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Definitions
In Angular Momentum
we defined angular-momentum
observables \(\hatb J\) by their commutation relations:
\[\hatb{J} \times \hatb{J} = i \hbar \hatb{J}\]
In order to interpret the outcomes of experiments related to the spin of particles,
three operators, denoted by \(\hat \sigma_i\), were formulated, that:
act in a 2-dimensional Hilbert space i.e. describe a two-level quantum system ;
obey a commutation relations very similar to the above mentioned ;
have eigenvalues \(\pm 1\).
These operators called Pauli operators are defined as
\[\begin{split}\hat \sigma_X =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\, ; \quad
\hat \sigma_Y =
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
\, ; \quad
\hat \sigma_Z =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}\end{split}\]
and they obey the commutation relation
\[\hatb{\sigma} \times \hatb{\sigma} = 2 i \hatb{\sigma}\]
Together with the identity operator \(\hat I\), they form a basis of the
space of operators acting in a 2-dimensional Hilbert space \(\mathcal H\),
i.e. any operator of \(\mathcal H\) can be expressed as linear combination of
\(\hat I\) and the \(\hat \sigma_i\).
An operator \(\hat a\),
that is defined as a linear combination of the \(\hat \sigma_i\),
can be written using a vector \(\bm n\):
\[\begin{split}\hat a
& = \bm{n} \cdot \hatb{\sigma} \\
& = n_X \hat \sigma_X + n_Y \hat \sigma_Y + n_Z \hat \sigma_Z\end{split}\]
The action of an operator \(\exp(-i \theta \, \bm{n} \cdot \hatb{\sigma})\)
on any state vector can be described in the Bloch sphere as a rotation of this vector
around the axis defined by \(\bm n\).