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In-Depth Information
a
b
8
5
4
7
3
6
2
5
1
4
0
3
−1
2
−2
1
−3
0
−4
−1
−5
0
20
40
60
80
100
120
140
160
180
200
0
20
40
60
80
100
120
140
160
180
200
x
x
Fig. 7.6
(
a
) Diagram of V.x/
0
where V.x/ is the harmonic potential of the QHO and
0
is
the associated eigenvalue (
b
) Linear drift force applied to the diffused particle as a result of the
harmonic potential V.x/
!
2
2
x
2
1
2
V.x/D
(7.58)
It is known that the ground mode of the QHO of Eq. (
7.48
) is a Gaussian function
[
56
,
164
], i.e.
0
.x/ D Ce
!x
2
(7.59)
2
2
1
while it can be proved easily that the associated eigenvalue is
0
D
2
!. A diagram
of V.x/
0
is given in Fig.
7.6
a.
For the diffusion constant holds
2
m
where
D
„
is Planck's constant and
1
2
m!
2
x
2
. Assuming the stationary p.d.f. of Eq. (
7.57
)
finally gives V.x/D
D C
2
e
!x
2
(7.60)
.x/ D
0
.x/
2
2
2
the force applied to the particle due to the harmonic potential V.x/ is given by
Eq. (
7.56
), and is found to be
@
0
.x/
@x
u
.x/ D
2
1
0
.x/
)
u
.x/ D!x
(7.61)
which means that the drift is a spring force applied to the particle and which aims at
leading it to an equilibrium position. The drift force is depicted in Fig.
7.6
b.
