(1) Photons are quanta of the EM field with energy E, = hf where h is Planck's constant and f is the frequency. What is the momentum of such a photon?

If the density of photons in space, all traveling in the x-direction and all with the same energy, is n (with units of m³), then express the energy density, Poynting vector, intensity, and momentum density of the EM field in terms of this number density. If a beam of such photons is absorbed by a sheet of area A what is the radiation force on this sheet in terms of the photon density? (2) Do Griffiths problem 9.18 (parallel or p polarization case), but with relative permittivity &r = 2.8. Use excel or some other software to produce good plots. (3) This is a qualitative essay question requiring several sentences plus maybe a cartoon. Sometimes at night, when there is a thermal inversion (i.e., higher temperatures at higher altitudes, but colder near the ground) sounds from long distances away can be heard. Explain this phenomenon, knowing how the speed of sound varies with temperature and knowing that something like Snell's law should apply to sound waves as well as EM waves. (4) The time-dependent Schrodinger's equation in 1D is: ih 24 at ħ² 2²4 2m əx² +V where y(x,t) is the 1D wavefunction and V is a 1D potential. Assume plane wave solutions in free space (V=0) of form: y(x,t) = wo exp{i(kx-cot)}. Where co(k) is a function of k; that is, the dispersion relation. (a) Find the dispersion relation co(k). (b) From (k) find the phase and group velocities of the wave. (c) Assuming this wavefunction represents a free particle what is its speed? What is its momentum? (d) Now assume that the particle can be represented by a wavepacket. The Fourier transform of the wave function is given by A(k) = Ak¹ rect(ko-Ak) where Ak is << ko. This is a square function. Find y(x,t) for this case. Describe it. What are the uncertainties in momentum and space (i.e., x). That is, is Heisenberg's uncertainty principle holding?