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