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{{Short description|Notation for conserved quantities in physics and chemistry}}
{{Redirect|Q-number|the Q-theory concept|Q-analog|the number format|Q (number format)}}
[[Image:Atomic orbitals n123 m-eigenstates.png|thumb|Single electron orbitals for hydrogen-like atoms with quantum numbers {{math|1=''n'' = 1, 2, 3}} (blocks), {{mvar|{{ell}}}} (rows) and {{mvar|m}} (columns). The spin {{mvar|s}} is not visible, because it has no spatial dependence.]]
{{Quantum mechanics|fundamentals}}
In [[Quantum mechanics|quantum physics]] and [[chemistry]], '''quantum numbers''' are quantities that characterize the possible states of the system. Quantum numbers are closely related to [[eigenvalues]] of [[Observable|observables]]. When the corresponding observable commutes with the Hamiltonian, the quantum number is said to be "[[Good quantum number|good]]", and acts as a [[constant of motion]] in the quantum dynamics.

To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the [[Principal quantum number|principal]], [[Azimuthal quantum number|azimuthal]], [[Magnetic quantum number|magnetic]], and [[Spin quantum number|spin]] quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the [[flavour (particle physics)|flavour]] of [[quarks]], which have no classical correspondence.

== History ==
{{also | History of quantum mechanics}}
===Electronic quantum numbers===
In the era of the [[old quantum theory]], starting from [[Max Planck]]'s proposal of quanta in his model of [[blackbody radiation]] (1900) and [[Albert Einstein]]'s adaptation of the concept to explain the [[photoelectric effect]] (1905), and until [[Erwin Schrödinger]] published his eigenfunction equation in 1926,<ref name="schrodinger">{{cite journal |author=Schrödinger, Erwin |year=1926 |title=Quantisation as an Eigenvalue Problem |journal=Annalen der Physik |volume=81 |issue=18 |pages=109–139 |bibcode=1926AnP...386..109S |doi=10.1002/andp.19263861802}}</ref> the concept behind quantum numbers developed based on atomic spectroscopy and theories from classical mechanics with extra ad hoc constraints.<ref name="Whittaker">{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|106}} Many results from atomic spectroscopy had been summarized in the [[Rydberg formula]] involving differences between two series of energies related by integer steps. The [[Bohr atom |model of the atom]], first proposed by [[Niels Bohr]] in 1913, relied on a single quantum number. Together with Bohr's constraint that radiation absorption is not classical, it was able to explain the [[Balmer series]] portion of Rydberg's atomic spectrum formula.<ref>{{Cite journal |last=Heilbron |first=John L. |date=June 2013 |title=The path to the quantum atom |url=https://www.nature.com/articles/498027a |journal=Nature |language=en |volume=498 |issue=7452 |pages=27–30 |doi=10.1038/498027a |issn=0028-0836}}</ref> 

As Bohr notes in his subsequent Nobel lecture, the next step was taken by [[Arnold Sommerfeld]] in 1915.<ref>[https://www.nobelprize.org/prizes/physics/1922/bohr/lecture/ Niels Bohr – Nobel Lecture]. NobelPrize.org. Nobel Prize Outreach AB 2024. Sun. 25 Feb 2024.</ref> Sommerfeld's atomic model added a second quantum number and the concept of quantized phase integrals to justify them.<ref>{{Cite book |last=Eckert |first=Michael |title=Arnold Sommerfeld: science, life and turbulent times 1868-1951 |last2=Eckert |first2=Michael |last3=Artin |first3=Tom |date=2013 |publisher=Springer |isbn=978-1-4614-7461-6 |location=New York}}</ref>{{rp|207}} Sommerfeld's model was still essentially two dimensional, modeling the electron as orbiting in a plane; in 1919 he extended his work to three dimensions using 'space quantization' in place of the quantized phase integrals.<ref name=Kragh2012Bohr>{{Cite book |last=Kragh |first=Helge |url=http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987 |title=Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925 |date=2012-05-17 |publisher=Oxford University Press |isbn=978-0-19-965498-7 |doi=10.1093/acprof:oso/9780199654987.003.0004}}</ref>{{rp|152}} [[Karl Schwarzschild]] and Sommerfeld's student, [[Paul Epstein]], independently showed that adding third quantum number gave a complete account for the [[Stark effect]] results. 

