Large Angular Momentum

Abstract

Quantum states of a spin $\tfrac{1}{2}$ (a qubit) are parametrized by the space ${\mathbf {CP}}^1 \sim S^2$, the Bloch sphere. A spin $j$ for a generic $j$ (a $2j+1$-state system) is represented instead by a point of a larger space, ${\mathbf {CP}}^{2j}$. Here we study the state of a single angular momentum/spin in the limit, $j \to \infty$. The special class of states $ | j, {\mathbf n}\rangle \in {\mathbf {CP}}^{2j} $, with spin oriented towards definite spatial directions ${\mathbf n} \in S^2$, i.e., $({\mathbf J}\cdot {\mathbf n} ) \, | j, {\mathbf n}\rangle = j\, |j, {\mathbf n}\rangle $, are found to behave as classical angular momenta, $j \, {\mathbf n}$, in this limit. Vice versa, general spin states in ${\mathbf {CP}}^{2j}$ do not become classical, even at large $j$. We discuss these questions, by analysing the Stern-Gerlach processes, the angular-momentum composition rule, and the rotation matrix. Our observations help to clarify better how classical mechanics emerges from quantum mechanics in this context (e.g., with unique trajectories for a particle carrying a large spin), and to make the widespread idea that large spins somehow become classical, more precise.