Chemistry Reference
In-Depth Information
3
(a)
2.5
2
1.5
1
0.5
0
0
0.25
0.5
0.75
1
3
(b)
2.5
2
1.5
1
0.5
0
−1
−0.5
0
φ
0.5
1
Figure 12.
Symmetries of the flow of the Grusin model. Panel (a) : Intersection on the antipodal
parallel of equation
θ
=
π
−
θ
(0) of two trajectories with the same cost. Panel (b) : symmetry with
respect to the axis
ϕ
=
0 of the flow. Numerical values are taken to be in the two cases
p
ϕ
(0)
=±
2,
p
θ
(0)
=±
5,
θ
(0)
=
π/
4 and
ϕ
(0)
=
0.
We assume that the initial and the target states belong, respectively, to
th
e two
spheres
S
i
and
S
f
. We can choose, for example,
1
/
√
2(
|
ψ
i
=|
2
and
|
ψ
f
=
|
1
+
|
). For the measurement process, the idea is to determine an observable
Q
for
which the system passes from
S
i
to
S
f
after a measurement. We thus see that the
introduction of measurements allows us to create a path from the initial state to
the target state. This is possible if all the eigenvectors of
Q
belong to
S
f
and form
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