Revision as of 06:54, 13 October 2008 by imported>Daniel Mietchen
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:
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f60e4a6ba92c7121a7fd9e2512ad579f383be70b)
Algebraic properties
![{\displaystyle \sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb3b09d66bfac21c67ba2302f43b37157c1ff85)
For i = 1, 2, 3:
![{\displaystyle {\mbox{det}}(\sigma _{i})=-1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c37bccefd319bf1e75917a59be667539ef02ffee)
![{\displaystyle {\mbox{Tr}}(\sigma _{i})=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6e8ded137e67c8c233b02c1d0d21f110d67b72)
![{\displaystyle {\mbox{eigenvalues}}=\pm 1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ff84755155216cc18562a7bf6d3b67f8f78f80)
Commutation relations
![{\displaystyle \sigma _{1}\sigma _{2}=i\sigma _{3}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f14a48c2491aa6784a709e23541bfd11f2e3fa7)
![{\displaystyle \sigma _{3}\sigma _{1}=i\sigma _{2}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a9de5c8ab1b7dfdb8d6225eb27c9fc254d5e0c)
![{\displaystyle \sigma _{2}\sigma _{3}=i\sigma _{1}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb2091f6c488d21063eb945309b3c784a64e925)
![{\displaystyle \sigma _{i}\sigma _{j}=-\sigma _{j}\sigma _{i}{\mbox{ for }}i\neq j\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/985a18c9f7bbb56e4775a22dfa9cd7115c7d4c05)
The Pauli matrices obey the following commutation and anticommutation relations:
![{\displaystyle {\begin{matrix}[\sigma _{i},\sigma _{j}]&=&2i\,\varepsilon _{ijk}\,\sigma _{k}\\[1ex]\{\sigma _{i},\sigma _{j}\}&=&2\delta _{ij}\cdot I\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/450cfd37e7d4f2865b026fdc3b145dbdb9be1d0b)
- where
is the Levi-Civita symbol,
is the Kronecker delta, and I is the identity matrix.
The above two relations can be summarized as:
![{\displaystyle \sigma _{i}\sigma _{j}=\delta _{ij}\cdot I+i\varepsilon _{ijk}\sigma _{k}.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f235da948df249950e6a4bc296b3a2e1b45ff0)