Coupled States of the System of Two Free Electrons on a Single Star

Abstract

A classification scheme of quantum states for the system of two free electrons in a cubic box, confined to a single star of quasi-momentuni is proposed within a Racah—Wigner type of approach. Coupling of angular momenta of the atomic case is here substituted by Mackey theorem for transitive representations, which provides a crystalline analogue of orbital angular niomentum — the resultant orbit of the geometric symmetry group. The action of the Pauli group is combined with that of the octahedral group which yields the connection between spin (i.e. the singlet or triplet pairing of electrons) and statistic of the positional factor. Resultant singlets and doublets — irreducible representations of the octahedral group, exhibit an ordinary Landau diamagnetic behaviour, whereas triplets are paramagnetic. A relation between the Mackey star and the star of resultant momentum is discussed.