A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized. This seemed to be confirmed when the results of the [[Stern-Gerlach]] experiment reported quantized results for silver atoms in an inhomogeneous magnetic field. The confirmation would turn out to be premature: more quantum numbers would be needed.<ref name="FriedrichHerschbach">{{Cite journal |last=Friedrich |first=Bretislav |last2=Herschbach |first2=Dudley |date=2003-12-01 |title=Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics |url=https://pubs.aip.org/physicstoday/article/56/12/53/632269/Stern-and-Gerlach-How-a-Bad-Cigar-Helped-Reorient |journal=Physics Today |language=en |volume=56 |issue=12 |pages=53–59 |doi=10.1063/1.1650229 |issn=0031-9228}}</ref>

The fourth and fifth quantum numbers of the atomic era arose from attempts to understand the [[Zeeman effect]]. Like the Stern-Gerlach experiment, the Zeeman effect reflects the interaction of atoms with a magnetic field; in a weak field the experimental results were called "anomalous", they diverged from any theory at the time. [[Wolfgang Pauli]]'s solution to this issue was to introduce another quantum number taking only two possible values, <math>\pm \hbar/2</math>.<ref name=Giulini>{{Cite journal |last=Giulini |first=Domenico |date=2008-09-01 |title=Electron spin or "classically non-describable two-valuedness" |url=https://www.sciencedirect.com/science/article/pii/S1355219808000269 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |volume=39 |issue=3 |pages=557–578 |doi=10.1016/j.shpsb.2008.03.005 |issn=1355-2198|arxiv=0710.3128 }}</ref> This would ultimately become the quantized values of the projection of [[spin (physics)|spin]], an intrinsic angular momentum quantum of the electron. In 1927 Ronald Fraser demonstrated that the quantization in the Stern-Gerlach experiment was due to the magnetic moment associated with the electron spin rather than its orbital angular momentum.<ref name=FriedrichHerschbach/> Pauli's success in developing the arguments for a spin quantum number without relying on classical models set the stage for the development of quantum numbers for elementary particles in the remainder of the 20th century.<ref name=Giulini/>

Bohr, with his [[Aufbau principle|Aufbau]] or "building up" principle, and Pauli with his [[Pauli exclusion principle|exclusion principle]] connected the atom's electronic quantum numbers in to a framework for predicting the properties of atoms.<ref>{{Cite book |last=Kragh |first=Helge |url=http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987 |title=Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925 |date=2012-05-17 |publisher=Oxford University Press |isbn=978-0-19-965498-7 |language=en |doi=10.1093/acprof:oso/9780199654987.003.0007}}</ref> When Schrödinger published his [[Schrodinger equation|wave equation]] and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics.

===Nuclear quantum numbers===
With successful models of the atom, the attention of physics turned to models of the nucleus. Beginning with Heisenberg's initial model of proton-neutron binding in 1932, [[Eugene Wigner]] introduced [[isospin]] in 1937, the first 'internal' quantum number unrelated to a symmetry in real space-time.<ref name=Brown1987>{{Cite book |last=Brown |first=L.M. |url=https://archive.org/details/festivalfestschr0000unse/page/40/mode/2up?q=isospin |title=Festi-Val: Festschrift for Val Telegdi; essays in physics in honour of his 65th birthday; [a symposium ... was held at CERN, Geneva on 6 July 1987] |date=1988 |publisher=North-Holland Physics Publ |isbn=978-0-444-87099-5 |editor-last=Winter |editor-first=Klaus |location=Amsterdam |language=en |chapter=Remarks on the history of isospin |editor-last2=Telegdi |editor-first2=Valentine L.}}</ref>{{rp|45}} 

===Connection to symmetry===
As quantum mechanics developed, abstraction increased and models based on symmetry and invariance played increasing roles. Two years before his work on the quantum wave equation, Schrödinger applied the symmetry ideas originated by [[Emmy Noether]] and [[Hermann Weyl]] to the electromagnetic field.<ref name=Baggott40>{{Cite book |last=Baggott |first=J. E. |title=The quantum story: a history in 40 moments |date=2013 |publisher=Oxford Univ. Press |isbn=978-0-19-956684-6 |edition=Impression: 3 |location=Oxford}}</ref>{{rp|198}} As [[quantum electrodynamics]] developed in the 1930's and 1940's, [[group theory]] became an important tool. By 1953 [[Chen Ning Yang]] had become obsessed with the idea that group theory could be applied to connect the conserved quantum numbers of nuclear collisions to symmetries in a field theory of nucleons.<ref name=Baggott40/>{{rp|202}} With [[Robert Mills (physicist)|Robert Mills]], Yang developed a [[non-abelian gauge theory]] based on the conservation of the nuclear [[isospin]] quantum numbers.

