Skip to main content

Mathematics for Physics

101 Conservation of Angular Momentum

A principle that is used to keep the total angular momentum of an isolated quantum system constant when there is no net external torque.

Conservation from rotational invariance. If the Hamiltonian is invariant under rotations, it commutes with the angular-momentum operators. This principle is used to treat \(\mathbf{J}\) as a constant of the motion.

The rotational conservation condition is

\[ [\mathbf{J},\hat{H}] = 0 \]

where

  • \(\mathbf{J}\) is the total angular-momentum operator.
  • \(\hat{H}\) is the Hamiltonian.

Conservation for a central potential. A central potential conserves orbital angular momentum. A central potential is a potential that depends only on the distance from the origin. This principle is used to label hydrogen states by \(\ell\) and \(m\).

Conservation of \(\mathbf{J}=\mathbf{L}+\mathbf{S}\). In the presence of spin, the conserved quantity is often \(\mathbf{J}=\mathbf{L}+\mathbf{S}\). This principle is used to couple orbital and spin angular momentum.

Angular-momentum selection rules. Radiative transitions must conserve angular momentum of the atom plus the photon. This principle is used to write dipole selection rules \(\Delta\ell=\pm 1\).

101.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — angular momentum and rotations.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — conservation of angular momentum.