I learn something about the Time evolution of Gaussian wave packet in free space. if the initial condition is a Dirac delta function at $t=0$, then the wave function is $$ \psi (x,t)=\frac{1}{\sqrt{2\pi it}}e^{-\frac{x^2}{2it}} $$
my question is it seems that the square of the modulus of
$$ \left |\psi (x,t) \right |^{2}=\frac{1}{{2\pi t}} $$
which has nothing to do with the independent variable x, I can not understand the result.
Your help would be highly appreciated!
