Assignment 8

Problem 1:

An electron is in a state with l = 2 in an atom. 
(a)  What is the magnitude of L?
(b)  What are the allowed quantum numbers j, and what is the magnitude of the vector J = L + S?

Problem 2:

Studies of the origin of the solar system suggest that sufficiently small particles might be blown out of the solar system by the force of sunlight.  To see how small such particles must be, compare the force of sunlight with the force of gravity, and find the particle radius r at which the two are equal.  Assume that the particles are spherical, act like perfect mirrors, and have a density 2 g/cm³.  
The solar luminosity is L = 3.83*10²⁶ Watts and the solar mass is M_s = 1.99*10³⁰ kg.

Why do you not need to worry about the distance from the Sun?

Problem 3:

Consider two particles with angular momenta
J = J₁ + J₂,  J_x = J_1x + J_2x,  J_y = J_1y + J_2y,  J_z = J_1z + J_2z.
J₁ and J₂ are the angular momentum operators of particle 1 and 2, respectively.
Show that the commutators [J²,J₁²] and [J_1z,J²] are zero and nonzero, respectively.  What does it mean in terms of measurements and Heisenberg's uncertainty principle?

Problem 4:

A plane-polarized electromagnetic wave propagates in free space along the +x axis.
At the position x = 0 the wave encounters a region of infinite extent in the y and z directions which is a low-density plasma of free electrons of number density n, mass m and charge -q_e.  For this plasma region, it is found that the current density j(t) and the electric field  E(t) = E₀exp(-iωt) are related by  j(t) = -nq_ev(t) = -(nq_e²/(imω))E(t).
(a)  Using Maxwell's equations, find the wave number k in the plasma region.  Assume ε = ε₀, μ = μ₀.
(b)  Describe the wave propagation in the plasma region for ω > ω_p and for ω < ω_p, where ω_p² = μ₀nq_e²c²/m = nq_e²/(ε₀m) and specifically determine the depth d of propagation in the plasma for the two cases.
(c)  Is there Joule-heating in the plasma?  If so, compute its value; if not explain why not.

Problem 5:

A sinusoidal electromagnetic wave with angular frequency ω is incident upon an interface at an angle θ with the normal as shown.

Assume ε₂ = ε(ω) is complex, ε(ω) = ε_r + iε_i, μ₁ = μ₂ = μ₀.
Then k_i² = ω²/c²,  k_t² = (ω²/c²)(ε_r + iε_i)/ε₀ = n²ω²/c².
Assume the wave is s-polarized, E = E(t)j.
(a)  Derive an expression for the components of the wave vector k_tx and k_tz in medium 2 when  ε_r = ε_i = ε₀ and θ = 30^o.
(b)  Boundary conditions for the tangential components of E and H yield the ratio E_r/E_i = (cosθ - ncosθ_t)/(cosθ + ncosθ_t).
Write down a complex expression for ncosθ_t and find the fraction of the incident energy absorbed by the medium.