 the Pauli's 2 x 2 sigma matrices - Maple Programming Help

Physics[Psigma] - the Pauli's 2 x 2 sigma matrices

 Calling Sequence Psigma[n]

Parameters

 n - an integer between 0 and 4, or an algebraic expression representing it, identifying a Pauli matrix

Description

 • The Psigma[n] command represents the three Pauli matrices; that is, the set of Hermitian and unitary matrices:

${\mathrm{Psigma}}_{1}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\mathrm{Psigma}}_{2}=\left[\begin{array}{cc}0& -I\\ I& 0\end{array}\right],{\mathrm{Psigma}}_{3}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

 where $I$ is the imaginary unit (to represent it with a lowercase $i$, see interface,imaginaryunit). Together with ${\mathrm{Psigma}}_{4}={\mathrm{Psigma}}_{0}$, representing the 2 x 2 identity matrix, the Pauli matrices form an orthogonal basis. The ${\mathrm{Psigma}}_{n}$ matrices are displayed as ${\mathrm{\sigma }}_{n}$.
 • When multiplied by the imaginary unit, these matrices are a realization of the Lie algebra of the SU(2) group, which is isomorphic to the Lie algebra of SO(3). So, the $I{\mathrm{\sigma }}_{i}$ are also a matrix realization of infinitesimal rotations in 3D space, hence serving as representation for the 3D angular momentum operator in Physics.
 • The Pauli matrices satisfy the commutation relations $\left[{\mathrm{\sigma }}_{i},{\mathrm{\sigma }}_{j}\right]=2I{\mathrm{\epsilon }}_{\mathrm{ijk}}{\mathrm{\sigma }}_{k}$, where ${\mathrm{\epsilon }}_{\mathrm{ijk}}$ is the Levi-Civita symbol, and $i,j,k$ range from 1 to 3. The ${\mathrm{\sigma }}_{i}$ also satisfy the anticommutation relations $\left\{{\mathrm{\sigma }}_{i},{\mathrm{\sigma }}_{j}\right\}=2{\mathrm{\delta }}_{\mathrm{ij}}$, where ${\mathrm{\delta }}_{\mathrm{ij}}$ is the Kronecker delta. Those two relations can be written as ${\mathrm{\sigma }}_{i}{\mathrm{\sigma }}_{j}=I{\mathrm{\epsilon }}_{\mathrm{ijk}}{\mathrm{\sigma }}_{k}+{\mathrm{\delta }}_{\mathrm{ij}}$.
 • For $i$ from 1 to 3, the Pauli matrices satisfy $\mathrm{Det}\left({\mathrm{\sigma }}_{i}\right)=-1,\mathrm{Trace}\left({\mathrm{\sigma }}_{i}\right)=0$, and ${\mathrm{\sigma }}_{i}^{2}=1$ (the 2 x 2 identity matrix), where Det represents the determinant, and Trace represents/computes the trace. In the context of the Physics package (see conventions), you can also use the index 0, as in ${\mathrm{Psigma}}_{0}$, and it will be automatically mapped into ${\mathrm{Psigma}}_{4}$.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)
 > ${\mathrm{Psigma}}_{1}$
 ${{\mathrm{\sigma }}}_{{1}}$ (2)
 > $\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\right)$
 $\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]$ (3)
 > ${\mathrm{Psigma}}_{1}{\mathrm{Psigma}}_{2}+{\mathrm{Psigma}}_{2}{\mathrm{Psigma}}_{1}$
 ${{\mathrm{\sigma }}}_{{1}}{}{{\mathrm{\sigma }}}_{{2}}{+}{{\mathrm{\sigma }}}_{{2}}{}{{\mathrm{\sigma }}}_{{1}}$ (4)
 > $\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\right)$
 ${\mathrm{.}}{}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]{,}\left[\begin{array}{cc}0& -I\\ I& 0\end{array}\right]\right){+}{\mathrm{.}}{}\left(\left[\begin{array}{cc}0& -I\\ I& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)$ (5)
 > $\mathrm{Library}:-\mathrm{PerformMatrixOperations}\left(\right)$
 $\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]$ (6)
 > ${\mathrm{Psigma}}_{1}{\mathrm{Psigma}}_{1}$
 ${{{\mathrm{\sigma }}}_{{1}}}^{{2}}$ (7)
 > $\mathrm{Trace}\left(\right)$
 ${2}$ (8)
 > ${\mathrm{Psigma}}_{1}{\mathrm{Psigma}}_{2}$
 ${{\mathrm{\sigma }}}_{{1}}{}{{\mathrm{\sigma }}}_{{2}}$ (9)
 > $\mathrm{Trace}\left(\right)$
 ${0}$ (10)
 > 

Compatibility

 • The Physics[Psigma] command was updated in Maple 2019.