In this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.
A photon with wavelength \lambda has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius R and a focal length f \gg R. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty \Delta \theta. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).
Suppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.
a. Find the standard deviation \Delta r of the distribution for r, the distance from the center of the telescope mirror to the point of reflection of the photon.
b. Use the uncertainty principle, \Delta r \Delta p_{r} \geq \hbar / 2, to place a bound on how accurately we can know the angle of the photon from the axis of the telescope. Give your answer in terms of R and \lambda. If you were unable to solve part a, you may also give your answer in terms of \Delta r.
c. Suppose we want to build a telescope that can tell with high probability whether a photon it detected from Alpha Centauri A came the left half or right half of the star. Approximately how large would a telescope have to be to achieve this? Alpha Centauri A is approximately 4 \times 10^{16} \mathrm{~m} from Earth and has a radius approximately 7 \times 10^{8} \mathrm{~m}. Assume visible light with \lambda=500 \mathrm{~nm}.