Let us once again confine our attention to the region,
The wave function for this region is given by
where
and
By Bloch's theorem, the full wave function must have the form,

Continuity of
and
at
requires that

and

The periodicity of
and continuity of
and
at
gives us

and

We have thus obtained four linear equations in
and
To derive the condition under which these equations have a nontrivial solution, we first eliminate
and
and then determine when the resulting
system has nontrivial solutions; this yields the condition,

where
is the width of the "barrier" parts of the potential. This, along with the equation,
yields the energy spectrum of the system.
If we take the limit
and
in such a way as to keep
finite, then we can obtain:

In this limit,

and

Our equations then reduce to, noting that
in this limit,

This is just the equation that we obtained for the Dirac comb potential; note that, here,
stands for the
in the Dirac comb problem described earlier.
Back to Motion in a Periodic Potential