Editing Schrödinger equation (section)

=== Hydrogen atom ===
[[File:Hydrogen Density Plots.png|thumb|[[Wave function]]s of the [[electron]] in a hydrogen atom at different [[energy level]]s. They are plotted according to solutions of the Schrödinger equation.]]
The Schrödinger equation for the electron in a [[hydrogen atom]] (or a hydrogen-like atom) is
<math display="block"> E \psi = -\frac{\hbar^2}{2\mu}\nabla^2\psi - \frac{q^2}{4\pi\varepsilon_0 r}\psi </math>
where <math> q </math> is the electron charge, <math> \mathbf{r} </math> is the position of the electron relative to the nucleus, <math> r = |\mathbf{r}| </math> is the magnitude of the relative position, the potential term is due to the [[Coulomb's law|Coulomb interaction]], wherein <math> \varepsilon_0 </math> is the [[permittivity of free space]] and
<math display="block"> \mu = \frac{m_q m_p}{m_q+m_p} </math>
is the 2-body [[reduced mass]] of the hydrogen [[Nucleus (atomic structure)|nucleus]] (just a [[proton]]) of mass <math> m_p </math> and the electron of mass <math> m_q </math>. The negative sign arises in the potential term since the proton and electron are oppositely charged. The reduced mass in place of the electron mass is used since the electron and proton together orbit each other about a common center of mass, and constitute a two-body problem to solve. The motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced mass.

The Schrödinger equation for a hydrogen atom can be solved by separation of variables.<ref>{{cite book|title=Physics for Scientists and Engineers – with Modern Physics |edition=6th |first1=P. A. |last1=Tipler |first2=G. |last2=Mosca |publisher=Freeman |year=2008 |isbn=978-0-7167-8964-2}}</ref> In this case, [[spherical polar coordinates]] are the most convenient. Thus,
<math display="block"> \psi(r,\theta,\varphi) = R(r)Y_\ell^m(\theta, \varphi) = R(r)\Theta(\theta)\Phi(\varphi),</math>
where {{math|''R''}} are radial functions and <math> Y^m_l (\theta, \varphi) </math> are [[spherical harmonic]]s of degree <math> \ell </math> and order <math> m </math>. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are:<ref>{{cite book|first=David J. |last=Griffiths |author-link=David J. Griffiths |title=Introduction to Elementary Particles|url=https://books.google.com/books?id=w9Dz56myXm8C&pg=PA162 | access-date=27 June 2011|year=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|pages=162–}}</ref>
<math display="block"> \psi_{n\ell m}(r,\theta,\varphi) = \sqrt {\left (  \frac{2}{n a_0} \right )^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^\ell L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^m(\theta, \varphi ) </math>
where
* <math> a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_q q^2} </math> is the [[Bohr radius]],
* <math> L_{n-\ell-1}^{2\ell+1}(\cdots) </math> are the [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomials]] of degree <math> n - \ell - 1 </math>,
* <math> n, \ell, m </math> are the [[principal quantum number|principal]], [[azimuthal quantum number|azimuthal]], and [[magnetic quantum number|magnetic]] [[quantum numbers]] respectively, which take the values <math>n = 1, 2, 3, \dots,</math> <math>\ell = 0, 1, 2, \dots, n - 1,</math> <math>m = -\ell, \dots, \ell.</math>