View source for Physics:Azimuthal quantum number

{{Short description|Quantum number denoting orbital angular momentum}}
[[File:HAtomOrbitals.png|thumb|The atomic orbital wavefunctions of a hydrogen atom: The azimuthal quantum number ({{mvar|ℓ}}) is denoted ''by letter'' at the top of each column. The principal quantum number ({{mvar|n}}) is shown at the right of each row.]]

In [[Physics:Quantum mechanics|quantum mechanics]], the '''azimuthal quantum number''' is a [[Physics:Quantum number|quantum number]] for an [[Physics:Atomic orbital|atomic orbital]] that determines its [[Physics:Angular momentum operator|orbital angular momentum]] and describes the shape of the orbital.  The [[Azimuth|azimuth]]al quantum number is the second of a set of quantum numbers that describe the unique [[Physics:Quantum state|quantum state]] of an [[Physics:Electron|electron]] (the others being the [[Physics:Principal quantum number|principal quantum number]] {{mvar|n}}, the [[Physics:Magnetic quantum number|magnetic quantum number]] {{mvar|m{{sub|ℓ}}}}, and the [[Physics:Spin quantum number|spin quantum number]] {{mvar|m{{sub|s}}}}).  It is also known as the '''orbital angular momentum quantum number''', '''orbital quantum number, subsidiary quantum number,''' or '''second quantum number''', and is symbolized as {{mvar|ℓ}} (pronounced ''ell'').

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

When solving to obtain the wave function, the [[Physics:Schrödinger equation|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|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) alternaive values for the reduced Planck's constant, {{math|''ħ''}}.]]

An electron's [[Physics:Angular momentum operator|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's constant, {{math|'''L'''}} is the orbital ''[[Physics:Angular momentum operator|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 [[Physics:Spectroscopic notation#Atomic and molecular orbitals|spectroscopy]]) denote the shape of the [[Physics:Atomic orbital|atomic orbital]]. The wavefunctions of these orbitals take the form of [[Physics:Spherical harmonic|spherical harmonic]]s, and so are described by [[Physics: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
! 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 [[Physics:Atomic orbital|atomic orbital]]; (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 [[Physics: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;"| '''g'''allant 
| 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 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 [[Physics:Principal quantum number|principal quantum number]] ''n'', the possible values of ''ℓ'' range from 0 to {{math|1=''n'' − 1}}; therefore, the {{math|1=''n'' = 1}} [[Physics: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 [[Physics:Hydrogen-like atom|simplistic one-electron model]] results in [[Physics:Energy level|energy level]]s depending on the principal number alone. In more complex atoms these energy levels [[Physics: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 [[Chemistry:Block (periodic table)|the block structure]] of the periodic table. No known atom possesses an electron having ''ℓ'' higher than three ('''f''') in its [[Physics:Ground state|ground state]].

The angular momentum quantum number, ''ℓ'', governs{{how|date=August 2019}} 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 [[Physics:Hydrogen atom|hydrogen atom]]:
{{unbulleted list | style = padding-left: 1.3em;
| <math>n = 1, L = 0</math>, [[Physics:Lyman series|Lyman series]] (ultraviolet)
| <math>n = 2, L = \sqrt{2}\hbar</math>, [[Physics:Balmer series|Balmer series]] (visible)
| <math>n = 3, L = \sqrt{6}\hbar</math>, Ritz–Paschen series (near infrared)
| <math>n = 4, L = 2\sqrt{3}\hbar</math>, Brackett series ([[Physics:Infrared#Commonly used sub-division scheme|short-wavelength infrared]])
| <math>n = 5, L = 2\sqrt{5}\hbar</math>, Pfund series ([[Physics:Infrared#Commonly used sub-division scheme|mid-wavelength infrared]]).
}}

== Addition of quantized angular momenta ==
{{further|Physics:Angular momentum coupling}}
Given a quantized total angular momentum <math>\mathbf{j}</math> which is the sum of two individual quantized angular momenta <math>\boldsymbol{\ell}_1</math> and <math>\boldsymbol{\ell}_2</math>,
<math display="block">\mathbf{j} = \boldsymbol{\ell}_1 + \boldsymbol{\ell}_2</math>
the [[Physics:Quantum number|quantum number]] <math>j</math> associated with its magnitude can range from <math>|\ell_1 - \ell_2|</math> to <math>\ell_1 + \ell_2</math> in integer steps
where <math>\ell_1</math> and <math>\ell_2</math> are quantum numbers corresponding to the magnitudes of the individual angular momenta.

=== Total angular momentum of an electron in the atom ===
[[File:LS coupling.svg|thumb|"Vector cones" of total angular momentum '''J''' (purple), orbital '''L''' (blue), and spin '''S''' (green). The cones arise due to quantum uncertainty between measuring angular momentum components (see vector model of the atom).]]

