Chemistry Reference
In-Depth Information
We would normally describe Schrödinger's equation as a second order
differential equation acting on the full wavefunction,
. We can also,
however, view eq. (E.4) as a second order differential equation involv-
ing the unknown function exp
ψ
n
k
(
r
)
(
i
k
0
·
r
)
u
n
k
(
r
)
. If we multiply both sides of
[−
(
−
)
·
]
eq. (E.4) from the left by exp
i
k
k
0
r
, we obtain amodified differential
equation from which to determine
E
n
k
:
e
−
i
(
k
−
k
0
)
·
r
H
0
e
i
(
k
−
k
0
)
·
r
e
i
k
0
·
r
u
n
k
e
−
i
(
k
−
k
0
)
·
r
E
n
k
e
i
(
k
−
k
0
)
·
r
e
i
k
0
·
r
u
n
k
[
]
(
(
r
))
=[
]
(
(
r
))
e
i
k
0
·
r
u
n
k
=
E
n
k
(
(
r
))
(E.5)
Between eqs (E.3) and (E.5), we have transformed from a
k
-dependent
wavefunction,
ψ
n
k
,toa
k
-dependent Hamiltonian, which we write as
H
q
,
where
q
=
k
−
k
0
. Equation (E.5) can be re-written as
e
−
i
q
·
r
e
i
q
·
r
2
2
m
∇
−
2
H
q
φ
n
k
(
r
)
=
+
V
(
r
)
φ
n
k
(
r
)
(E.6)
2
e
i
q
·
r
where
φ
(
r
)
=
exp
(
i
k
0
·
r
)
u
n
k
(
r
)
. We now expand the term
∇
φ
(
r
)
n
k
n
k
to obtain
2
2
m
∇
2
m
q
2
q
2
2
m
+
1
−
+
i
∇+
2
H
q
φ
(
r
)
=
·
V
(
r
)
φ
(
r
)
n
k
n
k
H
0
2
q
2
2
m
+
m
q
+
=
·
p
φ
(
r
)
(E.7)
n
k
where we have used eq. (E.1), and replaced
/
i
∇
by the momentum
operator,
p
, introduced in Chapter 1.
Equation (E.7) forms the basis of the
k
p
method. It reduces to the
standard form of Schrödinger's equation when
q
·
=
0, at the point
k
0
.
For many applications, we choose
k
0
=
point, where we generally
knowor can estimate the values of all the relevant zone centre energies,
E
n
0
.
We can then view
0, the
2
q
2
2
m
H
=
m
q
+
·
p
(E.8)
as a perturbation to the zone centre Hamiltonian,
H
0
, and use second order
perturbation theory to calculate the variation of the energy levels
E
n
k
with
wavevector
k
point.
For the case of a singly degenerate band, substituting eq. (E.8) into
eq. (C.23) gives the energy of the
n
th band in the neighbourhood of
k
(
=
q
)
close to the
=
0as
2
k
2
m
2
n
=
2
2
|
k
·
p
nn
|
+
m
k
p
nn
+
2
m
+
E
n
k
=
E
n
0
·
(E.9)
E
n
0
−
E
n
0
n
