Editing 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 [[quantum mechanics]], the '''azimuthal quantum number''' {{mvar|ℓ}} is a [[quantum number]] for an [[atomic orbital]] that determines its [[angular momentum operator|orbital angular momentum]] and describes aspects of the angular shape of the orbital. The [[azimuth]]al quantum number is the second of a set of quantum numbers that describe the unique [[quantum state]] of an [[electron]] (the others being the [[principal quantum number]] {{mvar|n}}, the [[magnetic quantum number]] {{mvar|''m''{{sub|ℓ}}}}, and the [[spin quantum number]] {{mvar|''m''{{sub|s}}}}).

For a given value of the principal quantum number {{mvar|n}} (''[[electron shell]]''), the possible values of {{mvar|ℓ}} are the integers from 0 to {{math|1=''n'' − 1}}. 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>.

For a given value of the azimuthal quantum number {{mvar|ℓ}}, the possible values of the magnetic quantum number {{mvar|''m''{{sub|ℓ}}}} are the integers from {{math|1=''m''{{sub|ℓ}}=−ℓ}} to {{math|1=''m''{{sub|ℓ}}=+ℓ}}, including 0. In addition, the spin quantum number {{mvar|''m''{{sub|s}}}} can take two distinct values. The set of orbitals associated with a particular value of&nbsp;{{mvar|ℓ}} are sometimes collectively called a ''subshell''.

While originally used just for isolated atoms, atomic-like orbitals play a key role in the configuration of electrons in compounds including gases, liquids and solids. The quantum number {{mvar|ℓ}} plays an important role here via the connection to the angular dependence of the [[spherical harmonics]] for the different orbitals around each atom.

== Nomenclature ==
The term "azimuthal quantum number" was introduced by [[Arnold Sommerfeld]] in 1915<ref>{{Cite book |last=Whittaker |first=Edmund Taylor |title=A history of the theories of aether and electricity |date=1989 |publisher=Dover |isbn=978-0-486-26126-3 |series=Dover classics of science and mathematics |location=New York}}</ref>{{rp|II:132}} as part of an ad hoc description of the energy structure of atomic spectra. Only later with the quantum model of the atom was it understood that this number, {{mvar|ℓ}}, arises from quantization of orbital angular momentum. Some textbooks<ref name=Schiff>{{Cite book |last=Schiff |first= Leonard  |title=Quantum mechanics |publisher= McGraw-Hill |year= 1949}}</ref>{{rp|199}} and the ISO standard 80000-10:2019<ref name="iso">{{Cite web |title=ISO Online Browsing Platform |url=https://www.iso.org/obp/ui/en/#iso:std:iso:80000:-10:ed-2:v1:en |access-date=2024-02-20 |at=10-13.3}}</ref> call {{mvar|ℓ}} the '''orbital angular momentum quantum number'''.

The energy levels of an atom in an external magnetic field depend upon the {{mvar|''m''{{sub|ℓ}}}} value so it is sometimes called the  magnetic quantum number.<ref>{{Cite book |last=Eisberg |first=Robert M. |title=Quantum physics of atoms, molecules, solids, nuclei, and particles |last2=Resnick |first2=Robert |date=2009 |publisher=Wiley |isbn=978-0-471-87373-0 |edition=2. ed., 37. [Nachdr.] |location=New York}}</ref>{{rp|240}}

The lowercase letter {{mvar|ℓ}}, is used to denote the orbital angular momentum of a single particle. For a system with multiple particles, the capital letter {{mvar|L}} is used.<ref name="iso"/>

== 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]]).
}}

== Addition of quantized angular momenta ==
{{further|Angular momentum coupling}}
Given a quantized total angular momentum <math>\mathbf{j}</math> that 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 [[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 component.]]

Due to the [[spin–orbit interaction]] in an atom, the orbital angular momentum no longer [[commutator|commutes]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], nor does the [[Spin (physics)|spin]]. These therefore change over time. However the [[total angular momentum]] {{math|'''J'''}} does commute with the one-electron Hamiltonian and so is constant. {{math|'''J'''}} is defined as
<math display="block">\mathbf{J} = \mathbf{L} + \mathbf{S}</math>
{{math|'''L'''}} being the [[angular momentum operator|orbital angular momentum]] and {{math|'''S'''}} the spin. The total angular momentum satisfies the same [[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 it follows that
<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 [[relativistic quantum chemistry]], often featuring in subscript in for deeper states near to the core for which spin-orbit coupling is important.

