‘The ‘Vector-Model’ wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta’

Abstract

Abstract We introduce the concept of an asymptotic spatial angular-momentum wavefunction, X m j φ , θ , χ = e im φ δ θ − θ m e i j + 1 2 χ , which treats j in the j m state as a three-dimensional entity; φ, θ, χ are the Euler angles, and θ m is the Vector-Model polar angle, given by cos θ m = m / j . A wealth of geometric information about j can be deduced from the eigenvalues and spatial transformation properties of the X m j φ , θ , χ . Specifically, the X m j φ , θ , χ wavefunction gives a computationally simple description of the sizes of the particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in j and m ), given by effective wavepacket angular uncertainty relations for Δ m Δ φ , Δ j Δ χ , and Δ φ Δ θ . The X m j φ , θ , χ also predicts the position of the particle-wavepacket angular motion in the orbital plane, so that the particle-wavepacket rotation can be experimentally probed through continuous and non-destructive j -rotation measurements. Finally, we use the X m j ( φ , θ , χ ) to determine geometrically well-known asymptotic expressions for Clebsch–Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, g = 2 , and the m -state-correlation matrix elements. Interestingly, for low j , even down to j = 1 / 2 , these expressions are either exact (the last two) or excellent approximations (the first two), showing that the X m j ( φ , θ , χ ) give a useful spatial description of quantum-mechanical angular momentum, and provide a smooth connection with classical angular momentum.