Master project of Karl Kästner
Computing the Energy Levels of a Confined Hydrogen Atom

Student: Karl Kästner (COSSE student, double degree with KTH Stockholm)

Research Group Numerical Mathematics Kees Vuik
Supervisor at TU-Delft Martin van Gijzen
Customer ESA ESTEC, Aerothermodynamics Section, Noordwijk (The Netherlands)
Supervisor at ESTEC D. Giordano
Project Start January 2012

In March 2012 the Interim Thesis and a presentation has been given.

The Master project has been finished in August 2012 by the completion of the Masters Thesis and a final presentation has been given.

For working address etc. we refer to our alumnipage.

Summary of the master project:

The confined one-electron atom is a popular model in theoretical chemistry and solid-state physics. In most studies, the simplest model of the spherical confinement is treated. The advantage of the spherical confinement model is that analytical expressions are known for the wave functions. However, in many physical situations the spherical confinement model is not realistic. In this study we consider a model for a hydrogen atom that is confined in a box. The energy levels of the hydrogen atom can be computed from the (non-dimensional) Schrödinger equation

$\displaystyle \frac{1}{2} \Delta \psi + \frac{1}{r} \psi = -\lambda \psi% \frac{1}{2} \Delta \psi + \frac{1}{r} \psi = {} & -\lambda \psi
$ (1)

In this equation, $ \Delta$ is the Laplace operator, $ \psi$ is the wave function, $ \lambda$ the energy level, and $ r$ the distance to the centre of mass. This equation is complemented by homogeneous boundary conditions. Equation 1 plus boundary conditions form an eigenvalue problem, in which $ \lambda$ is the eigenvalue and $ \psi$ the eigenfunction.

Discretisation of 1 leads to an algebraic eigenvalue problem of the form

$\displaystyle A \psi = \lambda \psi$ (2)

For realistic calculations, the size of this matrix can be prohibitive.

Hydrogen Wave Functions
Hydrogen Wave Functions

Research questions

The main research question is how to compute accurate approximations of a reasonable number of the smallest analytical eigenvalues $ \lambda$. To this end the following aspects will be studied:

The research will be carried out at the TU Delft, in close collaboration with the European Space Research and Technology Centre (ESTEC), located in Noordwijk.

Contact information: Kees Vuik

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