The spin rotation operator for spin one-half about a unit axis is
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Spin one-half, SU(2) and two-level dynamics
Problem
The spin rotation operator for spin one-half about a unit axis n^ is R^(n^,θ)=e−iθn^⋅σ/2, where σ=(σx,σy,σz) are the Pauli matrices. Using only the fact that (n^⋅σ)2=1, which follows from the Pauli algebra, derive a closed form for R^(n^,θ) by summing its Taylor series in θ.
((a)) Give the closed form.
((b)) Evaluate R^(n^,2π) and R^(n^,4π) for an arbitrary axis n^, writing each as a number times 1.
((c)) In SO(3), the matrix that rotates an ordinary three-dimensional vector by 2π about any axis is the identity, and so is the one for 4π. Reconcile this with your result in part (b), and state in one sentence what it tells you about the map from SU(2) to SO(3).