Good quantum number

In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , the are said to be good quantum numbers if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves. In other words, the eigenvalues are good quantum numbers if the corresponding operator 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 to be good is that commutes with the Hamiltonian . 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. We will work in the Heisenberg picture. Let be a hermitian operator and let be a complete, orthonormal basis of eigenstates with eigenvalues . The unitary time-translation operator is and so that . The main observation behind the proof is If commutes with the hamiltonian then the right side vanishes and we get that the are good quantum numbers. However, if we assume that the are good quantum numbers then the left hand side vanishes for all ; since the are complete, this implies that , which establishes the equivalence. The Ehrenfest Theorem gives the rate of change of the expectation value of operators. It reads as follows: Commonly occurring operators don't depend explicitly on time.