Editing Magnetic quantum number

{{short description|Number describing angular momentum along an axis}}
{{more citations needed|date=May 2016}}

In [[atomic physics]], a '''magnetic quantum number''' is a [[quantum number]] used to distinguish quantum states of an [[electron]] or other particle according to its [[angular momentum]] along a given axis in space. The '''orbital magnetic quantum number''' ({{mvar|m<sub>l</sub>}} or {{mvar|m}}{{efn|{{mvar|m}} is often used when only one kind of magnetic quantum number, such as {{mvar|m<sub>l</sub>}} or {{mvar|m<sub>j</sub>}}, is used in a text.}}) distinguishes the [[Atomic orbital|orbitals]] available within a given [[Electron shell|subshell]] of an atom. It specifies the component of the orbital angular momentum that lies along a given axis, conventionally called the ''z''-axis, so it describes the orientation of the orbital in space. The '''spin magnetic quantum number''' {{mvar|m<sub>s</sub>}} specifies the ''z''-axis component of the [[Spin (physics)|spin angular momentum]] for a particle having [[spin quantum number]] {{mvar|s}}. For an electron, {{mvar|s}} is {{frac|1|2}}, and {{mvar|m<sub>s</sub>}} is either +{{1/2}} or −{{1/2}}, often called "spin-up" and "spin-down", or α and β.<ref name="NIST 2019">{{cite web |last1=Martin |first1=W. C. |last2=Wiese |first2=W. L. |title=Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas |url=https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas |website=National Institute of Standards and Technology, Physical Measurement Laboratory |publisher=NIST |date=2019 |access-date=17 May 2023}}</ref><ref name="Atkins 1991">{{cite book | last=Atkins | first=Peter William | title=Quanta: A Handbook of Concepts |edition=2nd | publisher=Oxford University Press, USA | date=1991 | isbn=0-19-855572-5 | page=297}}</ref> The term ''magnetic'' in the name refers to the [[magnetic dipole moment]] associated with each type of angular momentum, so states having different magnetic quantum numbers shift in energy in a magnetic field according to the [[Zeeman effect]].<ref name="Atkins 1991" />

The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the [[principal quantum number]] ''n'',  the [[azimuthal quantum number|azimuthal (orbital) quantum number]] <math>\ell</math>, and the magnetic quantum numbers {{mvar|m<sub>l</sub>}} and {{mvar|m<sub>s</sub>}}. Electrons in a given subshell of an atom (such as s, p, d, or f) are defined by values of <math>\ell</math> (0, 1, 2, or 3). The orbital magnetic quantum number takes integer values in the range from <math>-\ell</math> to <math>+\ell</math>, including zero.<ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to quantum mechanics |date=2005 |publisher=Pearson Prentice Hall |isbn=0-13-111892-7 |edition=2nd |location=Upper Saddle River, NJ |pages=136–137 |oclc=53926857}}</ref> Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each. Each of these orbitals can accommodate up to two electrons (with opposite spins), forming the basis of the [[periodic table]].

Other magnetic quantum numbers are similarly defined, such as {{mvar|m<sub>j</sub>}} for the ''z''-axis component the [[Total angular momentum quantum number|total electronic angular momentum]] {{mvar|j}},<ref name="NIST 2019" /> and {{mvar|m<sub>I</sub>}} for the [[Spin quantum number#Nuclear spin|nuclear spin]] {{mvar|I}}.<ref name="Atkins 1991" /> Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as {{mvar|M<sub>L</sub>}} or {{mvar|m<sub>L</sub>}} for the total ''z''-axis orbital angular momentum of all the electrons in an atom.<ref name="Atkins 1991" />

==Derivation==
[[File:Atomic orbitals spdf m-eigenstates.png|thumb|These orbitals have magnetic quantum numbers <math>m_l=-\ell, \ldots,\ell</math> from left to right in ascending order. The <math>e^{m_li\phi}</math> dependence of the azimuthal component can be seen as a color gradient repeating <math>m_l</math> times around the vertical axis.]]
There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers <math>n</math>, <math>\ell</math>, <math>m_l</math>, and <math>m_s</math> specify the complete [[quantum state]] of a single electron in an atom called its [[wavefunction]] or orbital.  The [[Schrödinger equation]] for the wavefunction of an atom with one electron is a [[separable partial differential equation]].  (This is not the case for the neutral [[helium atom]] or other atoms with mutually interacting electrons, which require more sophisticated methods for solution<ref>{{cite web|url=http://farside.ph.utexas.edu/teaching/qmech/Quantum/node128.html|title=Helium atom|date=2010-07-20}}</ref>)  This means that the wavefunction as expressed in [[spherical coordinates]] can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:<ref>{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c3|title=Hydrogen Schrodinger Equation|website=hyperphysics.phy-astr.gsu.edu}}</ref>