As with any 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.
The [[eigenvector]]s of {{math|''j''}}, {{math|''s''}}, {{math|''m''<sub>''j''</sub>}} and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of {{math|''ℓ''}}, {{math|''s''}}, {{math|''m''<sub>''ℓ''</sub>}} and {{math|''m''<sub>''s''</sub>}}.

== Beyond isolated atoms ==
{{See also|Density functional theory|History of spectroscopy}}
[[File:Electron_energy_loss_spectroscopy_coreloss_lsmo.svg|thumb|Example of inner shell ionization edge (core loss) EELS data from La<sub>0.7</sub>Sr<sub>0.3</sub>MnO<sub>3</sub>, acquired in a [[Scanning transmission electron microscopy|scanning transmission electron microscope]].]]
The angular momentum quantum numbers strictly refer to isolated atoms. However, they have wider uses for atoms in solids, liquids or gases. The {{math|''ℓ m''}} quantum number corresponds to specific [[spherical harmonics]] and are commonly used to describe features observed in spectroscopic methods such as [[X-ray photoelectron spectroscopy]]<ref>{{Cite book |last=Hüfner |first=Stefan |url=http://link.springer.com/10.1007/978-3-662-09280-4 |title=Photoelectron Spectroscopy |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07520-9 |series=Advanced Texts in Physics |location=Berlin, Heidelberg |doi=10.1007/978-3-662-09280-4}}</ref> and [[electron energy loss spectroscopy]].<ref>{{Cite book |last=Egerton |first=R.F. |url=https://link.springer.com/10.1007/978-1-4419-9583-4 |title=Electron Energy-Loss Spectroscopy in the Electron Microscope |date=2011 |publisher=Springer US |isbn=978-1-4419-9582-7 |location=Boston, MA |language=en |doi=10.1007/978-1-4419-9583-4}}</ref> (The notation is slightly different, with [[X-ray notation]] where K, L, M are used for excitations out of electron states with <math>n=0, 1, 2</math>.)

The angular momentum quantum numbers are also used when the electron states are described in methods such as [[Kohn–Sham equations|Kohn–Sham density functional theory]]<ref>{{Cite journal |last=Kohn |first=W. |last2=Sham |first2=L. J. |date=1965 |title=Self-Consistent Equations Including Exchange and Correlation Effects |url=https://link.aps.org/doi/10.1103/PhysRev.140.A1133 |journal=Physical Review |language=en |volume=140 |issue=4A |pages=A1133–A1138 |doi=10.1103/PhysRev.140.A1133 |issn=0031-899X}}</ref> or with [[gaussian orbital]]s.<ref>{{Citation |last=Gill |first=Peter M.W. |title=Molecular integrals Over Gaussian Basis Functions |date=1994 |work=Advances in Quantum Chemistry |volume=25 |pages=141–205 |url=https://linkinghub.elsevier.com/retrieve/pii/S0065327608600192 |access-date=2024-02-20 |publisher=Elsevier |language=en |doi=10.1016/s0065-3276(08)60019-2 |isbn=978-0-12-034825-1|url-access=subscription }}</ref> For instance, in [[silicon]] the electronic properties used in [[semiconductor device]] are due to the p-like states with <math>l=1</math> centered at each atom, while many properties of [[transition metal]]s depend upon the d-like states with <math>l=2</math>.<ref>{{Cite book |last=Pettifor |first=D. G. |title=Bonding and structure of molecules and solids |date=1996 |publisher=Clarendon Press |isbn=978-0-19-851786-3 |edition=Reprint with corr |series=Oxford science publications |location=Oxford}}</ref>

== History ==
The azimuthal quantum number was carried over from the [[Bohr model|Bohr model of the atom]], and was posited by [[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 [[spectroscopy|spectroscopic analysis]] of atoms in combination with the [[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 |author-link=Robert Bruce Lindsay |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 [[Node (physics)|nodes]] crossing the nucleus, similar (in the lowest-energy state) to a skipping rope that oscillates in one large circle.

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

== References ==
{{reflist}}

== External links ==
* [http://galileo.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html Development of the Bohr atom]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html#c1 The azimuthal equation explained]

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