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Mathematics for Physics

105 Conservation of Momentum

A principle that is used to keep the total momentum of an isolated quantum system constant, where an isolated system feels no net external force.

\([P,H]=0\) for a translation-invariant potential. If the potential is independent of position, the momentum operator commutes with the Hamiltonian. This principle is used to treat \(\hat{p}\) as a constant of the motion.

The free-particle conservation condition is

\[ [P,H] = 0 \]

where

  • \(P\) is the momentum operator.
  • \(H\) is the Hamiltonian.

Free-particle plane waves. Plane-wave energy eigenfunctions are also momentum eigenfunctions. This principle is used to label free-particle states by \(p\) and \(E=p^{2}/2m\).

The free-particle energy is

\[ E = \dfrac{p^{2}}{2m} \]

where

  • \(E\) is the energy.
  • \(p\) is the momentum.
  • \(m\) is the mass.

Conservation of total momentum in an isolated collision. In a collision of an isolated pair, the total momentum of the two particles is conserved. This principle is used to analyze quantum scattering in the center-of-mass frame.

105.1 References

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