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Your textbook draws the Bohr atom with the electron on neat circles. Then the chemistry chapter talks about s, p, d and f subshells, and nobody explains the jump. A large part of that bridge was built by Arnold Sommerfeld, who asked a simple question: if planets move on ellipses, why should electrons only move on circles? This post covers what he did, works through the numbers behind his idea, and shows how it links to what you study for CBSE and JEE.
Who was Arnold Sommerfeld?
The MacTutor biography from the University of St Andrews gives his dates as 5 December 1868 in Königsberg to 26 April 1951 in Munich. He trained as a mathematician, worked as Felix Klein's assistant in Göttingen, and in 1906 became professor of theoretical physics in Munich, where he stayed for the rest of his career.
He was also a remarkable teacher. MacTutor names Peter Debye, Wolfgang Pauli, Werner Heisenberg and Hans Bethe among his doctoral students and says he supervised nearly 30 at Munich. All four later won Nobel Prizes: Heisenberg in 1932, Debye in chemistry in 1936, Pauli in 1945 and Bethe in 1967. Max Born, quoted by MacTutor, wrote that Sommerfeld "had the rare ability to have time to spare for his pupils", often on skiing trips where evening talk "invariably turned to mathematical physics".
What was wrong with circular orbits?
Bohr's model gave hydrogen's energy levels with one quantum number, n. It matched the Balmer lines well, as our post on Niels Bohr's atom model explains. But a spectroscope with high resolving power shows that each hydrogen line is really a close group of lines. One number could not explain that.
In his 1922 Nobel lecture, Bohr credited the step forward to Sommerfeld in 1915. Sommerfeld's main paper, Zur Quantentheorie der Spektrallinien, appeared in Annalen der Physik in 1916 and runs to 94 pages.
How did Sommerfeld's elliptical orbits work?
Sommerfeld let the electron move on ellipses and fixed them with two quantum numbers. Bohr's lecture describes the result: a principal quantum number n, "because it determines in the main the energy of the atom", and a subordinate quantum number k. Orbits with the same n have the same major axis; orbits with the same k have the same "parameter", set by the shortest chord through the focus.
He then added relativity. On an ellipse the electron speeds up near the nucleus, so its mass changes slightly, and the orbit slowly turns in its own plane. Bohr notes this turning is so slow that the electron makes about 40,000 revolutions before the orbit's closest point goes round once. Because of it, orbits with the same n but different k have slightly different energies, and each Balmer line splits. Bohr wrote that Sommerfeld's components "agree with the observations on the fine structure of hydrogen lines to within the limits of experimental error."
Worked example 1: the shapes for n = 3
Use two textbook facts about an inverse-square orbit. The semi-major axis grows as n², so a = n²a₀, where a₀ is the Bohr radius. Half that shortest chord, the semi-latus rectum ℓ, grows as the square of the angular momentum, which Sommerfeld set to kh/2π, so ℓ = k²a₀. For any ellipse, b² = aℓ.
b² = aℓ = n²a₀ × k²a₀, so b = nka₀ and b/a = k/n.
For n = 3, k can be 1, 2 or 3.
k = 3: b/a = 1, a circle of radius 9a₀.
k = 2: b/a = 2/3, a moderate ellipse.
k = 1: b/a = 1/3, a long thin ellipse that dives close to the nucleus.
That last orbit is where the electron moves fastest, which is why relativity matters most for it.
Worked example 2: why the splitting is so small
Sommerfeld's theory brought in a number now called the fine-structure constant, α. Its CODATA 2022 value from NIST is 7.2973525643 × 10⁻³, or about 1/137.036.
Speed in the first orbit. In the Bohr model this speed is exactly αc: 7.297 × 10⁻³ × 2.998 × 10⁸ = 2.19 × 10⁶ m/s. The NCERT Class 12 Atoms chapter works out the same 2.2 × 10⁶ m/s.
Size of the relativistic effect. The relativistic correction to the energy is of order v²/2c² = α²/2 = (7.297 × 10⁻³)² / 2 = 2.66 × 10⁻⁵.
So the split lines differ by a few parts in a hundred thousand. You need a very good spectroscope to see them, which is exactly what the observers had.
Which mistakes does this help you avoid?
Treating n as the whole story. In hydrogen the energy depends mainly on n. In many-electron atoms, 2s and 2p differ in energy, which is why the aufbau order exists.
Thinking orbits are real paths. The JEE (Main) 2026 syllabus asks for Bohr's model, its limitations and quantum numbers in chemistry. Sommerfeld's ellipses are history, not exam content, but they explain why there are two kinds of quantum number. Check the syllabus for your exam year.
When the link between energy levels, quantum numbers and subshells feels shaky, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can trace one idea back to the one before it.
What can a student take from Sommerfeld?
Borrow from another field. He took the ellipse from planetary motion and relativity from Einstein, and combined them.
Chase small discrepancies. A few parts in a hundred thousand looked like a detail. It became a new quantum number.
Teach to learn. His students went further than he did, and he gave them time.
Next step: redo worked example 1 for n = 4 and sketch all four orbits to scale on one page.
