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In-Depth Information
13.2.3
The Spin Eigenstates Define a Two-Level
Quantum System
The spin eigenstates correspond to two different energy levels. A neutral particle
is considered in a magnetic field of intensity B
z
. The particle's magnetic moment
M and the associated kinetic moment are collinear and are related to each other
through the relation
(13.6)
M D
The potential energy of the particle is
W DM
z
B
z
DB
z
z
(13.7)
Variable !
0
DB
z
is introduced, while parameter
z
is substituted by the spin's
measurement operator S
z
.
Thus the Hamiltonian H which describes the evolution of the spin of the
particle due to field B
z
becomes H
0
D !
0
S
z
, and the following relations between
eigenvectors and eigenvalues are introduced:
HjC >DC
„
!
2
jC >; Hj >D
„
!
2
j >
(13.8)
Therefore, one can distinguish 2 different energy levels (states of the quantum
system)
E
C
DC
„
!
0
2
E
D
„
!
0
2
(13.9)
By applying an external magnetic field the probability of finding the particle's
magnetic moment at one of the two eigenstates (spin up or down) can be changed.
This can be observed, for instance, in the Nuclear Magnetic Resonance (NMR)
model and is the objective of quantum control [
34
].
13.3
The Lindblad and Belavkin Description of Quantum
Systems
13.3.1
The Lindblad Description of Quantum Systems
It will be shown that the Lindblad and the Belavkin equation can be used in place
of Schrödinger's equation to describe the dynamics of a quantum system. These
equations use as state variable the probability density matrix Dj >< j,
