Chemistry Reference
In-Depth Information
of width
b
. We choose the origin at the centre of this barrier, so that the
potential is then symmetric about
x
=
0, with the right-hand well then
between
x
=
b
/
2and
b
/
2
+
a
, and the left-handwell between
x
=−
(
b
/
2
+
a
)
and
2. With this symmetry, the doublewellwavefunctionswill be either
even or odd about
x
−
b
/
=
0. The wavefunction for the lowest symmetric state
is illustrated by the dashed line in fig. 2.1, and can be written down in
terms of four unknown parameters
A
,
B
,
C
and
D
:
1 Within the central barrier, Schrödinger's equation takes the form of
eq. (1.46):
2
d
2
−
ψ
d
x
2
+
V
0
ψ
=
E
ψ
(2.1)
2
m
for which the even solution is
b
2
<
b
2
e
κ
x
e
−
κ
x
ψ(
x
)
=
A
(
+
)
−
x
<
(2.2)
2
.
2 Within the right-hand well, Schrödinger's equation is given by
eq. (1.44)
2
κ
=
(
−
)/
with
2
m
V
0
E
2
d
2
−
ψ
d
x
2
=
E
ψ
(2.3)
2
m
and we choose as our general solution
B
cos
k
x
C
sin
k
x
a
+
b
a
+
b
ψ(
x
)
=
−
+
−
2
2
a
b
2
<
b
2
x
<
+
(2.4)
where
k
2
2
, with the phase of the sine and cosine functions
chosen so that they are, respectively, odd and even about the well
centre,
=
/
2
mE
(
a
+
b
)/
2.
3
Schrödinger's equation has the same form in the right hand as in the
central barrier; and in order that
ψ
→
0as
x
→∞
, we choose
a
b
2
D
e
−
κ(
x
−
(
b
/
2
+
a
))
ψ(
x
)
=
x
>
+
(2.5)
Because of the symmetry, the wavefunctions in the left-hand well and
barrier depend on the same unknown coefficients
B
,
C
,and
D
.
