The energy levels of a hydrogen atom are given by the formula:
\[
E_n = -\frac{13.6 \, \text{eV}}{n^2}
\]
where \( n \) is the principal quantum number. The ground state corresponds to \( n = 1 \).
For the ground state (\( n = 1 \)):
\[
E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}
\]
To excite the electron from the ground state to a higher energy level, we need to calculate the energy differences between the ground state and higher states (\( n = 2, 3, 4, \ldots \)):
\[
E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -3.4 \, \text{eV}
\]
Energy required:
\[
\Delta E_{1 \to 2} = E_2 - E_1 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV}
\]
\[
E_3 = -\frac{13.6 \, \text{eV}}{3^2} = -1.5111 \, \text{eV}
\]
Energy required:
\[
\Delta E_{1 \to 3} = E_3 - E_1 = -1.5111 \, \text{eV} - (-13.6 \, \text{eV}) = 12.0889 \, \text{eV}
\]
\[
E_4 = -\frac{13.6 \, \text{eV}}{4^2} = -0.85 \, \text{eV}
\]
Energy required:
\[
\Delta E_{1 \to 4} = E_4 - E_1 = -0.85 \, \text{eV} - (-13.6 \, \text{eV}) = 12.75 \, \text{eV}
\]
\[
E_5 = -\frac{13.6 \, \text{eV}}{5^2} = -0.544 \, \text{eV}
\]
Energy required:
\[
\Delta E_{1 \to 5} = E_5 - E_1 = -0.544 \, \text{eV} - (-13.6 \, \text{eV}) = 13.056 \, \text{eV}
\]
\[
E_6 = -\frac{13.6 \, \text{eV}}{6^2} = -0.3778 \, \text{eV}
\]
Energy required:
\[
\Delta E_{1 \to 6} = E_6 - E_1 = -0.3778 \, \text{eV} - (-13.6 \, \text{eV}) = 13.2222 \, \text{eV}
\]
The photon energies that will excite the hydrogen atom from the ground state are:
\[
\boxed{10.2 \, \text{eV}, \, 12.0889 \, \text{eV}, \, 12.75 \, \text{eV}, \, 13.056 \, \text{eV}, \, 13.2222 \, \text{eV}}
\]
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