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Magnetic quantum number

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Number describing angular momentum along an axis
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Inatomic physics, amagnetic quantum number is aquantum number used to distinguish quantum states of anelectron or other particle according to itsangular momentum along a given axis in space. Theorbital magnetic quantum number (ml orm[a]) distinguishes theorbitals available within a givensubshell of an atom. It specifies the component of the orbital angular momentum that lies along a given axis, conventionally called thez-axis, so it describes the orientation of the orbital in space. Thespin magnetic quantum numberms specifies thez-axis component of thespin angular momentum for a particle havingspin quantum numbers. For an electron,s is1⁄2, andms is either +1⁄2 or −1⁄2, often called "spin-up" and "spin-down", or α and β.[1][2] The termmagnetic in the name refers to themagnetic 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 theZeeman effect.[2]

The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are theprincipal quantum numbern, theazimuthal (orbital) quantum numberℓ{\displaystyle \ell }, and the magnetic quantum numbersml andms. Electrons in a given subshell of an atom (such as s, p, d, or f) are defined by values ofℓ{\displaystyle \ell } (0, 1, 2, or 3). The orbital magnetic quantum number takes integer values in the range from−ℓ{\displaystyle -\ell } to+ℓ{\displaystyle +\ell }, including zero.[3] 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 theperiodic table.

Other magnetic quantum numbers are similarly defined, such asmj for thez-axis component thetotal electronic angular momentumj,[1] andmI for thenuclear spinI.[2] Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such asML ormL for the totalz-axis orbital angular momentum of all the electrons in an atom.[2]

Derivation

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These orbitals have magnetic quantum numbersml=−ℓ,…,ℓ{\displaystyle m_{l}=-\ell ,\ldots ,\ell } from left to right in ascending order. Theemliϕ{\displaystyle e^{m_{l}i\phi }} dependence of the azimuthal component can be seen as a color gradient repeatingml{\displaystyle m_{l}} times around the vertical axis.

There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbersn{\displaystyle n},ℓ{\displaystyle \ell },ml{\displaystyle m_{l}}, andms{\displaystyle m_{s}} specify the completequantum state of a single electron in an atom called itswavefunction or orbital. TheSchrödinger equation for the wavefunction of an atom with one electron is aseparable partial differential equation. (This is not the case for the neutralhelium atom or other atoms with mutually interacting electrons, which require more sophisticated methods for solution[4]) This means that the wavefunction as expressed inspherical coordinates can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:[5]

ψ(r,θ,ϕ)=R(r)P(θ)F(ϕ){\displaystyle \psi (r,\theta ,\phi )=R(r)P(\theta )F(\phi )}

The differential equation forF{\displaystyle F} can be solved in the formF(ϕ)=Aeλϕ{\displaystyle F(\phi )=Ae^{\lambda \phi }}. Because values of the azimuth angleϕ{\displaystyle \phi } differing by 2π{\displaystyle \pi }radians (360 degrees) represent the same position in space, and the overall magnitude ofF{\displaystyle F} does not grow with arbitrarily largeϕ{\displaystyle \phi } as it would for a real exponent, the coefficientλ{\displaystyle \lambda } must be quantized to integer multiples ofi{\displaystyle i}, producing animaginary exponent:λ=iml{\displaystyle \lambda =im_{l}}.[6] These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values ofml2{\displaystyle {m_{l}}^{2}} tend to decrease the magnitude ofP(θ),{\displaystyle P(\theta ),} and values ofml{\displaystyle m_{l}} greater than the azimuthal quantum numberℓ{\displaystyle \ell } do not permit any solution forP(θ).{\displaystyle P(\theta ).}

