As we know, all operators representing observables are Hermitian. In my undersatanding, this statement means that all operators representing observables are Hermitian if the system can be described by a wavefunction or a vector in L2. For example, the momentum operator p is Herminitian, because for any states described by Ψ and Φ, we have <Ψ|p|Φ> = ∫Ψ*(-ihdΦ/dx) dx = Ψ*(-ih)Φ + ∫(-ihdΨ/dx)*Φdx = <pΨ|Φ> where Ψ*(-ih)Φ = 0 is employed at -∞ and +∞. But the eigen function of p-operator (2πh)-1/2exp(ipx/h) does not satisfy = 0 at -∞ and +∞, and the eigen function of the momentum operator, (2πh)-1/2exp(ipx/h), is not a wave function describing quantum state. Am I right?