Chemistry Reference
In-Depth Information
ps
(
x
)
3
(e)
(d)
(
x
)
3
opw
(
x
)
3
(c)
(
x
)
2
(b)
(
x
)
1
(a)
V
(
x
)
0
3
6
Position (Å)
9
12
Figure 3.16
(a) The K-P potential with
a
=
b
=
6 Å, and barrier height
V
0
=
2.0 eV.
The solid lines in (b)-(d) show the exact wavefunctions for the three
lowest zone centre states in this potential. The exact wavefunction
ψ
3
(
x
)
for the third state (solid line in (d)) is well approximated by the dashed
line
φ
OPW
3
(
x
)
in (d), formed from a single plane wave (with wavevector
k
=
0, (e)) orthogonalised to the lowest state, as described in the text.
OPW
3
we first expand the left-hand side,
H
φ
(
x
)
, as the sum of several terms:
1
√
2
a
−
OPW
3
H
φ
(
x
)
=
H
H
α
1
ψ
1
(
x
)
2
d
2
d
x
2
1
√
2
a
+
1
√
2
a
−
α
=−
V
(
x
)
−
1
E
1
ψ
(
x
)
(3.46)
1
2
m
while the right-hand side is given by
E
1
√
2
a
−
α
OPW
3
E
φ
(
x
)
=
ψ
(
x
)
(3.47)
1
1
We can equate eqs (3.46) and (3.47), the two sides of the true Schrödinger
equation, and then re-arrange terms t
o g
ive a modified second order wave
equation, where we set
/
√
2
a
to give
PS
3
φ
(
x
)
=
1
2
d
2
d
x
2
+
√
2
a
−
PS
3
PS
3
V
(
x
)
+
α
(
E
3
−
E
1
)
ψ
(
x
)
φ
(
x
)
=
E
3
φ
(
x
)
1
1
2
m
