where the integral is over all k -space and ω ω = ω ω ( k ) = ℏ ℏ k 2 2 m (to ensure that the wave packet is a solution of the free particle Schrödinger equation). Here ψ ψ 0 is the value of the wave function at time 0 and ψ ψ ^ ^ 0 is the Fourier transform of ψ ψ 0 . (The Fourier transform ψ ψ ^ ^ 0 ( k ) is essentially the momentum wave function of the position wave function ψ ψ 0 ( r ) , but written as a function of k rather than p = ℏ ℏ k .)