Angular Momentum Operator

Operator

Angular momentum operators satisfy

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

Orbital angular momentum is

$$\mathbf L = \mathbf r\times\mathbf p,$$

while $\mathbf S$ denotes spin and $\mathbf J$ often denotes total angular momentum.

The standard eigenvalue equations are

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

Assumptions

• The operator acts in a representation of the rotation algebra.
• The meaning of $\mathbf J$, $\mathbf L$, and $\mathbf S$ is specified.
• The quantization axis is chosen, usually $z$.
• Domains matter for differential orbital operators.

Common Mistakes

• Trying to diagonalize $J_x$, $J_y$, and $J_z$ simultaneously.
• Confusing the operator $\mathbf J$ with the quantum number $j$.
• Mixing orbital and spin angular momentum without specifying the tensor-product space.
• Dropping powers of $\hbar$ in eigenvalues.

Canonical Links

• Angular Momentum Algebra
• Angular Momentum Formula Card
• Ladder Operators
• Angular Momentum Operator Symbol

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.