1.1Use the Hydrogen Energy Levels
For a hydrogen atom, what is the energy of the $n=2$ level in the Bohr model?
Solution
Use $E_n=-13.6\ \text{eV}/n^2$ for hydrogen ($Z=1$):
$$
E_2=-\frac{13.6}{2^2}\ \text{eV}=-3.40\ \text{eV}.
$$
Atomic and Molecular Physics Practice
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1.2Find a Photon Energy
An atom falls from $E_i=-3.40\ \text{eV}$ to $E_f=-13.6\ \text{eV}$. Find the emitted photon energy.
Solution
The photon carries the decrease in atomic energy:
$$
E_\gamma=E_i-E_f=10.2\ \text{eV}.
$$
1.3Apply a Dipole Selection Rule
Is an electric-dipole transition with $\Delta\ell=0$ allowed?
Solution
No. The electric-dipole rule is $\Delta\ell=\pm1$, so $\Delta\ell=0$ is forbidden in this approximation.
1.4Scale a Hydrogen-like Level
What is the ground-state energy of $\mathrm{He}^+$ in the hydrogen-like model?
Solution
Use $E_n=-13.6Z^2/n^2\ \text{eV}$ with $Z=2$ and $n=1$:
$$
E_1=-13.6(2)^2\ \text{eV}=-54.4\ \text{eV}.
$$
1.5Choose an Atomic Model
Can the hydrogen-like energy formula be used exactly for neutral helium? Explain.
Solution
No. Neutral helium has two electrons, so electron–electron repulsion and shielding alter the energy levels. The formula is exact only for a one-electron ion in the ideal Coulomb model.