Time: Wednesdays, 16:00. Venue: Department of Physics, Computer Lab

The seminar series addresses post-graduate and PhD students with a background in solid state physics. It is aimed to provide insight into the quantum mechanics of electronic states in solids within the band theory. Thereby the capabilities of the most simple model of a solid, the one-dimensional linear chain of Coulomb wells, are fully exploited. The seminar series starts with the basic aspects of a numerical Schrödinger equation solver. Simple well-known quantum mechanical problems like an electron in a potential well or the harmonic oscillator are treated and the Kronig-Penney model is discussed.

Then the atomic states of the 1D hydrogen atom are calculated. The example of the 1D hydrogen molecule ion is used to explain chemical bonding. In the next step a hydrogen crystal is constructed by aligning Coulomb wells periodically. Thereby the formation of bands from atomic states is studied. Finally localised electronic states in solids introduced by perturbations of the crystal potential (defects, surfaces, interfaces, etc.) are calculated.

The seminars are intended to be interactive. The source code of the computer program will be provided to the participants such that the they can reproduce the examples presented and simulate their own quantum mechanical problems.

- Introduction
- some properties of Schrödinger's equation
- how to solve Schrödinger's equation numerically
- the computer program
`solidsim` - Numerical solutions of well known problems in quantum mechanics
- free electron
- electron in a finite quantum well
- the harmonic oscillator
- coupled quantum wells
- the Kronig-Penney model
- From Atoms to Solids
- the 1D hydrogen atom
- Coulomb potential and its numerical difficulties
- wave functions and eigenstates
- discussion of the results and comparison with the analytic solutions for the 1D and the 3D hydrogen atom
- the 1D hydrogen molecule ion
- formation of molecule orbitals from atomic orbitals
- bonding and anti-bonding states
- determination of the bond length
- comparison with the 3D case (theory and experiment)
- 1D solid hydrogen
- 1D chain of Coulomb wells
- formation of bands from atomic orbitals
- band gaps
- occupation of the bands - fermi energy: metals, insulators and semiconductors
- insulating and metallic hydrogen
- influence of the lattice constant on the bandstructure
- influence of the Coulomb well charge on the bandstructure
- comparison with experimental data from solid hydrogen
- Localised electronic states in a 1D solid
- Point defects
- crystal symmetry is locally broken by

1. different lattice constant

2. different charge

3. both - electronic defect states in the bandgap(s)
- localised decaying wave functions
- donors and acceptors
- Surfaces and thin solid films
- the semi-infinite 1D chain of Coulomb wells - a simple model of a surface
- localised states at the surface
- the formation of surface states: a thought-experiment and its numerical implementation
- the vacuum - thin film - bulk structure
- Quantum wells and super lattices