4. What sort of particle is a photon? Let's explore its properties. (a) The energy of a photon is E=hf and its momentum is p = h/λ. What is the energy - momentum relation E(p)? Sketch it. (Electrons in graphene - a single atomic layer of graphite – have a similar energy-momentum relation.) (b) How does your answer to (a) compare with that for an ordinary particle, moving at speeds much less than c? If you double the momentum of a photon, what happens to its energy? If you double the momentum of a free electron, what happens to its energy? (c) Determine the frequency, wavelength, and photon energy (in eV) of photons for each of the following. (i) f = 850 MHz (a typical cell phone frequency) (ii) E 13 TeV (13 x 10¹² eV), the energy of the collision between two protons in the Large Hadron Collider (LHC) at CERN. (d) About 2% of the power input to an incandescent 60 W light bulb is converted to visible light photons. Assuming these photons all have a wavelength of 550 nm, how many photons/s are emitted? (e) Unlike ordinary particles like electrons and protons, photons can only move at the speed of light. According to Einstein's Special Theory of Relativity, the total energy of a particle is E² = mo²c² + p²c² = (mc²)², where p is the momentum, mo is the rest mass (when the particle is not moving) and m is the total (relativistic) mass that determines the momentum, p = mv. Using the formulas in (a), show that the rest mass of a photon must be zero. (f) What wavelength photon will have a relativistic mass m = p/c comparable to the rest mass of an electron? (This sets the wavelength scale for the Compton effect - and the EM wavelengths for which the Compton wavelength shift will be a large fraction of the incident wavelength. Can you see why?