Without loss of generality, we will assume that the means vanish, which just amounts to a shift of the origin of our coordinates. (A more general proof that does not make this assumption is given below.) This gives us the simpler form σ σ x 2 = ∫ ∫ − − ∞ ∞ ∞ ∞ x 2 ⋅ ⋅ | ψ ψ ( x ) | 2 d x σ σ p 2 = ∫ ∫ − − ∞ ∞ ∞ ∞ p 2 ⋅ ⋅ | φ φ ( p ) | 2 d p .