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Department of Applied Mathematics and Theoretical Physics; Centre for Theoretical Atomic, Molecular and Optical Physics
Queen's University Belfast

B-spline basis sets:

A flexible basis set for solving the Schrödinger equation.

B-spline basis sets have become a valuable tool for numerical calculations, especially in theoretical atomic physics. Their advantages can be exploited to perform accurate calculations in a systematic manner. Furthermore, B-splines form a highly flexible basis sets. The basis set can be adapted to the problem under investigation in a simple manner. These characteristics make the B-spline basis set highly suited for programming in a modern, modular manner. In order to introduce students to B-spline basis set, I am giving a short lecture series for Ph.D. students at the Queen's University of Belfast on the theory and application of B-spline basis sets. Background information on the following topics can be found online.

Table of contents:

  1. General introduction
  2. Mathematical Background
    1. Approximation by polynomials
    2. Piecewise Polynomials
    3. Introduction of continuity conditions
    4. The truncated power basis
  3. B-splines
    1. Definition of divided differences
    2. Definition of B-splines
    3. Recursion relation for B-splines
    4. B-splines form a basis set
      1. Proof of the Curry-Schönberg theorem
      2. Linear independence of B-splines
    5. Properties of B-splines
  4. Software
    1. The subroutine INTERV
    2. The subroutine BSPLVB
  5. Solving the Schrödinger equation in one dimension
    1. Reduction to a matrix equation
    2. Defining the knot set
    3. The overlap matrix
    4. The potential energy matrix
    5. The kinetic energy matrix
    6. Solving the matrix equation
    7. Application to the harmonic oscillator
    8. Parity
    9. Application to the hydrogen atom
  6. Solving the Schrödinger equation in two dimensions
    1. Confinement energies in T-shaped quantum wires
    2. Choice of knot set
    3. Energies
  7. Conclusions
  8. Further reading

Disclaimer: None of the information in these pages is necessarily correct. The information provided should be checked independently for accuracy.

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Page created by H.W. van der Hart, 30 January 2000 (9:43PM)