Editing Magnetic quantum number (section)

==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
|}