In quantum mechanics we assume the following:
- Each observable is associated with an Hermitian operator of a Hilbert space H. Its eigenvalues must be real and the eigenstates are orthogonal to each other, thus form a set of basis of H.
- Upon observation, one of the eigenvalues will be the quantity and the wave function will collapse onto one of the corresponding eigenstates.
Here I don’t want to discuss the deep insights, which I have no idea of. That’s why I take Copenhagen interpretation, so shut up and compute!
This article is written as a note of my understanding of physical and mathematical meaning of commutators, mainly the answer to the question: “why do commute operators have a common set of eigenstates?”