# 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

## Check your knowledge!

**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 H**_{2} 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 H
_{2} molecule ion. What happens to the electronic
states if their distance becomes zero? What happens for an infinite distance?