Good quantum number

rdf:langString Good quantum number

rdf:langString 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.