(6 pts) Derive the dielectric function \( \epsilon(\omega) \) of an ensemble of charged damped oscillators with mass \( m \), charge \( e \), resonant frequency \( \omega_0 \), and damping parameter \( \gamma \) distributed uniformly over volume with density \( N \). Start with the equation of motion for a damped oscillator driven by harmonic electric field.
(2 pts) Now assume there are oscillators of multiple types, each with its own frequency and damping parameter, such that type \( j \) makes up a fraction \( f_j \) of their total number. Generalize the results of item #1 and compare with Jackson (7.51).
(2 pts for the explanation) Review Fig. 7.8 and convince yourself that it reflects the behavior of \( \epsilon(\omega) \) with two types of oscillators.
(3 pts) Assume that the ensemble of oscillators contains a fraction \( f_0 \) of oscillators with zero resonant frequency (i.e., free particles). Show that their contribution to \( \epsilon(\omega) \) is equivalent to the result obtained from the Drude model in the tutorial on the dielectric response in metals. Find the explicit expressions for the static conductivity \( \sigma_0 \) and relaxation time \( \tau \) that follow from this equivalence.
(2 pts) Examine the expression for \( \sigma_0 \) and understand the origin of each factor appearing in it.
(4 pts) Assuming that only free electrons contribute to dielectric response of a metal, use the appropriate expression for \( \epsilon(\omega) \) to show that the integral \( \int^\infty_0 \omega \mathop{\mathrm{Im}}\epsilon(\omega)d\omega \) is equal to \( (\pi\sigma_0)/2\tau \). Use the result of item #4 to observe that the ratio \( \sigma_0/\tau \) does not depend on the relaxation time. Argue that the experimental measurement of the dielectric function can be used to obtain information about the density and effective mass of the charge carriers in a good metal.
(1 pts) Convince yourself that the most general linear connection between \( \mathbf{E} \) and \( \mathbf{D} \) that is local in space is given by Eq. (7.105) in Jackson. (Disregard all the text around it.)
(3 pts) What arguments could be used to justify the locality of Eq. (7.105) in space? Is there a small parameter?
(3 pts) Establish the connection between \( G(t) \) and \( \epsilon(\omega) \).
(4 pts) Section 7.10.B in Jackson takes the result of item #1 for an ensemble of identical oscillators and works out the expression for \( G(t) \) in Eq. (7.110). Fill in the missing steps.
(2 pts) Convince yourself that “memory” in the response extends to times of order \( \gamma^{-1} \), where \( \gamma \) is the damping parameter. Explain why this is reasonable.
(2 pts) Convince yourself that the response given by Eq. (7.110) is strictly causal.
(1 pts) Convince yourself that Eq. (7.111) is the strictly causal version of Eq. (7.105).
(2 pts) Explain why the response function (7.112) following from (7.111) is analytic in the upper half-plane when its argument is viewed as a complex number.
(6 pts) Follow the derivation of Kramers-Kronig relations (7.118-7.119) and (7.120) and fill in the missing steps. Pay special attention to the definition of the principal value of an integral.
(2 pts) Understand the assumptions and algebra leading to Eq. (7.121).
(10 pts) Solve problem 7.22(a) from Jackson. Make a plot. Does the real part of \( \epsilon(\omega) \) diverge anywhere? If yes, what does it tell us about the form of the imaginary part assumed in this problem?