Physical meaning of a commutator?

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The discussion centers on the physical meaning of a commutator in quantum mechanics. A commutator serves as a mathematical tool to determine whether two operators commute, which directly impacts the simultaneous measurability of observables. If the commutator of two operators equals zero, they can be measured simultaneously with definite values. Conversely, non-commuting operators are subject to Heisenberg's uncertainty principle, indicating limitations in precision when measuring paired observables.

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I know how to use a commutator as a mathematical formula but I really don't understand what it means. Can anyone explain it to me.

Is a commutator nothing more than a check to see if it commutes or not since I know that if you use a commutator with a wave function and the result equals zero then you can have a definite value for the observables (operators) simultaneously?



Thanks.
 
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https://www.physicsforums.com/showthread.php?t=131975

If two operators do not commute, then they cannot be measured simultaneously with infinite precision - the pair of observables has to obey Heisenberg's uncertainty principle.

If an operator commutes with the Hamiltonian, then its expectation value is constant.
 

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