==General properties==

Good quantum numbers correspond to [[eigenvalues]] of [[Operator (quantum mechanics)|operators]] that commute with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], quantities that can be known with precision at the same time as the system's energy. Specifically, observables that [[Commutator|commute]] with the Hamiltonian are [[simultaneously diagonalizable]] with it and so the eigenvalues <math>a</math> and the energy (eigenvalues of the Hamiltonian) are not limited by an [[Uncertainty principle|uncertainty relation]] arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a [[Basis (linear algebra)|basis]] state of the system, and can in principle be [[Measurement in quantum mechanics|measured]] together. Many observables have discrete [[Spectrum of an operator|spectra (sets of eigenvalues)]] in quantum mechanics, so the quantities can only be measured in discrete values.  In particular, this leads to quantum numbers that take values in [[Discrete mathematics|discrete sets of integers]] or [[half-integers]]; although they could approach [[infinity]] in some cases.

The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a [[quantum operator]] in the form of a [[Hamiltonian (quantum mechanics)|Hamiltonian]], {{mvar|H}}. There is one quantum number of the system corresponding to the system's energy; i.e., one of the [[eigenvalue]]s of the Hamiltonian. There is also one quantum number for each [[linear independence|linearly independent]] operator {{mvar|O}} that [[commutivity|commutes]] with the Hamiltonian. A [[complete set of commuting observables]] (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different [[Basis (linear algebra)|basis]] that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.

==Electron in a hydrogen-like atom==
Four quantum numbers can describe an electron energy level in a [[hydrogen-like atom]] completely:

*[[Principal quantum number]] ({{mvar|n}})
*[[Azimuthal quantum number]] ({{mvar|{{ell}}}})
*[[Magnetic quantum number]] ({{mvar|m<sub>{{ell}}</sub>}})
*[[Spin quantum number]] ({{mvar|m<sub>s</sub>}})
These quantum numbers are also used in the classical description of nuclear particle states (e.g. protons and neutrons).{{Citation needed|date=February 2024}} A quantum description of [[molecular orbitals]] requires other quantum numbers, because the symmetries of the molecular system are different.

=== Principal quantum number ===
The [[principal quantum number]] describes the [[electron shell]] of an electron. The value of {{mvar|n}} ranges from 1 to the shell containing the outermost electron of that atom, that is<ref>{{cite book|title=Concepts of Modern Physics |edition=4th |first=A. |last=Beiser |publisher=McGraw-Hill (International) |date=1987 |ISBN=0-07-100144-1}}{{page needed|date=November 2019}}</ref>
:{{math|1=''n'' = 1, 2, ...}}

For example, in [[caesium]] (Cs), the outermost [[valence (chemistry)|valence]] electron is in the shell with energy level 6, so an electron in caesium can have an {{mvar|n}} value from 1 to 6. The average distance between the electron and the nucleus increases with {{math|''n''}}.

=== Azimuthal quantum number ===
The [[azimuthal quantum number]], also known as the ''orbital angular momentum quantum number'', describes the [[electron shell#Subshells|subshell]], and gives the magnitude of the orbital [[angular momentum]] through the relation

:{{math|1=''L''<sup>2</sup> {{=}} ''ħ''<sup>2</sup> ''{{ell}}'' (''{{ell}}'' + 1).}}

In chemistry and spectroscopy, {{math|1=''{{ell}}'' = 0}} is called s orbital, {{math|1=''{{ell}}'' = 1}}, p orbital, {{math|1=''{{ell}}'' = 2}}, d orbital, and {{math|1=''{{ell}}'' = 3}}, f orbital.

