Are all eigenstates of observables orthogonal?

[AlonsoMcLaren]

Suppose psi1 and psi2 are eigenstates of observables O1 and O2

Suppose Value of O1 of psi1 = value of O1 of psi2

Therefore, <psi1|psi2>=1

Suppose value of O2 of psi1<>value of O2 of psi2

Therefore <psi1|psi2>=0

Contradiction!how to explain

 

[Avodyne]

Suppose psi1 and psi2 are eigenstates of observables O1 and O2

Suppose Value of O1 of psi1 = value of O1 of psi2

Therefore, <psi1|psi2>=1

This is not correct. Eigenvalues can be degenerate; that is, there can be more than one eigenstate for a particular eigenvalue. Example: in hydrogen, the energy only depends on the quantum number n, and not on the angular-momentum quantum numbers l and m. Eigenstates with the same value of n (and hence the same energy eigenvalue), but different values of l or m, are orthogonal.