Editing Atomic orbital (section)

== Hydrogen-like orbitals ==
{{Main|Hydrogen-like atom}}

The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the [[hydrogen atom]]. An atom of any other element [[ion]]ized down to a single electron (He<sup>+</sup>, Li<sup>2+</sup>, etc.) is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the [[eigenstates]] of the [[Hamiltonian operator]] for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a [[polynomial]] series, and [[Exponential function|exponential]] and [[trigonometric functions]]. (see [[hydrogen atom]]).

For atoms with two or more electrons, the governing equations can be solved only with the use of methods of iterative approximation. Orbitals of multi-electron atoms are ''qualitatively'' similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used.

A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: {{mvar|[[Principal quantum number|n]]|size=120%}}, {{mvar|[[Azimuthal quantum number|ℓ]]|size=120%}}, and {{mvar|[[magnetic quantum number|m<sub>ℓ</sub>]]|size=120%}}. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the [[periodic table]].

The stationary states ([[quantum state]]s) of a hydrogen-like atom are its atomic orbitals. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" ([[linear combination]]s) of multiple orbitals. See [[Linear combination of atomic orbitals molecular orbital method]].

The quantum number {{mvar|n}} first appeared in the [[Bohr model]] where it determines the radius of each circular electron orbit. In modern quantum mechanics however, {{mvar|n}} determines the mean distance of the electron from the nucleus; all electrons with the same value of ''n'' lie at the same average distance. For this reason, orbitals with the same value of ''n'' are said to comprise a "[[electron shell|shell]]". Orbitals with the same value of ''n'' and also the same value of&nbsp;{{mvar|ℓ}} are even more closely related, and are said to comprise a "[[electron subshell|subshell]]".