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:
\sigma_1 =
\begin{pmatrix}
0&1\
1&0
\end{pmatrix}


:
\sigma_2 =
\begin{pmatrix}
0&-i\
i&0
\end{pmatrix}


:
\sigma_3 =
\begin{pmatrix}
1&0\
0&-1
\end{pmatrix}



PROPERTIES


Identities

:
\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \begin{pmatrix} 1&0\0&1\end{pmatrix} = I
where ''I'' is the Identity Matrix .
:\sigma_1\sigma_2 = i\sigma_3\,\!
:\sigma_3\sigma_1 = i\sigma_2\,\!
:\sigma_2\sigma_3 = i\sigma_1\,\!
:\sigma_i\sigma_j = -\sigma_j\sigma_i\mbox{ for }i
e j\,\!


Additional properties

The Determinant s and Trace s of the Pauli matrices are:

:\begin{matrix}
\det (\sigma_i) &=& -1 & \ {Link without Title}
\operatorname{Tr} (\sigma_i) &=& 0 & \quad \hbox{for}\ i = 1, 2, 3
\end{matrix}

and as a result, the Eigenvalues of each matrix are ±1.

The Pauli matrices obey the following Commutation and Anticommutation relations:

:\begin{matrix}
\sigma_j &=& 2 i\, arepsilon_{i j k}\,\sigma_k \[1ex]
\{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I
\end{matrix}

where arepsilon_{ijk} is the Levi-Civita Symbol , \delta_{ij} is the Kronecker Delta , and I is the identity matrix. The above relations can be verified using

:\sigma_i \sigma_j = i arepsilon_{ijk} \sigma_k + \delta_{ij} \cdot I.

The above commutation relations are similar to those of the Lie Algebra su(2), and indeed su(2) may be identified with the Lie algebra of all real Linear Combination s of i times the Pauli matrices i \sigma_j, i.e. with the anti- Hermitian 2×2 matrices with Trace 0. In this sense, the Pauli matrices generate su(2). As a result, i \sigma_j can be seen as Infinitesimal Generator s of the corresponding Lie Group SU(2) .

The Lie algebra su(2) is Isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the Group of Rotation s in three-dimensional Space . In other words, i \sigma_j are a realization (and, in fact, the lowest-dimensional realization) of ''infinitesimal'' rotations in three-dimensional space.


PHYSICS


In Quantum Mechanics , i \sigma_j represent the generators of rotation acting on Non-relativistic particles with Spin ½. The State of the particles are represented as two-component Spinor s, which is the Fundamental Representation of SU(2). An interesting property of spin ½ particles is that they must be rotated by an angle of 4\pi in order to return to their original configuration. This is due to the fact that SU(2) and SO(3) are not globally isomorphic, even though their infinitesimal generators su(2) and so(3) are isomorphic. SU(2) is actually a "double cover" of SO(3), meaning that each element of SO(3) actually corresponds to ''two'' elements in SU(2). Also useful in the Quantum Mechanics of multiparticle systems, the general Pauli group Gn is defined to consist of all n-fold Tensor products of Pauli Matrices.

Together with the identity matrix I (which is sometimes written as σ0), the Pauli matrices form a Basis for the real Vector Space of 2 × 2 complex Hermitian matrices. This basis is equivalent to the Quaternion s, and when used as the basis for the spin-½ rotation operator is the same as the corresponding quaternion rotation representation.


SEE ALSO




REFERENCES