Case Study 1: The Hydrogen Spectrum — From Fraunhofer Lines to Stellar Composition

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Case Study 1: The Hydrogen Spectrum — From Fraunhofer Lines to Stellar Composition

The Dark Lines That Changed Everything

In 1802, the English chemist William Hyde Wollaston noticed dark lines in the solar spectrum --- gaps in the rainbow of colors produced when sunlight passes through a prism. He published a brief note and moved on. Twelve years later, the Bavarian optician Joseph von Fraunhofer, working to calibrate the optical properties of glass, built far superior instruments and systematically catalogued over 570 of these dark lines, labeling the most prominent with letters A through K. Fraunhofer had no theory for why these lines existed. He simply recorded their positions with extraordinary precision.

The explanation came in 1859, when Gustav Kirchhoff and Robert Bunsen demonstrated that each chemical element absorbs and emits light at characteristic wavelengths. The dark lines in the solar spectrum --- now called Fraunhofer lines --- are caused by elements in the Sun's cooler outer atmosphere absorbing specific wavelengths from the continuous spectrum produced by the hotter interior.

Among the most prominent Fraunhofer lines are the C and F lines, which correspond to wavelengths of 656.3 nm (red) and 486.1 nm (blue-green). These are the first and second lines of the Balmer series of hydrogen. Their presence in the solar spectrum told scientists, decades before quantum mechanics, that hydrogen was abundant in the Sun.

Balmer's Empirical Formula

In 1885, the Swiss mathematician Johann Jakob Balmer, a schoolteacher with a hobby of finding numerical patterns, noticed that the wavelengths of the four visible hydrogen lines could be fit by a remarkably simple formula:

$$\lambda = B\frac{n^2}{n^2 - 4}, \quad n = 3, 4, 5, 6$$

where $B = 364.56\;\text{nm}$ is a constant. Balmer was 60 years old when he published this result, and it was his first and essentially only contribution to physics. But it was a contribution of extraordinary importance.

Line	$n$	$\lambda$ (Balmer)	$\lambda$ (measured)	Color
H-$\alpha$	3	656.3 nm	656.3 nm	Red
H-$\beta$	4	486.1 nm	486.1 nm	Blue-green
H-$\gamma$	5	434.0 nm	434.0 nm	Violet
H-$\delta$	6	410.2 nm	410.2 nm	Violet

The agreement was exact to within experimental uncertainty. Balmer's formula was purely empirical --- he had no idea why it worked.

In 1888, Johannes Rydberg generalized Balmer's formula to:

$$\frac{1}{\lambda} = R_\infty\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

This predicted not just the Balmer series ($n_f = 2$) but entire families of spectral lines for other values of $n_f$. The predictions were confirmed spectacularly:

Lyman series ($n_f = 1$): discovered in the ultraviolet by Theodore Lyman in 1906
Paschen series ($n_f = 3$): discovered in the infrared by Friedrich Paschen in 1908
Brackett series ($n_f = 4$): discovered by Frederick Brackett in 1922
Pfund series ($n_f = 5$): discovered by August Pfund in 1924

Each discovery confirmed the Rydberg formula with increasing precision, but the question remained: why does this formula work?

Bohr's Explanation (1913)

Niels Bohr provided the first theoretical explanation in 1913 by postulating quantized orbits with angular momentum $L = n\hbar$. His model yielded $E_n = -13.6\;\text{eV}/n^2$, from which the Rydberg formula follows immediately. This was a triumph --- but as we discussed in Section 5.5, the Bohr model had deep conceptual problems.

The full explanation came only with Schrodinger's equation in 1926. The energy eigenvalues $E_n = -13.6\;\text{eV}/n^2$, derived rigorously in Section 5.4 of this chapter, reproduce the Rydberg formula exactly. The Rydberg constant itself is expressed in terms of fundamental constants:

$$R_\infty = \frac{m_e e^4}{8\epsilon_0^2 h^3 c} = 1.097\,373\,156\,816(11) \times 10^7\;\text{m}^{-1}$$

This is one of the most precisely known constants in physics, measured to 12 significant figures.

Stellar Classification: Reading the Stars

The hydrogen spectrum is the primary tool for classifying stars. The Harvard spectral classification system (O, B, A, F, G, K, M) is fundamentally based on which hydrogen lines are visible and how strong they are.

Type A stars (like Sirius and Vega, with surface temperatures around 10,000 K) show the strongest Balmer absorption lines. Why? At this temperature, the Boltzmann distribution puts a significant fraction of hydrogen atoms in the $n = 2$ state, which can then absorb Balmer-series photons. At lower temperatures (type K and M stars), almost all hydrogen is in the ground state, and Balmer absorption is weak. At higher temperatures (type O and B stars), hydrogen is largely ionized, so again Balmer absorption weakens.

