Angular momentum component commutator

Core idea

Overview

Only one component of angular momentum can be specified sharply together with the total magnitude.

When to use: Shows that different angular-momentum components do not commute.

Why it matters: Only one component of angular momentum can be specified sharply together with the total magnitude.

Walkthrough

Derivation

Derivation of Angular momentum component commutator

Shows that different angular-momentum components do not commute.

The symbols use the standard quantum-chemistry convention for this topic.
The expression is used within the model named in the entry.

1

Start from the model

Interpret the displayed relation as a rule, definition, or operator statement.

[\hat{L}_{i}, \hat{L}_{j}] = i ℏ ϵ_{ij k} \hat{L}_{k}

2

Identify the physical pieces

Only one component of angular momentum can be specified sharply together with the total magnitude.

[\hat{L}_{i}, \hat{L}_{j}] = i ℏ ϵ_{ij k} \hat{L}_{k}

3

Use the result carefully

Apply the expression only when the assumptions of the model are satisfied.

[\hat{L}_{i}, \hat{L}_{j}] = i ℏ ϵ_{ij k} \hat{L}_{k}

Result

[\hat{L}_{i}, \hat{L}_{j}] = i ℏ ϵ_{ij k} \hat{L}_{k}

Source: Chemistry LibreTexts, Rotational Motions of Rigid Molecules; Chemistry LibreTexts, Selection Rule for the Rigid Rotator

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h ee q u a t i o nin v o l v es t h e L e v i - C i v i t a sy mb o l, w hi c hi s anin d e x - ba se d t e n sor coe f f i c i e n t r a t h er t hana s t an d a r d a l g e b r ai c v a r iab l e . T h e v a r iab l es L_{i}, L_{j}, an d L_{k} a r eo p er a t or s w i t hina co mm u t a t or a l g e b r a, mak in g i t im p oss ib l e t o p er f or m s t an d a r d a l g e b r ai c i so l a t i o n (e . g ., so l v in g f or o n eo p er a t or in t er m so f t h eo t h er s) a s t h ey a r e p a r t o f an o n - co mm u t a t i v eo p er a t or mani f o l d .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Only one component of angular momentum can be specified sharply together with the total magnitude.

main expression

Shows that different angular-momentum components do not commute.

Only one component of angular momentum can be specified sharply together with the total magnitude.

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 is used to demonstrate the non-commutative nature of angular momentum operators in quantum mechanics, where the units of the operators and Planck's constant dictate the units of the resulting commutator.

Common confusion

Students may sometimes overlook the units of the operators and Planck's constant, incorrectly assuming the commutator might be dimensionless.

Dimension note

While the Levi-Civita symbol is dimensionless, the angular momentum operators and Planck's constant carry physical units, making the commutator have units of angular momentum.

Unit systems

$L_{i}$ J s - Angular momentum components have units of energy times time (Joule-seconds) or equivalently, mass times length squared times inverse time (kg m^2 s^-1).

hbarJ s - The reduced Planck constant (ħ) has the same units as angular momentum.

$e p s i l o n_{i} j k$ dimensionless - The Levi-Civita symbol is a tensor that is dimensionless.

One free problem

Practice Problem

What is [Lx, Ly] using the angular-momentum commutator?

Solve for: $[\hat{L}_i, \hat{L}_j]

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 Atomic orbitals are usually labeled by L^2 and Lz, not by all three components at once, Angular momentum component commutator is used to calculate $[\hat{L}_i, \hat{L}_j] 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

For cyclic components, [Lx, Ly] = i hbar Lz.
The Levi-Civita symbol tracks the cyclic sign.

Avoid these traps

Common Mistakes

Assuming Lx, Ly, and Lz can all be known exactly together.
Dropping the factor of i hbar.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Shows that different angular-momentum components do not commute.

Only one component of angular momentum can be specified sharply together with the total magnitude.

Assuming Lx, Ly, and Lz can all be known exactly together. Dropping the factor of i hbar.

In Atomic orbitals are usually labeled by L^2 and Lz, not by all three components at once, Angular momentum component commutator is used to calculate $[\hat{L}_i, \hat{L}_j] from the measured values. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

For cyclic components, [Lx, Ly] = i hbar Lz. The Levi-Civita symbol tracks the cyclic sign.

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
NIST CODATA
IUPAC Gold Book
Quantum Mechanics, by David J. Griffiths
Principles of Quantum Mechanics, by R. Shankar
Griffiths, David J. Introduction to Quantum Mechanics

Overview

Derivation

Start from the model

Identify the physical pieces

Use the result carefully

Rearrangements

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Angular momentum operator

Angular momentum magnitude commutator

Spherical harmonics

Associated Legendre polynomial

Frequently Asked Questions

Sources