From a computational perspective there is a commonality to the types of problems that arise when constructing computer simulations of coherent and collective quantum states. The purpose of this part of the mini-course is to provide the background on the computational techniques and details that are particularly useful to be aware of when developing simulations of problems involving quantum states. While the focus problem will be the solution of the Schroedinger-Poisson equations, the information presented generalizes to computational tasks associated with other mathematical models, e.g. Density Functional Theory (DFT) or Full Configuration Interaction.

Syllabus

I. General Overview

1. The mathematical structure of the 3D problem

2. Task dimension reduction through the use of analytical models

3. The computational structure of 1D, 2D and 3D simulations

4. Focus problem: Schroedinger-Poisson equation in a layered semi-conductor

II. Numerical solution of the 1D Schroedinger-Poisson equation

1. Electrostatics

(a) Non-uniform finite-volume discretizations

(b) Solution of the discrete equations

2. Eigensystem

(a) Non-uniform finite-volume discretizations

(b) Computation of eigenvectors and eigenvalues

3. Self-Consistent iteration

(a) Fixed point iteration

(b) \Evolving" to a solution

4. The sample simulation (Matlab code)

II. 2D and 3D Problems

1. Electrostatics

(a) Discretization of 2D and 3D problems

(b) Solution of the discrete equations

2. Eigensystem

(a) Discretization of 2D and 3D problems

(b) Computation of eigensystems using subspace methods

III. Improvements and extensions