Theoretical predictions from a 1D model of a solid: Bandstructure, defects, interfaces and quantum wells

Seminar series given at the University of Pretoria by Matthias Schmidt in 2014.
Time: Wednesdays, 16:00. Venue: Department of Physics, Computer Lab

Seminar 1, Introduction
• How is Schrödinger's equation integrated numerically?
• What is the particularity of a solution of Schrödinger's equation which is an eigenstate? How is this used in order to find the eigenstates of a quantum mechanical problem numerically?
Seminar 2, Getting started
• What are the command-line arguments which must be passed to solidsim in order to
• run a simulation?
• print the potential values to a file (1001 steps, range -10< x < 10)?
• How can the potentials for a quantum well and a harmonic oscillator, respectively, be defined?
• Why is the free electron in a computer actually an electron in an infinite quantum well?
Seminar 3, Coupled Quantum Wells and the Kronig-Penney model
• What happens to the eigenstates if two identical finite quantum wells are in close vicinity?
• How do the energy gaps calculated within the Kronig-Penney model depend on the barrier height?
• How do the energy gaps calculated within the Kronig-Penney model depend on the lattice constant?
• How does the finite x interval used by solidsim for the Kronig-Penney model simulations influence the wave-functions?
• Why does every lattice cell contribute exactly one eigenstate to one energy band? Hint: coupled quantum wells.
Seminar 4, 1D Hydrogen Atom and H2 Molecule Ion
• Why does the Coulomb potential bear a numerical difficulty? How can that difficulty be overcome?
• How do the energy levels of the 1D hydrogen atom compare to those calculated in the 3D case?
• In the 3D case the eigenstates of the H-atom are degenerate. What about the 1D H-atom?
• Consider two H-atoms in close vicinity forming an H2 molecule ion. What happens to the electronic states if their distance becomes zero? What happens for an infinite distance?