Good quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian, quantities that can be known with precision at the same time as the system's energy. Specifically, observables that commute with the Hamiltonian are simultaneously diagonalizable with it and so the eigenvalues a and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together. Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete values. In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers ; although they could approach infinity in some cases.