Phys 498CQM Lecture 28
Review of Course

Summary - Follows closely the outline in Lecture 0

1. Click here for Extended Outline of Course
   
2. Components of course
   • Homework
   • Project: A major part of the course
   • Exam(s)
3. Question asked at the beginning: Why Computational Quantum Mechanics?
4. Discussion in class of the answers for various types of problems:
   • Single-particle problems
     • Solution of Schrodinger Eq. in general potentials, where no analytical solutions exist
     • Time-dependent evolution
   • Many-body problems
     • Mean field independent-particle approximations lead to coupled non-linear equations which are coupled effective single-body equations.
       Examples: Hartree-Fock and Density Functional solutions for atoms, molecules, solids, and nuclear matter; BCS theory of superconductivity
     • Efficient methods for realistic problems, e.g. introduction to "Car-Parrinello" simulations of materials
     • Many-body calculations (very few examples can be solved analytically!)
       • Exact diagonalization for small systems
       • Quantum Monte Carlo simulations of many particles
       • Greens Function Quasiparticle Approaches
       • Prototypical Many-body problems - Hubbard, Anderson/Kondo
     • Field Theory strongly coupled systems like QCD very difficult
       • Strongly coupled systems like QCD very difficult
       • Usually solved computationally on a lattice
       • Only simple examples considered here
     • Visualization of solutions
   • Computational issues
     • What do we need to solve?
       • Differential Equations, Integrals
       • Finding roots (zeros)
       • Matrix (eigenvalue) problems
       • Many-particle equations
     • What do we want?
       • Stability, robustness of solution
       • Meaningful comparison with experiment (error estimates!)
       • Visualization