Contents
Introduction The Setup Quantum Numbers Energy Levels Wave Functions Fine Structure Hyperfine Splitting The Lamb Shift Misconceptions FAQ Sources Introduction
The hydrogen atom — one proton and one electron — is the cleanest test bed in quantum mechanics. Its Schrödinger equation can be solved exactly, yielding the famous Rydberg formula for spectral lines and a complete set of energy eigenstates labeled by quantum numbers (n, ℓ, m, ms ). The agreement between theory and experiment is extraordinary; refinements through fine structure, hyperfine splitting, Lamb shift, and QED corrections continue at the parts-per-trillion level today.
This article walks through the standard quantum-mechanical solution, the quantum numbers, the wave functions, and the small corrections that turn approximate agreement into precision agreement with experiment.
The Setup
An electron of mass m and charge −e moves in the Coulomb potential V(r) = −e²/(4πε₀r) of a proton. Treating the proton as infinitely heavy (a good first approximation), the Schrödinger equation becomes:
[−(ℏ²/2m)∇² − e²/(4πε₀r)] ψ = Eψ
Spherical symmetry allows separation of variables: ψ(r,θ,φ) = R(r)Yℓm (θ,φ), where Yℓm are spherical harmonics. The radial equation has analytic solutions in terms of associated Laguerre polynomials [1 ].
Quantum Numbers
Each state is characterized by four quantum numbers:
n (principal): 1, 2, 3, ... Determines energy.ℓ (orbital angular momentum): 0, 1, ..., n−1.m (magnetic): −ℓ, −ℓ+1, ..., ℓ.ms (spin): ±½.
Notation
ℓ = 0 is called s, ℓ = 1 is p, ℓ = 2 is d, ℓ = 3 is f. So 1s is n=1, ℓ=0; 2p is n=2, ℓ=1; etc. The degeneracy of energy level n (ignoring spin) is n² (from ℓ = 0 to n−1 with 2ℓ+1 m values each).
Energy Levels
The bound-state energies are:
En = −13.6 eV / n²
or more precisely, En = −me e⁴/(8ε₀²h²n²) = −RH /n², with RH = 13.605693 eV (the Rydberg energy).
The Rydberg Formula
Transitions between levels emit or absorb photons with frequencies:
ν = R[1/nf ² − 1/ni ²]
This reproduces the Lyman, Balmer, Paschen, Brackett, and Pfund spectral series of hydrogen. Schrödinger's first major success in 1926 was deriving this formula from his equation [2 ].
Ionization Energy
Removing the electron from the ground state (n = 1) requires 13.6 eV — the ionization energy of hydrogen, accurately measured and matching theory to many decimal places.
Wave Functions
The full wave functions are products of radial and angular parts:
ψnℓm (r, θ, φ) = Rnℓ (r) Yℓm (θ, φ)
Ground State (1s)
ψ100 = (1/√π) (1/a₀)^(3/2) e^(−r/a₀), with a₀ = 4πε₀ℏ²/(me e²) = 5.29 × 10⁻¹¹ m, the Bohr radius. Spherically symmetric; exponential decay from the nucleus.
2s and 2p
2s has one radial node. 2p has angular nodes (the three p orbitals point along x, y, z, with lobes of opposite sign).
Higher n
For larger n, ℓ, more nodes and more complex angular structure. The orbital shapes (s spherical, p dumbbell, d four-lobe, f eight-lobe) are standard in chemistry.
Fine Structure
The simple Schrödinger solution ignores relativistic effects. Including them gives the "fine structure" — small splittings of the energy levels [3 ].
Three Corrections
Relativistic kinetic energy correction: T = p²/2m + p⁴/(8m³c²) + ...
Spin-orbit interaction: the electron's spin couples to its orbital motion in the proton's field.
Darwin term: a relativistic correction concentrated at the origin.
All three corrections are of order α² × En , where α ≈ 1/137 is the fine-structure constant. Combined, they shift energy levels by amounts that lift some degeneracies and produce the observed multiplet structure.
Total Angular Momentum
Fine structure mixes states with the same total angular momentum j = ℓ ± ½. States are labeled by n, ℓ, j (and mj ). The 2p level splits into 2p½ and 2p3/2 .
