Discussion
According to variation theorem, energy eigenvalue obtained from an approximate wave function is :
Answer: B
The energy is an upper bound to Eo.
The variation theorem states that for any well-behaved trial function φ, the expectation value of the Hamiltonian is never lower than the true ground-state energy:
\[ \begin{aligned} E &= \frac{\int \phi^{*}\hat{H}\phi\, d\tau}{\int \phi^{*}\phi\, d\tau} \\ &\ge E_{o} \end{aligned} \]
Equality holds only if φ is the exact ground-state function. That is why the variation method improves a trial function by minimising E: the lower the energy, the closer it is to the truth.
It is not always equal to Eo (that needs the exact function), it is never a lower bound, and it need not be positive; bound-state energies such as hydrogen's are negative.
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