In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, given a particular Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

  and an operator  with corresponding eigenvalues and eigenvectors given by then the physical quantities are said to be "good quantum numbers" if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves.

Hence, if:


then we require

for all eigenvectors in order to call a good quantum number.

A necessary and sufficient condition for q to be good is that commute

Commutativity

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

 with the Hamiltonian . Proof:

Assume . If is an eigenvector of , then we have (by definition) that , and so :

It can be said that a good classical analogue is that good quantum numbers are the equivalent of conserved quantities in classical mechanics, as their values do not change over time.In non-relativistic treatment,l and s are good quantum numbers but
in relativistic quantum mechanics they are no longer good quantum numbers as L and S do not commute with H(in Dirac theory). J=L+S is a good q.no. in relativistic quantum mechanics as J commutes with H.