I know that this topic doesn't take much attention of most of you as there are more interesting topics about paradoxes of physics but I need a little bit guidance about reproducing the band structure diagrams shown in Kittel ,
170p in 8th Edition :)
So we have different wave solution for V
0 region and 0 region,
\psi_{1}(x)=Ae^{ikx}+Be^{-ikx}\ \ \ \ \ \ \ for \ \ \ \ \ \ \ \ 0<x<a
\psi_{2}(x)=Ce^{qx}+De^{-qx}\ \ \ \ \ \ \ for \ \ \ \ \ \ \ \ \-b<x<0
After we apply the boundary conditions (usual QM boundary condiions in in square well potentials ),
\psi_{1}(0) = \psi_{2}(0) and \psi_{1}'(0) = \psi_{2}'(0)
\psi_{1}(a) = \psi_{2}(-b) and \psi_{1}'(a) = \psi_{2}'(-b)
Then, we have four equations
A+B=C+D
ik(A-B)=q(C-D)
Ae^{ika}+Be^{-ika}=(Ce^{-qb}+De^{qb})e^{i\alpha(a+b)}
ik(Ae^{ika}-Be^{-ika})=q(Ce^{-qb}-De^{qb})e^{i\alpha(a+b)}
Determinant of the coefficients of this equation system mush vanish to have solutions. Than determinant yields,
([q^{2}-k^{2}]/2qk) sinh(qb)sin(ka) + cosh(qb)cosh(ka) = cos (\alpha(a+b))
Finally my issues
- How can I define the allowed and forbidden energy values?
- to be continued...
e^{i\alpha(a+b)} ---->The term \alpha came from Bloch's Theorem and (a+b) came from application of transitional symmetry operation to Bloch form wavefunction!