Double-group theory on the half-shell and the two-level system. I. Rotation and half-integral spin states

Abstract

A geometrical construction by Hamilton is used to simplify the quantum mechanics of half-integral spin. A slide rule is described which can be used to (a) compute products of half-integral or integral spin rotation operators, (b) convert between the Euler-angle and ’’axis-angle’’ rotation operator parameters, and (c) calculate the time evolution of a spin-1/2 state for a constant Hamiltonian operator. A type of nomogram is developed which suggests ways to simplify the ’’double-group’’ theory of half-integral spin in molecular point symmetry, as well as the ’’ordinary’’ group theory for integral spin systems. Cubic and icosahedral symmetry group characters are derived for half-integral spin operators.