Specifically, it is not difficult to compute an exact formula for the uncertainty Δ Δ ψ ψ ( t ) X as a function of time, where X is the position operator. Working in one spatial dimension for simplicity, we have: ( Δ Δ ψ ψ ( t ) X ) 2 = t 2 m 2 ( Δ Δ ψ ψ 0 P ) 2 + 2 t m ( ⟨ 1 2 ( X P + P X ) ⟩ ψ ψ 0 − − ⟨ X ⟩ ψ ψ 0 ⟨ P ⟩ ψ ψ 0 ) + ( Δ Δ ψ ψ 0 X ) 2 , where ψ ψ 0 is the time-zero wave function. The expression in parentheses in the second term on the right-hand side is the quantum covariance of X and P .