Today we mentioned what happens when we have a weak potential.
All states shift their energy in the same way.
This effect shows up when we look at the absolute value of the Fermi energy or chemical potential. It measures the depth of the bottom of the conduction band with respect to the vacuum zero. Changes in the energy which depend on n or are of second order. This is not true when we have a degeneracy. In one dimension this only occurs at the brillouin zone boundary or at the center, but in three dimensions there are more possibilities.