The value of {{mvar|{{ell}}}} ranges from 0 to {{math|''n'' − 1}}, so the first p orbital ({{math|1=''{{ell}}'' = 1}}) appears in the second electron shell ({{math|1=''n'' = 2}}), the first d orbital ({{math|1=''{{ell}}'' = 2}}) appears in the third shell ({{math|1=''n'' = 3}}), and so on:<ref>{{cite book|title=Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry|volume=1|first=P. W.|last=Atkins|publisher=Oxford University Press|date=1977|ISBN=0-19-855129-0}}{{page needed|date=February 2019}}</ref>

:{{math|1=''{{ell}}'' = 0, 1, 2,..., ''n'' − 1}}

A quantum number beginning in {{mvar|n}} = 3,''{{ell}}'' = 0, describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an [[atomic orbital]] and strongly influences [[chemical bond]]s and [[bond angle]]s. The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, {{math|1=''{{ell}}'' = 1}} and thus the amount of angular nodes in a p orbital is 1.

=== Magnetic quantum number ===
The [[magnetic quantum number]] describes the specific [[atomic orbital|orbital]] within the subshell, and yields the ''projection'' of the orbital [[angular momentum]] ''along a specified axis'':

:{{math|1=''L<sub>z</sub>'' = ''m<sub>{{ell}}</sub> ħ''}}

The values of {{mvar|m<sub>{{ell}}</sub>}} range from {{math|−''{{ell}}''}} to {{mvar|{{ell}}}}, with integer intervals.{{sfn|Eisberg|Resnick|1985}}{{page needed|date=February 2019}}

The s subshell ({{math|1=''{{ell}}'' = 0}}) contains only one orbital, and therefore the {{math|m<sub>{{ell}}</sub>}} of an electron in an s orbital will always be 0. The p subshell ({{math|1=''{{ell}}'' = 1}}) contains three orbitals, so the {{mvar|m<sub>{{ell}}</sub>}} of an electron in a p orbital will be −1, 0, or 1. The d subshell ({{math|1=''{{ell}}'' = 2}}) contains five orbitals, with {{mvar|m<sub>{{ell}}</sub>}} values of −2, −1, 0, 1, and 2.

=== Spin magnetic quantum number ===
The [[spin magnetic quantum number]] describes the intrinsic [[Spin (physics)|spin angular momentum]] of the electron within each orbital and gives the projection of the spin angular momentum {{mvar|S}} along the specified axis:

:{{math|1=''S<sub>z</sub>'' = ''m<sub>s</sub> ħ''}}.

In general, the values of {{mvar|m<sub>s</sub>}} range from {{math|−''s''}} to {{mvar|s}}, where {{mvar|s}} is the spin quantum number, associated with the magnitude of particle's intrinsic spin angular momentum:<ref>{{cite book|title=Quantum Mechanics |edition=2nd |first1=Y. |last1=Peleg |first2=R. |last2=Pnini |first3=E. |last3=Zaarur |first4=E. |last4=Hecht |series=Schuam's Outlines |publisher=McGraw Hill (USA) |date=2010 |ISBN=978-0-07-162358-2}}{{page needed|date=February 2019}}</ref>

:{{math|1=''m<sub>s</sub>'' =  −''s'', −''s'' + 1, −''s'' + 2, ..., ''s'' − 2, ''s'' − 1, ''s''}}.

An electron state has spin number {{math|1=''s'' = {{sfrac|1|2}}}}, consequently {{mvar|m<sub>s</sub>}} will be +{{sfrac|1|2}} ("spin up") or -{{sfrac|1|2}} "spin down" states. Since electron are [[fermions]] they obey the [[Pauli exclusion principle]]: each electron state must have different quantum numbers.  Therefore every orbital will be occupied with at most two electrons, one for each spin state.

