The isospin functions ζ and η are orthornormal, in analogy with the regular spin functions α and β. Unlike the exact orthornormality of α and β, however, the orthornormality of ζ and η is an approximate property, which nonetheless is highly accurate when the conning (gate) potentials vary slowly over the distance dened by the lattice constant a of the material. This follows from the fact [73] that the Bloch functions that multiply the envelope functions in the eective-mass approach are varying rapidly in space with a period determined by the material's lattice constant a, whereas the envelope functions vary slowly over the much larger extent dened by the size of the nanostructure. Specically, the many-body wave function is written as