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The '''hydrogen atom''' is the simplest atomic system, consisting of a single electron bound to a proton by the Coulomb interaction. It is the only atom in quantum mechanics that admits a fully exact analytical solution of the Schrödinger equation, making it a fundamental model for understanding atomic structure, spectroscopy, and quantum theory.<ref>[https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/University_Physics_III_-_Optics_and_Modern_Physics_%28OpenStax%29/08%3A_Atomic_Structure/8.02%3A_The_Hydrogen_Atom The Hydrogen Atom (OpenStax/LibreTexts)]</ref>
The '''hydrogen atom''' is the simplest atomic system, consisting of a single electron bound to a proton by the Coulomb interaction. It is the only atom in quantum mechanics that admits a fully exact analytical solution of the Schrödinger equation, making it a fundamental model for understanding atomic structure, spectroscopy, and quantum theory.<ref>[https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/University_Physics_III_-_Optics_and_Modern_Physics_%28OpenStax%29/08%3A_Atomic_Structure/8.02%3A_The_Hydrogen_Atom The Hydrogen Atom (OpenStax/LibreTexts)]</ref>
[[File:Hydrogen_atom_energy_levels_and_spectral_lines.jpg|thumb|400px|Energy levels and spectral series of the hydrogen atom showing Lyman (ultraviolet), Balmer (visible), and Paschen, Brackett, Pfund (infrared) transitions.]]
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[[File:Hydrogen_atom_energy_levels_and_spectral_lines.jpg|thumb|280px|Quantum Hydrogen atom.]]
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== Schrödinger equation and Coulomb potential ==
== Schrödinger equation and Coulomb potential ==

Revision as of 14:00, 17 May 2026


← Back to Atomic and spectroscopy Book I

The hydrogen atom is the simplest atomic system, consisting of a single electron bound to a proton by the Coulomb interaction. It is the only atom in quantum mechanics that admits a fully exact analytical solution of the Schrödinger equation, making it a fundamental model for understanding atomic structure, spectroscopy, and quantum theory.[1]

Quantum Hydrogen atom.

Schrödinger equation and Coulomb potential

The electron in a hydrogen atom is described by the time-independent Schrödinger equation in a central Coulomb potential:

[−ℏ22m∇2−e24πε0r]ψ(𝐫)=Eψ(𝐫)

Because the potential depends only on the radial coordinate r, the equation is separable in spherical coordinates.[2]

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Quantum numbers

The solutions are characterized by three quantum numbers:

  • Principal quantum number: n=1,2,3,…
  • Orbital angular momentum: ℓ=0,1,…,n−1
  • Magnetic quantum number: m=−ℓ,…,ℓ

These arise from the separation of variables into radial and angular parts.[3]

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Energy levels

The allowed energy levels depend only on the principal quantum number:

En=−13.6eVn2

This degeneracy is a consequence of the underlying symmetry of the Coulomb potential.[4]

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Wavefunctions and orbitals

The hydrogen wavefunctions are products of radial functions and spherical harmonics:

ψnℓm(r,θ,ϕ)=Rnℓ(r)Yℓm(θ,ϕ)

These define the familiar atomic orbitals:

  • s-orbitals (ℓ=0) — spherical symmetry
  • p-orbitals (ℓ=1) — directional lobes
  • d-orbitals (ℓ=2) — more complex structures

[5]

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Angular momentum

The orbital angular momentum is quantized:

L2=ℏ2ℓ(ℓ+1),Lz=ℏm

The hydrogen atom also includes electron spin, introducing total angular momentum when relativistic effects are considered.[6]

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Spectral lines

Transitions between energy levels produce photons with energy:

E=hν=Ei−Ef

This gives rise to discrete spectral series:

  • Lyman series (n→1) — ultraviolet
  • Balmer series (n→2) — visible
  • Paschen, Brackett, Pfund — infrared

The wavelengths satisfy the Rydberg formula:

1λ=RH(1nf2−1ni2)

where RH is the Rydberg constant.[7][8]

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Fine and hyperfine structure

More accurate treatments include:

  • Fine structure — relativistic corrections and spin–orbit coupling
  • Hyperfine structure — interaction between electron and nuclear spin

These effects lift degeneracies and produce small spectral splittings.[9][10]

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Importance in quantum mechanics

The hydrogen atom plays a central role because:

  • It provides an exact solution of the Schrödinger equation
  • It explains atomic spectra quantitatively
  • It reveals hidden symmetries (e.g., Runge–Lenz vector)
  • It serves as the starting point for multi-electron approximations

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See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ The Hydrogen Atom (OpenStax/LibreTexts)
  2. ↑ Schrödinger Theory of Hydrogen
  3. ↑ Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers
  4. ↑ Atomic Data for Hydrogen (NIST)
  5. ↑ The Schrödinger Wave Equation for the Hydrogen Atom
  6. ↑ The Hydrogen Atom (OpenStax/LibreTexts)
  7. ↑ Bohr's Theory of the Hydrogen Atom (OpenStax)
  8. ↑ NIST Fundamental Physical Constants
  9. ↑ Fine Structure of Hydrogen
  10. ↑ Hyperfine Structure


Author: Harold Foppele


Source attribution: Quantum Hydrogen atom

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