Chapter 8, Optical Propertles of Solids Video Solutions, Many-Particle Physics

Problem 1

Derive the finite temperature form of the free-polaron absorption (8.1.15) assuming Maxwell-Boltzmann statistics. To order $\alpha$, an exact result can be obtained in terms of Bessel functions $K_1\left(\downarrow \beta\left|\omega \pm \omega_0\right|\right)$.

Check back soon!

Problem 2

Derive (8.1.23).

Check back soon!

Problem 3

Consider the force-force correlation function for scattering from an impurity. Discuss whether the multiple scattering from an impurity can be represented by a $T$ matrix or similar series.

Check back soon!

Problem 4

Derive the Wilson-Butcher formula (8.1.20) from the golden rule. Find the matrix element $\langle f| \hat{\varepsilon} \cdot \mathbf{p}|i\rangle$ by writing both initial $|i\rangle$ and final $|f\rangle$ Bloch states to first order in the potential $V_{\mathbf{G}}$.

Check back soon!

Problem 5

Consider a Hamiltonian which is the homogeneous electron gas plus the crystal potential $\sum V_{\mathbf{G}} \varrho(\mathbf{G})$. Discuss the summation of terms which occurs in higher order when evaluating the force-force correlation function.

Check back soon!

Problem 6

Derive (8.1.42) from the correlation function (8.1.29). Remember that each vertex where $i \dot{\omega}$ enters must have a phonon line, but other internal lines are $W(0)=v_{\mathrm{q}}+V_{\mathrm{ph}}$.

Check back soon!

Problem 7

Derive (8,1.43).

Check back soon!

Problem 8

Derive a formula for $\operatorname{Re}\left[\sigma_0(\omega)\right]$ at $T=0$ which results from $M_b(4, t \omega)$ in (8.1.43). Show that both integrals $d u$ and $d u^{\prime}$ can be done analytically for a phonon Green's function $D_\lambda(\mathbf{q}, u)=-2 \omega_\lambda /\left(\omega_\lambda{ }^2-u^4-i \delta\right)$ and derive the formulas.

Check back soon!

Problem 9

Derive (8.1.44). Then evaluate this expression at zero temperature, and derive a formula for $\operatorname{Re}\left[\sigma_4(\omega)\right]$ which is a single-frequency integral over algebraic combinations of the two self-energy functions $\Sigma(\mathbf{k}, \omega)$ and $S(\mathbf{k}, \omega)$.

Check back soon!

Problem 10

Where do the final state interactions for the Coulomb scattering of the electron and hole appear in formula (8.1.20) for interband transitions?

Check back soon!

01:30

Problem 11

Show that the rate of photon emission in an insulator with a nonequilibrium distribution of electrons and holes is governed by the matrix element $\left|w_0(\mathbf{k})\right|^2$ as in (8.2.18). Assume the initial state of the system at zero temperature is $|k\rangle$ $=d^{\dagger} a_k{ }^{\dagger}|0\rangle$, and use arguments analogous to those following (8.2.19).

Ajay Singhal

Numerade Educator

Problem 12

Write out the correlation function $\pi^{(1)}$ in (8.2.8), with one vertex diagram, for the frequency-dependent screening function (6.3.7). Do the Matsubara summations.

Check back soon!

07:07

Problem 13

Show there is an infrared divergence in the X-ray response resulting from the piezoelectric electron-phonon interaction (1.3.7) in insulators when used in the response function (8.3.23).

Ameer Said

Numerade Educator

Problem 14

Solve the Tomonaga model for the orthogonality catastrophe. Work in a spherical coordinate system, where the potential is at the center of a large sphere of radius $R$. Normalize the wave function in this sphere. Keep only $s$-wave terms. This is a one-dimensional problem in the radial variable.

Check back soon!

03:28

Problem 15

Consider the $X$-ray edge problem with an interaction term $H_{e d}$ in (1.4.19) between the conduction electrons and the core hole. What is the contribution of this term to the orthogonality index $\propto$ from the cumlant $F_3(t)$ (Girvin and Hopfield, 1976)?

Adriano Chikande

Numerade Educator

01:23

Problem 16

Show that the imaginary part of the retarded correlation function (7.1.7) has the following sum rule relating to the average kinetic energy:

$$
\int_{-\infty}^{\infty} \frac{d \omega}{2 \pi} n_B(-\omega) \operatorname{Im}[\pi(\omega)]=\frac{e^2 n_0}{3 m} E_{K, \mathbf{E}}
$$

Next, show for Fröhlich polarons that the ground state energy can also be related to the average kinetic energy:

$$
E_0(\alpha)=-2 \int_0^\alpha \frac{d \alpha^{\prime}}{\alpha^{\prime}} E_{\mathrm{K}, \mathrm{E},( }\left(\alpha^{\prime}\right)
$$
Thus the ground state energy $E_0(\alpha)$ can be related to the conductivity. Use (8.1.13) to obtain an expression for $E_0(\alpha)$ (Lemmens, DeSitter, and Devreese).

Dading Chen

Numerade Educator