Atomic Spectra

The Hydrogen Atom Spectrum: From Discrete Lines to Energy Levels

The story of atomic spectra begins with a simple observation that shattered classical physics. When hydrogen gas is heated or electrically excited, it does not emit a continuous rainbow of colours. Instead, it glows with a few distinct, sharp colours — a line spectrum. This is not what classical electromagnetism predicts. According to Maxwell's theory, an accelerating electron should radiate energy continuously, spiralling into the nucleus and producing a continuous spectrum. Nature disagrees. The hydrogen spectrum is the fingerprint of quantum behaviour, and it was the key that unlocked the structure of the atom.

The experimental setup is straightforward. A discharge tube containing hydrogen gas at low pressure is connected to a high voltage. The gas glows, and the light passes through a narrow slit, then through a prism or diffraction grating. What appears on the screen is not a band of colours but a series of sharp, isolated lines at specific wavelengths. These lines are not random — they fall into distinct families called spectral series.

The Balmer Series: A Pattern in the Visible

The most famous series lies partly in the visible region. In 1885, Johann Balmer, a Swiss schoolteacher, found a simple formula that reproduced the wavelengths of the four visible hydrogen lines (Hα, Hβ, Hγ, Hδ) with astonishing accuracy. He expressed it as:

where $n = 3, 4, 5, 6$ for the four visible lines, and $R_H$ is a constant now called the Rydberg constant for hydrogen. The modern value is $R_H = 1.097 \times 10^7 \text{ m}^{-1}$.

The four lines correspond to:

• $n = 3$: Hα (red, 656.3 nm)
• $n = 4$: Hβ (blue-green, 486.1 nm)
• $n = 5$: Hγ (violet, 434.1 nm)
• $n = 6$: Hδ (violet, 410.2 nm)

The Generalised Rydberg Formula: All Series

Balmer's success prompted a search for other series. They were found — in the ultraviolet and infrared — and all fit a generalised form. The Swedish spectroscopist Johannes Rydberg wrote the universal formula:

where $n_f$ is the final principal quantum number (the lower energy level) and $n_i$ is the initial principal quantum number (the higher energy level), with $n_i > n_f$. Each choice of $n_f$ defines a spectral series:

The Physical Meaning: Energy Level Transitions

The Rydberg formula is not just a numerical curiosity. It tells us something profound about the atom. Rewrite it in terms of the wavenumber $\tilde{\nu} = 1/\lambda$ (the number of waves per unit length):

Now multiply both sides by $hc$ (Planck's constant times the speed of light). Since $E = hc/\lambda = hc\tilde{\nu}$, we get:

This is the energy of the emitted photon. It equals the difference between two atomic energy levels:

where $E_i$ is the energy of the initial state (higher energy) and $E_f$ is the energy of the final state (lower energy). Comparing the two expressions, we see that the energy of a state with principal quantum number $n$ must be:

The negative sign means the electron is bound to the nucleus — you must add energy to remove it. The ground state ($n=1$) has the most negative energy, and as $n$ increases, the energy approaches zero (the ionisation limit).

This is the famous hydrogen energy level formula. The constant $13.6 \text{ eV}$ is the ionisation energy of hydrogen — the minimum energy needed to remove the electron from the ground state.

The Rydberg Constant in Terms of Fundamental Constants

Bohr's model derived the Rydberg constant from first principles:

where $m_e$ is the electron mass, $e$ is the elementary charge, $\epsilon_0$ is the permittivity of free space, $h$ is Planck's constant, and $c$ is the speed of light. Plugging in the numbers gives $R_H = 1.097 \times 10^7 \text{ m}^{-1}$, in excellent agreement with experiment. …