Editing Hydrogen atom (section)

==== 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>.