Assignment 7

Problem 1:

Consider a highly excited He atom.  One electron is in a state with n1 = 4, l1 = 3 and the other in a state with n2 = 5,  l2 = 4
(a)  Find the possible values of l (total orbital angular momentum quantum number) for the system.
(b)  Find the possible values of s (total spin angular momentum quantum number) for the system.
(c)  Find the possible values of j (total angular momentum quantum number) for the system.
(d)  How many distinct angular momentum states with j = 1 are there?

Solution:

Problem 2:

(a)  Write down Maxwell's equations for a conducting medium with conductivity σ, permittivity ε0 and permeability μ0.
(b)  A plane wave of low frequency ω << σ/ε0 is propagating in the z-direction inside the conducting medium.
Let E = E0exp(i(kz - ωt)),  B = B0exp(i(kz - ωt)), where E0, B0, and k are complex.
Use Maxwell's equations to show that k = k1 + ik2, k1 ≈ k2 ≈ (μ0ωσ/2)½.
Calculate the ratio of the complex amplitude of the two fields, E0/B0 (magnitude and phase).
(c)  Calculate the energy flux (time averaged Poynting vector) in the conducting medium.

Solution:

Problem 3:

Consider a two-level system |Φa>, |Φb> with <Φij> = δij.  Show that an "entangled", two-particle state of the form
α|Φa(1)>|Φb(2)> + β|Φb(1)>|Φa(2)>, α, β ≠ 0,
CANNOT be written as a product state |ψr(1)>|ψs(2)> for any one particle states |ψr> and |ψs>.

Solution:

Problem 4:

A beam of monochromatic light of wavelength λ in vacuum is incident normally on a nonmagnetic dielectric film of refractive index n and thickness d.  Calculate the fraction of the incident energy that is reflected.

Solution: