The Schrödinger equation

• Time evolution of the wave function ψ(x,t):
  • in suitable units given by
    • 
  • with the Hamilton operator
    • 
  • where
• Interpretation of ψ:
  • normed solutions in L₂(Rⁿ) describe a particle, i.e.
    • 
  • probability p(B) to find the particle in region
    • 
• Numerical solution:
  • finite differences in x and t
  • based on algorithm by DeRaedt (4^th order in dx and dt)
  • "perfectly matched layer" → no boundary reflections