Ballentine (Quantum Mechanics: A Modern Development 2nd edition, page 290) writes the attached in his introduction of the variational method. My question is about his very last line: why does $H$ being Hermitian imply that stationarity will then require the functional $\Lambda$ to be real at the stationary $\phi,\psi$, and why does this in turn require $\phi = \psi$? Is Ballentine perhaps saying that, in general, we use the variational method (10.110) in such a way that we are searching for minimization of expectation values, and so we enforce $\phi = \psi$ on (10.110)? I am not sure because Ballentine is seeming to suggest that this is something we arrive at as a conclusion, rather than as an assumption.