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Mathematics for Physics

100 Conservation Laws

A set of statements that is used to identify quantities that stay constant in time when their operators commute with the Hamiltonian.

Conservation from \([A,H]=0\). If an operator \(A\) has no explicit time dependence and commutes with \(\hat{H}\), then \(\langle A\rangle\) is constant. This principle is used to identify conserved energy, momentum, angular momentum, and charge.

The conservation condition is

\[ [A,\hat{H}] = 0 \implies \dfrac{d\langle A\rangle}{dt} = 0 \]

where

  • \(A\) is an observable operator.
  • \(\hat{H}\) is the Hamiltonian.
  • \(t\) is time.

Energy from time translation. Time-translation symmetry implies conservation of energy. This principle is used to treat \(\hat{H}\) itself as the conserved energy when \(\hat{H}\) does not depend on time.

Momentum from spatial translation. Spatial-translation symmetry implies conservation of momentum. This principle is used to treat \(\hat{p}\) as conserved for a free particle or a translation-invariant potential.

Angular momentum from rotation. Rotational symmetry implies conservation of angular momentum. This principle is used to label atomic states by \(\ell\) and \(m\).

Selection rules. Transitions that would violate a conserved quantity are forbidden. This principle is used to write selection rules.

Note: Also called constants of the motion.

100.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — symmetries and conservation laws.
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. — Ehrenfest theorem and constants of the motion.