Hyperfine Splitting
The proton has its own spin and magnetic moment. The interaction between electron magnetic moment and proton magnetic moment splits the levels further — the hyperfine structure [4 ].
The 21 cm Line
The ground state of hydrogen splits into two by the hyperfine interaction. The energy difference corresponds to a transition at 1420 MHz, or wavelength 21 cm. This is the famous "21 cm line" used in radio astronomy to map atomic hydrogen throughout the Milky Way and beyond.
Scale of Hyperfine Splitting
Hyperfine corrections are of order (me /mp ) α² × En , smaller than fine structure by the electron-to-proton mass ratio (~10⁻³). For the ground state of hydrogen, the splitting is ~6 × 10⁻⁶ eV.
The Lamb Shift
In 1947, Willis Lamb measured a small energy difference between the 2s½ and 2p½ states of hydrogen that the Dirac equation predicted to be degenerate [5 ]. The splitting (~1057 MHz) is the Lamb shift .
The shift comes from quantum electrodynamics — specifically, the interaction of the electron with vacuum fluctuations of the electromagnetic field. Bethe (1947) gave a non-relativistic calculation [6 ]; modern QED calculates the shift to parts per 10⁹ accuracy.
The Lamb shift was a major triumph of quantum electrodynamics. Lamb won the 1955 Nobel Prize for the measurement; Bethe, Tomonaga, Schwinger, and Feynman shared various prizes for the theoretical framework.
Common Misconceptions
"The Schrödinger equation gives all hydrogen physics"
It gives the gross structure. Fine structure, hyperfine, and Lamb shift require relativistic QED.
"Electrons orbit in circles around the nucleus"
Bohr's old model. The quantum picture has electrons in orbitals — probability distributions, not classical paths.
"Energy levels are determined by n alone"
For the non-relativistic Schrödinger equation, yes. With relativistic corrections, energy depends on both n and j (and slightly on ℓ via QED corrections).
"The 21 cm line is a 21-cm wavelength oscillation of the atom"
It's the wavelength of the photon emitted in the hyperfine transition. The atom itself is much smaller (~angstrom).
FAQ
Why is the Rydberg formula so accurate?
Because the Coulomb potential gives an exactly solvable problem in non-relativistic QM. Small corrections from relativity, vacuum effects, and reduced-mass effect are all calculable and tiny.
What's the size of a hydrogen atom?
The Bohr radius a₀ = 5.29 × 10⁻¹¹ m is the characteristic scale. The 1s orbital has ⟨r⟩ = 1.5 a₀. Higher states are larger; the n-th state has ⟨r⟩ ~ n²a₀.
How is the hydrogen spectrum measured today?
With frequency combs, precision spectroscopy reaches parts per 10¹⁵ in the 1s-2s transition. The hydrogen spectrum is the most precisely measured atomic spectrum.
What's the proton radius puzzle?
Muonic hydrogen measurements gave a proton radius slightly different from electronic measurements. The discrepancy is now being resolved with better theory and additional experiments [7 ].
Are there bound states with negative ℓ?
No — ℓ ≥ 0 is required by the geometry. The principal quantum number n = ℓ + 1 + nr , where nr ≥ 0 is the radial quantum number.
Sources
Griffiths, D. J. (2018). Introduction to Quantum Mechanics , 3rd ed.
Schrödinger, E. (1926). "Quantisierung als Eigenwertproblem." Annalen der Physik .
Bethe, H. A., Salpeter, E. E. (1957). Quantum Mechanics of One- and Two-Electron Atoms . Springer.
Bethe, H. A., Salpeter, E. E. (1957). Op. cit.
Lamb, W. E., Retherford, R. C. (1947). "Fine structure of the hydrogen atom by a microwave method." Physical Review , 72(3), 241.
Bethe, H. A. (1947). "The electromagnetic shift of energy levels." Physical Review , 72(4), 339.
Antognini, A., et al. (2013). "Proton structure from the measurement of 2S-2P transition frequencies of muonic hydrogen." Science , 339(6118), 417–420.