Commutators and compatible observables
The commutator measures the difference between doing two operator actions in opposite orders.
Definition
[L, M] = LM - MLIf [L, M] = 0, the operators commute. For observables, commuting is the mathematical signal that they can be simultaneously specified by a shared basis of eigenvectors.
Plain-language picture
For a single spin, σz and σx do not commute. A state that is definite for one is not generally definite for the other. For two separate spins, Alice’s σz and Bob’s τz do commute because they act on different factors of the tensor product space.
Complete set of commuting observables
In larger systems, one observable may not be enough to label a basis state. A complete set of commuting observables provides enough compatible labels to identify basis vectors uniquely.
Link to uncertainty
Nonzero commutators imply lower bounds on products of uncertainties. The famous [X, P] = iħ gives Δx Δp ≥ ħ/2.
Common pitfalls
- Do not think commuting means the numerical measurement results are equal; it means operation order is irrelevant.
- Do not assume every pair of Hermitian operators can be measured sharply together.
- Do not forget degeneracy: commuting with one operator may not fully label the states unless the set is complete.