Pauli spin matrices

From Citizendium, the Citizens' Compendium

Jump to: navigation, search


This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
http://en.citizendium.org/wiki?title=Pauli_spin_matrices&printable=yes
This is a draft article, under development and not meant to be cited but you can help to improve it. These unapproved articles are subject to a disclaimer.

The Pauli spin matrices (named after physicist Wolfgang Ernst Pauli) are a set of unitary Hermitian matrices which form an orthogonal basis (along with the identity matrix) for the real Hilbert space of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 2 matrices. They are usually denoted:

\sigma_x=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y=\begin{pmatrix} 0 & -\mathit{i} \\ \mathit{i} & 0 \end{pmatrix}, \quad \sigma_z=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

Algebraic properties

\sigma_x^2=\sigma_y^2=\sigma_z^2=I

For i = 1, 2, 3:

\mbox{det}(\sigma_i)=-1\,
\mbox{Tr}(\sigma_i)=0\,
\mbox{eigenvalues}=\pm 1\,

Commutation relations

\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\ne j\,\!

The Pauli matrices obey the following commutation and anticommutation relations:

\begin{matrix} [\sigma_i, \sigma_j] &=& 2 i\,\varepsilon_{i j k}\,\sigma_k \\[1ex] \{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I \end{matrix}
where \varepsilon_{ijk} is the Levi-Civita symbol, \delta_{ij} is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:

\sigma_i \sigma_j = \delta_{ij} \cdot I + i \varepsilon_{ijk} \sigma_k. \,
Retrieved from "http://en.citizendium.org/wiki/Pauli_spin_matrices"
Views
Personal tools