Sinc Collocation Methods for Solving Quantum Mechanical Problems

• Author / Creator

  Gaudreau, Philippe J

• In quantum mechanics, the Schrödinger equation is the staple for investi- gating and understanding quantum phenomena. Adjunct with the Schrödinger equation, the mathematical and physical laboratory that are anharmonic os- cillator potentials provide a powerful tool for modelling complex quantum systems. In this work, we successfully apply the the double exponential Sinc- collocation method (DESCM) to the classical anharmonic potential for the numerical evaluation of energy eigenvalues. The DESCM was able to achieve unprecedented accuracy even in the case of multiple wells. This great suc- cess has lead us to our current research endeavours. In our current work, we wish to adapt the DESCM to the rational-anharmonic potential as well as the Coulombic-anharmonic potential. The rational-anharmonic potential has several complex singularities which impede the convergence of the DE- SCM. As a result, we investigate conformal mappings in order to relocate these complex singularities accelerating the convergence of the DESCM. The Coulombic-anharmonic potential has singularities at the end points of its do- main which can affect the numerical stability of the DESCM. Subsequently, we investigate methods to remedy these numerical problems. Additionally, we wish to exploit the added symmetrical properties of the matrices generated by the DESCM in the presence of even potentials. We have been able to show that this added symmetry results in centrosymmetry. This added symmetry can be exploited to reduce the complexity of the DESCM by half.

• Subjects / Keywords

  • Sinc Collocation Methods
  • Anharmonic Oscillators
  • Double Exponential Transformation
  • Schrödinger equation

• Graduation date

  Spring 2017

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3TD9NM4Q

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.