# Quantum number
A quantum number is a value of a quantized physical quantity that labels a quantum state of a system, such as an electron in an atom or a subatomic particle. In atomic physics, a one-electron state is defined by the quantum numbers \( n\,l\,m_l\,m_s \) or \( n\,l\,j\,m_j \), where \( n \) is the principal quantum number and \( l \) the orbital angular momentum quantum number[1](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10). Each number arises because a bound quantum system allows only discrete values of energy, angular momentum, and their projections, and the set of numbers uniquely identifies one state.
| Key fact | Detail |
|---|---|
| Principal quantum number \( n \) | Positive integers 1, 2, 3, ...; sets the electron's total energy, \( E_n = -E_0/n^2 \) with \( E_0 = 13.6 \) eV for hydrogen[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom) |
| Orbital quantum number \( l \) | Integers 0, 1, ..., \( n-1 \); magnitude of orbital angular momentum \( L = \sqrt{l(l+1)}\,\hbar \)[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom) |
| Projection quantum numbers | \( m_l = -l, ..., +l \) with \( L_z = m_l\hbar \); electron spin \( s = 1/2 \) always, with \( m_s = \pm 1/2 \)[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom) • [3](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules) |
| Total angular momentum | \( j = l \pm 1/2 \) for one electron, from coupling orbital and spin angular momenta[1](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10) |
| State count | For hydrogen with \( n = 3 \): 9 orbital states (1 + 3 + 5), all at \( -1.51 \) eV[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom) |
| Statistical weight | A level of total angular momentum \( J \) has statistical weight \( g = 2J + 1 \)[4](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html) |
| Particle-physics numbers | Additive quantum numbers such as baryon number (\( B = 1/3 \) for quarks, 0 for leptons) and lepton number label particles and constrain decays[5](https://cds.cern.ch/record/2910436/files/2409.09211.pdf) |
## The atomic quantum numbers
**Principal quantum number.** \( n \) takes the positive integers and is associated with the total energy of the electron. In hydrogen, \( E_n = -E_0/n^2 \) with \( E_0 = 13.6 \) eV, the same dependence as in the [Bohr model](https://www.edgepedia.org/bohr-model)[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom). Because the Coulomb interaction between nucleus and electron dominates, the largest energy separations in hydrogen-like ions are between levels of different \( n \)[1](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10).
**Orbital angular momentum.** For a given \( n \), \( l \) runs from 0 to \( n-1 \), and the magnitude of the orbital angular momentum is \( L = \sqrt{l(l+1)}\,\hbar \). This differs from Bohr's rule \( L = n\hbar \)[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom). States of different \( l \) are written with the spectroscopic letters s, p, d, f for \( l = 0, 1, 2, 3 \), from the early spectral-line classifications sharp, principal, diffuse, and fundamental; after f the letters continue alphabetically (g, h, ...)[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom).
**Projections and spin.** The magnetic quantum number \( m_l \) gives the z-component of orbital angular momentum, \( L_z = m_l\hbar \), with \( m_l = -l, ..., +l \); the z-axis is chosen by the experimenter, for example along an external magnetic field[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom). [Electron spin](https://www.edgepedia.org/electron-spin) is independent of \( n \), \( l \), and \( m_l \), always having \( s = 1/2 \), with spin projection \( m_s = +1/2 \) (spin up) or \( -1/2 \) (spin down)[3](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules). Only one magnitude of intrinsic spin is allowed for a given particle type: protons and neutrons, like electrons, have \( s = 1/2 \), photons have \( s = 1 \), and pions have \( s = 0 \)[3](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules).
**Coupled angular momentum.** Denoted \( j \), this quantum number gives the total angular momentum that results when an electron's orbital and spin angular momenta are combined, \( j = l + s \), hence \( j = l \pm 1/2 \)[1](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10).
