We are working with the periodic potential
The general solution in the region
Now lets look at the boundary conditions at . Continuity of the wave function gives
The discontinuity in the first derivative is
Substituting from the first equation
Plugging this equation for back into the equation above for we get
We now have two pairs of equations for the coefficients in terms of the coefficients.
Using the second pair of equations to eliminate the coefficients, we have
Now we can eliminate all the coefficients.
Jim Branson 2013-04-22