Elements of Modern Physics
270
n = – N/2, ± (1 + N/2), ± (2 + N/2), ..., N,
2nd Brillouin zone etc. (8.25)
where for simplicity of notation it has been assumed that N is an even number.
It is important to note that each Brillouin zone has 2N allowed states (including
a factor of 2 for the electron spin).
It is assumed that in the presence of the interaction, the wave function is a
superposition of mainly two plane waves,
ψ(x) =
1/2
1/2
ikx
ik x
e
e
A
B
L
L
′
+
(8.26)
where for the sake of definiteness we take k′ ≤ k. We substitute this expression
in Eq. (8.22), multiply the equation by e
– ikx
or e
– ik
′
x
and integrate to obtain
2 2
1
2
2
1
1
2
2
1
2
2
+
=
2π
− ′ =
′
+
=
k A
V B EA
m
k k
k
a
B
V A EB
m
(8.27)
In these equations, the interaction connects only the states which satisfy
the condition
k – k′ =
2
a
π .
This is related to the fact that for scattering by a one-dimensional lattice, a
Bragg maximum is obtained for precisely the same condition [see Eq. (8.15)].
The interaction which gives rise to the scatting, is also responsible for mixing
the two plane-wave states in Eq. (8.26). Solving the two homogeneous equations
gives
E =
1/2
2
4
2
2
2
2
2
1
2
(
)
1
(
)
4
4
16
2
− ′
+ ′ ±
+
k
k
k
k
V
m
m
(8.28)
B
A
=
1/2
2
2
2
2
2
2
1
2
1
2
(
)
(
)
1
4
4
16
2
2
′ −
− ′
±
+
k
k
k
k
V
V
m
m
where k′ = k –
2
a
π . Clearly, the changes introduced, by the perturbation V 1 are
significant mainly for | k | ≈ | k′ | and hence for k ≈ π/a. For k < π/a, the negative
sign corresponds to | A | ≥ | B |, which is therefore associated with the 0 ≤ k ≤ π/
a branch of the solutions, while for k > π/a, the positive sign corresponds to
270
n = – N/2, ± (1 + N/2), ± (2 + N/2), ..., N,
2nd Brillouin zone etc. (8.25)
where for simplicity of notation it has been assumed that N is an even number.
It is important to note that each Brillouin zone has 2N allowed states (including
a factor of 2 for the electron spin).
It is assumed that in the presence of the interaction, the wave function is a
superposition of mainly two plane waves,
ψ(x) =
1/2
1/2
ikx
ik x
e
e
A
B
L
L
′
+
(8.26)
where for the sake of definiteness we take k′ ≤ k. We substitute this expression
in Eq. (8.22), multiply the equation by e
– ikx
or e
– ik
′
x
and integrate to obtain
2 2
1
2
2
1
1
2
2
1
2
2
+
=
2π
− ′ =
′
+
=
k A
V B EA
m
k k
k
a
B
V A EB
m
(8.27)
In these equations, the interaction connects only the states which satisfy
the condition
k – k′ =
2
a
π .
This is related to the fact that for scattering by a one-dimensional lattice, a
Bragg maximum is obtained for precisely the same condition [see Eq. (8.15)].
The interaction which gives rise to the scatting, is also responsible for mixing
the two plane-wave states in Eq. (8.26). Solving the two homogeneous equations
gives
E =
1/2
2
4
2
2
2
2
2
1
2
(
)
1
(
)
4
4
16
2
− ′
+ ′ ±
+
k
k
k
k
V
m
m
(8.28)
B
A
=
1/2
2
2
2
2
2
2
1
2
1
2
(
)
(
)
1
4
4
16
2
2
′ −
− ′
±
+
k
k
k
k
V
V
m
m
where k′ = k –
2
a
π . Clearly, the changes introduced, by the perturbation V 1 are
significant mainly for | k | ≈ | k′ | and hence for k ≈ π/a. For k < π/a, the negative
sign corresponds to | A | ≥ | B |, which is therefore associated with the 0 ≤ k ≤ π/
a branch of the solutions, while for k > π/a, the positive sign corresponds to