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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Atomic and spectroscopy}}
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.]]

== Schrödinger equation and Coulomb potential ==

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

<math>
\left[-\frac{\hbar^2}{2m}\nabla^2 - \frac{e^2}{4\pi \varepsilon_0 r} \right]\psi(\mathbf{r}) = E \psi(\mathbf{r})
</math>

Because the potential depends only on the radial coordinate <math>r</math>, the equation is separable in spherical coordinates.<ref>[https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Walet%29/12%3A_Quantum_Mechanics_of_the_Hydrogen_Atom/12.03%3A_Schrodinger_Theory_of_the_Hydrogen_Atom/12.3.01%3A_Schrodinger_Theory_of_Hydrogen Schrödinger Theory of Hydrogen]</ref>

---

== Quantum numbers ==

The solutions are characterized by three quantum numbers:

* Principal quantum number: <math>n = 1, 2, 3, \dots</math>
* Orbital angular momentum: <math>\ell = 0, 1, \dots, n-1</math>
* Magnetic quantum number: <math>m = -\ell, \dots, \ell</math>

These arise from the separation of variables into radial and angular parts.<ref>[https://chem.libretexts.org/Courses/Manchester_University/Manchester_University_Physical_Chemistry_I_%28CHEM_341%29/06%3A_One-Electron_Atoms_and_Ions/6.02%3A_Hydrogen_Atomic_Orbitals_Depend_upon_Three_Quantum_Numbers Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers]</ref>

---

== Energy levels ==

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

<math>
E_n = -\frac{13.6\,\text{eV}}{n^2}
</math>

This degeneracy is a consequence of the underlying symmetry of the Coulomb potential.<ref>[https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable1.htm Atomic Data for Hydrogen (NIST)]</ref>

---

== Wavefunctions and orbitals ==

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

<math>
\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\,Y_{\ell}^{m}(\theta,\phi)
</math>

These define the familiar atomic orbitals:

* <math>s</math>-orbitals (<math>\ell = 0</math>) — spherical symmetry
* <math>p</math>-orbitals (<math>\ell = 1</math>) — directional lobes
* <math>d</math>-orbitals (<math>\ell = 2</math>) — more complex structures

<ref>[https://chem.libretexts.org/Courses/University_of_California_Davis/Chem_107B%3A_Physical_Chemistry_for_Life_Scientists/Chapters/4%3A_Quantum_Theory/4.10%3A_The_Schr%C3%B6dinger_Wave_Equation_for_the_Hydrogen_Atom The Schrödinger Wave Equation for the Hydrogen Atom]</ref>

---

== Angular momentum ==

The orbital angular momentum is quantized:

<math>
L^2 = \hbar^2 \ell(\ell+1), \quad L_z = \hbar m
</math>

The hydrogen atom also includes electron spin, introducing total angular momentum when relativistic effects are considered.<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>

---

== Spectral lines ==

Transitions between energy levels produce photons with energy:

<math>
E = h\nu = E_i - E_f
</math>

This gives rise to discrete spectral series:

* '''Lyman series''' (<math>n \to 1</math>) — ultraviolet
* '''Balmer series''' (<math>n \to 2</math>) — visible
* '''Paschen, Brackett, Pfund''' — infrared

The wavelengths satisfy the Rydberg formula:

<math>
\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)
</math>

where <math>R_H</math> is the Rydberg constant.<ref>[https://openstax.org/books/college-physics-ap-courses-2e/pages/30-3-bohrs-theory-of-the-hydrogen-atom Bohr's Theory of the Hydrogen Atom (OpenStax)]</ref><ref>[https://physics.nist.gov/cuu/Constants/index.html NIST Fundamental Physical Constants]</ref>

---

== 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.<ref>[https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/11%3A_Time-Independent_Perturbation_Theory/11.07%3A_Fine_Structure_of_Hydrogen Fine Structure of Hydrogen]</ref><ref>[https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Walet%29/12%3A_Quantum_Mechanics_of_the_Hydrogen_Atom/12.05%3A_Smaller_Effects/12.5.01%3A_Hyperfine_Structure Hyperfine Structure]</ref>

---

== 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

---

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Hydrogen atom|1}}

Physics:Quantum Hydrogen atom - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template