The Vector-Model Wavefunction: Spatial Description and Wavepacket Formation of Quantum-Mechanical Angular Momenta

In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum j. In contrast, here we show that a spatial wavefunction, j_m (ϕ,θ,χ)= e^i m ϕδ (θ - θ_m) e^i(j+1/2)χ, which treats j in the |jm> state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; ϕ, θ, χ are the Euler angles, and cos θ_m=(m/|j|) is the Vector-Model polar angle. The j_m (ϕ,θ,χ) gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in j and m) which predicts the effective wavepacket angular uncertainty relations for Δ m Δϕ, Δ j Δχ, and ΔϕΔθ, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive j-rotation measurements. We also use the j_m (ϕ,θ,χ) to determine 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 j_m (ϕ,θ,χ) gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.