So I have been trying very hard for the last day or so to solve the Kronig-Penney model for finite barriers to no avail with Mathematica. The unfortunate bit is that it doesn't seem like my 4x4 matrix is wrong and that I just can't seem to get Mathematica to simplify out the result nicely. It is particularly infuriating because I have exactly the same steps written down at this point as a solution guide (Kronig-Penney Solution). But take note of what's said in the first paragraph of page 2. The only interesting solutions that exist are ones for which the coefficients of A, B, C, D (for equations C5 to C8) have a determinant that is equal to zero. They then quote the result below stating that you can arrive at the result if you are determined.
After setting all of our equations equal to zero the matrix that we are to take the determinant of should look something like this: (K = x, gamma = y)
m = {{Exp[I*x*b],
Exp[-I*x*b], -Exp[y*b], -Exp[-y*b]}, {I*x*Exp[I*x*b], -I*x*
Exp[-I*x*b], -y*Exp[I*b], y*Exp[-I*b]}, {Exp[I*x*a],
Exp[-I*x*a], -Exp[I*k*a], -Exp[I*k*a]}, {I*x*Exp[I*x*a], -I*x*
Exp[-I*x*a], -y*Exp[I*k*a], y*Exp[I*k*a]}}
But no matter how many times I take the determinant, FullSimplify, expand, reduce or divide out any terms that are outside the parentheses because the det(m) = 0 I just can't seem to get the result quoted in the solution. I really don't want to just quote a result from online so I would really appreciate a hand in understanding how I can solve this.
Thank you