Editing Atomic orbital (section)

=== Complex orbitals ===
[[File:Electronic_levels.svg|thumb|450px|alt=Electronic levels|Energetic levels and sublevels of polyelectronic atoms]]
In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure [[spherical harmonic]]. The quantum numbers, together with the rules governing their possible values, are as follows:

The [[principal quantum number]] {{mvar|n}} describes the energy of the electron and is always a [[positive integer]]. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of ''n''; these orbitals together are sometimes called ''[[electron shells]]''.

The [[azimuthal quantum number]] {{mvar|ℓ}} describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where {{mvar|n}} is some integer {{math|''n''<sub>0</sub>}}, {{mvar|ℓ}} ranges across all (integer) values satisfying the relation <math>0 \le \ell \le n_0-1</math>. For instance, the {{math|1=''n'' = 1}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and the {{math|1=''n'' = 2}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and <math>\ell=1</math>. The set of orbitals associated with a particular value of&nbsp;{{mvar|ℓ}} are sometimes collectively called a ''subshell''.

The [[magnetic quantum number]], <math>m_\ell</math>, describes the projection of the orbital angular momentum along a chosen axis. It determines the magnitude of the current circulating around that axis and the orbital contribution to the [[electron magnetic moment|magnetic moment of an electron]] via the [[Magnetic moment#Ampèrian loop model|Ampèrian loop]] model.<ref>{{Cite book |last=Greiner |first=Walter |url=http://archive.org/details/quantummechanics0001grei |title=Quantum mechanics : Introduction |date=1994 |publisher=Springer-Verlag |isbn=978-0-387-58080-7 |edition=2nd corrected |location=New York, Berlin, Heidelberg |pages=163}}</ref> Within a subshell <math>\ell</math>, <math>m_\ell</math> obtains the integer values in the range <math>-\ell \le m_\ell \le \ell </math>.

The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of <math>m_\ell</math> available in that subshell. Empty cells represent subshells that do not exist.

{| class="wikitable"
|-
!
! {{math|1=''ℓ'' = 0 (s)}}
! {{math|1=''ℓ'' = 1 (p)}}
! {{math|1=''ℓ'' = 2 (d)}}
! {{math|1=''ℓ'' = 3 (f)}}
! {{math|1=''ℓ'' = 4 (g)}}
! ...
|-
! {{math|1=''n'' = 1}}
| <math>m_\ell=0</math>
| || || || || ...
|-
! {{math|1=''n'' = 2}}
| 0 || −1, 0, 1
| || || || ...
|-
! {{math|1=''n'' = 3}}
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2
| || || ...
|-
! {{math|1=''n'' = 4}}
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2 || −3, −2, −1, 0, 1, 2, 3
| || ...
|-
! {{math|1=''n'' = 5}}
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2 || −3, −2, −1, 0, 1, 2, 3 || −4, −3, −2, −1, 0, 1, 2, 3, 4
| ...
|-
! ...
| ... || ... || ... || ... || ... || ...
|}

Subshells are usually identified by their <math>n</math>- and <math>\ell</math>-values. <math>n</math> is represented by its numerical value, but <math>\ell</math> is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with <math>n=2</math> and <math>\ell=0</math> as a '2s subshell'.

Each electron also has angular momentum in the form of [[Spin (physics)|quantum mechanical spin]] given by spin ''s'' = {{sfrac|1|2}}. Its projection along a specified axis is given by the [[spin magnetic quantum number]], ''m<sub>s</sub>'', which can be +{{sfrac|1|2}} or −{{sfrac|1|2}}. These values are also called "spin up" or "spin down" respectively.

The [[Pauli exclusion principle]] states that no two electrons in an atom can have the same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, ({{mvar|n}}, {{mvar|ℓ}}, {{mvar|m}}), these two electrons must differ in their spin projection ''m<sub>s</sub>''.

The above conventions imply a preferred axis (for example, the ''z'' direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing {{math|1=''m'' = +1}} from {{math|1=''m'' = −1}}. As such, the model is most useful when applied to physical systems that share these symmetries. The [[Stern–Gerlach experiment]]{{Emdash}}where an atom is exposed to a magnetic field{{Emdash}}provides one such example.<ref>{{cite journal|last1=Gerlach |first1=W. |last2=Stern |first2=O. |title=Das magnetische Moment des Silberatoms|journal=[[Zeitschrift für Physik]]|volume=9|issue=1 |pages=353–355|year=1922|doi=10.1007/BF01326984|bibcode = 1922ZPhy....9..353G |s2cid=126109346 }}</ref>