The product of the standard deviations is therefore σ σ x σ σ p = ℏ ℏ 2 n 2 π π 2 3 − − 2 . For all n = 1 , 2 , 3 , … … , the quantity n 2 π π 2 3 − − 2 is greater than 1, so the uncertainty principle is never violated. For numerical concreteness, the smallest value occurs when n = 1 , in which case σ σ x σ σ p = ℏ ℏ 2 π π 2 3 − − 2 ≈ ≈ 0.568 ℏ ℏ > ℏ ℏ 2 .