- Yes, eigenfunctions of self-adjoint operators are always orthogonal. See [Spectral theorem](https://en.wikipedia.org/wiki/Spectral_theorem) for more details. Under some very general conditions on the potential, the eigenfunctions *are always orthogonal*. See also [Sturm–Liouville theory](https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory).

 - For the discussion of why there are transitions between eigenstates, see [Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?](http://physics.stackexchange.com/questions/295365/why-do-excited-states-decay-if-they-are-eigenstates-of-hamiltonian-and-should-no/).