Question about schrodinger free equation

Question

Left $u$ be a solution to free linear schrodinger equation $u_t=i\Delta u$. The momentum is defined as $$P(t)=Im \int_{\mathbb{R}^N} \overline{u}(x,t)\nabla u(x,t) dx $$ I want to prove the following equality $$ 4\pi\int _ {\mathbb{R}^N} |\widehat{u}(y,t)|^2y\ dy = 2P(t)$$ Where $\widehat{u}$ is the Fourier transform of $u$. I could not even start... please, any hint or idea would be very much appreciated. Thanks in advance.

Answer 1

I assume you define the Fourier transform by $$\mathcal{F}(u(x))(y) = \int_\mathbb{R^d} e^{ix \cdot y} u(x)\;dx.$$ In this case, you have the Fourier isomorphism identity $$\langle u, v \rangle = 2\pi \langle \hat u, \hat v \rangle$$ where $\langle \cdot, \cdot \rangle$ is the usual $L^2$ inner product $$\langle u, v \rangle = \int_{\mathbb{R}^d} u \overline{v}\;dx$$ If we also make use of the identity $\mathcal{F}(\nabla u(x))(y) = iy \hat u(y)$, then we find that $$P(t) = \Im \langle \nabla u, u \rangle = 2\pi \Im \langle iy \hat u, \hat u \rangle = 2\pi\Im i \int_{\mathbb{R}^d} y|\hat u(y, t)|^2\;dy = 2\pi \int_{\mathbb{R}^d} y |\hat u(y,t)|^2\;dy$$ at which point mutliplying by 2 gives the desired result.