Angular momentum magnitude commutator
Shows that any one angular-momentum component commutes with the total squared angular momentum.
This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.
Core idea
Overview
This is why quantum states can be labeled by both l and one component quantum number m.
When to use: Shows that any one angular-momentum component commutes with the total squared angular momentum.
Why it matters: This is why quantum states can be labeled by both l and one component quantum number m.
Walkthrough
Derivation
Derivation of Angular momentum magnitude commutator
Shows that any one angular-momentum component commutes with the total squared angular momentum.
- The symbols use the standard quantum-chemistry convention for this topic.
- The expression is used within the model named in the entry.
Start from the model
Interpret the displayed relation as a rule, definition, or operator statement.
Identify the physical pieces
This is why quantum states can be labeled by both l and one component quantum number m.
Use the result carefully
Apply the expression only when the assumptions of the model are satisfied.
Result
Source: Chemistry LibreTexts, Rotational Motions of Rigid Molecules; Chemistry LibreTexts, Selection Rule for the Rigid Rotator
Free formulas
Rearrangements
Solve for
Solve for reason
The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.
Why it behaves this way
Intuition
This is why quantum states can be labeled by both l and one component quantum number m.
Signs and relationships
- positive terms: Positive terms usually represent kinetic energy, barriers, or magnitudes.
- negative terms: Negative terms usually represent attractive interactions or energy lowering when present.
Free study cues
Insight
Canonical usage
This equation demonstrates that the components of angular momentum commute with the total squared angular momentum, implying that these quantities can be simultaneously measured.
Common confusion
Students may confuse the concept of commutation with the idea that the quantities themselves are dimensionless.
Dimension note
The equation itself is a statement about operators and their commutation relations, not a calculation yielding a dimensionless quantity.
Unit systems
One free problem
Practice Problem
Can and Lz have simultaneous eigenfunctions?
Solve for: $[\hat{L}_i, \hat{L}^2]
Hint: Focus on what the formula is telling you physically.
The full worked solution stays in the interactive walkthrough.
Where it shows up
Real-World Context
In a spherical harmonic can be an eigenfunction of L^2 and Lz at the same time, Angular momentum magnitude commutator is used to calculate $[\hat{L}_i, \hat{L}^2] from the measured values. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.
Study smarter
Tips
- commutes with Lx, Ly, and Lz.
- Usually Lz is chosen as the measured component.
Avoid these traps
Common Mistakes
- Confusing this with the nonzero commutator between different components.
- Thinking all three components commute because each commutes with .
Common questions
Frequently Asked Questions
Shows that any one angular-momentum component commutes with the total squared angular momentum.
Shows that any one angular-momentum component commutes with the total squared angular momentum.
This is why quantum states can be labeled by both l and one component quantum number m.
Confusing this with the nonzero commutator between different components. Thinking all three components commute because each commutes with L^2.
In a spherical harmonic can be an eigenfunction of L^2 and Lz at the same time, Angular momentum magnitude commutator is used to calculate $[\hat{L}_i, \hat{L}^2] from the measured values. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.
L^2 commutes with Lx, Ly, and Lz. Usually Lz is chosen as the measured component.
References
Sources
- Chemistry LibreTexts, Rotational Motions of Rigid Molecules; Chemistry LibreTexts, Selection Rule for the Rigid Rotator
- Chemistry LibreTexts, Rotational Motions of Rigid Molecules
- Chemistry LibreTexts, Selection Rule for the Rigid Rotator
- Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
- Sakurai, J. J., & Napolitano, Jim. (2017). Modern Quantum Mechanics (2nd ed.). Cambridge University Press.
- Griffiths, David J. Introduction to Quantum Mechanics
- Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics