In quantum mechanics, observable variables are represented by operators, and thus can be replaced by matrix in a certain basis. If we have H|n>=E(n)|n>, where |n> are eigenfunctions of Hamilton matrix. Here is the problem: what's the physical meaning of <m|P|n>, namely off-diagonal elements of another observable operator, such as momentum? I have known that sometimes this has something to do with quantum transition. But I want to know is there any universal understanding of off-diagonal elements? Thanks!