Hermitian operators degenerate

It can be proved that the eigenvalues of hermitian operators are real. Accordingly, observables are represented by hermitian operators, as that guarantees that the outcome of observations is real. Another property of those operators is that their non-degenerate eigenfunctions are orthogonal, that is... 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

This definition is, however, legal since all the eigenvalues of a Hermitian H are real and thus for any nonreal z the operator in the brackets is not degenerate and thus can be inverted. On the other hand the following holds ... 

Each of the six terms of the hyperpolarizability tensor so formed transforms into one of the six counterpart terms in (2ca, go, — ), the tensor for degenerate downconversion, on performing the combined operations of Hermitian conjugation and time reversal (the radiation tensor for downconversion is also obtained by performing the same procedure on p,- ). For example, the last term of Pi(/ ) (—2 , ), in the order that logically follows from Eqs. (82) and (83), behaves as follows ... 

Since the (25 -f 1)(2L -f 1) degenerate wave functions may be chosen to be real and since the expectation value of a Hermitian imaginary operator vanishes for real wave functions, we obtain for the spin-orbit operator... 

---