Physical meaning of a commutator?

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A commutator is a mathematical tool used to determine whether two operators commute, which affects 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 indicate limitations on precision, aligning with Heisenberg's uncertainty principle. Additionally, if an operator commutes with the Hamiltonian, its expectation value remains constant over time. Understanding these concepts is crucial for grasping the implications of quantum mechanics.
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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.
 
Thread 'Correct statement about size of wire to produce larger extension'

Thread 'Correct statement about size of wire to produce larger extension'

The answer is (B) but I don't really understand why. Based on formula of Young Modulus: $$x=\frac{FL}{AE}$$ The second wire made of the same material so it means they have same Young Modulus. Larger extension means larger value of ##x## so to get larger value of ##x## we can increase ##F## and ##L## and decrease ##A## I am not sure whether there is change in ##F## for first and second wire so I will just assume ##F## does not change. It leaves (B) and (C) as possible options so why is (C)...
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