Good quantum number

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} angle =q_{j}|q_{j} angle } , then the numbers (or the eigenvalues) q j {displaystyle q_{j}} are said to be 'good quantum numbers' if every eigenvector | q j ⟩ {displaystyle |q_{j} angle } remains an eigenvector of O {displaystyle O} with the same eigenvalue as time evolves. 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} angle =q_{j}|q_{j} angle } , then the numbers (or the eigenvalues) q j {displaystyle q_{j}} are said to be 'good quantum numbers' if every eigenvector | q j ⟩ {displaystyle |q_{j} angle } remains an eigenvector of O {displaystyle O} with the same eigenvalue as time evolves. Hence, if: O | q j ⟩ = O ∑ k c k ( 0 ) | e k ⟩ = q j | q j ⟩ {displaystyle O|q_{j} angle =Osum _{k}c_{k}(0)|e_{k} angle =q_{j}|q_{j} angle }