# Hydrogen atom

The hydrogen atom is the neutral atom consisting of a single proton and a single electron bound together by the Coulomb attraction between their opposite charges. It is the simplest atom in nature and the reference system for atomic physics: its energy levels, spectral lines, and corrections to them can be calculated and measured with higher accuracy than those of any other atom.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup>

| Key fact | Detail |
|---|---|
| Composition | One proton (charge +e) and one electron (charge −e); the proton is approximately 1800 times more massive than the electron, so the center of mass lies very close to the proton<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup> |
| Ground state | 1s, term symbol \( ^{2}S_{1/2} \); ionization energy 13.598434599702 eV (109678.7717 cm⁻¹)<sup>[2](https://physics.nist.gov/cgi-bin/ASD/ie.pl?at_num_out=on&biblio=on&e_out=0&el_name_out=on&level_out=on&shells_out=on&spectra=H-DS+i&unc_out=on&units=1)</sup><sup> • </sup><sup>[3](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable1.htm)</sup> |
| Energy formula | \( E_n = -13.6\ \text{eV}/n^{2} \); for n = 3 the energy is −1.51 eV<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup> |
| Rydberg constant | \( R_H = 1.09737 \times 10^{7}\ \text{m}^{-1} \) for hydrogen<sup>[4](https://openstax.org/books/university-physics-volume-3/pages/6-4-bohrs-model-of-the-hydrogen-atom)</sup> |
| Size | The radial probability of the ground-state electron peaks at the Bohr radius, about 0.529 Å (52.9 pm); the mean radius is 1.5 times larger<sup>[5](https://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html)</sup> |
| Strongest line | Lyman-alpha at 1215.66824 Å, with transition probability \( A_{ki} = 6.2648 \times 10^{8}\ \text{s}^{-1} \)<sup>[6](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable3.htm)</sup> |
| Hyperfine line | The 21 cm line at 1420.40575 MHz, with a transition probability of \( 2.6 \times 10^{-15}\ \text{s}^{-1} \)<sup>[7](https://arxiv.org/abs/2108.00115)</sup> |
| Isotopes | ¹H (mass 1.007825, abundance 99.985%, spin 1/2) and ²H or deuterium (mass 2.0140, abundance 0.015%, spin 1)<sup>[3](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable1.htm)</sup> |

## The Bohr model and the spectrum

Hydrogen's emission spectrum shows only four visible lines, at 656 nm (red, the H-α line), 486 nm (aqua), 434 nm (blue), and 410 nm (violet), with infinitely many further lines in the ultraviolet and infrared.<sup>[4](https://openstax.org/books/university-physics-volume-3/pages/6-4-bohrs-model-of-the-hydrogen-atom)</sup> Johann Balmer found an empirical formula for the visible series in 1885, and the general [Rydberg formula](https://www.edgepedia.org/rydberg-formula),

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_f^{2}} - \frac{1}{n_i^{2}} \right), \]

summarized all the observed lines, with \( n_f = 1 \) giving the Lyman series, \( n_f = 2 \) the [Balmer series](https://www.edgepedia.org/balmer-series), \( n_f = 3 \) the Paschen series, and higher values the Brackett, Pfund, and Humphreys series.<sup>[4](https://openstax.org/books/university-physics-volume-3/pages/6-4-bohrs-model-of-the-hydrogen-atom)</sup> Nobody could explain why the formula worked until [Niels Bohr](https://www.edgepedia.org/niels-bohr), a Danish physicist, proposed in 1913 the first model of atomic structure that correctly explained the hydrogen spectrum. In Bohr's model the electron moves in a perfectly circular orbit around the proton under the Coulomb force, but only at fixed radii labeled by a single quantum number \( n \), with energy \( E(n) = -R/n^{2} \) in the \( n \)-th stationary state.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup><sup> • </sup><sup>[8](https://en.wikisource.org/wiki/1926_Encyclop%C3%A6dia_Britannica/Atom)</sup> Transitions between these states emit or absorb photons at exactly the Rydberg-formula wavelengths. Bohr received the 1922 [Nobel Prize in Physics](https://www.edgepedia.org/nobel-prize-in-physics) for his investigation of atomic structure.<sup>[9](https://www.livescience.com/16362-nobel-prize-physics-list.html)</sup>

