Chemistry Reference
In-Depth Information
∞
∞
V
0
b
L
Position
Figure C.2
Infinite square well of width
L
, with a barrier of height
V
0
and width
b
added in its centre.
to estimate the energy of that state,
W
(
1
)
k
,as
d
V
ψ
∗
(
0
)
k
ψ
(
0
)
k
(
r
)
H
(
r
)
W
(
1
)
k
=
(C.6)
d
V
ψ
∗
(
0
)
k
)ψ
(
0
)
k
(
r
(
r
)
H
where the denominator in eq. (C.6) is equal to 1. Replacing
H
by
H
0
+
and splitting the integral in eq. (C.6) into two parts gives
W
(
1
)
k
ψ
∗
(
0
)
k
ψ
(
0
)
k
ψ
∗
(
0
)
k
H
ψ
(
0
)
k
=
d
V
(
r
)
H
0
(
r
)
+
d
V
(
r
)
(
r
)
E
(
0
)
k
ψ
∗
(
0
)
k
H
ψ
(
0
)
k
=
+
d
V
(
r
)
(
r
)
(C.7)
The energy levels
E
(
0
k
of the Hamiltonian
H
0
thus provide the zeroth-
order guess for the energy levels of the full Hamiltonian
H
, while
E
(
1
)
k
=
d
V
ψ
∗
(
0
)
k
H
ψ
(
0
)
k
(
)
(
)
r
r
gives the first order correction to the estimated
energy.
C.2.1 Example: double square well with infinite outer barriers
To illustrate the application of first order perturbation theory, we consider
an infinite squarewell in the region 0
<
x
<
L
, towhich is added a potential
barrier of height
V
0
and width
b
between
L
/
2
−
b
/
2 and
L
/
2
+
b
/
2 (fig. C.2).
The energy levels
E
(
0
n
and wavefunctions
ψ
(
0
)
(
x
)
of the unperturbed well
n
are given by (eq. (1.11))
h
2
n
2
8
mL
2
E
(
0
)
=
(C.8a)
n