The detailed analysis is quantitative. Using the Saha equation (for ionization) and the Boltzmann distribution (for excitation), astrophysicists can determine:

Temperature: From the relative strengths of different spectral lines
Surface gravity: From the pressure broadening of spectral lines
Chemical composition: From the presence and strength of specific elemental lines
Radial velocity: From the Doppler shift of known lines
Rotation rate: From the Doppler broadening of lines

All of this begins with the hydrogen spectrum that we derived in this chapter.

The 21-cm Line: Mapping the Galaxy

One of the most important spectral lines in all of astrophysics is not in the Balmer series at all. The 21-cm line (1420 MHz) corresponds to the hyperfine transition in the hydrogen ground state --- a spin-flip of the electron relative to the proton. This transition is "forbidden" (it violates electric dipole selection rules), so the spontaneous emission lifetime is about 10 million years. Nevertheless, there is so much hydrogen in the galaxy that the 21-cm line is easily observable.

Radio astronomers use the 21-cm line to map the distribution and velocity of neutral hydrogen throughout the Milky Way and other galaxies. This technique revealed the spiral structure of our galaxy in the 1950s, long before optical observations could do so (because radio waves pass through the dust that blocks visible light).

The hyperfine transition is a quantum mechanical effect that depends on the interaction between the electron spin and the proton's magnetic moment. We will not derive it here (it requires the formalism of Chapters 13 and 18), but it is worth noting: this single spectral line, arising from a tiny energy splitting ($5.9 \times 10^{-6}\;\text{eV}$) in the hydrogen ground state, has revealed more about the large-scale structure of the universe than perhaps any other measurement.

The Cosmic Microwave Background and Recombination

Looking even further back in cosmic history, the hydrogen spectrum plays a starring role in one of the most important events in the early universe: recombination. About 380,000 years after the Big Bang, the universe cooled enough (to about 3,000 K) for protons and electrons to combine into neutral hydrogen atoms. The photons released during this process --- the last photons to scatter off free electrons --- form the cosmic microwave background (CMB), discovered by Penzias and Wilson in 1965.

The physics of recombination depends critically on the hydrogen energy levels we derived in this chapter. The process is not simply all atoms capturing electrons at once; it is a complex interplay of photoionization, recombination to excited states, and cascading transitions down through the hydrogen levels. The fact that recombination preferentially proceeds through the $n = 2$ level (because Lyman-$\alpha$ photons are quickly reabsorbed) creates a characteristic signature in the CMB that can be detected by precision cosmological experiments.

Discussion Questions

Balmer discovered his formula for hydrogen wavelengths in 1885, 28 years before Bohr's model and 41 years before Schrodinger's equation. What does the success of a purely empirical formula --- without any physical explanation --- tell us about the relationship between pattern recognition and understanding in science?
The 21-cm line has a spontaneous emission lifetime of about $10^7$ years, yet it is one of the most commonly observed lines in radio astronomy. How is this possible? What does this tell you about the conditions in interstellar space?
The hydrogen spectrum can reveal a star's temperature, composition, velocity, and rotation rate. Each of these measurements relies on different physical effects (Boltzmann distribution, transition probabilities, Doppler shift, broadening mechanisms). For each, identify which aspects of the hydrogen atom solution from this chapter are most directly relevant.
The Rydberg constant is known to 12 significant figures. What would it mean if a future, more precise measurement disagreed with the theoretical prediction? What kind of new physics could cause such a discrepancy? (See Case Study 2 for a real example of this kind of tension.)

Quantitative Exercises

CS1.1. Compute the wavelengths of H-$\alpha$ through H-$\delta$ using the Rydberg formula. Express your answers in nanometers and verify agreement with the table above.

CS1.2. At what temperature is the ratio $N_2/N_1$ (population of $n = 2$ relative to $n = 1$, including degeneracy) equal to 0.01? This gives a rough estimate of the temperature needed to observe strong Balmer absorption.

CS1.3. The 21-cm line frequency is $\nu = 1420.405\;\text{MHz}$. If a galaxy is receding from us at $v = 1000\;\text{km/s}$, what frequency would we observe for this line? What is the fractional shift $\Delta\nu/\nu$?

CS1.4. The Sun's surface temperature is approximately 5,778 K. Compute the fraction of hydrogen atoms in the $n = 2$ state using the Boltzmann factor. Why are the Balmer lines still visible in the solar spectrum despite this small fraction?