**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(k_{F}),
total kinetic energy per particle and Fermi energy E_{K}/N(E_{F}),
and total kinetic energy per volume and density E_{K}/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 E_{A} and E_{B}. 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.