What does it mean to say that two operators commute? What is...

What does it mean to say that two operators commute? What is the significance of two observables having mutually commuting operators? Given that the commutator $[P, Q] \neq 0$ for some observables $P$ and $Q$, does it follow that for all $|\psi\rangle \neq 0$ we have $[P, Q[|\psi\rangle \neq 0 ?$

What does it mean to say that two operators commute? What is the significance of two observables having mutually commuting operators?
Given that the commutator $[P, Q] \neq 0$ for some observables $P$ and $Q$, does it follow that for all $|\psi\rangle \neq 0$ we have $[P, Q[|\psi\rangle \neq 0 ?$

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Key Concepts

Commuting Operators

This concept refers to two operators whose order of application does not affect the outcome; that is, for operators A and B, they commute if AB = BA. This property is important because it often implies that the operators have a set of common eigenstates and can be simultaneously diagonalized.

Commutator

The commutator of two operators, defined as [A, B] = AB - BA, quantifies the degree to which the operators fail to commute. If the commutator is zero for all vectors, the operators commute, whereas a non-zero commutator indicates non-commutativity, which has significant implications for the corresponding observables.

Simultaneous Observability and Diagonalization

When two operators commute, the observables they represent can be simultaneously measured with certainty, and their associated eigenstates form a common basis. This allows for the precise determination of both quantities in the same system without interference from one another.

Operator Property vs. State-Specific Evaluations

The property of a commutator being non-zero is an intrinsic feature of the operators and does not imply that its expectation value is non-zero for every individual state. There may exist specific states for which the expectation value of the commutator vanishes, even though the overall operator commutation relation does not vanish.

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