A spin operator generates rotations about the -axis exactly as generates…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator

Problem

A spin operator S^z\hat S_z generates rotations about the zz-axis exactly as p^\hat p generates translations: rotation by angle θ\theta is R^z(θ)=e−iθS^z/ℏ\hat R_z(\theta) = e^{-i\theta \hat S_z/\hbar}. Under this rotation a spin vector operator transforms as an ordinary vector rotated in the xyxy-plane (the same rule that governs Larmor precession), R^z(θ)†S^x R^z(θ)=S^xcos⁡θ−S^ysin⁡θ.\hat R_z(\theta)^\dagger \hat S_x\, \hat R_z(\theta) = \hat S_x\cos\theta - \hat S_y\sin\theta. (a) Differentiate both sides of this relation with respect to θ\theta and set θ=0\theta = 0, using exactly the method used for T^(a)†x^ T^(a)\hat T(a)^\dagger \hat x\, \hat T(a), to derive [S^z,S^x][\hat S_z, \hat S_x]. (b) The classical Poisson bracket of angular-momentum components obeys {Lx,Ly}PB=Lz\{L_x,L_y\}_{\rm PB} = L_z and its cyclic permutations. Write down the cyclic permutation that corresponds to part (a), and confirm that [S^z,S^x]/iℏ[\hat S_z,\hat S_x]/i\hbar reproduces it.

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