Question #43554

A monochromatic beam of photons of wavelength $\lambda$ is incident normally on a transmission diffraction grating consisting of $N$ parallel slits, with a constant slit spacing $d$. The diffraction pattern is observed on a screen at a distance $L \gg N d$. A particular photon is detected exactly at the center of the $m$-th order principal maximum. Let $p_y$ be the exact transverse momentum of this specific photon just before it hits the screen. According to the Heisenberg uncertainty principle, the spatial confinement of the photon's wavefunction across the entire grating width $W = Nd$ imposes a fundamental minimum uncertainty in its transverse momentum, $\Delta p_{y,\text{min}}$, given by $\Delta y \Delta p_y \ge \frac{h}{4\pi}$ (taking $\Delta y = W$). Assuming the photon's total momentum magnitude is conserved, which of the following gives the correct value of $p_y$ and the ratio $R = \frac{p_y}{\Delta p_{y,\text{min}}}$? (Assume all quantities are expressed in standard SI units, and $h$ represents Planck's constant).