Editing Schrödinger equation (section)

=== Separation of variables ===
If the Hamiltonian is not an explicit function of time, Schrödinger's equation reads: 
<math display="block">i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ - \frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right ] \Psi(\mathbf{r},t).</math> The operator on the left side depends only on time; the one on the right side depends only on space.
Solving the equation by [[separation of variables]] means seeking a solution of the form of a product of spatial and temporal parts<ref>{{Cite journal|last=Singh|first=Chandralekha|author-link=Chandralekha Singh|date=March 2008|title=Student understanding of quantum mechanics at the beginning of graduate instruction|url=http://aapt.scitation.org/doi/10.1119/1.2825387|journal=American Journal of Physics|language=en|volume=76|issue=3|pages=277–287|doi=10.1119/1.2825387|arxiv=1602.06660 |bibcode=2008AmJPh..76..277S |s2cid=118493003 |issn=0002-9505}}</ref>
<math display="block">\Psi(\mathbf{r},t)=\psi(\mathbf{r})\tau(t),</math>
where <math>\psi(\mathbf{r})</math> is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and <math>\tau(t)</math> is a function of time only. Substituting this expression for <math>\Psi</math> into the time dependent left hand side shows that <math>\tau(t)</math> is a phase factor:
<math display="block"> \Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i{E t/\hbar}}.</math>
A solution of this type is called ''stationary,'' since the only time dependence is a phase factor that cancels when the probability density is calculated via the Born rule.<ref name=Shankar1994/>{{rp|143ff}}

The spatial part of the full wave function solves the equation<ref name="Adams Sigel Mlynek 1994 pp. 143–210">{{cite journal | last1=Adams | first1=C.S | last2=Sigel | first2=M | last3=Mlynek | first3=J | title=Atom optics | journal=Physics Reports | publisher=Elsevier BV | volume=240 | issue=3 | year=1994 | issn=0370-1573 | doi=10.1016/0370-1573(94)90066-3 | pages=143–210| bibcode=1994PhR...240..143A | doi-access=free }}</ref>
<math display="block"> \nabla^2\psi(\mathbf{r}) + \frac{2m}{\hbar^2} \left [E - V(\mathbf{r})\right ] \psi(\mathbf{r}) = 0,</math>
where the energy <math>E</math> appears in the phase factor.

This generalizes to any number of particles in any number of dimensions (in a time-independent potential): the [[standing wave]] solutions of the time-independent equation are the states with definite energy, instead of a probability distribution of different energies. In physics, these standing waves are called "[[stationary state]]s" or "[[energy eigenstate]]s"; in chemistry they are called "[[atomic orbital]]s" or "[[molecular orbital]]s". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels. The energy eigenstates form a basis: any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is an example of the [[spectral theorem]], and in a finite-dimensional state space it is just a statement of the completeness of the eigenvectors of a [[Hermitian matrix]].

Separation of variables can also be a useful method for the time-independent Schrödinger equation. For example, depending on the symmetry of the problem, the [[Cartesian coordinates|Cartesian axes]] might be separated, as in
<math display="block">\psi(\mathbf{r}) = \psi_x(x)\psi_y(y)\psi_z(z),</math>
or [[spherical coordinates|radial and angular coordinates]] might be separated:
<math display="block">\psi(\mathbf{r}) = \psi_r(r)\psi_\theta(\theta)\psi_\phi(\phi).</math>