Pauli matrices
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The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Wolfgang Pauli. They are:
- <math>
\sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} </math>
- <math>
\sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} </math>
- <math>
\sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} </math>
Contents |
Properties
Identities
- <math>
\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I</math> where I is the identity matrix.
- <math>\sigma_1\sigma_2 = i\sigma_3\,\!</math>
- <math>\sigma_3\sigma_1 = i\sigma_2\,\!</math>
- <math>\sigma_2\sigma_3 = i\sigma_1\,\!</math>
- <math>\sigma_i\sigma_j = -\sigma_j\sigma_i\mbox{ for }i\ne j\,\!</math>
Additional properties
The determinants and traces of the Pauli matrices are:
- <math>\begin{matrix}
\det (\sigma_i) &=& -1 & \\ \operatorname{Tr} (\sigma_i) &=& 0 & \quad \hbox{for}\ i = 1, 2, 3 \end{matrix}</math>
and as a result, the eigenvalues of each matrix are ±1.
The Pauli matrices obey the following commutation and anticommutation relations:
- <math>\begin{matrix}
[\sigma_i, \sigma_j] &=& 2 i\,\epsilon_{i j k}\,\sigma_k \\ \{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I \end{matrix}</math>
where εijk is the Levi-Civita symbol, δij is the Kronecker delta, and I is the identity matrix. The above relations can be verified using
- <math>\sigma_i \sigma_j = i \epsilon_{ijk} \sigma_k + \delta_{ij} \cdot I</math>.
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 combinations of i times the Pauli matrices iσj, i.e. with the anti-Hermitian 2×2 matrices with trace 0. In this sense, the Pauli matrices generate su(2). As a result, iσj can be seen as infinitesimal generators 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 rotations in three-dimensional space. In other words, iσj are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space.
Physics
In quantum mechanics, iσj represent the generators of rotation acting on non-relativistic particles with spin ½. The state of the particles are represented as two-component spinors, 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 4π 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 quaternions, and when used as the basis for the spin-½ rotation operator is the same as the corresponding quaternion rotation representation.
See also
References
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