So the propagation kernel is the (future) time evolution of a delta function, and it is continuous, in a sense: it goes to the initial delta function at small times. If the initial wave function is an infinitely narrow spike at position y, ψ ψ 0 ( x ) = δ δ ( x − − y ) , it becomes the oscillatory wave, ψ ψ t ( x ) = 1 2 π π i t e i ( x − − y ) 2 / 2 t .