Good quantum number

In quantum mechanics, the eigenvalue ${\displaystyle q}$ of an observable${\displaystyle O}$ is said to be a good quantum number if the observable ${\displaystyle O}$ is a constant of motion. In other words, the quantum number is good if the corresponding observable commutes with the Hamiltonian. If the system starts from the eigenstate with an eigenvalue${\displaystyle q}$, it remains on that state as the system evolves in time, and the measurement of ${\displaystyle O}$ always yields the same eigenvalue ${\displaystyle q}$.

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.