Applied Quantum Mechanics Homeworks

Keivan Esfarjani, Sharif University of Technology

Saturdays (17:00-18:00) and Tuesdays (16:00-18:00) 

Series 1: The free electron gas and kinetic energy functionals

Calculate in 1,2 and 3 dimensions for the non-interacting free electron gas the relation between density and Fermi wave vector r(kF), total kinetic energy per particle and Fermi energy EK/N(EF), and total kinetic energy per volume and density EK/V(r).

Series 2: Thomas-Fermi model

For a single electron in a two-dimensional harmonic well, calculate the ground state density by using the Thomas-Fermi approximation. Deduce the total energy as well as the potential and kinetic energy within this model and compare to the exact results.

Series 3: Quantum wells

Calculate the bound states of the one dimensional quantum well of finite width and height: give the relation defining the bound state energies.

Series 4: Ionic versus covalent bonding

Calculate the eigenvalues and eigenstates of an asymmetric dimer molecule with different onsite energies EA and EB. Deduce the probabilities of having the lowest state electron on each of the atoms.

Series 5: Perturbation theory I

Calculate in the limit of small hopping integral compared to the difference between the onsite energies the shift in the eigenvalues of the asymmetric dimer model.

Series 6: Perturbation theory II

Using perturbation, calculate the correction to the energy eigenvalues and wavefunctions of the N atom chain with one s orbital per atom.