Editing Pauli matrices (section)

== Algebraic properties ==

{| class="wikitable floatright" style="text-align: center;"
|+ [[Cayley table]]; the entry shows the value of the row times the column.
! ×
! style=width:3em | <math>\sigma_x</math>
! style=width:3em | <math>\sigma_y</math>
! style=width:3em | <math>\sigma_z</math>
|-
! <math>\sigma_x</math>
| <math>I</math> || <math>i \sigma_z</math> || <math>-i \sigma_y</math>
|-
! <math>\sigma_y</math>
| <math>-i \sigma_z</math> || <math>I</math> || <math>i \sigma_x</math>
|-
! <math>\sigma_z</math>
| <math>i \sigma_y</math> || <math>-i \sigma_x</math> || <math>I</math>
|}

All three of the Pauli matrices can be compacted into a single expression:
:<math>
  \sigma_j = 
    \begin{pmatrix}
      \delta_{j3}                   &  \delta_{j1} - i\,\delta_{j2}\\
      \delta_{j1} + i\,\delta_{j2}  & -\delta_{j3}
    \end{pmatrix},
</math>
where  {{mvar|δ{{sub|jk}} }} is the [[Kronecker delta]], which equals {{math|+1}} if {{math|''j'' {{=}} ''k''}} and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of {{math|''j'' {{=}} 1, 2, 3,}} in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are [[Involutory matrix|''involutory'']]:
:<math>\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\,\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I,</math>
where {{mvar|I}} is the [[identity matrix]].

The [[determinant]]s and [[trace of a matrix|trace]]s of the Pauli matrices are
:<math>\begin{align}
               \det \sigma_j &= -1, \\
  \operatorname{tr} \sigma_j &= 0,
\end{align}</math>
from which we can deduce that each matrix {{mvar|σ{{sub|j}} }} has [[eigenvalues]] +1 and −1.

With the inclusion of the identity matrix {{mvar|I}} (sometimes denoted {{math|''σ''{{sub|0}}}}), the Pauli matrices form an orthogonal basis (in the sense of [[Hilbert–Schmidt operator|Hilbert–Schmidt]]) of the [[Hilbert space]] <math>\mathcal{H}_2</math> of {{math|2 × 2}} Hermitian matrices over <math>\mathbb{R}</math>, and the Hilbert space <math>\mathcal{M}_{2,2}(\mathbb{C})</math> of all [[complex number|complex]] {{math|2 × 2}} matrices over <math>\mathbb{C}</math>.

=== Commutation and anti-commutation relations ===
==== Commutation relations====
The Pauli matrices obey the following [[commutator|commutation]] relations:
:<math>[\sigma_j, \sigma_k] = 2 i \varepsilon_{j k l}\,\sigma_l, </math>
where the [[Levi-Civita symbol]] {{math|''ε{{sub|jkl}}''}} is used.

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra <math>(\mathbb{R}^3, \times) \cong \mathfrak{su}(2) \cong \mathfrak{so}(3) .</math>

==== Anticommutation relations====
They also satisfy the [[anticommutator|anticommutation]] relations:
:<math>\{\sigma_j, \sigma_k\} = 2 \delta_{j k}\,I,</math>

where <math>\{\sigma_j, \sigma_k\}</math> is defined as <math>\sigma_j \sigma_k + \sigma_k \sigma_j,</math> and {{math|''δ{{sub|jk}}''}} is the [[Kronecker delta]]. {{mvar|I}} denotes the {{math|2 × 2}} identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the [[Clifford algebra]] for <math>\mathbb{R}^3,</math> denoted <math>\mathrm{Cl}_3(\mathbb{R}) .</math>

The usual construction of generators <math>\sigma_{jk} = \tfrac{1}{4} [\sigma_j, \sigma_k]</math> of <math>\mathfrak{so}(3)</math> using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.

A few explicit commutators and anti-commutators are given below as examples:
{| style="text-align:left;"
! Commutators
! Anticommutators
|-
| <math>\begin{align}
\left[\sigma_1, \sigma_1\right] &= 0 \\
\left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \\
\left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \\
\left[\sigma_3, \sigma_1\right] &= 2i\sigma_2 
\end{align}</math>{{quad}}
| <math>\begin{align}
\left\{\sigma_1, \sigma_1\right\} &= 2I \\
\left\{\sigma_1, \sigma_2\right\} &= 0  \\
\left\{\sigma_2, \sigma_3\right\} &= 0  \\
\left\{\sigma_3, \sigma_1\right\} &= 0
\end{align}</math>
|}

=== Eigenvectors and eigenvalues ===
Each of the ([[Hermitian matrix|Hermitian]]) Pauli matrices has two [[eigenvalues]]: {{math|+1}} and {{math|−1}}. The corresponding [[Normalisable wavefunction|normalized]] [[eigenvectors]] are
:<math>\begin{align}
  \psi_{x+} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\  1 \end{bmatrix}, &
  \psi_{x-} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \\
  \psi_{y+} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\  i \end{bmatrix}, &
  \psi_{y-} &= \frac{1}\sqrt{2} \begin{bmatrix} 1 \\ -i \end{bmatrix}, \\
  \psi_{z+} &=                  \begin{bmatrix} 1 \\  0 \end{bmatrix}, &
  \psi_{z-} &=                  \begin{bmatrix} 0 \\  1 \end{bmatrix}.
\end{align}</math>