Guided reading
This page puts the target result first: the compact commutator of angular-momentum components. The definitions and Levi-Civita notation then show the algebraic path behind that result.
Main commutator
The section then obtains the compact relation
\[[\hat L_u,\hat L_v]=i\hbar\epsilon_{uvw}\hat L_w\]
where \(u,v,w\) represent Cartesian directions.
Definition used in the proof
Angular momentum begins as a vector product and is then written in Cartesian components.
\[\vec L=\vec r\times\vec p\]
\[\vec L=L_x\hat x+L_y\hat y+L_z\hat z\]
For operators, the vector product is represented by the Levi-Civita symbol:
\[\hat L_i=\epsilon_{ijk}\hat x_j\hat p_k\]
This form, together with \([\hat x_i,\hat p_j]=i\hbar\delta_{ij}\), leads to the compact commutator above (Main Commutator).
Levi-Civita notation
The Levi-Civita symbol has values \(+1\), \(-1\), or \(0\), depending on the permutation of its three indices.
\[\epsilon_{ijk}=\begin{cases}+1, & \text{even permutations},\\ -1, & \text{odd permutations},\\ 0, & i=j\ \text{or}\ i=k\ \text{or}\ j=k.\end{cases}\]
The reduced identity used in the commutator manipulation is
\[\epsilon_{ijk}\epsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}\]
Einstein notation is used: a repeated index indicates summation over that index.
Cyclic components
\[[\hat L_x,\hat L_y]=i\hbar\hat L_z\]
\[[\hat L_z,\hat L_x]=i\hbar\hat L_y\]
\[[\hat L_y,\hat L_z]=i\hbar\hat L_x\]
Physical consequence
Because these components do not commute, it is impossible to have simultaneous and precise knowledge of two angular-momentum components.
In matrix language, the component matrices cannot be diagonalized simultaneously because they do not share the same vector basis.
Exercise-ready boundary
This page supports guided exercises on: the angular-momentum commutator, the Levi-Civita notation used in its proof and the direct physical meaning.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed for a compact calculation or explanation.
- Long-form material: complete proofs, long demonstrations, extended examples and the full exercise set remain in the textbook.
- Boundary: do not introduce results, notation or physical claims outside the sequence presented here and in Chapter 6.
Practice anchors
Use these anchors only within the material introduced on this page.
- Focus: the angular-momentum commutator and Levi-Civita notation.
- Conceptual check: connect the nonzero commutator to what can be measured simultaneously.
- Key equation 1:
\[[\hat L_u,\hat L_v]=i\hbar\epsilon_{uvw}\hat L_w\]
- Key equation 2:
\[\hat L_i=\epsilon_{ijk}\hat x_j\hat p_k\]
- Typical task: use the compact notation to identify a component commutator and state its measurement consequence.
Original book and previews: Source note: Original auxiliary summary for this book-app, based on Chapter 6 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for the complete presentation, proofs, examples and exercises.