler case of identical nucleons or maximum isospin, T = n/2. Some Hamiltonians may be expressed in that case in terms of the space coordinates of the particles and their momenta. Some explicit dependence on spin operators which they may have, can be eliminated in some cases. If such dependence occurs through the scalar product s ₁ · s ₂, the interactions can be expressed in terms of the spin exchange operator (10.64). Due to the Pauli principle, the action of P 12 σ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq2793.tif"/> on any allowed wave function may be obtained by applying to it − P 12 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq2794.tif"/> , the Majorana operator which exchanges the space coordinates of particles 1 and 2. If the dependence on the spins is more complicated, as in the case of spin orbit interaction or tensor forces, this is no longer the case. The Hamiltonians we consider in this section are those which contain central interactions only. They may be expressed by operators acting only on space coordinates and thus are invariant under permutations of the space coordinates of the particles. It follows that any eigenstate, which is a function of space coordinates, is transformed by a permutation to 576another eigenstate with the same eigenvalue. Among all eigenstates which belong to the same eigenvalue we can define a set of functions ϕ_i which form a basis. All other functions are linear combinations of these basis functions. Apart from possible accidental degeneracies, these functions ϕ_i form a basis of an irreducible representation of the group of permutations. In the beginning of Section 26 we saw examples of such considerations. In the following, we shall use a similar approach. The steps we take are very similar to those taken in Section 26.