Suppose I have a 2 state system s.t. $H\psi_1=E_1\psi_1$ and $H\psi_2=E_2\psi_2$.
And I have $\phi_+={1\over \sqrt2}(\psi_1+\psi_2)$ and $\phi_-={1\over \sqrt2}(\psi_1-\psi_2)$
Would I be right to think then that $\langle H\rangle$ for both states + and - are equal to ${1\over 2}(E_1+E_2)$? Isn't this weird?? Thanks.