Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S3 / 1 / 4 / Solution

For an aligned narrower slit placed immediately after the first, uniform illumination gives transmission probability $\boxed{d/w}$. Conditional on passage the same normalized formula applies with $w$ replaced by $d$ and central-lobe momentum scale $h/d$. Position rms is $d/\sqrt{12}$, while the exact hard-edge rms $\Delta p_x$ is infinite. Thus the source's finite “uncertainty” can only mean a characteristic diffraction width; the rigorous Heisenberg inequality is not violated, and smoothing the edges restores finite rms moments satisfying it.

A 10 keV X-ray has $\lambda \simeq 0.124$ nm. The ideal position-filter probability and momentum formula at fixed aperture widths are unchanged, since the Fourier transform involves $h$ and the aperture geometry. Its larger total momentum makes the angular diffraction width smaller, approximately $\lambda /d$. Real slit-material transmission and absorption are additional experimental issues.