Angular Momentum Convention Translator

Angular-momentum formulas are convention-sensitive: signs, phases, and basis order all matter.

Default Convention

Use

$$[J_i,J_j]=i\hbar\sum_k\epsilon_{ijk}J_k, \qquad \epsilon_{xyz}=+1,$$

and

$$J_\pm=J_x\pm iJ_y.$$

The ladder action is

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

Translation Table

Convention Issue	Default Choice	What to Check
basis order	descending $m$ in tables	matrix rows and columns
spherical-harmonic phase	Condon-Shortley	signs of $Y_{\ell,\pm m}$
ladder signs	$J_+=J_x+iJ_y$	whether $J_+$ raises $m$
spin operators	$S_i=(\hbar/2)\sigma_i$ for spin-$1/2$	whether a table uses $\sigma_i$ or $S_i$
Clebsch-Gordan coefficients	Condon-Shortley	phases of singlet and coupled states
Wigner symbols	standard 3-j relation	phase factor in coefficient conversion

Common Mistakes

• Comparing a $J_y$ matrix written in the opposite basis order.
• Dropping $\hbar$ in eigenvalues while keeping it in commutators.
• Confusing $\ell$ for orbital angular momentum with $j$ for total angular momentum.
• Copying Clebsch-Gordan coefficients without checking the phase convention.

Canonical Links

• Angular Momentum Conventions
• Spin Matrices
• Spherical Harmonics
• Clebsch-Gordan Coefficients
• Wigner Symbols

References

• 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.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.