The
Kronig-Penney model was presented in lecture. The potential is composed of
barriers of width
*
b
*
and height
*
U
_{
0
}
*
and wells of width

(a)
Beginning with Schrodinger's equation, prove that inside the wells ψ
_{
1
}
(
*
x
*
)
=
*
A
*
e
*
^{
iαx
}
*
+

(b)
Write down four equations for the four coefficients (A, B, C, D), using continuity
of ψ(
*
x
*
) and its derivatives at
*
x
*
= 0 and periodicity of ψ(
*
x
*
)
and its derivatives at
*
x
*
=
*
−b
*
. Find the matrix
**
Q
**
such that

(c)
By setting the determinant of
*
Q
*
= 0 and taking the limit of the
delta function potential, one can show that cos(

(d) For
the same problem find the band gap at
*
k
*
= π/a.