  In 2D topological insulators (TIs) based on semiconductor quantum wells such
as HgTe/CdTe or InAs/GaSb/AlSb, spin polarized edge states have been predicted
with a massless Dirac like dispersion. In a hard wall treatment based on the 4
x 4 BHZ Hamiltonian and open boundary conditions (OBCs), the wave function is
weakly confined near the edge, with which it makes no contact. In contrast,
standard boundary conditions for the wave function and its derivative (SBCs)
lead to strong confinement with a peak amplitude at the edge. Unfortunately,
weak confinement exhibits unphysical behavior related to a spurious gap
solution that is included in the OBC wave function. This is confirmed by the
gap solutions of the parent multiband Hamiltonian from which the smaller
Hamiltonian is derived, which exhibit physical behavior and do not satisfy
OBCs. Unlike OBCs or other approaches based on phenomenological boundary
conditions, SBCs treat the wall explicitly. Using a basis of empty crystal free
electron states for the vacuum with the same symmetry as the TI states, it is
shown that a large wall band gap overlapping that of the TI can only be
achieved by including a thin passivation layer. For passivation materials such
as silicon dioxide where the mid gap energy is nearly degenerate with that of
the TI, the Dirac point is very close to mid gap and virtually independent of
the TI band asymmetry. The treatment also demonstrates that a significant shift
of the dispersion may be introduced by interface band mixing. The shift is
largest at the Dirac point and decreases monotonically with edge state wave
vector, vanishing when the edge states merge with the bulk band edges.
