Editing Eigenfunction (section)

===Schrödinger equation===
In [[quantum mechanics]], the [[Schrödinger equation]]
<math display="block">i \hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = H \Psi(\mathbf{r},t)</math>
with the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]]
<math display="block"> H = -\frac{\hbar^2}{2m}\nabla^2+ V(\mathbf{r},t)</math>
can be solved by separation of variables if the Hamiltonian does not depend explicitly on time.{{sfn|Davydov|1976|p=51}} In that case, the [[wave function]] {{math|1=Ψ('''r''',''t'') = ''φ''('''r''')''T''(''t'')}} leads to the two differential equations,
{{NumBlk||<math display="block"> H\varphi(\mathbf{r}) = E\varphi(\mathbf{r}),</math>|{{EquationRef|2}}}}
{{NumBlk||<math display="block"> i\hbar \frac{\partial T(t)}{\partial t} = ET(t).</math>|{{EquationRef|3}}}}
Both of these differential equations are eigenvalue equations with eigenvalue {{mvar|E}}. As shown in an earlier example, the solution of Equation {{EqNote|3}} is the exponential
<math display="block"> T(t) = e^{{-iEt}/{\hbar}}.</math>

Equation {{EqNote|2}} is the time-independent Schrödinger equation. The eigenfunctions {{mvar|φ<sub>k</sub>}} of the Hamiltonian operator are [[stationary state]]s of the quantum mechanical system, each with a corresponding energy {{mvar|E<sub>k</sub>}}. They represent allowable energy states of the system and may be constrained by boundary conditions.

The Hamiltonian operator {{mvar|H}} is an example of a Hermitian operator whose eigenfunctions form an orthonormal basis. When the Hamiltonian does not depend explicitly on time, general solutions of the Schrödinger equation are linear combinations of the stationary states multiplied by the oscillatory {{math|''T''(''t'')}},{{sfn|Davydov|1976|p=52}} <math display="inline"> \Psi(\mathbf{r},t) = \sum_k c_k \varphi_k(\mathbf{r}) e^{{-iE_kt}/{\hbar}} </math> or, for a system with a continuous spectrum,
<math display="block"> \Psi(\mathbf{r},t) = \int dE \, c_E \varphi_E(\mathbf{r}) e^{{-iEt}/{\hbar}}.</math>

The success of the Schrödinger equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.