Let $\phi\in \mathcal S(\mathbb R)$ and consider the uni-dimensional global Cauchy problem for the Schrödinger equation of a free particle, that is
$$(*)\qquad iu_t+u_{xx}=0\quad \text{and}\quad u(x, 0)=\phi(x),\quad x\in \mathbb R, t>0.$$ I must find a solution and prove the following equality,
$$(**)\qquad\lim_{t\to +\infty}\int_{|x|<t}|u(x,t)|^2\,dx=\int_{|\xi|<1/2}|\hat{\phi}(\xi)|^2\,d\xi.$$
I've already found a solution (using Fourier transform), a fundamental solution for $(*)$ is $$E(x, t)=\frac{1}{(4\pi i t)^{1/2}} e^{-\frac{x^2}{4it}},$$ and so a solution for $(*)$ will be $$u(x, t)=\big( E(-, t)*\phi \big)(x)=\frac{1}{(4\pi i t)^{1/2}} \int_{\mathbb R}e^{\frac{i(x-y)^2}{4t}}\phi(y)\, dy.$$ However, I can't prove $(**)$, can you help me with that? (It must not be very hard, but I don't see a clean way to prove this)
Also, if possible, can you give me a non-technical physical interpretation of $(**)$ ? Regards