Editing Azimuthal quantum number (section)

== Relation to atomic orbitals ==
There are four quantum numbers{{mdash}}''n'', ''ℓ'', ''m''<sub>''ℓ''</sub>, ''m''<sub>''s''</sub>{{mdash}} connected with the energy states of an isolated atom's electrons. These four numbers specify the unique and complete quantum state of any single [[electron]] in the [[atom]], and they combine to compose the electron's [[wavefunction]], or ''[[atomic orbital#Electron properties|orbital]]''.

When solving to obtain the wave function, the [[Schrödinger equation]] resolves into three equations that lead to the first three quantum numbers, meaning that the three equations are interrelated. The ''azimuthal quantum number'' arises in solving the polar part of the wave equation{{mdash}}relying on the [[spherical coordinate system]], which generally works best with models having sufficient aspects of [[spherical symmetry]].

[[File:Vector model of orbital angular momentum.svg|left|thumb|Azimuthal quantum number: Illustrating (five) alternative ''orbital angular momentum'' shapes  as "cones"{{mdash}}here portraying (five) alternative values for the reduced Planck constant, {{math|''ħ''}}.]]

An electron's angular momentum, {{math|''L''}}, is related to its quantum number {{math|''ℓ''}} by the following equation:
<math display="block">\mathbf{L}^2\Psi = \hbar^2 \ell(\ell + 1) \Psi,</math>
where {{math|''ħ''}} is the [[reduced Planck constant]], {{math|'''L'''}} is the orbital ''angular momentum operator'' and <math>\Psi</math> is the wavefunction of the electron. The quantum number {{math|''ℓ''}} is always a non-negative integer: 0, 1, 2, 3, etc. (Notably, {{math|'''L'''}} has no real meaning except in its use as the angular momentum operator; thus, it is standard practice to use the quantum number {{math|''ℓ''}} when referring to angular momentum).

Atomic orbitals have distinctive shapes, (see top graphic) in which letters, '''s''', '''p''', '''d''', '''f''', etc., (employing a convention originating in [[Spectroscopic notation#Atomic and molecular orbitals|spectroscopy]]) denote the shape of the atomic orbital. The wavefunctions of these orbitals take the form of [[spherical harmonic]]s, and so are described by [[associated Legendre polynomials|Legendre polynomials]]. The several orbitals relating to the different (integer) values of ''ℓ'' are sometimes called '''sub-shells'''{{mdash}}referred to by lowercase [[Latin letters]] chosen for historical reasons{{mdash}}as shown in the table "Quantum subshells for the azimuthal quantum number".
{{Clear}}
{| class="wikitable"
|+ Quantum subshells for the azimuthal quantum number
|-
! style="text-align: center; vertical-align: bottom;"| Azimuthal<br/>quantum<br/>number&nbsp;(''ℓ'')
! style="text-align: center; vertical-align: bottom;"| Historical<br/>letter
! style="text-align: center; vertical-align: bottom;"| Historical<br/>name<ref>{{Cite book |last=Whittaker |first=E. T. |title=A history of the theories of aether & electricity |date=1989 |publisher=Dover Publications |isbn=978-0-486-26126-3 |location=New York}}</ref>{{rp|II:133}}
! style="text-align: center; vertical-align: bottom;"| Maximum<br/>electrons
! style="text-align: left; vertical-align: bottom;"| Shape
|-
| style="text-align: center;"| 0 
| style="text-align: center;"| '''s''' 
| style="text-align: left;"| '''s'''harp
| style="text-align: center;"| 2 
| Spherical (see [[:File:Spherical Harmonics.png|this picture of spherical harmonics, top row]]).
|-
| style="text-align: center;"| 1 
| style="text-align: center;"| '''p''' 
| style="text-align: left;"| '''p'''rincipal
| style="text-align: center;"| 6 
| Three [[atomic orbital|dumbbell-shaped]] polar-aligned orbitals; one lobe on each pole of the x, y, and z axes (on both + and − axes).
|-
| style="text-align: center;"| 2 
| style="text-align: center;"| '''d''' 
| style="text-align: left;"| '''d'''iffuse
| style="text-align: center;"| 10
| Nine dumbbells and one doughnut, or "Unique shape #1" (see [[:File:Spherical Harmonics.png|this picture of spherical harmonics, third row center]]).
|-
| style="text-align: center;"| 3 
| style="text-align: center;"| '''f''' 
| style="text-align: left;"| '''f'''undamental
| style="text-align: center;"| 14 
| "Unique shape #2" (see [[:File:Spherical Harmonics.png|this picture of spherical harmonics, bottom row center]]).
|-
| style="text-align: center;"| 4 
| style="text-align: center;"| '''g''' 
| style="text-align: left;"| 
| style="text-align: center;"| 18  
|-
| style="text-align: center;"| 5 
| style="text-align: center;"| '''h''' 
| style="text-align: center;"|  
| style="text-align: center;"| 22  
|-
| style="text-align: center;"| 6 
| style="text-align: center;"| '''i''' 
| style="text-align: center;"|  
| style="text-align: center;"| 26 
|-
| colspan=5 style="text-align: left; text-align: center;" | The letters after the '''g''' sub-shell follow in alphabetical order{{mdash}}excepting letter&nbsp;''j'' and those already used.
|}

