Translation operator. Central potentials
PROFESSOR: The fact is that angular momentum is an observable, and as such it deserves attention. There is an active way of thinking of observables, and we have not developed it that much in this course. But for example, with a momentum operator you've learned that the momentum operator can give you the differential operator. It's a derivative, and derivatives tell you how to move, how a function varies. So with the momentum operator, for example, you have the momentum operator p hat, which is h bar over i d dx. And you could ask the question of, OK, so the momentum operator moves or takes a derivative, does the momentum operator move a function? Does it generate a translation? And the answer is, yes. That's another way of thinking of the momentum operator as a generator of translations. But how does it do it? This is a Hermitian operator, and it takes a derivative. It doesn't translate the function. But there is a universal trick that if you exponentiate ...