=== The Aufbau principle and Hund's Rules ===
{{Main | Aufbau principle | Hund's rules}}
A multi-electron atom can be modeled qualitatively as a hydrogen like atom with higher nuclear charge and correspondingly more electrons. The occupation of the electron states in such an atom can be predicted by the Aufbau principle and Hund's empirical rules for the quantum numbers.  The Aufbau principle fills orbitals based on their principal and azimuthal quantum numbers (lowest <math>n+l</math> first, with lowest <math>n</math> breaking ties; Hund's rule favors unpaired electrons in the outermost orbital). These rules are empirical but they can be related to electron physics.<ref name=Jolly>{{cite book |last1=Jolly |first1=William L. |title=Modern Inorganic Chemistry |edition=1st |publisher=McGraw-Hill |date=1984 |pages=[https://archive.org/details/trent_0116300649799/page/10 10–12] |isbn=0-07-032760-2 |url=https://archive.org/details/trent_0116300649799/page/10 }}</ref>{{rp|10}}<ref>{{Cite book |last=Levine |first=Ira N. |title=Physical chemistry |date=1983 |publisher=McGraw-Hill |isbn=978-0-07-037421-8 |edition=2|location=New York}}</ref>{{rp|260}}

==Spin-orbit coupled systems==
When one takes the [[spin–orbit interaction]] into consideration, the {{mvar|L}} and {{mvar|S}} operators no longer [[Commutativity|commute]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], and the eigenstates of the system no longer have well-defined orbital angular momentum and spin. Thus another set of quantum numbers should be used. This set includes<ref>{{cite book|title=Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry |volume=1 |first=P. W. |last=Atkins |publisher=Oxford University Press |date=1977 |ISBN=0-19-855129-0}}{{page needed|date=February 2019}}</ref><ref name="Atkins 1977">{{cite book|title=Molecular Quantum Mechanics Part III: An Introduction to Quantum Chemistry |volume=2 |first=P. W. |last=Atkins |publisher=Oxford University Press |date=1977}}{{ISBN missing}}{{page needed|date=February 2019}}</ref>

{{ordered list
|1= The [[total angular momentum quantum number]]:
:{{math|''j'' {{=}} {{abs|''{{ell}}'' ± ''s''}}}}
which gives the total [[angular momentum]] through the relation
:{{math|''J''<sup>2</sup> {{=}} ''ħ''<sup>2</sup> ''j'' (''j'' + 1)}}
|2= The [[Azimuthal quantum number#Total angular momentum of an electron in the atom|projection of the total angular momentum]] along a specified axis:
:{{math|1=''m<sub>j</sub>'' = −''j'', −''j'' + 1, −''j'' + 2, ..., ''j'' − 2, ''j'' − 1, ''j''}}
analogous to the above and satisfies
:{{math|1=''m<sub>j</sub>'' = ''m<sub>{{ell}}</sub>'' + ''m<sub>s</sub>''}} and {{math|{{abs|''m<sub>{{ell}}</sub>'' + ''m<sub>s</sub>''}} ≤ ''j''}}
|3=[[Parity (physics)|Parity]]
This is the [[eigenvalue]] under reflection: positive (+1) for states which came from even {{mvar|{{ell}}}} and negative (−1) for states which came from odd {{mvar|{{ell}}}}. The former is also known as '''even parity''' and the latter as '''odd parity''', and is given by
:{{math|1=''P'' = (−1)<sup>''{{ell}}''</sup>}}
}}

For example, consider the following 8 states, defined by their quantum numbers:
:{| style="border: none; border-spacing: 1em 0" class="wikitable"
! 
! {{math|''n''}}
! {{math|''{{ell}}''}}
! {{math|''m<sub>{{ell}}</sub>''}}
! {{math|''m<sub>s</sub>''}}
| rowspan=9  style="border:0px;" |
! {{math|''{{ell}}'' + ''s''}}
! {{math|''{{ell}}'' − ''s''}}
! {{math|''m<sub>{{ell}}</sub>'' + ''m<sub>s</sub>''}}
<!--                             N     L      ML      MS     L+S     L-S        ML+MS                          -->
|-align=right
!  (1) 
|  2 ||  1 ||   1  ||  +{{sfrac|1|2}} || {{sfrac|3|2}} || <s>{{sfrac|1|2}}</s>  ||  {{sfrac|3|2}}
|-align=right
!  (2) 
|  2 ||  1 ||   1  ||  −{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  {{sfrac|1|2}}
|-align=right
!  (3) 
|  2 ||  1 ||   0  ||  +{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  {{sfrac|1|2}}
|-align=right
!  (4) 
|  2 ||  1 ||   0  ||  −{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  −{{sfrac|1|2}}
|-align=right
!  (5) 
|  2 ||  1 ||  −1  ||  +{{sfrac|1|2}} || {{sfrac|3|2}} || {{sfrac|1|2}}  ||  −{{sfrac|1|2}}
|-align=right
!  (6) 
|  2 ||  1 ||  −1  ||  −{{sfrac|1|2}} || {{sfrac|3|2}} || <s>{{sfrac|1|2}}</s>  ||  −{{sfrac|3|2}}
|-align=right
!  (7) 
|  2 ||  0 ||   0  ||  +{{sfrac|1|2}} || {{sfrac|1|2}} || −{{sfrac|1|2}} ||  {{sfrac|1|2}}
|-align=right
!  (8) 
|  2 ||  0 ||   0  ||  −{{sfrac|1|2}} || {{sfrac|1|2}} || −{{sfrac|1|2}} ||  −{{sfrac|1|2}}
|}

