  We prove that $$ \|X(|u|^2)\|_{L^3_{t,\ell}}\leq
C\|f\|_{L^2(\mathbb{R}^2)}^2, $$ where $u(x,t)$ is the solution to the linear
time-dependent Schr\"odinger equation on $\mathbb{R}^2$ with initial datum $f$,
and $X$ is the (spatial) X-ray transform on $\mathbb{R}^2$. In particular, we
identify the best constant $C$ and show that a datum $f$ is an extremiser if
and only if it is a gaussian. We also establish bounds of this type in higher
dimensions $d$, where the X-ray transform is replaced by the $k$-plane
transform for any $1\leq k\leq d-1$. In the process we obtain sharp $L^2(\mu)$
bounds on Fourier extension operators associated with certain high-dimensional
spheres, involving measures $\mu$ supported on natural "co-$k$-planarity" sets.
