Coupling of Equivalent Particles in a Field of Given Symmetry

A system of N identical particles is considered, which are subject to a common external field V ( x , y , z ) and interact with each other as given by a spinfree symmetric operator V 1 . V is assumed to be invariant against rotations of a group \(G\). When these particles are assigned to orbitals of a degenerate bound level γ of \(F\), a single configuration γ N is obtained. Interaction V 1 splits this configuration into a number of states, each of which is characterized by an irreducible representation \(\) of \(G\) and an irreducible representation ∑ of the symmetric permutation group. A general formula is derived by which the number n (\(\), ∑) of states with representations \(\) and ∑ arising from a configuration γ N is easily obtained. The application of this method is discussed for the following cases: (1) Russell-Saunders' coupling of electrons in an atom, (2) Coupling of electrons assigned to a degenerate molecular orbital, or electrons in an ion subject to a strong ligand field, (3) Coupling of nucleons in a nucleus for the case in which the interaction is assumed independent of ordinary spin as well as of isotopic spin (Wigner's case I). The method is shown to be applicable slso to the case of j – j coupling.