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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 numbers''' are a set of values that describe certain properties of particles. They are used to characterize the energy, angular momentum, and other properties of particles. The four main quantum numbers are the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s). These numbers help to define the allowed energy levels and spatial distribution of electrons in an atom.

Formally, quantum numbers describe values of [[conserved quantity|conserved quantities]] in the dynamics of a [[quantum system]]. 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<ref group="note">specifically, observables <math>\widehat{A}</math>  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.</ref>—and their corresponding eigenspaces. 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.

An important aspect of quantum mechanics is the [[Quantization (physics)|quantization]] of many observable quantities of interest.<ref group="note">Many observables have discrete [[Spectrum of an operator|spectra (sets of eigenvalues)]] in quantum mechanics, so the quantities can only be measured in discrete (often integer) values.</ref>  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. This distinguishes quantum mechanics from [[classical mechanics]] where the values that characterize the system such as mass, charge, or momentum, all range continuously. Quantum numbers often describe specifically the [[energy level]]s of electrons in atoms, but other possibilities include [[angular momentum]], [[Spin (physics)|spin]], etc. An important family is [[flavour (particle physics)|flavour quantum numbers]] – [[internal symmetry|internal]] quantum numbers which determine the type of a particle and its interactions with other particles through the [[fundamental force]]s.  Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.

==Quantum numbers needed for a given system==
{{Main|Quantum system}}

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 an atom==
Four quantum numbers can describe an electron in an 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>}})

The [[spin–orbit interaction|spin–orbital interaction]], however, relates these numbers. Thus, a complete description of the system can be given with fewer quantum numbers, if orthogonal choices are made for these basis vectors.

===Specificity===
Different electrons in a system will have different quantum numbers. For example, the highest occupied orbital electron, the actual differentiating electron (i.e. the electron that differentiates an element from the previous one), or the differentiating electron according to the [[Aufbau principle#Madelung energy ordering rule|''aufbau'' approximation]]. In [[lanthanum]], as a further illustration, the electrons involved are in the 6s; 5d; and 4f orbitals, respectively. In this case the principal quantum numbers are 6, 5, and 4.

===Common terminology===
The model used here describes electrons using four quantum numbers, {{mvar|n}}, {{mvar|{{ell}}}}, {{mvar|m<sub>{{ell}}</sub>}}, {{mvar|m<sub>s</sub>}}, given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of [[molecular orbitals]] require other quantum numbers, because the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and its symmetries are different.

====Principal quantum number====
{{Main|Principal quantum number}}
{{See also|electron shell}}
The principal quantum number describes the [[electron shell]], or energy level, 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.

For particles in a time-independent potential (see [[Schrödinger equation#Time independent|Schrödinger equation]]), it also labels the {{mvar|n}}th eigenvalue of Hamiltonian ({{mvar|H}}), that is, the energy {{mvar|E}}, with the contribution due to angular momentum (the term involving {{math|'''J'''<sup>2</sup>}}) left out. So this number depends only on the distance between the electron and the nucleus (that is, the radial coordinate {{math|'''r'''}}). The average distance increases with {{math|'''n'''}}.  Hence quantum states with different principal quantum numbers are said to belong to different shells.

====Azimuthal quantum number====
{{Main|Azimuthal quantum number}}
{{See also|electron shell#Subshells}}
The azimuthal quantum number, also known as the '''angular momentum quantum number''' or '''orbital 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.

'''Shape''' of orbital is also given by azimuthal quantum number.

====Magnetic quantum number====
{{Main|Magnetic quantum number}}
{{See also|atomic orbital}}
The magnetic quantum number describes the specific [[atomic orbital|orbital]] (or "cloud") within that 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 (in some systems, depicted as three "dumbbell-shaped" clouds), 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 quantum number====
{{Main|Spin quantum number}}
{{See also|Spin (physics)}}
The spin quantum number describes the intrinsic spin [[angular momentum]] of the electron within each orbital and gives the projection of the [[Angular momentum operator#Spin angular momentum|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 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 has spin number {{math|1=''s'' = {{sfrac|1|2}}}}, consequently {{mvar|m<sub>s</sub>}} will be ±{{sfrac|1|2}}, referring to "spin up" and "spin down" states. Each electron in any individual orbital must have different quantum numbers because of the [[Pauli exclusion principle]], therefore an orbital never contains more than two electrons.