Each of the different angular momentum states can take 2(2''ℓ''&nbsp;+&nbsp;1) electrons. This is because the third quantum number ''m''<sub>ℓ</sub> (which can be thought of loosely as the [[angular momentum quantization|quantized]] projection of the angular momentum vector on the z-axis) runs from &minus;''ℓ'' to ''ℓ'' in integer units, and so there are 2''ℓ''&nbsp;+&nbsp;1 possible states. Each distinct ''n'', ''ℓ'', ''m''<sub>ℓ</sub> orbital can be occupied by two electrons with opposing spins (given by the quantum number ''m''<sub>s</sub>&nbsp;=&nbsp;±{{1/2}}), giving 2(2''ℓ''&nbsp;+&nbsp;1) electrons overall. Orbitals with higher ''ℓ'' than given in the table are perfectly permissible, but these values cover all atoms so far discovered.

For a given value of the [[principal quantum number]] ''n'', the possible values of ''ℓ'' range from 0 to {{math|1=''n'' − 1}}; therefore, the {{math|1=''n'' = 1}} [[electron shell|shell]] only possesses an s subshell and can only take 2&nbsp;electrons, the {{math|1=''n'' = 2}} shell possesses an '''s''' and a '''p''' subshell and can take 8&nbsp;electrons overall, the {{math|1=''n'' = 3}} shell possesses '''s''', '''p''', and '''d''' subshells and has a maximum of 18&nbsp;electrons, and so on.

A [[hydrogen-like atom|simplistic one-electron model]] results in [[energy level]]s depending on the principal number alone. In more complex atoms these energy levels [[energy level splitting|split]] for all {{math|''n'' > 1}}, placing states of higher ''ℓ'' above states of lower ''ℓ''. For example, the energy of 2p is higher than of 2s, 3d occurs higher than 3p, which in turn is above 3s, etc. This effect eventually forms [[block (periodic table)|the block structure]] of the periodic table. No known atom possesses an electron having ''ℓ'' higher than three ('''f''') in its [[ground state]].

The angular momentum quantum number, ''ℓ'' and the corresponding spherical harmonic govern the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as the midpoint between crest and trough, which has zero magnitudes. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number ''ℓ'' takes the value of 0. In a '''p''' orbital, one node traverses the nucleus and therefore ''ℓ'' has the value of 1. <math>L</math> has the value <math>\sqrt{2}\hbar</math>.

Depending on the value of ''n'', there is an angular momentum quantum number ''ℓ'' and the following series. The wavelengths listed are for a [[hydrogen atom]]:
{{unbulleted list | style = padding-left: 1.3em;
| <math>n = 1, L = 0</math>, [[Lyman series]] (ultraviolet)
| <math>n = 2, L = \sqrt{2}\hbar</math>, [[Balmer series]] (visible)
| <math>n = 3, L = \sqrt{6}\hbar</math>, [[Paschen series|Ritz–Paschen series]] ([[near infrared]])
| <math>n = 4, L = 2\sqrt{3}\hbar</math>, [[Brackett series]] ([[infrared#Commonly used sub-division scheme|short-wavelength infrared]])
| <math>n = 5, L = 2\sqrt{5}\hbar</math>, [[Pfund series]] ([[infrared#Commonly used sub-division scheme|mid-wavelength infrared]]).
}}