Large Angular Momentum

Abstract

Quantum states of a spin $\frac{1}{2}$ (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 $|j,\mathbf{n}⟩\in {\mathbf{CP}}^{2j}$, with spin oriented towards definite spatial directions $\mathbf{n}\in S^2$, i.e., $(\mathbf{J}\cdot \mathbf{n})|j,\mathbf{n}⟩=j|j,\mathbf{n}⟩$, 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.

arXiv comment29 pages, 10 figures

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Cited by in corpus (1)

• Deformed angular momentum algebra within the real Hilbert space