The model also accounted for the finite mass of the nucleus: replacing the electron mass with the reduced mass \( m_r = m_e/(1 + m_e/M_N) \) scales the [Rydberg constant](https://www.edgepedia.org/rydberg-constant) for any hydrogen-like ion, which is why deuterium's spectral lines are slightly shifted from hydrogen's.<sup>[10](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=842564)</sup>

## The quantum mechanical description

Schrödinger's equation for the electron in the proton's potential \( U(r) = -k e^{2}/r \), where \( k = 1/4\pi\epsilon_0 \), is solved most naturally in spherical coordinates because the potential is spherically symmetric and time independent.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup> Its solutions, the wave functions \( \psi_{nlm} \), are characterized by three quantum numbers:

- **Principal quantum number n** (1, 2, 3, …) fixes the energy, \( E_n = -13.6\ \text{eV}/n^{2} \), and the general size of the orbital.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup><sup> • </sup><sup>[11](https://openstax.org/books/chemistry-atoms-first-2e/pages/3-3-development-of-quantum-theory)</sup>
- **Orbital quantum number l** (0 to n − 1) fixes the magnitude of the orbital angular momentum, \( L = \sqrt{l(l+1)}\,\hbar \), and the orbital's shape. This differs from Bohr's rule \( L = n\hbar \). States are written s, p, d, f for l = 0, 1, 2, 3, from the old spectroscopic names sharp, principal, diffuse, and fundamental.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup>
- **Magnetic quantum number m** (−l to +l) fixes the z-component, \( L_z = m\hbar \), that is, the orbital's orientation.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup>

Because the energy depends only on n, all states with the same n have the same energy, a degeneracy that is lifted in atoms with more than one electron by electron–electron interactions.<sup>[11](https://openstax.org/books/chemistry-atoms-first-2e/pages/3-3-development-of-quantum-theory)</sup> For n = 3 there are 1 + 3 + 5 = 9 allowed states, all at −1.51 eV.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup>

The squared wave function gives the probability density for finding the electron, and the electron does not follow a well-defined path as in the [Bohr model](https://www.edgepedia.org/bohr-model); the uncertainty principle makes such a path unknowable.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup> The number of radial nodes, radii where the probability density is zero, is n − l − 1, so the 1s state has none, the 2s state one spherical node, and the 3s state two.<sup>[11](https://openstax.org/books/chemistry-atoms-first-2e/pages/3-3-development-of-quantum-theory)</sup> For the ground state the radial probability density is

\[ P(r)\,dr = \frac{4}{a_0^{3}}\, r^{2} e^{-2r/a_0}\, dr, \]

which peaks at the [Bohr radius](https://www.edgepedia.org/bohr-radius) \( a_0 \approx 0.529 \) Å; the average radius is \( 1.5\,a_0 \) because the distribution's extended tail pulls the mean outward.<sup>[1](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)</sup><sup> • </sup><sup>[5](https://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html)</sup>

## Fine structure and the Lamb shift

Bohr's levels with the same n are not exactly merged. Dirac's relativistic theory separates levels with different total angular momentum j, splitting the single Balmer-α line (the n = 3 to n = 2 transition) into five components of various intensities; the two strongest make the line appear as a doublet.<sup>[10](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=842564)</sup> The measured doublet separations, however, were consistently smaller than Dirac's prediction. Pasternack showed the discrepancy could be explained by an upward shift of the 2S₁/₂ level by about 0.03 cm⁻¹, roughly 1 GHz, and Willis Lamb's experiments proved this shift real; it is explained by quantum electrodynamics, the theory developed first by Bethe. The relativistic mass-correction term that also breaks the l-degeneracy contributes only about 2 kHz to the 2P₁/₂–2S₁/₂ transition, far below the roughly 1 GHz Lamb shift itself.<sup>[10](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=842564)</sup> Lamb received the 1955 Nobel Prize in Physics for his discoveries concerning the fine structure of the hydrogen spectrum.<sup>[9](https://www.livescience.com/16362-nobel-prize-physics-list.html)</sup>

Because hydrogen is the simplest atom to interpret theoretically, its fine and hyperfine structure has been a fruitful source of new ideas in physics, including precision determinations of constants such as the fine structure constant and the proton charge radius.<sup>[12](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication343.pdf)</sup><sup> • </sup><sup>[10](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=842564)</sup>

