Using the standard rules for creation and annihilation operators on the energy eigenstates, a † † | n ⟩ ⟩ = n + 1 | n + 1 ⟩ ⟩ a | n ⟩ ⟩ = n | n − − 1 ⟩ ⟩ , the variances may be computed directly, σ σ x 2 = ℏ ℏ m ω ω ( n + 1 2 ) σ σ p 2 = ℏ ℏ m ω ω ( n + 1 2 ) . The product of these standard deviations is then σ σ x σ σ p = ℏ ℏ ( n + 1 2 ) ≥ ≥ ℏ ℏ 2 .