This shows that a ψ ψ n and a † † ψ ψ n are also eigenstates of the Hamiltonian, with eigenvalues E n − − ℏ ℏ ω ω and E n + ℏ ℏ ω ω respectively. This identifies the operators a and a † † as "lowering" and "raising" operators between adjacent eigenstates. The energy difference between adjacent eigenstates is Δ Δ E = ℏ ℏ ω ω .