Rotation Operator

Operator

A rotation by angle $\theta$ about unit axis $\hat{\mathbf n}$ is represented by

$$U(\hat{\mathbf n},\theta) = \exp \left( -\frac{i}{\hbar}\theta\, \hat{\mathbf n}\cdot\mathbf J \right).$$

For a scalar wavefunction under a spatial rotation $R$,

$$(U(R)\psi)(\mathbf r) = \psi(R^{-1}\mathbf r).$$

For spin $1/2$,

$$U(\hat{\mathbf n},\theta) = \exp \left( -\frac{i\theta}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

Assumptions

• The rotation convention is active on states.
• $\mathbf J$ is the total rotation generator acting on all relevant degrees of freedom.
• Orbital and spin parts must both be included when both are present.
• Spinor rotations use $SU(2)$, which double covers $SO(3)$.

Common Mistakes

• Rotating coordinates, vectors, and states with inconsistent active/passive conventions.
• Forgetting the spin contribution to total angular momentum.
• Treating $U$ and $-U$ as different physical rotations for spinor rays.
• Assuming finite rotations commute because infinitesimal axes are easy to label.

Canonical Links

• Unitary Symmetries
• Angular Momentum Operator
• Spin Rotations
• SU(2) versus SO(3)

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015.