The [[quantum state]]s in the system can be described as linear combination of these 8 states. However, in the presence of [[spin–orbit interaction]], if one wants to describe the same system by 8 states that are [[eigenvector]]s of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:
:{| class="wikitable"
! {{math|''j''}} ||  {{math|1=''m<sub>j</sub>''}} || parity || 
|-
| {{sfrac|3|2}}|| align=right |   {{sfrac|3|2}}||  align=right | odd  || coming from state (1) above
|-
| {{sfrac|3|2}}|| align=right |   {{sfrac|1|2}}||  align=right | odd  || coming from states (2) and (3) above
|-
| {{sfrac|3|2}}|| align=right |  −{{sfrac|1|2}}||  align=right | odd  || coming from states (4) and (5) above
|-
| {{sfrac|3|2}}|| align=right |  −{{sfrac|3|2}}||  align=right | odd  || coming from state (6) above
|-
| {{sfrac|1|2}}|| align=right |   {{sfrac|1|2}}||  align=right | odd  || coming from states (2) and (3) above
|-
| {{sfrac|1|2}}|| align=right |  −{{sfrac|1|2}}||  align=right | odd  || coming from states (4) and (5) above
|-
| {{sfrac|1|2}}|| align=right |   {{sfrac|1|2}}||  align=right | even || coming from state (7) above
|-
| {{sfrac|1|2}}|| align=right |  −{{sfrac|1|2}}||  align=right | even || coming from state (8) above
|}

==Atomic nuclei==
In [[Atomic nucleus|nuclei]], the entire assembly of [[proton]]s and [[neutron]]s ([[nucleon]]s) has a resultant [[angular momentum]] due to the angular momenta of each nucleon, usually denoted {{mvar|I}}. If the total angular momentum of a neutron is {{math|1=''j''<sub>n</sub> = ''{{ell}}'' + ''s''}} and for a proton is {{math|1=''j''<sub>p</sub> = ''{{ell}}'' + ''s''}} (where {{mvar|s}} for protons and neutrons happens to be {{sfrac|1|2}} again (''see note'')), then the '''nuclear angular momentum quantum numbers''' {{mvar|I}} are given by:

:{{math|1=''I'' = {{abs|''j''<sub>n</sub> − ''j''<sub>p</sub>}}, {{abs|''j''<sub>n</sub> − ''j''<sub>p</sub>}} + 1, {{abs|''j''<sub>n</sub> − ''j''<sub>p</sub>}} + 2, ..., (''j''<sub>n</sub> + ''j''<sub>p</sub>) − 2, (''j''<sub>n</sub> + ''j''<sub>p</sub>) − 1, (''j''<sub>n</sub> + ''j''<sub>p</sub>)}}

''Note: ''The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and  proton are half-integer multiples.  It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, {{mvar|I}}, of any odd-A nucleus and integer values for any even-A nucleus.

