Editing Quantum number (section)

==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 display=block>n = 1, 2, \ldots</math>

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 {{mvar|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 display=block>L^2 = \hbar^2 \ell(\ell + 1).</math>

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 display=block>\ell = 0, 1, 2, \ldots, n-1</math>

A quantum number beginning in {{math|1=''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 display=block>L_z = m_\ell \hbar</math>

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 display=block>S_z = m_s \hbar</math>

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 display=block>m_s = -s, -s+1, -s+2, \cdots, s-2, s-1, s</math>

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}} first, with lowest {{mvar|n}} 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}}