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I'm looking for possible ways to construct $J$-matrices. $J$-matrices are the higher-order version of Pauli matrices. Pauli matrices are suited for spin-1/2 systems, while J-matrices can be for any spin system, 1 or 3/2 or 2 or anything else.

My favorite way:

My favorite way for constructing J-matrices is using Wigner D-matrices. Starting from the eigenvalues of the $J_z$ operator (assuming $z$ is the quantization axis), which are very trivial to systematically derive. For example, for a spin-1 system:

$$J_z=\left(\begin{matrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0\text{} & 0 & -1 \end{matrix}\right)$$

Now we rotate this operator with Wigner D-matrices to get a hypothetical vector that points along $z$ to get once along $x$ and once along $y$. This will create $J_x$ and $J_y$.

However, this assumes that I know the Wigner D-matrices, which I don't (because my problem is computational, and I don't want to calculate the Wigner D-matrices).

So my question is: Is there a simpler way to derive the $J$ matrices? The simplest way I would love is if there's a way to derive them from the Pauli-matrices.

The Quantum Physicist
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1 Answers1

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As a matter of fact, there is a much easier way to derive those matrices. What you are really after is the matrix representation of the $\mathfrak{su}(2)$ Lie algebra generators $J_x$, $J_y$ and $J_z$ in the irreducible $\mathrm{SU}(2)$ representation with spin $j$, given by

$$ J_z|j,m\rangle = m |j,m\rangle,\quad J_\pm|j,m\rangle = C_\pm(j,m) |j,m\pm 1\rangle, $$ with $$ J_\pm=J_x \pm i J_y $$ and $$ C_\pm(j,m)=\sqrt{(j\mp m)(j\pm m+1)}. $$ The matrix representations of $J_z$ and $J_\pm$ in this basis are $$ J_z=\begin{pmatrix} j\\ & j-1\\ && \ddots\\ &&& -j+1\\ &&&& -j \end{pmatrix} $$ $$ J_+= \begin{pmatrix} 0 & C_+(j,j-1)\\ & 0 & C_+(j,j-2)\\ && \ddots\\ &&& 0 & C_+(j,-j)\\ &&&& 0 \end{pmatrix} $$ $$ J_-= \begin{pmatrix} 0 \\ C_-(j,j) & 0\\ && \ddots\\ && C_-(j,-j+2) &0\\ &&& C_-(j,-j+1) &0 \end{pmatrix}, $$ and you can recover $J_x$ and $J_y$ as $$ J_x=\frac{J_+ + J_-}{2},\quad J_y=\frac{J_+ - J_-}{2i}. $$ For example, with $j=1$ you would get $$ J_z=\begin{pmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0& 0 & -1 \end{pmatrix},\quad J_x=\frac{\sqrt{2}}{2}\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 1\\ 0& 1 & 0 \end{pmatrix},\quad J_y=\frac{\sqrt{2}}{2}\begin{pmatrix}0 & -i & 0\\ i & 0 & -i\\ 0& i & 0 \end{pmatrix}. $$

Giuseppe Sellaroli
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