Consider an arbitrary unnormalized wavefunction ψ, expanded in terms of eigenstates of Hamiltonian H, where
\(H\left|\phi_n\right\rangle= \varepsilon_n\left|\phi_n\right\rangle, n=0,1,2, \ldots\)
\(\varepsilon_0 \leq \varepsilon_1 \leq \varepsilon_2 \text { etc.. } \)
\(ψ =\sum_n a_n\left|\phi_n\right\rangle\)
The correct option, which definitely holds for any set of an, is
1
\(\frac{\sum_n\left|a_n\right|^2 \varepsilon_n}{\sum_n\left|a_n\right|^2}<\varepsilon_0\)
2
\(\frac{\sum_n\left|a_n\right|^2 \varepsilon_n}{\sum_n\left|a_n\right|^2} \geq \varepsilon_0\)
3
\(\frac{\sum_n a_n \varepsilon_n}{\Sigma\left|a_n\right|} \geq \varepsilon_0\)
4
\(\frac{\sum_n a_n \varepsilon_n}{\sum_n\left|a_n\right|^2}<\varepsilon_0\)