Which operators commute

Two operators commute if the following equation is true:. If the same answer is obtained subtracting the two functions will equal zero and the two operators will commute.

If two operators commute, then they can have the same set of eigenfunctions. For example, the operations brushing-your-teeth and combing-your-hair commute, while the operations getting-dressed and taking-a-shower do not. This theorem is very important. If two operators commute and consequently have the same set of eigenfunctions, then the corresponding physical quantities can be evaluated or measured exactly simultaneously with no limit on the uncertainty.

As mentioned previously, the eigenvalues of the operators correspond to the measured values. Although it will not be proven here, there is a general statement of the uncertainty principle in terms of the commutation property of operators.

Consequently, both a and b cannot be eigenvalues of the same wavefunctions and cannot be measured simultaneously to arbitrary precision. Operators are very common with a variety of purposes. They also help to explain observations made in the experimentally. An example of this is the relationship between the magnitude of the angular momentum and the components. However the components do not commute themselves. In quantum mechanics the measuring process plays an important role.

It will alter the state of the system it's supposed to measure. If we are going to perform two experiments one after other then there is a possibility that some of the information is changed.

A commutator is a mathematical construct that tells us whether two operators commute or not. Hence it means the commutator is not equal to zero.

Otherwise it's zero. Which means that the two observables can be simultaneously measured. So a commutator tells us if we can measure two physical observables at the same time which are called compatible observables or not. If we know the value of the commutator then it tells how the measurements are going to alter things. It gives more information such as the uncertainty.

Physically it means, it does not matter in which temporal order you measure the two commuting observables. It means you can in principle measure both quantities to arbitrary precision at the same time. If they didnt commute then this would be impossible by the uncertainty principle. You may consider commutativity of different variables as physical independence, something like separated independent variables:. When two qm operators do not commute, it means that we are missing stuff in Nature.

That is quantum mechanics is a theory of measurement but not of Nature because of non-commutation. Hence this means that the stuff we miss cannot be described by quantum mechanics, and this leads to the conclusion that qm is not a complete description of Nature. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?

Learn more. Ask Question. Asked 10 years, 6 months ago. Active 6 years ago. Viewed 81k times. Improve this question. If two operators do not commute any measurement made of one of these with a certain accuracy will result in an uncertainty of the expectation value of the second operator - e.

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I'm not sure how that can be the case. In relativity, the order of measurements may in general depend of the frame of reference. Compatible observables do not obey uncertainty relationships, as far as I'm informed. This is equivalent to saying they are in a way independent maybe linked through other inequalities or other boundary conditions of sorts.