- #1

sophiatev

- 39

- 4

"P and H are compatible observables, and hence we can find a complete set of functions that are simultaneous eigenstates of both. That is to say, we can find solutions to the Schrodinger equation that are either symmetric (eigenvalue +1) or antisymmetric (eigenvalue -1) under exchange"

I understand why commuting (or as he calls them, compatible) observables share a common eigenbasis. What I don't see is why P, the exchange operator, and H, the Hamiltonian, need to commute for the second sentence to be true. If P is an observable, then supposedly it is a Hermitian operator whose eigenstates span the L2 Hilbert space. It's true that for some Hermitian operators (like the position one, for example), the eigenstates of the operator do not themselves lie in L2 Hilbert space. That being said, I can imagine other ways of showing that the eigenstates of P lie in L2 Hilbert space and thus constitute valid solutions to Schrodinger's equation. I feel like there's something else he's trying to show here, but I'm not exactly sure what. What is he trying to show by stating the P and H commute?