  The effective Hamiltonian for two dimensional quantum wells with rough
interfaces is formally derived. Two new terms are generated. The first term is
identified to the local energy level fluctuations, which was introduced
phenomenologically in the literature for interface roughness scattering but is
now shown to be valid only for an infinite potential well or Hamiltonians with
one single length scale. The other term is shown to modulate the wavefunction
and cause fluctuations in the charge density. This will further reduce the
electron mobility to the magnitude that is close to the experimental result.
