Good quantum number

In quantum mechanics, given a particular Hamiltonian ${\displaystyle H}$ and an operator ${\displaystyle O}$ with corresponding eigenvalues and eigenvectors given by ${\displaystyle O|q_{j}\rangle =q_{j}|q_{j}\rangle }$, the ${\displaystyle q_{j}}$ are said to be good quantum numbers if every eigenvector ${\displaystyle |q_{j}\rangle }$ remains an eigenvector of ${\displaystyle O}$ with the same eigenvalue as time evolves.

In other words, the eigenvalues ${\displaystyle q_{j}}$ are good quantum numbers if the corresponding operator ${\displaystyle O}$ is a constant of motion. Good quantum numbers are often used to label initial and final states in experiments. For example, in particle colliders:

1. Particles are initially prepared in approximate momentum eigenstates; the particle momentum being a good quantum number for non-interacting particles.

2. The particles are made to collide. At this point, the momentum of each particle is undergoing change and thus the particles’ momenta are not a good quantum number for the interacting particles during the collision.

3. A significant time after the collision, particles are measured in momentum eigenstates. Momentum of each particle has stabilized and is again a good quantum number a long time after the collision.

Theorem: A necessary and sufficient condition for the ${\displaystyle q_{j}}$ to be good is that ${\displaystyle O}$ commutes with the Hamiltonian ${\displaystyle H}$.

A proof of this theorem is given below. Note that the theorem holds even if the spectrum is continuous; the proof is slightly more difficult (but no more illuminating) in that case.