We can compute both, they just won't be equal.if 2 hermitian operator A, B is commute, then AB=BA, the expectation value <.|AB|.>=<.|BA|.>. how about if A and B is non commute operator? so we can not calculate the exp value <.|AB|.> or <.|BA|.>?
In some cases (when the commutation is a projector to a particular eigen space) they might be equal.We can compute both, they just won't be equal.
Yes, of course. But in general they won't be.In some cases (when the commutation is a projector to a particular eigen space) they might be equal.
Generally we can't, because if the vector psi is in Dom(AB), it may not be in Dom(BA).We can compute both, they just won't be equal.
Then the expectation is zero. Both operators map from Hilbert Space to Hilbert Space in this context. And you can always take an inner product between two vectors in Hilbert Space. They might be orthogonal, because they belong to different sub-spaces, but then the inner product is trivially zero and that's your expectation value.Generally we can't, because if the vector psi is in Dom(AB), it may not be in Dom(BA).
In nearly all physical situations (without weird boundary conditions) you can compute both. One example, of weird boundary conditions, is particle in infinite square well where [itex]D(\hat{x}\hat{p}^\dagger) \neq D({\hat{p}^\dagger\hat{x}})[/itex] and you can't compute both (always).conclusion is if AB is non commute. we can only compute 1 of the expectation value, either <|AB|> or <|BA|>?