Chemistry Reference
In-Depth Information
Fig. 2.3
Schematic illustration of a symmetric double minimum potential. a H
ij
= 0 b H
ij
= 0
2.1.2 Tunneling in a Symmetric Double Minimum Potential
Symmetry is important to understand diverse concepts and laws of nature. Here I
focus on the role of symmetry in H transfer. The symmetry of the potential
landscape is a key to the mechanism involving quantum tunneling. The simplest,
but a ubiquitous system in physics and chemistry is a symmetric double minimum
potential (Fig.
2.3
), where we can consider two wave functions |W
1
[ and
|W
2
[ with a specific Hamiltonian H
i
(i = 1 or 2) in each potential well where the
eigenenergy is given by H
i
W = E
0
W
i
. If each of the two states is independent, the
wave packet is localized in each of the potential well for all time. Now we take into
account the transfer of a wave packet between two wells and introduce the tran-
sition matrix H
ij
i
;
j
¼
1or 2; i
6¼ ð Þ
that transfers the wave packet into another
well. As long as the matrix element H
ij
¼
\W
i
j
H
ij
j
W
j
[
¼
0
;
each of the two
states degenerates into the eigenstate with its eigenenergy of E
0
(Fig.
2.3
a).
However, this situation is varied if H
ij
= 0 (Fig.
2.3
b), where |W
1
[ and |W
2
[ are
no longer the eigenstate of the system and the eigenfunctions are given by the
symmetric (gerade) and anti-symmetric (ungerade) linear combination of |W
1
[ and
|W
2
[ .
j
g [
¼
1
p ðj
W
1
[
þj
W
2
[
Þ; j
u [
¼
1
2
p ðj
W
1
[
j
W
2
[
Þ
2
The corresponding energy levels are split into
E
k
¼
E
0
j
H
12
j
where k is g (gerade) or u (ungerade) and the lower or upper energy levels belongs
to the symmetric or anti-symmetric state. |H
12
| corresponds to the energy splitting.
Given that the system is initially in the state of |W
1
[ , the system shows a periodic
motion between |W
1
[ and |W
2
[ states because the |W
1
[ is no longer an eigenstate
of the system. Then the probabilities finding the wave packet in |W
1
[ and
|W
2
[ change according to
P
1
¼
cos
2
ð
H
12
j
t
Þ;
P
2
¼
sin
2
ð
H
12
j
t
Þ
