A spin-half particle starts in the eigenstate of

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Spin one-half, SU(2) and two-level dynamics

Problem

A spin-half particle starts in the eigenstate ∣↑⟩|\uparrow\rangle of σ^z\hat\sigma_z. The rotation operator about y^\hat y, R^(y^,θ)=cos⁡(θ/2)1−isin⁡(θ/2)σ^y\hat{\mathcal R}(\hat y,\theta) = \cos(\theta/2)\mathbb1 - i\sin(\theta/2)\hat\sigma_y, acts on it with θ=π/3\theta=\pi/3. ((a)) Write the resulting spinor in the {∣↑⟩,∣↓⟩}\{|\uparrow\rangle,|\downarrow\rangle\} basis. ((b)) Find the probability that a subsequent measurement of σ^z\hat\sigma_z gives +1+1.

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