For the generator of rotations about is

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

The Structure of Quantum Mechanics, Ch Angular momentum from the rotation group

Problem

For j=12j=\tfrac12 the generator of rotations about xx is J^x=ℏ2σx\hat J_x=\tfrac\hbar2\sigma_x, with σx=(0110)\sigma_x=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix} in the basis m=12,−12m=\tfrac12,-\tfrac12. ((a)) Sum the exponential series to find the 2×22\times2 matrix of R^(x^,β)=e−iβJ^x/ℏ\hat{\mathcal R}(\hat x,\beta)=e^{-i\beta\hat J_x/\hbar}. ((b)) A spin prepared in ∣12,12⟩|\tfrac12,\tfrac12\rangle is rotated by β=π/2\beta=\pi/2 about xx. Find the probabilities that S^z\hat S_z then gives +ℏ/2+\hbar/2 and −ℏ/2-\hbar/2. ((c)) Show that R^(x^,2π)=−1\hat{\mathcal R}(\hat x,2\pi)=-\mathbb 1, the same as for a full turn about yy.

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