Quantum mechanics - eigenvalue problem

 

• from T.C. Scott, R.A. Moore, M.B. Monagan, G.J. Fee, E.R. Vrscay, J. of Comp. Phys., Vol. 87, No. 2, p.366-395, (1990).

• Rayleigh-Schrodinger perturbation theory

• Hamiltonian, solvable part and perturbative part

  
  
  

• eigenvalues (energies) and eigenfunctions are expanded into power series in

  
  
  

• eigenvalue problem

  
  
  

• hierarchy of equations

  
  
  

• normalizations

  
  
  

• separable problems, second order ordinary differential equations (ODEs), inhomogenous ODE

  
  
  

• one solution of the homogeneous equation and another linearly independent solution in which the degree decreases
  
  

• particular solution of the inhomogenous equation, variation of parameters

  
  
  

• the general solution of the inhomogenous equation is

  
  
  

• one particle Dirac equation

  
  
  

• Hamiltonian

  
  
  

• for hydrogen-like atoms

  
  
  

• two particle Dirac equation

• necessary algebraic operations

  • indefinite integrals

  • definite integrals

  • pattern matching, zero recognition

  • manipulation and simplification of large sums