for Spin 1/2, we have Pauli matrix as in wiki. So what's the generalized 3-by-3 Pauli matrix for spin 1 or even larger spin? Is there a generalization method?
$\begingroup$
$\endgroup$
2
-
2$\begingroup$ Related: physics.stackexchange.com/q/79348/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Sep 2, 2016 at 13:10
-
2$\begingroup$ Searching for "spin 1 matrices" yields a plethora of usable results. This question shows no research effort. $\endgroup$– ACuriousMind ♦Commented Sep 3, 2016 at 13:56
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The spin operator (or Pauli matrices ) is in general the angular momentum operator $J_{x}, \, J_{y}, \, J_{z}$ in which they satisfy the commutation relation \begin{align*} [J_{i} , \, J_{j}] = i \hbar \epsilon_{ijk} J_{k} \quad \mathrm{and} \quad [J^2 , \, J_{i}] = 0 \end{align*} and they operate on the state as \begin{align*} J^2 | j, m> \quad &= \quad j(j+1) \hbar^2 \; |j,m> \\ J_{\pm} |j , m > \quad &= \quad \hbar \sqrt{j(j+1) \mp m( m \pm 1)} \; |j,m \pm 1> \end{align*} where $J_{\pm } = J_{x} \pm i J_{y} $. $ j $ is the angular momentum quantum number and $m$ is the z-direction (usually) angular momentum quantum number. Note that $j$ can be positive integer (and zero) or half-integer and $ m = -j , \, -j+1 , \, \cdots , j-1 , \, j$
For the case of spin $1/2$, we put $j = 1/2$ and the allowed $m = -1/2$ and $1/2$. Then apply the rules above to obtain the matrix elements for the spin (or angular momentum) operator
