Assignment 8

Problem 1:

An electromagnetic wave with circular frequency ω propagates in a medium of dielectric constant ε, magnetic permeability μ, and conductivity σ.
(a)  Show that there is a plane wave solution in which the amplitude of the E and B fields decreases exponentially along the direction of propagation, and find the characteristic decay length.
(b)  Simplify by assuming that σ is great enough so that σ/(εω) >> 1.

Problem 2:

The HeNe lasers in our undergraduate laboratories produce unpolarized light. 
(a)  Describe how you can produce a beam of right-hand or left-hand polarized light from such a HeNe laser.  How much of the original light intensity do you loose?
(b)  Explain how you can change the polarization axis of a linearly polarized laser beam without loosing intensity.

Problem 3:

Consider a composite system made of two non-identical spin ½ particles.  For t < 0 the Hamiltonian does not depend on time and can be taken to be zero.  For t > 0 the Hamiltonian is given by  H = (4∆/ħ2)S1∙S2, where ∆ is a constant.  Suppose that the system is in the state |+-> for t ≤ 0.  Find, as a function of time, the probability for being in each of the states |++>, |+->, |-+>, and |-->,
(a)  by solving the problem exactly, using |Ψ(t)> = U(t, t0)Ψ(t0)> and
(b)  by solving the problem assuming the validity of first-order time-dependent perturbation theory with H as a perturbation which is switched on a t = 0.
(b)  Under what conditions does the perturbation calculation disagree with the exact solution and why?

Problem 4:

Submit this problem on Canvas as Assignment 8.  If you used an AI as a Socratic tutor, submit a copy of your session leading to your solution and reflect on your session.  If you did not need any help or worked with another student, explain your reasoning, do not just write down formulas. 

A pair of magnetic ions with individual spins s1 and s2 interact through the scaled Hamiltonian H = Cs1∙s2.  Let s(s + 1) be the eigenvalue of s2, where s = s1 + s2 is the total angular momentum.  Note ħ = 1.
(a)  Show that the total angular momentum is a conserved quantity.
(b)  Find the ground state energy E0 and the energy of the highest state Emax.
(c)  Now suppose a magnetic field is applied and the new Hamiltonian is H', where
H' = H - b(s1z + s2z).  What is the residual symmetry of the new Hamiltonian H', and what are the associated "good quantum numbers"?