The hydrogen atom was one of the first systems for which a simple quantum model successfully explained line spectra.
In the Bohr model, the electron is allowed to occupy only certain quantized energy levels labeled by the principal
quantum number \(n = 1,2,3,\dots\). For hydrogen-like ions, meaning a nucleus of charge \(+Ze\) with only one electron,
the allowed energies scale with \(Z^2\), so ions such as He⁺ and Li²⁺ have the same structure as hydrogen but with
larger energy spacings.
Bohr energy levels.
\[
\begin{aligned}
E_n &= -\frac{13.6\,Z^2}{n^2}\ \mathrm{eV}
\end{aligned}
\]
The negative sign means the electron is bound to the nucleus. The lowest level \(n=1\) is the ground state, while
higher values of \(n\) correspond to excited states. As \(n\) increases, the energy approaches zero from below,
which corresponds to ionization.
Transitions between levels
If the electron moves from an initial level \(n_i\) to a final level \(n_f\), the atomic energy changes by
Atomic energy change.
\[
\begin{aligned}
\Delta E_{\text{atom}} &= E_f - E_i
\end{aligned}
\]
When \(n_i > n_f\), the electron drops to a lower level and the atom emits a photon. That is an emission transition.
When \(n_f > n_i\), the electron must absorb a photon to move upward, so the transition is absorption. In either case,
the photon energy is the magnitude of the atomic energy change.
Photon energy.
\[
\begin{aligned}
E_{\gamma} &= \left|\Delta E_{\text{atom}}\right|
\end{aligned}
\]
The corresponding wavelength can be obtained from the Rydberg formula, which is especially useful for hydrogen-like
atoms and ions:
Rydberg relation.
\[
\begin{aligned}
\frac{1}{\lambda} &= R_{\infty} Z^2 \left|\frac{1}{n_f^2} - \frac{1}{n_i^2}\right|
\end{aligned}
\]
Once the wavelength is known, the photon frequency follows from
\[
\begin{aligned}
f &= \frac{c}{\lambda}
\end{aligned}
\]
This structure explains why hydrogen spectra are discrete. Only specific differences between quantized levels are
allowed, so only specific photon wavelengths appear.
Hydrogen series
Spectral lines are grouped into named series according to the lower energy level involved. In emission, the electron
falls into that lower level. In absorption, the same wavelength is associated with a jump away from that lower level.
| Series |
Lower level |
Typical region |
Example line |
| Lyman |
\(n = 1\) |
Ultraviolet |
Lyα: \(2 \to 1\) |
| Balmer |
\(n = 2\) |
Visible / near UV |
Hα: \(3 \to 2\) |
| Paschen |
\(n = 3\) |
Infrared |
Paα: \(4 \to 3\) |
| Brackett |
\(n = 4\) |
Infrared |
\(5 \to 4\) |
Sample calculation: the Balmer Hα line
A classic example is the hydrogen transition from \(n_i = 3\) to \(n_f = 2\). This is a Balmer-series line and is
usually labeled Hα.
Step 1. Compute the two energy levels.
\[
\begin{aligned}
E_3 &= -\frac{13.6}{3^2} \\
&= -\frac{13.6}{9} \\
&\approx -1.51\ \mathrm{eV}
\end{aligned}
\]
\[
\begin{aligned}
E_2 &= -\frac{13.6}{2^2} \\
&= -\frac{13.6}{4} \\
&= -3.40\ \mathrm{eV}
\end{aligned}
\]
Step 2. Find the atomic energy change.
\[
\begin{aligned}
\Delta E_{\text{atom}} &= E_2 - E_3 \\
&= -3.40 - (-1.51) \\
&\approx -1.89\ \mathrm{eV}
\end{aligned}
\]
The negative sign shows that the atom loses energy, so this is an emission transition. The emitted photon therefore
has energy \(1.89\ \mathrm{eV}\).
Step 3. Use the Rydberg formula for the wavelength.
\[
\begin{aligned}
\frac{1}{\lambda} &= R_{\infty}\left(\frac{1}{2^2} - \frac{1}{3^2}\right) \\
&= R_{\infty}\left(\frac{1}{4} - \frac{1}{9}\right) \\
&= R_{\infty}\left(\frac{5}{36}\right)
\end{aligned}
\]
\[
\begin{aligned}
\lambda &\approx 656.3\ \mathrm{nm}
\end{aligned}
\]
That value lies in the visible red part of the spectrum, which is why the Hα line is one of the best-known hydrogen
spectral lines in astronomy and spectroscopy.
Why the factor \(Z^2\) matters
For hydrogen-like ions, every energy level becomes more negative by the factor \(Z^2\). That means the spacing between
levels grows rapidly as \(Z\) increases. As a result, the same pair of quantum numbers produces a larger photon energy
and a shorter wavelength for He⁺ than for H, and still shorter wavelengths for Li²⁺ and higher ions.
This simple scaling is one of the reasons the Bohr and Rydberg formulas remain so useful in introductory atomic
physics: they provide a direct bridge between quantized energy levels and the observed spectrum.
Advanced note
The formulas above use the ideal Rydberg constant for an infinitely heavy nucleus. At university level, one may improve
the model by including reduced-mass corrections, fine structure, spin effects, and full quantum-mechanical wave
functions. Those refinements slightly adjust the wavelengths, but the Bohr picture remains an excellent first model
for understanding hydrogen-like spectra.
| Concept |
Main relation |
Meaning |
| Level energy |
\(E_n = -13.6 Z^2 / n^2\ \mathrm{eV}\) |
Quantized Bohr energy of a hydrogen-like atom |
| Atomic change |
\(\Delta E_{\text{atom}} = E_f - E_i\) |
Change in the atom’s own energy |
| Photon energy |
\(E_{\gamma} = |\Delta E_{\text{atom}}|\) |
Energy carried by emitted or absorbed radiation |
| Rydberg formula |
\(1/\lambda = R_{\infty} Z^2 |1/n_f^2 - 1/n_i^2|\) |
Transition wavelength relation |
| Frequency |
\(f = c/\lambda\) |
Photon oscillation rate |
| Series |
Named by the lower level |
Lyman, Balmer, Paschen, and higher series |