The normalized hydrogen atom 1 s wave function is given by $\psi_{1 \mathrm{s}}=\frac{1}{\sqrt{\pi}}\left(\frac{\zeta}{\mathrm{a}_{0}}\right)^{2} \mathrm{e}^{-\frac{\sigma_{0}}{\mathrm{a}_{0}}}$ where $\zeta=1$ and energy is -0.5 au . If we use a normalized wave function of the above form with $\zeta \neq 1$, the average value of energy of the ground state of hydrogen atom is:
(a)Greater than - 0.5 au
(b)Equal to - 0.5 au
(c)Less than -0.5 au
(d)Equal to $\zeta$ times -0.5 au .
Answer
Answer: A ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
By the variation theorem any trial function other than the exact one gives an energy above the exact value: ⟨E⟩ > −0.5 au for ζ ≠ 1.