Interpretation of the conformal symmetry of Schrodinger equation

Consider the linear Schrodinger equation
$$
begin{cases}
ipartial_t u + Delta u =0,
u|_{t=0}=u_0,
end{cases},
tinmathbb R,xin mathbb R^n, uin mathbb C.
$$
If $v$ is a solution to the problem, then so is
$$
u(t,x):=frac{1}{(1+t)^{n/2}} v(t/(1+t), x/(1+t)) expleft(ifrac{|x|}{4(1+t)}right).
$$

My question is: do we have a nice physical interpretation of this result? How do people come up with this at the first place?

NOTE: Of course we can prove dissipation of wave packets by this symmetry, but I wish to have something more specific: this formula certainly provides much more information than the general rate of spread of probability amplitudes.