Once again, restricting attention to one dimension, the solution to the Schrödinger equation satisfying the initial condition ψ ψ ( x , 0 ) = 2 / π π 4 exp ⁡ ⁡ ( − − x 2 + i k 0 x ) , representing a wave packet localized in space at the origin, is seen to be ψ ψ ( x , t ) = 2 / π π 4 1 + 2 i t e − − 1 4 k 0 2 e − − 1 1 + 2 i t ( x − − i k 0 2 ) 2 = 2 / π π 4 1 + 2 i t e − − 1 1 + 4 t 2 ( x − − k 0 t ) 2 e i 1 1 + 4 t 2 ( ( k 0 + 2 t x ) x − − 1 2 t k 0 2 ) .