Spin Matrices

This table gives the standard angular-momentum matrices for spin representations most often used in quantum mechanics.

Convention

The basis is ordered by descending $m$:

$$\lvert j,j\rangle,\ \lvert j,j-1\rangle,\ldots,\lvert j,-j\rangle.$$

The ladder operators act as

$$J_\pm\lvert j,m\rangle =\hbar\sqrt{j(j+1)-m(m\pm1)}\, \lvert j,m\pm1\rangle.$$

Then

$$J_x=\frac{J_++J_-}{2}, \qquad J_y=\frac{J_+-J_-}{2i}.$$

Spin One-Half

For $j=1/2$,

$$J_i=\frac{\hbar}{2}\sigma_i.$$

Use Pauli Matrices for the full $2\times2$ table.

Spin One

In the basis $\lvert1,1\rangle,\lvert1,0\rangle,\lvert1,-1\rangle$,

Operator	Matrix divided by $\hbar$
$J_z$	$\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}$
$J_x$	$\frac{1}{\sqrt2}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}$
$J_y$	$\frac{1}{\sqrt2}\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}$
$J_+$	$\sqrt2\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}$
$J_-$	$\sqrt2\begin{pmatrix}0&0&0\\1&0&0\\0&1&0\end{pmatrix}$

General Matrix Elements

Matrix Element	Value
$\langle j,m'\rvert J_z\lvert j,m\rangle$	$\hbar m\,\delta_{m'm}$
$\langle j,m'\rvert J_+\lvert j,m\rangle$	$\hbar\sqrt{j(j+1)-m(m+1)}\,\delta_{m',m+1}$
$\langle j,m'\rvert J_-\lvert j,m\rangle$	$\hbar\sqrt{j(j+1)-m(m-1)}\,\delta_{m',m-1}$

Common Mistakes

• Ordering the basis from $m=-j$ to $m=j$ while using matrices written for descending $m$.
• Dropping the factor of $\hbar$.
• Confusing dimensionless Pauli matrices with spin operators.
• Changing the phase convention before comparing Clebsch-Gordan coefficients.

Canonical Links

• Angular Momentum Algebra
• Ladder-Operator Action
• Pauli Matrices
• Spin-Half Hilbert Space
• Higher Spin Systems

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.