01 Jun 2026

Angular Momentum Operators

Commutation relations, ladder operators, and the algebraic structure of angular momentum in quantum mechanics.

msc semester-i quantum-mechanics angular-momentum commutation-relations

Angular momentum in quantum mechanics is not only a vector quantity. It is an operator algebra. The components of angular momentum do not commute with each other, and this non-commutativity is the reason why only one component, usually $L_z$, can be specified sharply along with $L^2$.

Orbital angular momentum

For a particle with position operator $\mathbf r$ and momentum operator $\mathbf p$, the orbital angular momentum operator is

\[\mathbf L=\mathbf r\times \mathbf p.\]

In Cartesian components,

\[L_x=yp_z-zp_y,\qquad L_y=zp_x-xp_z,\qquad L_z=xp_y-yp_x.\]

The canonical commutation relations are

\[[x_i,p_j]=i\hbar\delta_{ij},\qquad [x_i,x_j]=0,\qquad [p_i,p_j]=0.\]

Using these, one obtains the angular momentum algebra

\[\boxed{ [L_x,L_y]=i\hbar L_z,\qquad [L_y,L_z]=i\hbar L_x,\qquad [L_z,L_x]=i\hbar L_y. }\]

In compact notation,

\[\boxed{ [L_i,L_j]=i\hbar\epsilon_{ijk}L_k. }\]

Here $\epsilon_{ijk}$ is the Levi-Civita symbol.

Total angular momentum algebra

The same algebra is satisfied by any angular momentum operator $\mathbf J$:

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

This includes orbital angular momentum $\mathbf L$, spin angular momentum $\mathbf S$, and total angular momentum

\[\mathbf J=\mathbf L+\mathbf S.\]

Commutation with $J^2$

Define

\[J^2=J_x^2+J_y^2+J_z^2.\]

Although the components of $\mathbf J$ do not commute with each other, $J^2$ commutes with every component:

\[\boxed{ [J^2,J_x]=[J^2,J_y]=[J^2,J_z]=0. }\]

Therefore $J^2$ and one chosen component, conventionally $J_z$, can have simultaneous eigenstates.

Ladder operators

Define the raising and lowering operators

\[J_+=J_x+iJ_y,\qquad J_-=J_x-iJ_y.\]

They satisfy

\[\boxed{ [J_z,J_\pm]=\pm\hbar J_\pm }\]

and

\[\boxed{ [J_+,J_-]=2\hbar J_z. }\]

The operators $J_+$ and $J_-$ change the magnetic quantum number $m$ without changing $j$.

Useful identities

The ladder operators are related to $J^2$ by

\[J^2=J_-J_+ + J_z^2+\hbar J_z\]

and

\[J^2=J_+J_- + J_z^2-\hbar J_z.\]

Equivalently,

\[J_+J_-=J^2-J_z^2+\hbar J_z,\] \[J_-J_+=J^2-J_z^2-\hbar J_z.\]

These identities are used to derive the allowed eigenvalues of angular momentum.

Main points

• Angular momentum components do not commute.
• The operator $J^2$ commutes with all components of $\mathbf J$.
• The pair $(J^2,J_z)$ can be diagonalized simultaneously.
• Ladder operators move between states with different $m$ values.
• The algebra is the same for orbital, spin, and total angular momentum.

Practice questions

1. Starting from $L_x=yp_z-zp_y$, derive $[L_x,L_y]=i\hbar L_z$.
2. Prove that $[J^2,J_z]=0$.
3. Derive $[J_z,J_\pm]=\pm\hbar J_\pm$.
4. Show that $[J_+,J_-]=2\hbar J_z$.
5. Explain why $J_x$, $J_y$, and $J_z$ cannot all be measured sharply in the same state.