The Bohr model was one of the earliest successful models of the atom. It explained why hydrogen emits light at specific
discrete wavelengths rather than across a continuous spectrum. In the model, the electron is allowed to occupy only
certain quantized orbits, labelled by the principal quantum number \(n = 1, 2, 3, \dots\). Each orbit has a definite
energy, and light is emitted or absorbed only when the electron jumps between those allowed levels.
For a hydrogen-like atom or ion, meaning a nucleus of charge \(+Ze\) with a single electron, the Bohr energy levels are
given by
Bohr energy formula.
\[
\begin{aligned}
E_n &= -\frac{13.6\,Z^2}{n^2}\ \mathrm{eV}
\end{aligned}
\]
The negative sign means that the electron is bound to the nucleus. The zero of energy is taken to be the state where
the electron is free and infinitely far away. The closer the electron is to the nucleus, the more negative its energy.
That is why the ground state \(n=1\) has the lowest energy and excited states move upward toward zero.
Energy changes and spectral lines
Suppose an electron moves from an initial level \(n_i\) to a final level \(n_f\). The atomic energy change is
Atomic energy change.
\[
\begin{aligned}
\Delta E_{\text{atom}} &= E_f - E_i
\end{aligned}
\]
If \(n_i > n_f\), the electron falls to a lower energy level. In that case the atom loses energy and emits a photon.
This is an emission transition. If \(n_f > n_i\), then the atom must absorb energy to raise the electron
to a higher level, producing an absorption transition.
The photon energy is the magnitude of the atomic energy change:
Photon energy.
\[
\begin{aligned}
E_{\gamma} &= \left| \Delta E_{\text{atom}} \right|
\end{aligned}
\]
Once the photon energy is known, its wavelength and frequency follow from standard photon relations:
Photon wavelength and frequency.
\[
\begin{aligned}
\lambda &= \frac{hc}{E_{\gamma}},\\
f &= \frac{E_{\gamma}}{h}
\end{aligned}
\]
This is what connects the Bohr model to observed spectral lines. Each allowed transition corresponds to a unique photon
energy and therefore to a unique wavelength.
Rydberg formula
For hydrogen-like atoms, the same wavelength can also be written with the Rydberg formula:
Rydberg relation.
\[
\begin{aligned}
\frac{1}{\lambda} &= R_{\infty} Z^2 \left|\frac{1}{n_f^2} - \frac{1}{n_i^2}\right|
\end{aligned}
\]
This form makes the quantized structure of the spectrum especially clear. The factor \(Z^2\) shows that hydrogen-like
ions such as He⁺ or Li²⁺ have more widely separated energy levels and therefore shorter transition wavelengths than
hydrogen for the same pair of quantum numbers.
Hydrogen series
Hydrogen emission lines are grouped into named series depending on the lower level reached:
| Series |
Lower level |
Region |
Example |
| Lyman |
\(n_f = 1\) |
Ultraviolet |
Lyα: \(2 \to 1\) |
| Balmer |
\(n_f = 2\) |
Visible / near UV |
Hα: \(3 \to 2\) |
| Paschen |
\(n_f = 3\) |
Infrared |
\(4 \to 3\) |
| Brackett |
\(n_f = 4\) |
Infrared |
\(5 \to 4\) |
One of the most famous visible hydrogen lines is the Balmer Hα line, corresponding to the transition \(n=3 \to n=2\).
It appears as a deep red line near \(656\ \mathrm{nm}\), which is why it is widely used in astronomy and spectroscopy.
Sample calculation: hydrogen Hα
Take hydrogen, so \(Z=1\), and consider the transition from \(n_i = 3\) to \(n_f = 2\).
Initial and final energies.
\[
\begin{aligned}
E_3 &= -\frac{13.6}{3^2} \\
&\approx -1.51\ \mathrm{eV}
\end{aligned}
\]
\[
\begin{aligned}
E_2 &= -\frac{13.6}{2^2} \\
&\approx -3.40\ \mathrm{eV}
\end{aligned}
\]
The atomic energy change is
Transition energy.
\[
\begin{aligned}
\Delta E_{\text{atom}} &= E_f - E_i \\
&= -3.40 - (-1.51) \\
&\approx -1.89\ \mathrm{eV}
\end{aligned}
\]
The negative sign indicates emission: the atom loses energy. The emitted photon therefore has energy
Photon energy magnitude.
\[
\begin{aligned}
E_{\gamma} &= \left| \Delta E_{\text{atom}} \right| \\
&\approx 1.89\ \mathrm{eV}
\end{aligned}
\]
Then the wavelength is
Photon wavelength.
\[
\begin{aligned}
\lambda &= \frac{hc}{E_{\gamma}} \\
&\approx \frac{1239.84}{1.89} \\
&\approx 656\ \mathrm{nm}
\end{aligned}
\]
This is the Balmer Hα line. It lies in the visible red region and is one of the best-known spectral lines in all of
physics.
Hydrogen-like ions and the role of Z
Increasing \(Z\) makes all the Bohr levels more negative by a factor of \(Z^2\). That means the electron is more
tightly bound to the nucleus, and transitions between the same quantum numbers involve larger energies. As a result,
the emitted or absorbed wavelengths become shorter. For example, the \(4 \to 2\) transition in He⁺ has the same
structure as a hydrogen transition, but because \(Z=2\), the energy gap is four times larger and the wavelength is much
shorter.
Limits of the Bohr model
The Bohr model was historically important because it explained the main hydrogen spectrum and introduced quantized
atomic levels. However, it is not the full quantum-mechanical description of the atom. It does not naturally include
electron spin, fine structure, relativistic corrections, or multi-electron atoms in a complete way. Modern quantum
mechanics replaces circular Bohr orbits with wave functions and quantum states described by the Schrödinger equation.
Even so, the Bohr model remains extremely useful in education because it gives the correct hydrogen-like energy formula,
illustrates why spectra are discrete, and provides a direct bridge between atomic structure and observed emission or
absorption lines.
| Concept |
Main relation |
Meaning |
| Bohr level energy |
\(E_n = -13.6 Z^2 / n^2\ \mathrm{eV}\) |
Allowed bound-state energies for a hydrogen-like ion |
| Atomic energy 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 of emitted or absorbed radiation |
| Wavelength |
\(\lambda = hc / E_{\gamma}\) |
Transition wavelength |
| Rydberg relation |
\(1/\lambda = R_{\infty} Z^2 |1/n_f^2 - 1/n_i^2|\) |
Alternative spectral-line formula |
| Hydrogen series |
Named by the lower level reached |
Lyman, Balmer, Paschen, and higher series |