Editing Hydrogen atom (section)

=== Schrödinger equation ===
The Schrödinger equation is the standard quantum-mechanics model; it allows one to calculate the stationary states and also the time evolution of quantum systems.  Exact analytical answers are available for the nonrelativistic hydrogen atom. Before we go to present a formal account, here we give an elementary overview.

Given that the hydrogen atom contains a nucleus and an electron, quantum mechanics allows one to predict the probability of finding the electron at any given radial distance <math>r</math>. It is given by the square of a mathematical function known as the "[[wave function|wavefunction]]", which is a solution of the Schrödinger equation. The lowest energy equilibrium state of the hydrogen atom is known as the ground state. The ground state wave function is known as the <math>1\mathrm{s}</math> wavefunction. It is written as:
<math display="block">\psi_{1 \mathrm{s}} (r) = \frac{1}{\sqrt{\pi} a_0^{3 / 2}} \mathrm{e}^{-r / a_0}.</math>

Here, <math>a_0</math> is the numerical value of the Bohr radius. The probability density of finding the electron at a distance <math>r</math> in any radial direction is the squared value of the wavefunction:
<math display="block">| \psi_{1 \mathrm{s}} (r) |^2 = \frac{1}{\pi a_0^3} \mathrm{e}^{-2 r / a_0}.</math>

The <math>1 \mathrm{s}</math> wavefunction is spherically symmetric, and the surface area of a shell at distance <math>r</math> is <math>4 \pi r^2</math>, so the total probability <math>P(r) \, dr</math> of the electron being in a shell at a distance <math>r</math> and thickness <math>dr</math> is
<math display="block">P (r) \, \mathrm dr = 4 \pi r^2 | \psi_{1 \mathrm{s}} (r) |^2 \, \mathrm dr.</math>

It turns out that this is a maximum at <math>r = a_0</math>. That is, the Bohr picture of an electron orbiting the nucleus at radius <math>a_0</math> corresponds to the most probable radius. Actually, there is a finite probability that the electron may be found at any place <math>r</math>, with the
probability indicated by the square of the wavefunction. Since the probability of finding the electron ''somewhere'' in the whole volume is unity, the integral of <math>P(r) \, \mathrm dr</math> is unity. Then we say that the wavefunction is properly normalized.

As discussed below, the ground state <math>1 \mathrm{s}</math> is also indicated by the [[Quantum number#Electron in an atom|quantum numbers]] <math>(n = 1, \ell = 0, m = 0)</math>. The second lowest energy states, just above the ground state, are given by the quantum numbers <math>(2, 0, 0)</math>, <math>(2, 1, 0)</math>, and <math>(2, 1, \pm 1)</math>. These <math>n = 2</math> states all have the same energy and are known as the <math>2 \mathrm{s}</math> and <math>2 \mathrm{p}</math> states. There is one <math>2 \mathrm{s}</math> state:
<math display="block">\psi_{2, 0, 0} = \frac{1}{4 \sqrt{2 \pi} a_0^{3 / 2}} \left( 2 - \frac{r}{a_0} \right) \mathrm{e}^{-r / 2 a_0},</math>
and there are three <math>2 \mathrm{p}</math> states:
<math display="block">\psi_{2, 1, 0} = \frac{1}{4 \sqrt{2 \pi} a_0^{3 / 2}} \frac{r}{a_0} \mathrm{e}^{-r / 2 a_0} \cos \theta,</math>
<math display="block">\psi_{2, 1, \pm 1} = \mp \frac{1}{8 \sqrt{\pi} a_0^{3/2}} \frac{r}{a_0} \mathrm{e}^{-r / 2 a_0} \sin \theta ~ e^{\pm i \varphi}.</math>

An electron in the <math>2 \mathrm{s}</math> or <math>2 \mathrm{p}</math> state is most likely to be found in the second Bohr orbit with energy given by the Bohr formula.

==== 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).

==== Results of Schrödinger equation ====
The solution of the Schrödinger equation (wave equation) for the hydrogen atom uses the fact that the [[Coulomb's law|Coulomb potential]] produced by the nucleus is [[isotropic]] (it is radially symmetric in space and only depends on the distance to the nucleus). Although the resulting [[energy eigenfunctions]] (the ''orbitals'') are not necessarily isotropic themselves, their dependence on the [[Spherical coordinate system|angular coordinates]] follows completely generally from this isotropy of the underlying potential: the [[eigenstates]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the [[angular momentum operator]]. This corresponds to the fact that angular momentum is conserved in the [[orbital motion (quantum)|orbital motion]] of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum [[quantum number]]s, <math>\ell</math> and <math>m</math> (both are integers). The angular momentum quantum number <math>\ell = 0, 1, 2, \ldots</math> determines the magnitude of the angular momentum. The magnetic quantum number <math>m = -\ell, \ldots, +\ell</math> determines the projection of the angular momentum on the (arbitrarily chosen) <math>z</math>-axis.

