Consider the linear Schrodinger equation $$ \begin{cases} i\partial_t u + \Delta u =0,\\ u|_{t=0}=u_0, \end{cases}, t\in\mathbb R,x\in \mathbb R^n, u\in \mathbb C. $$ If $v$ is a solution to the problem, then so is $$ u(t,x):=\frac{1}{(1+t)^{n/2}} v(t/(1+t), x/(1+t)) \exp\left(i\frac{|x|}{4(1+t)}\right). $$
My question is: do we have a nice physical interpretation of this result? How do we come up with this?