Parity with the number {{mvar|I}} is used to label nuclear angular momentum states, examples for some isotopes of [[hydrogen]] (H), [[carbon]] (C), and [[sodium]] (Na) are;<ref name="Krane 1988">{{cite book|title=Introductory Nuclear Physics |first=K. S. |last=Krane |date=1988 |publisher=John Wiley & Sons |ISBN=978-0-471-80553-3}}{{page needed|date=February 2019}}</ref>

:{|
| style="text-align:right;" | {{nuclide|link=yes|Hydrogen|1}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|9}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>−</sup> || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|20}} || {{mvar|I}} = 2<sup>+</sup>
|-
| style="text-align:right;" | {{nuclide|link=yes|Hydrogen|2}} || {{mvar|I}} = 1<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|10}} || {{mvar|I}} = 0<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|21}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>+</sup>
|-
| style="text-align:right;" | {{nuclide|link=yes|Hydrogen|3}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|11}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>−</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|22}} || {{mvar|I}} = 3<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|12}} || {{mvar|I}} = 0<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|23}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|13}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>−</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|24}} || {{mvar|I}} = 4<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|14}} || {{mvar|I}} = 0<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|25}} || {{mvar|I}} = ({{sfrac|5|2}})<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|15}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|26}} || {{mvar|I}} = 3<sup>+</sup>
|-
|}

The reason for the unusual fluctuations in {{mvar|I}}, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of [[NMR]] spectroscopy in [[organic chemistry]],<ref name="Atkins 1977" /> and [[MRI]] in [[nuclear medicine]],<ref name="Krane 1988" /> due to the [[nuclear magnetic moment]] interacting with an external [[magnetic field]].

==Elementary particles==
{{For|a more complete description of the quantum states of elementary particles|Standard model|Flavour (particle physics)}}

[[Elementary particle]]s contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are [[quantum state]]s of the [[standard model]] of [[particle physics]], and hence the quantum numbers of these particles bear the same relation to the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of this model as the quantum numbers of the [[Bohr atom]] does to its [[Hamiltonian (quantum mechanics)|Hamiltonian]]. In other words, each quantum number denotes a symmetry of the problem. It is more useful in [[quantum field theory]] to distinguish between spacetime and internal symmetries.

Typical quantum numbers related to [[spacetime symmetries]] are [[Spin (physics)|spin]] (related to rotational symmetry), the [[parity (physics)|parity]], [[C-parity]] and [[T-parity]] (related to the [[Poincaré symmetry]] of [[spacetime]]). Typical '''internal symmetries'''{{clarify|date=August 2016}} are [[lepton number]] and [[baryon number]] or the [[electric charge]]. (For a full list of quantum numbers of this kind see the article on [[flavour (particle physics)|flavour]].)

==Multiplicative quantum numbers==
Most conserved quantum numbers are additive, so in an elementary particle reaction, the ''sum'' of the quantum numbers should be the same before and after the reaction. However, some, usually called a ''[[Parity (physics)|parity]]'', are multiplicative; i.e., their ''product'' is conserved. All [[multiplicative quantum number]]s belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing ([[Involution (mathematics)|involution]]).

==See also==
{{Portal|Physics}}
* [[Electron configuration]]
{{Clear}}

==References==
{{Reflist}}

==Further reading==
*{{cite book | author=Dirac, Paul A. M. | author-link=Paul Dirac | title=Principles of Quantum Mechanics | publisher=Oxford University Press |year=1982 |isbn=0-19-852011-5}}
*{{cite book | author=Griffiths, David J. | author-link=David J. Griffiths | title=Introduction to Quantum Mechanics |edition=2nd| publisher=Prentice Hall | year=2004 | isbn=0-13-805326-X | url-access=registration | url=https://archive.org/details/introductiontoel00grif_0 }}
*{{cite book |author1=Halzen, Francis |authorlink1 = Francis Halzen |author2=Martin, Alan D. | authorlink2 = Alan Martin (physicist)|name-list-style=amp | title=Quarks and Leptons: An Introductory Course in Modern Particle Physics |url=https://archive.org/details/quarksleptonsint0000halz |url-access=registration | publisher=John Wiley & Sons |year=1984 |isbn=0-471-88741-2}}
* {{cite book| title = Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles | edition = 2nd
 | last1 = Eisberg | first1 = Robert Martin
 | last2 = Resnick | first2 = Robert
 | author-link2 = Robert Resnick
 | year = 1985
 | publisher = John Wiley & Sons
 | url = https://archive.org/details/quantumphysicsof00eisb | via = [[Internet Archive]]
 | isbn = 978-0-471-87373-0
}}

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[[Category:Quantum measurement| ]]
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[[Category:Quantum numbers]]
[[Category:Dimensionless numbers]]