Chemistry Reference
In-Depth Information
and
2
L
sin
n
π
x
ψ
(
0
)
(
x
)
=
(C.8b)
n
L
Substituting eq.
(C.8b) in eq.
(C.7),
the estimated first order shift
in the energy levels is given by
L
/
2
+
b
/
2
V
0
sin
2
n
d
x
2
L
π
x
E
(
1
)
=
n
L
L
/
2
−
b
/
2
V
0
b
sin
n
n
+
1
L
+
(
−
)
π
1
b
=
(C.9)
n
π
L
That is,
E
(
1
n
varies linearly with the height of the perturbing potential,
V
0
. The solid lines in fig. C.3 show how the true energy levels vary with
the barrier height
V
0
in the case where the barrier width
b
is half of the
total infinite well width,
b
2, while the straight dashed lines show the
estimated variation using first order perturbation theory. It can be seen that
first order perturbation theory is indeed useful for small perturbations,
but rapidly becomes less accurate as
V
0
increases. The accuracy of the
perturbation estimate can, however, be extended to larger values of
V
0
by going to second order perturbation theory, which will then introduce
further correction terms to the estimated energy of order
V
0
.
=
L
/
12
8
4
0
0
2 4
Barrier height (
V
0
/
E
1
)
6
8
Figure C.3
Variation of the confined state energy for the three lowest energy levels
in an infinite square well, as a function of the magnitude of the perturbing
potential,
V
0
introduced in fig. C.2. The barrier width
b
is set equal to
half of the total infinite well width,
b
=
L
/
2, and
V
0
is plotted in units of
h
2
/
8
mL
2
(the ground state energy,
E
1
). Solid line: exact solution; dashed
(straight) lines: using first order perturbation theory; dotted (parabolic)
lines: using second order perturbation theory.
