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| {{quantum mechanics}}
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| The '''azimuthal quantum number''' is a [[quantum number]] for an [[atomic orbital]] that determines its [[angular momentum operator|orbital angular momentum]] and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique [[quantum state]] of an electron (the others being the [[principal quantum number]], following [[spectroscopic notation]], the [[magnetic quantum number]], and the [[spin quantum number]]). It is also known as the '''orbital angular momentum''' quantum number, '''orbital quantum number''' or '''second quantum number''', and is symbolized as '''ℓ''' (in Unicode: {{unichar|2113|SCRIPT SMALL L|html=}}).
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| == Derivation ==
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| [[File:Vector model of orbital angular momentum.svg|250px|right|thumb|Illustration of quantum mechanical orbital angular momentum.]]
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| Connected with the energy states of the electrons of an atom is a set of four quantum numbers: ''n'', ''ℓ'', ''m''<sub>''ℓ''</sub>, and ''m''<sub>''s''</sub>. These specify the complete and unique [[quantum state]] of a single electron in an atom, and make up its [[wavefunction]] or [[atomic orbital|orbital]]. The wavefunction of the [[Schrödinger wave equation]] reduces to three equations that when solved, lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The azimuthal quantum number arose in the solution of the polar part of the wave equation as shown below. To aid understanding of this concept of the [[azimuth]], it may also prove helpful to review [[spherical coordinate system]]s, and/or other alternative mathematical coordinate systems besides the [[cartesian coordinate system]]. Generally, the spherical coordinate system works best with spherical models, the [[cylindrical coordinate system|cylindrical system]] with cylinders, the cartesian with general volumes, etc.
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| An atomic electron's [[angular momentum operator|angular momentum]], ''L'', is related to its quantum number ''ℓ'' by the following equation:
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| :<math>\mathbf{L}^2\Psi = \hbar^2{\ell(\ell+1)}\Psi</math>
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| where ''ħ'' is the [[reduced Planck's constant]], '''L'''<sup>2</sup> is the [[angular momentum operator|orbital angular momentum operator]] and <math>\Psi</math> is the wavefunction of the electron. The quantum number ''ℓ'' is always a nonnegative integer: 0,1,2,3, etc. (see [[angular momentum quantization]]). While many introductory textbooks on quantum mechanics will refer to '''L''' by itself, '''L''' has no real meaning except in its use as the angular momentum operator. When referring to angular momentum, it is best to simply use the quantum number ''ℓ''.
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| [[File:HAtomOrbitals.png|frame|right|The [[atomic orbital]] wavefunctions of a [[hydrogen atom]]. The [[principal quantum number]] is at the right of each row and the azimuthal quantum number is denoted by letter at top of each column.]]
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| Atomic orbitals have distinctive shapes denoted by letters. In the illustration, the letters s, p, and d describe the shape of the [[atomic orbital]].
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| Their [[wavefunctions]] take the form of [[spherical harmonic]]s, and so are described by [[associated Legendre polynomials|Legendre polynomials]]. The various orbitals relating to different values of ℓ are sometimes called '''sub-shells''', and (mainly for historical reasons) are referred to by letters, as follows:
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| :{| class="wikitable"
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| ! ℓ !! Letter !! Max electrons !! Shape !! Name
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| | 0 || s || 2 || [[sphere]] || '''s'''harp
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| | 1 || p || 6 || two [[dumbbell]]s || '''p'''rincipal
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| | 2 || d || 10 || four dumbbells or unique shape one || '''d'''iffuse
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| | 3 || f || 14 || eight dumbbells or unique shape two || '''f'''undamental
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| | 4 || g || 18 || ||
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| | 5 || h || 22 || ||
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| | 6 || i || 26 || ||
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| |}
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| The letters after the ''f'' sub-shell just follow ''f'' in alphabetical order except those already used. One [[mnemonic]] to remember the sequence ''S''. ''P''. ''D''. ''F''. ''G''. ''H''. ... is "'''S'''ober '''P'''hysicists '''D'''on't '''F'''ind '''G'''iraffes '''H'''iding '''I'''n '''K'''itchens '''L'''ike '''M'''y '''N'''ephew". A few other mnemonics are '''S'''mart '''P'''eople '''D'''on't '''F'''ail, '''S'''illy '''P'''eople '''D'''rive '''F'''ast, '''s'''illy '''p'''rofessors '''d'''ance '''f'''unny, '''S'''cott '''p'''icks '''d'''ead '''f'''lowers, '''s'''ome '''p'''oor '''d'''umb '''f'''ool! etc.
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| Each of the different angular momentum states can take 2(2''ℓ'' + 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 −''ℓ'' to ''ℓ'' in integer units, and so there are 2''ℓ'' + 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>''), giving 2(2''ℓ'' + 1) electrons overall. Orbitals with higher ''ℓ'' than given in the table are perfectly permissible, but these values cover all atoms so far discovered.
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| For a given value of the [[principal quantum number]] ''n'', the possible values of ''ℓ'' range from 0 to ''n'' − 1; therefore, the ''n'' = 1 shell only possesses an s subshell and can only take 2 electrons, the ''n'' = 2 shell possesses an s and a p subshell and can take 8 electrons overall, the ''n'' = 3 shell possesses s, p and d subshells and has a maximum of 18 electrons, and so on. Generally speaking, the maximum number of electrons in the ''n''th energy level is 2''n''<sup>2</sup>.
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| The angular momentum quantum number, ''ℓ'', governs 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 magnitude. 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. ''L'' has the value √2''ħ''.
