**Abstract**

Many introductory courses on many-body physics start by noticing that it is unfeasible to exactly solve the many particle Schrodinger equation, and then immediately introduce approximate methods such as Hartree-Fock and Density Functional Theory (DFT).

In this seminar, I will instead study one of the simplest conceivable many-particle problems: Two particles in a one-dimensional box that interact through a Coulomb-like repulsion. The low dimensionality of this problem permits an exact solution of the problem. Some general features of this solution will be discussed.

Subsequently, the Hartree-Fock solution of the same problem will be compared with the exact results. I will highlight some successes and failures of the

method for this specific problem. Next, I will show how the exact Kohn-Sham DFT potential and orbitals can be obtained from the knowledge of the exact groundstate electron density. By solving for the `optimized effective (exchange) potential', it is furthermore possible to separate this exact potential in its Hartree, exchange and correlation contributions. The effect of each is these terms on the orbitals will be discussed.