Principles of Quantum Mechanics · Problem 12.5.5

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · finite-rotations

Problem

*(Hard).* Using results from the previous exercise and Eq. (12.5.23), show that (1)

D^{(1/2)}[R] = \exp(-i\hat{\theta} \cdot \mathbf{J}^{(1/2)}/\hbar) = \cos(\theta/2) I^{(1/2)} - (2i/\hbar)\sin(\theta/2)\hat{\theta} \cdot \mathbf{J}^{(1/2)}

(2)

D^{(1)}[R] = \exp(-i\theta_x J_x^{(1)}/\hbar) = (\cos\theta_x - 1)\left(\frac{J_x^{(1)}}{\hbar}\right)^2 - i\sin\theta_x \left(\frac{J_x^{(1)}}{\hbar}\right) + I^{(1)}