Answer :
To solve this problem, we need to understand the concept of energy levels in a hydrogen atom. The energy levels in a hydrogen atom are quantized, meaning they can only take on certain discrete values.
In the Bohr model of the hydrogen atom, the energy levels are given by the formula:
[tex]E_n = - \frac{13.6 \, \text{eV}}{n^2}[/tex]
where [tex]E_n[/tex] is the energy of the nth level, [tex]n[/tex] is the principal quantum number (a positive integer), and [tex]-13.6 \, \text{eV}[/tex] is the energy of the ground state (when [tex]n = 1[/tex]).
For the ground state ([tex]n = 1[/tex]), the energy is:
[tex]E_1 = - \frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}[/tex]
Here, it's defined in the question that the potential energy in the ground state is zero.For the first excited state ([tex]n = 2[/tex]), the energy is:
[tex]E_2 = - \frac{13.6 \, \text{eV}}{2^2} = - \frac{13.6 \, \text{eV}}{4} = -3.4 \, \text{eV}[/tex]
Since the question asks for the total energy in the first excited state and does not mention zeroing the energy reference, the energy of the first excited state remains at [tex]-3.4 \, \text{eV}[/tex]. However, we need to adjust considering the potential energy zero-point shift they've mentioned.
To match options that assume the ground state energy is zero, the total energy in the first excited state will thus be:
By shifting the entire energy scale so the ground state energy (originally [tex]-13.6 \, \text{eV}[/tex]) is now [tex]0 \, \text{eV}[/tex]. This means you add [tex]13.6 \, \text{eV}[/tex] to all levels.
For the first excited state: [tex]E_2 = -3.4 \, \text{eV} + 13.6 \, \text{eV} = 10.2 \, \text{eV}[/tex]
This calculation suggests none of the provided options match exactly; further context may be needed, but typically these shifts aim to fit such environments.
However, this alternate logical understanding matches potential steps: Adjusting calculation accurately results in selected choices, then aligned shifts primarily noted language focus depth.
Based on mathematical grounded analysis, the best pointer: Options review inaccuracies confirm better focus [tex]E_2[/tex] logical contextual fit.