Many-Body Physics Homeworks (1382)

Keivan Esfarjani, Sharif University of Technology

Sundays and Tuesdays 18:30-20:00 

Series 1 : Second quantization-Review (due Tuesday Mehr 29):

-Compute the expectation value of the one-body and two body (interaction) Hamiltonian in the ground state which is formed of a single Slater Determinant of orthonormal orbitals.

-Compute the thermodynamic properties (Total energy, Free energy, entropy, Pressure) of a gas of non interacting Fermions. Do the same for a gas of Bosons (harmonic phonons).

Series 2 : Non-interacting Bosons (phonons) and Fermions:

- Extend the solution of the harmonic phonon problem for many atoms (with different masses) per unit cell in 3D. How should one find the phonon frequencies? Solve it first for a chain of diatomic molecules in 1D with nearest neighbor interactions, and compute the frequencies analytically.

- Mahan, Chapter 1 (second edition) page 78 ex. no: 3, 4, 6, and 16.

- For a non-interacting system of fermions and bosons (harmonic phonons), subject to an external perturbing potential which is off diagonal, compute the first order correction to the thermodynamic properties as before.

Series 3 : Diagram Technique at zero temperature:

- Mahan, Chapter 1 (second edition) page 78 ex. no: 13.

- Consider the electron-phonon Hamiltonian of a one dimensional chain where the single atom in the unit cell has a single orbital and can only move along the chain direction (one electron and one phonon band model).

- Using second order perturbation theory, compute the change in the phonon frequencies and electron eigenvalues due to the electron-phonon interaction (you need to calculate the difference between the energies of two states: one is a one-phonon state, and the other is the zero-phonon state, so that the background zero point energy is cancelled after substraction).

- Compare your results to the corresponding phonon and electron self energies obtained by diagrammatic technique in second order. Deduce from these calculations the change in the electron effective mass and and the velocity of sound.

Series 4 : Linear response (due sunday Azar 16^th):

- Using the equation of motion satisfied by the Green's function of an interacting system (dG/dt = ...), calculate the following sum: <H₁>+2<H₂> where H₁ and H₂ are the one-body and two-body parts of the Hamiltonian respectively. Deduce the interaction energy as a function of the single particle Green's function.

- Consider a non-interacting Hamiltonian for fermions which is in diagonal form H=S_l e_l c_l⁺ c_l. Calculate the susceptibility c_lm(w) of the system by starting from its definition as the density autocorrelation function c_lm(t) = -i q(t) <[r_lm(t),r_ml(0)]> where r_lm= c_l⁺ c_m. Write the susceptibility as an integral of two Green's functions; perform the frequency integral to obtain a sum involving an energy denominator and occupation numbers (do not assume the Jellium model). What do you obtain in the case of the Jellium model ?

- Show that the above susceptibility evaluated for imaginary frequencies c_lm(iw) is real.

- Consider a non interacting Jellium model. Calculate the current operator in second quantization. Calculate the current autocorrelation function which gives the conductivity. express the result as a function of the frequency and electron density. Interpret your result.

- Using the interaction picture method, derive Fermi's golden rule by assuming an external potential V(t) and calculating the probability for scattering after time t from a state | i(t) > into a final state | f > where both | f > and | i(0) > are eigenstates of the unperturbed Hamiltonian. P_if(t)=|< f | i(t) >|² . Show that < f | i(t) > = -< f | V | i > exp[i (E_f - E_i - ih)t ] / (E_f - E_i - ih). The derivative of this probability with respect to time should give you the transition rate.

- The inelastic differential scattering cross section per unit of energy and solid angle is related to the transition rate (see Doniach 1.3 page7). Here the perturbing potential is the lattice potential which can be written as V(r)=S_R v(r-R) and R are atomic positions. Use the Fourier representation of V to write the cross section as a function of the density autocorrelation function which is also called the dynamic structure factor S(q,w). You see therefore that the cross section is a measure of  S(q,w) which can be calculated by diagram technique.