Nederlands

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

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

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

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

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:

• Discretisation with the Finite Difference method.
  Specific research questions are:
  • What is an optimal scaling of the parameters?
  • How should the singularity at $r = 0$ be taken into account?
  • What is the order of convergence of the eigenvalues?
  
  
  
• Solution algorithms for the algebraic eigenvalue problem.
  We will consider the following two iterative solution algorithms for solving the eigenvalue problem: Lanczos method and Jacobi-Davidson method. Specific research questions are:
  • Lanczos: What is the problems size that can be handled. How reliable are the computed eigenvalues? Can the robustness of the algorithm be improved, for example by selective reorthogonalisation?
  • Jacobi-Davidson: What is the problem size that can be handled? How reliable are the computed eigenpairs? What is a good preconditioner for this problem?
  
  
  
• Implementation
  An initial implementation will be made using Matlab, with which the above research questions will be studied. Based on the outcomes of this study an algorithm will be selected for implementation on a GPU. Specific research questions are:
  • Which algorithm is most promising?
  • How can this algorithm be efficiently implemented on a GPU?
  • How reliable are the computations?
  
  
  
• Evaluation
  The final stage of the research is the evaluation of the software. This stage will evaluate to what extend the main research question has been answered: which realistic problems can be solved with the developed software?

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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