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!)