## Many-electron atoms and term notation
In the central field approximation for a many-electron atom, wave functions are built from products of one-electron states, and electrons with the same \( n \) belong to the shell for that number[1](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10). In light atoms the dominant coupling scheme is [LS coupling](https://www.edgepedia.org/ls-coupling): the orbital angular momenta of the electrons couple to a total \( L = \sum_i l_i \), the spins to a total \( S = \sum_i s_i \). A particular combination of \( S \) and \( L \) is a spectroscopic term, written \( ^{2S+1}L \), where \( 2S+1 \) is the multiplicity; coupling \( S \) and \( L \) gives the total angular momentum \( J = S + L \), and the level is denoted \( ^{2S+1}L_J \)[6](https://physics.nist.gov/Pubs/AtSpec/node06.html). In helium-like ions, for example, the 1s\( nl \) terms are singlets (\( S = 0 \)) or triplets (\( S = 1 \))[6](https://physics.nist.gov/Pubs/AtSpec/node06.html).
A level with total angular momentum \( J \) has \( 2J+1 \) sublevels; this statistical weight \( g = 2J+1 \) enters the tabulation of transition probabilities and line intensities[4](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html).
## Selection rules
Quantum numbers also govern which transitions between states are allowed. For electric dipole (E1) transitions, the rigorous rules are \( \Delta J = 0, \pm 1 \) (except \( 0 \leftrightarrow 0 \)), \( \Delta M = 0, \pm 1 \) (except \( 0 \leftrightarrow 0 \) when \( \Delta J = 0 \)), and a change of parity, with one electron jumping and \( \Delta l = \pm 1 \) while \( \Delta n \) is arbitrary. In LS coupling, \( \Delta S = 0 \) additionally applies. [Magnetic dipole](https://www.edgepedia.org/magnetic-dipole) (M1) and electric quadrupole (E2) transitions, classed as forbidden, follow different rules: M1 requires no parity change and no change of electron configuration, and E2 allows \( \Delta J = 0, \pm 1, \pm 2 \) with further exclusions[7](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy).
## Quantum numbers in particle physics
Beyond atoms, additive quantum numbers label elementary particles and decide which reactions are possible. Baryon number assigns \( B = 1/3 \) to every quark and \( B = 0 \) to leptons, and lepton number is carried separately by each lepton generation; these conservation laws are used to identify which decays of an unstable particle are forbidden[5](https://cds.cern.ch/record/2910436/files/2409.09211.pdf). Intrinsic spin divides particles into half-integral and integral classes, a distinction with crucial physical consequences[3](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules).
## By the numbers
The state counting follows a simple pattern. For hydrogen with principal quantum number \( n \), the number of orbital states is \( \sum_{l=0}^{n-1} (2l+1) = n^2 \); for \( n = 3 \) this gives 9 states, all degenerate at \( -1.51 \) eV because the energy depends only on \( n \)[2](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom). Including spin doubles the count for each \( (n, l, m_l) \) set. The finest quantum-number effect in ordinary atomic physics is the hydrogen hyperfine splitting of the 1s ground level, 1420.4057517667(10) MHz, produced by the interaction of the proton and electron magnetic moments; it is the source of the famous 21 cm line of radio astronomy[1](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10).
## References
1. [Atomic Spectroscopy - Atomic States, Shells, and Configurations, NIST](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-10)
2. [8.1 The Hydrogen Atom, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/8-1-the-hydrogen-atom)
3. [30.8 Quantum Numbers and Rules, College Physics 2e, OpenStax](https://openstax.org/books/college-physics-2e/pages/30-8-quantum-numbers-and-rules)
4. [NIST Atomic Spectra Database - Spectral Lines Help File](https://physics.nist.gov/PhysRefData/ASD/Html/lineshelp.html)
5. [Chapter 8. The Standard Model, CERN Document Server](https://cds.cern.ch/record/2910436/files/2409.09211.pdf)
6. [Atomic Spectroscopy - Helium-like Ions; LS Coupling, NIST](https://physics.nist.gov/Pubs/AtSpec/node06.html)
7. [Atomic Spectroscopy - Spectral Lines, NIST](https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › Quantum numbers*
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