These can be obtained via the relationships a i j † † = ⟨ ψ ψ i | a † † | ψ ψ j ⟩ and a i j = ⟨ ψ ψ i | a | ψ ψ j ⟩ . The eigenvectors ψ ψ i are those of the quantum harmonic oscillator, and are sometimes called the "number basis".