In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found. It is only here that the details of the <math>1 / r</math> Coulomb potential enter (leading to [[Laguerre polynomials]] in <math>r</math>). This leads to a third quantum number, the principal quantum number <math>n = 1, 2, 3, \ldots</math>. The principal quantum number in hydrogen is related to the atom's total energy.

Note that the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to <math>n - 1</math>, i.e., <math>\ell = 0, 1, \ldots, n - 1</math>.

Due to angular momentum conservation, states of the same <math>\ell</math> but different <math>m</math> have the same energy (this holds for all problems with [[rotational symmetry]]). In addition, for the hydrogen atom, states of the same <math>n</math> but different <math>\ell</math> are also [[degenerate energy levels|degenerate]] (i.e., they have the same energy). However, this is a specific property of hydrogen and is no longer true for more complicated atoms which have an (effective) potential differing from the form <math>1 / r</math> (due to the presence of the inner electrons shielding the nucleus potential).

Taking into account the [[Spin (physics)|spin]] of the electron adds a last quantum number, the projection of the electron's spin angular momentum along the <math>z</math>-axis, which can take on two values. Therefore, any [[eigenstate]] of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any [[quantum superposition|superposition]] of these states. This explains also why the choice of <math>z</math>-axis for the directional [[quantization (physics)|quantization]] of the angular momentum vector is immaterial: an orbital of given <math>\ell</math> and <math>m'</math> obtained for another preferred axis <math>z'</math> can always be represented as a suitable superposition of the various states of different <math>m</math> (but same <math>\ell</math>) that have been obtained for <math>z</math>.

==== Mathematical summary of eigenstates of hydrogen atom ====
{{Main|Hydrogen-like atom}}

In 1928, [[Paul Dirac]] found [[Dirac equation|an equation]] that was fully compatible with [[special relativity]], and (as a consequence) made the wave function a 4-component "[[Dirac spinor]]" including "up" and "down" spin components, with both positive and "negative" energy (or matter and antimatter). The solution to this equation gave the following results, more accurate than the Schrödinger solution.

=====Energy levels=====
The energy levels of hydrogen, including [[fine structure]] (excluding [[Lamb shift]] and [[hyperfine structure]]), are given by the [[Fine-structure constant|Sommerfeld fine-structure]] expression:<ref name="Sommerfeld">{{cite book | first=Arnold |last= Sommerfeld |title=Atombau und Spektrallinien | trans-title=Atomic Structure and Spectral Lines | publisher=Friedrich Vieweg und Sohn| location=Braunschweig|year=1919| isbn=3-87144-484-7}} [https://archive.org/stream/atombauundspekt00sommgoog German] [https://archive.org/details/AtomicStructureAndSpectralLines English]</ref>

<math display="block">\begin{align}
E_{j \, n} = {} & -\mu c^2 \left[ 1 - \left( 1 + \left[ \frac{\alpha}{n - j - \frac{1}{2} + \sqrt{\left( j + \frac{1}{2} \right)^2 - \alpha^2}} \right]^2 \right)^{-1 / 2} \right] \\
\approx {} & -\frac{\mu c^2 \alpha^2}{2 n^2} \left[ 1 + \frac{\alpha^2}{n^2} \left( \frac{n}{j + \frac{1}{2}} - \frac{3}{4} \right) \right],
\end{align}</math>

where <math>\alpha</math> is the [[fine-structure constant]] and <math>j</math> is the [[total angular momentum quantum number]], which is equal to <math>\left| \ell \pm \tfrac{1}{2} \right|</math>, depending on the orientation of the electron spin relative to the orbital angular momentum.<ref>{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |title=Physical Chemistry |date=2006 |publisher=W. H. Freeman |isbn=0-7167-8759-8 |page=[https://archive.org/details/atkinsphysicalch00pwat/page/349 349] |edition=8th |url=https://archive.org/details/atkinsphysicalch00pwat/page/349 }}</ref> This formula represents a small correction to the energy obtained by Bohr and Schrödinger as given above. The factor in square brackets in the last expression is nearly one; the extra term arises from relativistic effects (for details, see [[#Features going beyond the Schrödinger solution]]). It is worth noting that this expression was first obtained by [[Arnold Sommerfeld|A. Sommerfeld]] in 1916 based on the relativistic version of the [[Old quantum theory|old Bohr theory]]. Sommerfeld has however used different notation for the quantum numbers.