Relationship between Quantum Numbers
OrbitalValuesNumber of Values forml{\displaystyle m_{l}}[7]Electrons per subshell
sℓ=0,ml=0{\displaystyle \ell =0,\quad m_{l}=0}12
pℓ=1,ml=−1,0,+1{\displaystyle \ell =1,\quad m_{l}=-1,0,+1}36
dℓ=2,ml=−2,−1,0,+1,+2{\displaystyle \ell =2,\quad m_{l}=-2,-1,0,+1,+2}510
fℓ=3,ml=−3,−2,−1,0,+1,+2,+3{\displaystyle \ell =3,\quad m_{l}=-3,-2,-1,0,+1,+2,+3}714
gℓ=4,ml=−4,−3,−2,−1,0,+1,+2,+3,+4{\displaystyle \ell =4,\quad m_{l}=-4,-3,-2,-1,0,+1,+2,+3,+4}918

As a component of angular momentum

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Illustration of quantum mechanical orbital angular momentum. The cones and plane represent possible orientations of the angular momentum vector forℓ=2{\displaystyle \ell =2} andml=−2,−1,0,1,2{\displaystyle m_{l}=-2,-1,0,1,2}. Even for the extreme values ofml{\displaystyle m_{l}}, thez{\displaystyle z}-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 numberml{\displaystyle m_{l}} refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called thez{\displaystyle z}-direction orquantization axis.Lz{\displaystyle L_{z}}, the magnitude of the angular momentum in thez{\displaystyle z}-direction, is given by the formula:[7]

Lz=mlℏ{\displaystyle L_{z}=m_{l}\hbar }.

This is a component of the atomic electron's total orbital angular momentumL{\displaystyle \mathbf {L} }, whose magnitude is related to the azimuthal quantum number of its subshellℓ{\displaystyle \ell } by the equation:

L=ℏℓ(ℓ+1){\displaystyle L=\hbar {\sqrt {\ell (\ell +1)}}},

whereℏ{\displaystyle \hbar } is thereduced Planck constant. Note that thisL=0{\displaystyle L=0} forℓ=0{\displaystyle \ell =0} and approximatesL=(ℓ+12)ℏ{\displaystyle L=\left(\ell +{\tfrac {1}{2}}\right)\hbar } for highℓ{\displaystyle \ell }. It is not possible to measure the angular momentum of the electron along all three axes simultaneously. These properties were first demonstrated in theStern–Gerlach experiment, byOtto Stern andWalther Gerlach.[8]

Effect in magnetic fields

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The quantum numberml{\displaystyle m_{l}} refers, loosely, to the direction of theangular momentumvector. The magnetic quantum numberml{\displaystyle m_{l}} 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 ofml{\displaystyle m_{l}} are equivalent. The magnetic quantum number determines the energy shift of anatomic orbital due to an external magnetic field (theZeeman effect) — hence the namemagnetic quantum number. However, the actualmagnetic 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 vectorL{\displaystyle \mathbf {L} } parallel to the field, a phenomenon known asLarmor precession.

See also

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Notes

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  1. ^m is often used when only one kind of magnetic quantum number, such asml ormj, is used in a text.

References

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  1. ^abMartin, W. C.; Wiese, W. L. (2019)."Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas".National Institute of Standards and Technology, Physical Measurement Laboratory. NIST. Retrieved17 May 2023.
  2. ^abcdAtkins, Peter William (1991).Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297.ISBN 0-19-855572-5.
  3. ^Griffiths, David J. (2005).Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. pp. 136–137.ISBN 0-13-111892-7.OCLC 53926857.
  4. ^"Helium atom". 2010-07-20.
  5. ^"Hydrogen Schrodinger Equation".hyperphysics.phy-astr.gsu.edu.
  6. ^"Hydrogen Schrodinger Equation".hyperphysics.phy-astr.gsu.edu.
  7. ^abHerzberg, Gerhard (1950).Molecular Spectra and Molecular Structure (2 ed.). D van Nostrand Company. pp. 17–18.
  8. ^"Spectroscopy: angular momentum quantum number". Encyclopædia Britannica.
Quantum numbers
Ground-state configurations
Electron filling
Electron pairing
Bonding participation
Electron counting rules
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