Due to the [[Physics:Spin–orbit interaction|spin–orbit interaction]] in the atom, the orbital angular momentum no longer [[Commutator|commutes]] with the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]], nor does the [[Physics:Spin|spin]]. These therefore change over time. However the [[Physics:Total angular momentum|total angular momentum]] {{math|'''J'''}} does commute with the one-electron Hamiltonian and so is constant. {{math|'''J'''}} is defined through
<math display="block">\mathbf{J} = \mathbf{L} + \mathbf{S}</math>
{{math|'''L'''}} being the [[Physics:Angular momentum operator|orbital angular momentum]] and {{math|'''S'''}} the spin. The total angular momentum satisfies the same [[Physics:Angular momentum operator#Commutation relations|commutation relations as orbital angular momentum]], namely
<math display="block">[J_i, J_j ] = i \hbar \varepsilon_{ijk} J_k</math>
from which follows
<math display="block">\left[J_i, J^2 \right] = 0</math>
where {{math|''J''<sub>''i''</sub>}} stand for {{math|''J''<sub>''x''</sub>}}, {{math|''J''<sub>''y''</sub>}}, and {{math|''J''<sub>''z''</sub>}}.

The quantum numbers describing the system, which are constant over time, are now {{math|''j''}} and {{math|''m''<sub>''j''</sub>}}, defined through the action of {{math|'''J'''}} on the wavefunction <math>\Psi</math>
<math display="block">\begin{align}
\mathbf{J}^2\Psi &= \hbar^2 j(j+1) \Psi \\[1ex]
\mathbf{J}_z\Psi &= \hbar m_j\Psi
\end{align}</math>

So that {{math|''j''}} is related to the norm of the total angular momentum and {{math|''m''<sub>''j''</sub>}} to its projection along a specified axis. The ''j'' number has a particular importance for [[Chemistry:Relativistic quantum chemistry|relativistic quantum chemistry]], often featuring in subscript in [[Chemistry:Extended periodic table#Electron configurations|electron configuration of superheavy elements]].

As with any [[Physics:Angular momentum operator|angular momentum in quantum mechanics]], the projection of {{math|'''J'''}} along other axes cannot be co-defined with {{math|''J''<sub>z</sub>}}, because they do not commute.

==== Relation between new and old quantum numbers ====
{{math|''j''}} and {{math|''m''<sub>''j''</sub>}}, together with the [[Physics:Parity|parity]] of the [[Physics:Quantum state|quantum state]], replace the three quantum numbers {{math|''ℓ''}}, {{math|''m''<sub>''ℓ''</sub>}} and {{math|''m''<sub>''s''</sub>}} (the projection of the [[Physics:Spin|spin]] along the specified axis). The former quantum numbers can be related to the latter.

Furthermore, the eigenvectors of {{math|''j''}}, {{math|''s''}}, {{math|''m''<sub>''j''</sub>}} and parity, which are also eigenvectors of the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]], are linear combinations of the eigenvectors of {{math|''ℓ''}}, {{math|''s''}}, {{math|''m''<sub>''ℓ''</sub>}} and {{math|''m''<sub>''s''</sub>}}.

== List of angular momentum quantum numbers ==
* Intrinsic (or spin) angular momentum quantum number, or simply [[Physics:Spin quantum number|spin quantum number]]
* orbital angular momentum quantum number (the subject of this article)
* [[Physics:Magnetic quantum number|magnetic quantum number]], related to the orbital momentum quantum number
* [[Physics:Total angular momentum quantum number|total angular momentum quantum number]]

== History ==
The azimuthal quantum number was carried over from the [[Physics:Bohr model|Bohr model of the atom]], and was posited by [[Biography:Arnold Sommerfeld|Arnold Sommerfeld]].<ref>{{cite book|last=Eisberg|first=Robert|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles | year=1974|publisher=John Wiley & Sons Inc|location=New York|isbn=978-0-471-23464-7|pages=114–117}}</ref> The Bohr model was derived from [[Physics:Spectroscopy|spectroscopic analysis]] of the atom in combination with the [[Biography:Ernest Rutherford|Rutherford]] atomic model.  The lowest quantum level was found to have an angular momentum of zero.  Orbits with zero angular momentum were considered as oscillating charges in one dimension and so described as "pendulum" orbits, but were not found in nature.<ref>{{cite journal|title=Note on "pendulum" orbits in atomic models |journal=Proc. Natl. Acad. Sci. |year=1927 |volume=13 |pages=413–419 |pmid=16587189 |author=R.B. Lindsay |issue=6 |doi=10.1073/pnas.13.6.413 |bibcode=1927PNAS...13..413L |pmc=1085028|doi-access=free }}</ref> In three-dimensions the orbits become spherical without any [[Physics:Node|nodes]] crossing the nucleus, similar (in the lowest-energy state) to a skipping rope that oscillates in one large circle.

== See also ==
* [[Physics:Angular momentum operator|Angular momentum operator]]
* [[Physics:Introduction to quantum mechanics|Introduction to quantum mechanics]]
* [[Particle in a spherically symmetric potential]]
* [[Physics:Angular momentum coupling|Angular momentum coupling]]
* [[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]]

== References ==
<references/>

==External links==
* [http://galileo.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html Development of the Bohr atom]
* [http://www.src.wits.ac.za/pages/teaching/Connell/phys284/2005/lecture-03/lecture_03/node7.html Detailed explanation of the Orbital Quantum Number l]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html#c1 The azimuthal equation explained]

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