## The 21 cm line

The ground state is split by the interaction between the electron's and the proton's magnetic moments, a hyperfine splitting of \( \Delta E = 5.9 \times 10^{-6} \) eV corresponding to a photon of about 21 cm wavelength at 1420.40575 MHz. The transition from the parallel-spin (triplet) state to the anti-parallel (singlet) state is strongly forbidden, with a transition probability of \( 2.6 \times 10^{-15}\ \text{s}^{-1} \), but the abundance of hydrogen in the universe makes the line one of the most important tools in observational astronomy.<sup>[7](https://arxiv.org/abs/2108.00115)</sup> Ewen and Purcell detected it in 1951, and by 1952 the first maps of neutral hydrogen in the galaxy had revealed the [Milky Way](https://www.edgepedia.org/milky-way)'s spiral structure; Doppler shifts of the line are used to measure gas velocities and map matter distribution.<sup>[7](https://arxiv.org/abs/2108.00115)</sup> Redshifted 21 cm radiation is now pursued as a probe of the cosmic dark ages and the epoch of reionization.<sup>[7](https://arxiv.org/abs/2108.00115)</sup>

## Precision spectroscopy

The NIST tabulation gives the ionization energy as 13.598434599702 eV with an uncertainty of 1.2 × 10⁻¹¹ eV, a measure of how tightly hydrogen's levels are now known.<sup>[2](https://physics.nist.gov/cgi-bin/ASD/ie.pl?at_num_out=on&biblio=on&e_out=0&el_name_out=on&level_out=on&shells_out=on&spectra=H-DS+i&unc_out=on&units=1)</sup> Resolved fine-structure wavelengths are tabulated for the strongest lines, including Lyman-alpha at 1215.66824 Å and 1215.67364 Å for its two components, Balmer-alpha components at 6562.7110 to 6562.8518 Å, and Balmer-beta at 4861.2786 to 4861.3615 Å.<sup>[6](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable3.htm)</sup> Hydrogen spectroscopy combined with theory enters the CODATA adjustment of fundamental constants, and transition frequencies in hydrogen and deuterium depend on the bound-state proton and deuteron charge radii, which are also measured independently in muonic hydrogen.<sup>[13](https://physics.nist.gov/cuu/pdf/CODATA_JPCRD2016.pdf)</sup>

## References

1. [8.1 The Hydrogen Atom, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)
2. [NIST Atomic Ionization Energies Output, Hydrogen](https://physics.nist.gov/cgi-bin/ASD/ie.pl?at_num_out=on&biblio=on&e_out=0&el_name_out=on&level_out=on&shells_out=on&spectra=H-DS+i&unc_out=on&units=1)
3. [Atomic Data for Hydrogen (H), NIST Handbook of Basic Atomic Spectroscopic Data](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable1.htm)
4. [6.4 Bohr's Model of the Hydrogen Atom, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/6-4-bohrs-model-of-the-hydrogen-atom)
5. [Hydrogen Ground State Properties, HyperPhysics](https://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html)
6. [Persistent Lines of Neutral Hydrogen (H I), NIST](https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable3.htm)
7. [Moazzenzadeh & Firouzjaee (2021). Searching for the hydrogen 21 cm line in cosmos. arXiv:2108.00115](https://arxiv.org/abs/2108.00115)
8. [Atom, Encyclopædia Britannica 13th edition (1926), Wikisource](https://en.wikisource.org/wiki/1926_Encyclop%C3%A6dia_Britannica/Atom)
9. [Nobel Prize in Physics: 1901–Present, Live Science](https://www.livescience.com/16362-nobel-prize-physics-list.html)
10. [Kramida, A.E. (2010). A Critical Compilation of Experimental Data on Spectral Lines and Energy Levels of Hydrogen, Deuterium, and Tritium. Atomic Data and Nuclear Data Tables 96, 586–644](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=842564)
11. [3.3 Development of Quantum Theory, Chemistry: Atoms First 2e, OpenStax](https://openstax.org/books/chemistry-atoms-first-2e/pages/3-3-development-of-quantum-theory)
12. [Precision Measurement and Fundamental Constants, NBS Special Publication 343](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication343.pdf)
13. [CODATA Recommended Values of the Fundamental Physical Constants: 2014, Journal of Physical and Chemical Reference Data](https://physics.nist.gov/cuu/pdf/CODATA_JPCRD2016.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Atomic and molecular physics › Atomic structure and spectra › Electronic structure of atoms*

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