====Rules====
There are no universal fixed values for {{mvar|m<sub>{{ell}}</sub>}} and {{mvar|m<sub>s</sub>}}. Rather, the {{mvar|m<sub>{{ell}}</sub>}} and {{mvar|m<sub>s</sub>}} values are [[arbitrary]]. The only restrictions on the choices for these constants is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p orbital could be described as {{math|1=''m<sub>{{ell}}</sub>'' = −1}} or {{math|1=''m<sub>{{ell}}</sub>'' = 0}} or {{math|1=''m<sub>{{ell}}</sub>'' = 1}}, but the {{mvar|m<sub>{{ell}}</sub>}} value of the next unpaired electron in that orbital must be different; yet, the {{mvar|m<sub>{{ell}}</sub>}} assigned to electrons in other orbitals again can be {{math|1=''m<sub>{{ell}}</sub>'' = −1}} or {{math|1=''m<sub>{{ell}}</sub>'' = 0}} or {{math|1=''m<sub>{{ell}}</sub>'' = 1}}).

These rules are summarized as follows:

:{| class="wikitable"
! scope="col" | Name
! scope="col" | Symbol
! scope="col" | Meaning
! scope="col" | Range of values
! scope="col" | Value examples
|-
| [[Principal quantum number]] || {{mvar|n}} || shell || {{math|1 ≤ ''n''}} || {{math|1=''n'' = 1, 2, 3, …}}
|-
| [[Azimuthal quantum number]] ([[angular momentum]])|| {{mvar|{{ell}}}} || subshell (s orbital is listed as 0, p orbital as 1 etc.) || {{math|0 ≤ ''{{ell}}'' ≤ ''n'' − 1}} || for {{math|1=''n'' = 3}}: <br /> {{math|1=''{{ell}}'' = 0, 1, 2}} (s, p, d)
|-
| [[Magnetic quantum number]] (projection of [[angular momentum]])|| {{mvar|m<sub>{{ell}}</sub>}}|| Orbital (orientation of the orbital) || {{math|−''{{ell}}'' ≤ ''m<sub>{{ell}}</sub>'' ≤ ''{{ell}}''}} || for {{math|1=''{{ell}}'' = 2}}: <br /> {{math|1=''m<sub>{{ell}}</sub>'' = −2, −1, 0, 1, 2}}
|-
| [[Spin quantum number]] || {{mvar|m<sub>s</sub>}}|| spin of the electron (−{{sfrac|1|2}} = "spin down", {{sfrac|1|2}} = "spin up") || {{math|−''s'' ≤ ''m<sub>s</sub>'' ≤ ''s''}} || for an electron {{math|1=''s'' = {{sfrac|1|2}}}}, <br /> so {{math|1=''m<sub>s</sub>'' = −{{sfrac|1|2}}, +{{sfrac|1|2}}}}
|}

Example: The quantum numbers used to refer to the outermost [[valence (chemistry)|valence]] [[electron]]s of a [[carbon]] (C) [[atom]], which are located in the 2p [[atomic orbital]], are; {{math|1=''n'' = 2}} (2nd electron shell), {{math|1=''{{ell}}'' = 1}} (p orbital [[electron shell#Subshells|subshell]]), {{math|1=''m<sub>{{ell}}</sub>'' = 1, 0, −1}}, {{math|1=''m<sub>s</sub>'' = {{sfrac|1|2}}}} (parallel spins).

Results from [[spectroscopy]] indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to [[Hund's rules]], which addresses the [[Pauli exclusion principle]]. A fourth quantum number, which represented spin with two possible values, was added as an ''[[ad hoc]]'' assumption to resolve the conflict; this supposition would later be explained in detail by relativistic quantum mechanics and from the results of the renowned [[Stern–Gerlach experiment]].

===Background===
Many different models have been proposed throughout the [[history of quantum mechanics]], but the most prominent system of nomenclature spawned from the Hund-Mulliken [[molecular orbital]] theory of [[Friedrich Hund]], [[Robert S. Mulliken]], and contributions from [[Erwin Schrödinger|Schrödinger]], [[John C. Slater|Slater]] and [[John Lennard-Jones]]. This system of nomenclature incorporated [[Niels Bohr|Bohr]] energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on [[spectroscopy]] and [[Hund's rules]].<ref>Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, {{ISBN|0-19-855148-7}}</ref>

==Total angular momenta numbers==
===Total angular momentum of a particle===
{{Further|Clebsch–Gordan coefficients}}
{{See also|Azimuthal quantum number#Total angular momentum of an electron in the atom}}
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 their eigenvalues therefore change over time. 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
|}

===Nuclear angular momentum quantum numbers===
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 [[wikt:internal|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 numbers 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]]
* [[Multiplicative quantum number]]
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== Notes ==
{{Reflist|group=note}}

==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 (2nd ed.) | 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
}}

==External links==
*[http://hyperphysics.phy-astr.gsu.edu/hbase/qunoh.html Quantum numbers for the hydrogen atom]
*[http://www.physics.byu.edu/faculty/durfee/courses/Summer2009/physics222/AtomicQuantumNumbers.pdf Lecture notes on quantum numbers]
*[http://pdg.lbl.gov/ The particle data group]

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