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| 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]]:
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| :''n'' = 1, ''L'' = 0, [[Lyman series]] (ultraviolet)
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| :''n'' = 2, ''L'' = √2''ħ'', [[Balmer series]] (visible)
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| :''n'' = 3, ''L'' = √6''ħ'', [[Paschen series|Ritz-Paschen series]] ([[near infrared]])
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| :''n'' = 4, ''L'' = 2√3''ħ'', [[Brackett series]] ([[infrared#Commonly_used_sub-division_scheme|short-wavelength infrared]])
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| :''n'' = 5, ''L'' = 2√5''ħ'', [[Pfund series]] ([[infrared#Commonly_used_sub-division_scheme|mid-wavelength infrared]]).
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| == Addition of quantized angular momenta ==
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| {{details|Angular momentum coupling}}
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| Given a quantized total angular momentum <math>\vec{\jmath}</math> which is the sum of two individual quantized angular momenta <math>\vec{\ell_1}</math> and <math>\vec{\ell_2}</math>,
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| :<math>\vec{\jmath} = \vec{\ell_1} + \vec{\ell_2}</math>
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| 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
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| where <math>\ell_1</math> and <math>\ell_2</math> are quantum numbers corresponding to the magnitudes of the individual angular momenta.
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| === Total angular momentum of an electron in the atom ===
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| [[File:LS coupling.svg|250px|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]].]] | |
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| Due to the [[spin-orbit interaction]] in the atom, the orbital angular momentum no longer [[Commutativity|commutes]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], nor does the [[Spin (physics)|spin]]. These therefore change over time. However the [[total angular momentum]] '''J''' does [[Commutativity|commute]] with the Hamiltonian and so is constant. '''J''' is defined through
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| :<math>\vec{J} = \vec{L} + \vec{S}</math>
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| '''L''' being the [[angular momentum operator|orbital angular momentum]] and '''S''' the spin. The total angular momentum satisfies the same [[Angular momentum operator#Commutation relations|commutation relations as orbital angular momentum]], namely
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| :<math>[J_i, J_j ] = i \hbar \epsilon_{ijk} J_k</math>
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| from which follows
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| :<math>\left[J_i, J^2 \right] = 0</math> | |
| where ''J''<sub>i</sub> stand for ''J''<sub>x</sub>, ''J''<sub>y</sub>, and ''J''<sub>z</sub>.
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| The [[quantum number]]s describing the system, which are constant over time, are now ''j'' and ''m''<sub>''j''</sub>, defined through the action of '''J''' on the [[wavefunction]] <math>\Psi</math>
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| :<math>\mathbf{J}^2\Psi = \hbar^2{j(j+1)}\Psi</math> | |
| :<math>\mathbf{J}_z\Psi = \hbar{m_j}\Psi</math>
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| So that ''j'' is related to the norm of the total angular momentum and ''m''<sub>''j''</sub> to its projection along a specified axis.
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| As with any [[Angular momentum operator|angular momentum in quantum mechanics]], the projection of '''J''' along other axes cannot be co-defined with ''J''<sub>z</sub>, because they do not [[Commutativity|commute]].
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| ==== Relation between new and old quantum numbers ==== | |
| {{details|Quantum number#Quantum numbers with spin-orbit interaction}}
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| ''j'' and ''m''<sub>''j''</sub>, together with the [[Parity (physics)|parity]] of the [[quantum state]], replace the three [[quantum numbers]] ''ℓ'', ''m''<sub>''ℓ''</sub> and ''m''<sub>''s''</sub> (the projection of the [[Spin (physics)|spin]] along the specified axis). The former quantum numbers can be related to the latter.
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| Furthermore, the [[eigenvector]]s of ''j'', ''m''<sub>''j''</sub> and parity, which are also [[eigenvector]]s of the [[Hamiltonian (quantum mechanics)|Hamiltonian]], are linear combinations of the [[eigenvector]]s of ''ℓ'', ''m''<sub>''ℓ''</sub> and ''m''<sub>''s''</sub>.
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| == List of angular momentum quantum numbers ==
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| * Intrinsic (or spin) angular momentum quantum number, or simply [[spin quantum number]]
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| * orbital angular momentum quantum number (the subject of this article)
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| * [[magnetic quantum number]], related to the orbital momentum quantum number
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| * [[total angular momentum quantum number]]
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| == History ==
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| The [[azimuthal]] quantum number was carried over from the [[Bohr atom|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 the atom 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.<ref>{{cite journal|title=Note on "pendulum" orbits in atomic models|journal=Proc. Nat. Acad. Sci.|year=1927|volume=13|pages=413-419|url=http://www.pubmedcentral.gov/picrender.fcgi?tool=pmcentrez&blobtype=pdf&artid=1085028|author=R.B. Lindsay|authorlink=Robert Bruce Lindsay}}</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.
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| == See also ==
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| * [[Angular momentum operator]]
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| * [[Basic quantum mechanics]]
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| * [[Particle in a spherically symmetric potential]]
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| * [[Quantum number]]
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| ** [[Magnetic quantum number]]
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| ** [[Principal quantum number]]
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| ** [[Spin quantum number]]
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| ** [[Total angular momentum quantum number]]
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| * [[Angular momentum coupling]]
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| * [[Clebsch–Gordan coefficients]]
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| == References ==
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| <references/>
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| == External links ==
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| * [http://galileo.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html Development of the Bohr atom]
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| * [http://itl.chem.ufl.edu/ao_pict/ao_pict.html Pictures of atomic orbitals]
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| * [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]
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| * [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html#c1 The azimuthal equation explained]
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| {{electron configuration navbox}}
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| [[Category:Atomic physics]]
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| [[Category:Rotational symmetry]]
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| [[de:Quantenzahl#Nebenquantenzahl]]
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