Angular Momentum Conventions

The default convention uses $J_i$ for generic angular momentum, $L_i$ for orbital angular momentum, $S_i$ for spin angular momentum, and $\mathbf J=\mathbf L+\mathbf S$ or sums over subsystems for total angular momentum when appropriate.

Angular momentum components satisfy

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

with $\epsilon_{xyz}=+1$. The standard simultaneous eigenbasis diagonalizes $J^2$ and $J_z$.

Eigenstate Labels

Angular momentum eigenstates are written

$$\lvert j,m\rangle,$$

with

$$J^2\lvert j,m\rangle =\hbar^2j(j+1)\lvert j,m\rangle,$$

and

$$J_z\lvert j,m\rangle =\hbar m\lvert j,m\rangle.$$

For fixed $j$,

$$m=-j,-j+1,\ldots,j.$$

The labels $j$ and $m$ are dimensionless. The eigenvalues carry the factors of $\hbar$.

Raising and Lowering Operators

The default convention is

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

Their action is

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

This convention implies that $J_+$ raises $m$ by one and $J_-$ lowers $m$ by one.

Orbital Angular Momentum

For orbital angular momentum, use $\ell$ rather than $j$ when the distinction is helpful:

$$L^2\lvert \ell,m\rangle =\hbar^2\ell(\ell+1)\lvert \ell,m\rangle, \qquad L_z\lvert \ell,m\rangle =\hbar m\lvert \ell,m\rangle.$$

Spherical harmonics use the Condon–Shortley phase convention unless a page explicitly declares otherwise. This matters for signs in ladder operations, Clebsch–Gordan coefficients, and tables.

Coupled and Uncoupled Labels

For two angular momenta, uncoupled states are written

$$\lvert j_1,m_1\rangle\otimes\lvert j_2,m_2\rangle,$$

or compactly as $\lvert j_1m_1;j_2m_2\rangle$ after declaration. Coupled states are written

$$\lvert j_1,j_2;J,M\rangle$$

when the component angular momenta need to remain visible, or $\lvert J,M\rangle$ when the context is unambiguous.

Common Mistakes

• Confusing the dimensionless labels $j,m$ with the eigenvalues $\hbar^2j(j+1)$ and $\hbar m$.
• Reversing the signs of $J_+$ and $J_-$.
• Forgetting that only one component, conventionally $J_z$, is diagonalized with $J^2$.
• Mixing spherical-harmonic phase conventions across sources.
• Using $\ell$ for spin angular momentum when $s$ or $j$ would be clearer.

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.

Exercises

1. For $j=3/2$, list the allowed $m$ values.

Solution

The allowed values run from $-j$ to $j$ in integer steps:

$$m=-\frac32,-\frac12,\frac12,\frac32.$$

2. What does $J_+$ do to $\lvert j,j\rangle$?

Solution

The coefficient contains

$$j(j+1)-j(j+1)=0.$$

Therefore

$$J_+\lvert j,j\rangle=0.$$