Editing Hydrogen atom (section)

==== Wavefunction ====
The [[Hamiltonian mechanics|Hamiltonian]] of the hydrogen atom is the radial kinetic energy operator plus the Coulomb electrostatic potential energy between the positive proton and the negative electron. Using the time-independent Schrödinger equation, ignoring all spin-coupling interactions and using the [[reduced mass]] <math>\mu = m_e M/(m_e + M)</math>, the equation is written as:
<math display="block">\left( -\frac{\hbar^2}{2 \mu} \nabla^2 - \frac{e^2}{4 \pi \varepsilon_0 r} \right) \psi (r, \theta, \varphi) = E \psi (r, \theta, \varphi)</math>

Expanding the [[Laplace operator|Laplacian]] in spherical coordinates:
<math display="block">-\frac{\hbar^2}{2 \mu} \left[ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \psi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \psi}{\partial \varphi^2} \right] - \frac{e^2}{4 \pi \varepsilon_0 r} \psi = E \psi</math>

This is a [[Separation of variables|separable]], [[partial differential equation]] which can be solved in terms of special functions. When the wavefunction is separated as product of functions <math>R(r)</math>, <math>\Theta(\theta)</math>, and <math>\Phi(\varphi)</math> three independent differential functions appears<ref>{{Cite web|title=Solving Schrödinger's equation for the hydrogen atom :: Atomic Physics :: Rudi Winter's web space|url=https://users.aber.ac.uk/ruw/teach/327/hatom.php|access-date=2020-11-30 |website=users.aber.ac.uk}}</ref> with A and B being the separation constants:
* radial: <math display="block">\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) + \frac{2\mu r^2}{\hbar^2} \left(E+\frac{e^2}{4\pi\varepsilon_0r}\right)R - AR = 0</math>
* polar: <math display="block">\frac{\sin\theta}{\Theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)+A\sin^2\theta- B = 0</math>
* azimuth: <math display="block">\frac{1}{\Phi} \frac{d^2\Phi}{d\varphi^2}+B=0.</math>

The normalized position [[wavefunction]]s, given in [[spherical coordinates]] are:
<math display="block"> \psi_{n \ell m}(r, \theta, \varphi) = \sqrt{{\left( \frac{2}{n a^*_0} \right)}^3 \frac{(n - \ell - 1)!}{2 n (n + \ell)!}} \mathrm{e}^{-\rho / 2} \rho^{\ell} L_{n - \ell - 1}^{2 \ell + 1}(\rho) Y_\ell^m (\theta, \varphi)</math>

[[Image:Hydrogen eigenstate n4 l3 m1.png|thumb|right|3D illustration of the eigenstate <math>\psi_{4, 3, 1}</math>. Electrons in this state are 45% likely to be found within the solid body shown.]]
where:
* <math>\rho = {2 r \over {n a^*_0}}</math>,
* <math>a^*_0</math> is the [[reduced Bohr radius]], <math>a^*_0 = {{4 \pi \varepsilon_0 \hbar^2} \over {\mu e^2}}</math>,
* <math>L_{n-\ell-1}^{2\ell+1}(\rho) </math> is a [[Laguerre polynomial#Generalized Laguerre polynomials|generalized Laguerre polynomial]] of degree <math>n - \ell - 1</math>, and
* <math>Y_\ell^m (\theta, \varphi)</math> is a [[spherical harmonic]] function of degree <math>\ell</math> and order <math>m</math>. 

Note that the generalized Laguerre polynomials are defined differently by different authors. The usage here is consistent with the definitions used by Messiah,<ref>{{cite book |last=Messiah|first=Albert| title=Quantum Mechanics| date=1999|publisher=Dover| location=New York|isbn=0-486-40924-4 |pages=1136}}</ref> and Mathematica.<ref>[http://reference.wolfram.com/mathematica/ref/LaguerreL.html LaguerreL]. Wolfram Mathematica page</ref> In other places, the Laguerre polynomial includes a factor of <math>(n + \ell) !</math>,<ref>Griffiths, p. 152</ref> or the generalized Laguerre polynomial appearing in the hydrogen wave function is <math>L_{n + \ell}^{2 \ell + 1} (\rho)</math> instead.<ref>{{cite book|last=Condon and Shortley | title=The Theory of Atomic Spectra| date=1963 |publisher=Cambridge |location=London |pages=441}}</ref>

The quantum numbers can take the following values:
* <math>n = 1, 2, 3, \ldots</math> ([[principal quantum number]])
* <math>\ell = 0, 1, 2, \ldots, n - 1</math> ([[azimuthal quantum number]])
* <math>m=-\ell, \ldots, \ell</math> ([[magnetic quantum number]]).

Additionally, these wavefunctions are ''normalized'' (i.e., the integral of their modulus square equals 1) and [[Orthogonal functions|orthogonal]]:
<math display="block">\int_0^{\infty} r^2 \, dr \int_0^{\pi} \sin \theta \, d\theta \int_0^{2 \pi} d\varphi \, \psi^*_{n \ell m} (r, \theta, \varphi) \psi_{n' \ell' m'} (r, \theta, \varphi) = \langle n, \ell, m | n', \ell', m' \rangle = \delta_{n n'} \delta_{\ell \ell'} \delta_{m m'},</math>
where <math>| n, \ell, m \rangle</math> is the state represented by the wavefunction <math>\psi_{n \ell m}</math> in [[Dirac notation]], and <math>\delta</math> is the [[Kronecker delta]] function.<ref>Griffiths, Ch. 4 p. 89</ref>

The wavefunctions in momentum space are related to the wavefunctions in position space through a Fourier transform
<math display="block">\varphi (p, \theta_p, \varphi_p) = (2 \pi \hbar)^{-3 / 2} \int \mathrm{e}^{-i \vec{p} \cdot \vec{r} / \hbar} \psi (r, \theta,\varphi) \, dV,</math>
which, for the bound states, results in<ref>{{cite book | first=B. H. | last=Bransden | author2=Joachain, C. J.  | title=Physics of Atoms and Molecules | publisher=[[Longman]] | date=1983 | isbn=0-582-44401-2| page = Appendix 5
}}</ref>
<math display="block">\varphi (p, \theta_p, \varphi_p) = \sqrt{\frac{2}{\pi} \frac{(n - \ell - 1)!}{(n + \ell)!}} n^2 2^{2 \ell + 2} \ell! \frac{n^\ell p^\ell}{(n^2 p^2 + 1)^{\ell + 2}} C_{n - \ell - 1}^{\ell + 1} \left( \frac{n^2 p^2 - 1}{n^2 p^2 + 1} \right) Y_\ell^m (\theta_p, \varphi_p),</math>
where <math>C_N^\alpha (x)</math> denotes a [[Gegenbauer polynomial]] and <math>p</math> is in units of <math>\hbar / a^*_0</math>.

The solutions to the Schrödinger equation for hydrogen are [[analytical expression|analytical]], giving a simple expression for the hydrogen [[energy levels]] and thus the frequencies of the hydrogen [[spectral line]]s and fully reproduced the Bohr model and went beyond it. It also yields two other quantum numbers and the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states, thus explaining the [[anisotropic]] character of atomic bonds.

The Schrödinger equation also applies to more complicated atoms and [[molecule]]s. When there is more than one electron or nucleus the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made.

Since the Schrödinger equation is only valid for non-relativistic quantum mechanics, the solutions it yields for the hydrogen atom are not entirely correct. The [[Dirac equation]] of relativistic quantum theory improves these solutions (see below).