:<math> \psi(r,\theta,\phi) = R(r)P(\theta)F(\phi)</math>

The differential equation for <math>F</math> can be solved in the form <math> F(\phi) = A e ^{\lambda\phi} </math>.  Because values of the azimuth angle <math>\phi</math> differing by 2<math>\pi</math> [[radians]] (360 degrees) represent the same position in space, and the overall magnitude of <math>F</math> does not grow with arbitrarily large <math>\phi</math> as it would for a real exponent, the coefficient <math>\lambda</math> must be quantized to integer multiples of <math>i</math>, producing an [[imaginary exponent]]: <math>\lambda = i m_l</math>.<ref>{{Cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html|title=Hydrogen Schrodinger Equation|website=hyperphysics.phy-astr.gsu.edu}}</ref>  These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of <math>{m_l}^2</math> tend to decrease the magnitude of <math>P(\theta),</math> and values of <math>m_l</math> greater than the azimuthal quantum number <math>\ell</math> do not permit any solution for <math>P(\theta). </math>

{| class="wikitable"
|-
! colspan="4" | '''Relationship between Quantum Numbers'''
|-
! Orbital
! Values
! Number of Values for <math>m_l</math><ref name=h50>{{cite book|last1=Herzberg|first1=Gerhard|title=Molecular Spectra and Molecular Structure|date=1950|publisher=D van Nostrand Company|pages=17–18|edition=2}}</ref>
! Electrons per subshell
|-
! s
|<math>\ell=0,\quad m_l=0</math>|| 1 || 2
|-
! p
|<math>\ell=1,\quad m_l=-1,0,+1</math>|| 3 || 6
|-
! d
|<math>\ell=2,\quad m_l=-2,-1,0,+1,+2</math>|| 5 || 10
|-
! f
|<math>\ell=3,\quad m_l = -3,-2,-1,0,+1,+2,+3</math>|| 7 || 14
|-
! g
|<math>\ell=4,\quad m_l = -4,-3,-2,-1,0,+1,+2,+3,+4</math>|| 9 || 18
|}

== As a component of angular momentum ==
[[File:Vector model of orbital angular momentum.svg|250px|right|thumb|Illustration of quantum mechanical orbital angular momentum. The cones and plane represent possible orientations of the angular momentum vector for <math>\ell = 2</math> and <math>m_l = -2, -1, 0, 1, 2</math>. Even for the extreme values of <math>m_l</math>, the <math>z</math>-component of this vector is less than its total magnitude.]]
The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number <math>m_l</math> refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the <math>z</math>-direction or [[quantization axis]]. <math>L_z</math>, the magnitude of the angular momentum in the <math>z</math>-direction, is given by the formula:<ref name=h50/>

:<math>L_z = m_l \hbar</math>.

This is a component of the atomic electron's total orbital angular momentum <math>\mathbf{L}</math>, whose magnitude is related to the azimuthal quantum number of its subshell <math>\ell</math> by the equation:

:<math>L = \hbar \sqrt{\ell (\ell + 1)}</math>,

where <math>\hbar</math> is the [[reduced Planck constant]]. Note that this <math>L = 0</math> for <math>\ell = 0</math> and approximates <math>L = \left( \ell + \tfrac{1}{2} \right) \hbar</math> for high <math>\ell</math>. It is not possible to measure the angular momentum of the electron along all three axes simultaneously. These properties were first demonstrated in the [[Stern–Gerlach experiment]], by [[Otto Stern]] and [[Walther Gerlach]].<ref>{{cite web|url=http://www.britannica.com/science/spectroscopy/Types-of-electromagnetic-radiation-sources#ref620216|title=Spectroscopy: angular momentum quantum number|publisher=Encyclopædia Britannica}}</ref>

== Effect in magnetic fields ==
The quantum number <math>m_l</math> refers, loosely, to the direction of the  [[angular momentum]] [[Vector (geometric)|vector]].  The magnetic quantum number <math>m_l</math> only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of <math>m_l</math> are equivalent.  The magnetic quantum number determines the energy shift of an [[atomic orbital]] due to an external magnetic field (the [[Zeeman effect]]) &mdash; hence the name ''magnetic'' quantum number.  However, the actual [[magnetic dipole moment]] of an electron in an atomic orbital arises not only from the electron angular momentum but also from the electron spin, expressed in the spin quantum number.

Since each electron has a magnetic moment in a magnetic field, it will be subject to a torque which tends to make the vector <math>\mathbf{L}</math> parallel to the field, a phenomenon known as [[Larmor precession]].

==See also==
* [[Quantum number]]
** [[Azimuthal quantum number]]
** [[Principal quantum number]]
** [[Spin quantum number]]
** [[Total angular momentum quantum number]]
* [[Electron shell]]
* [[Basic quantum mechanics]]
* [[Bohr atom]]
* [[Schrödinger equation]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

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[[Category:Atomic physics]]
[[Category:Rotational symmetry]]
[[Category:Quantum numbers]]