Editing Eigenfunction (section)

==Applications==
===Vibrating strings===
[[File:Standing wave.gif|thumb|270px|The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of oscillation.]]
Let {{math|''h''(''x'', ''t'')}} denote the transverse displacement of a stressed elastic chord, such as the [[vibrating string]]s of a [[string instrument]], as a function of the position {{mvar|x}} along the string and of time {{mvar|t}}. Applying the laws of mechanics to [[infinitesimal]] portions of the string, the function {{mvar|h}} satisfies the [[partial differential equation]]
<math display="block" alt="the second partial derivative of h with respect to t equals c squared times the second partial derivative of h with respect to x">\frac{\partial^2 h}{\partial t^2} = c^2\frac{\partial^2 h}{\partial x^2},</math>
which is called the (one-dimensional) [[wave equation]]. Here {{mvar|c}} is a constant speed that depends on the tension and mass of the string.

This problem is amenable to the method of [[separation of variables]]. If we assume that {{math|''h''(''x'', ''t'')}} can be written as the product of the form {{math|''X''(''x'')''T''(''t'')}}, we can form a pair of ordinary differential equations:
<math display="block" alt="d square big X over d x squared equals negative of omega over c quantity squared times big X, and d squared big T over d t squared equals negative omega squared times T">\frac{d^2}{dx^2}X=-\frac{\omega^2}{c^2}X, \qquad \frac{d^2}{dt^2}T = -\omega^2 T.</math>

Each of these is an eigenvalue equation with eigenvalues <math display="inline">-\frac{\omega^2}{c^2}</math> and {{math|−''ω''<sup>2</sup>}}, respectively. For any values of {{mvar|ω}} and {{mvar|c}}, the equations are satisfied by the functions
<math display="block">X(x) = \sin\left(\frac{\omega x}{c} + \varphi\right), \qquad T(t) = \sin(\omega t + \psi),</math>
where the phase angles {{mvar|φ}} and {{mvar|ψ}} are arbitrary real constants.

If we impose boundary conditions, for example that the ends of the string are fixed at {{math|1=''x'' = 0}} and {{math|1=''x'' = ''L''}}, namely {{math|1=''X''(0) = ''X''(''L'') = 0}}, and that {{math|1=''T''(0) = 0}}, we constrain the eigenvalues. For these boundary conditions, {{math|1=sin(''φ'') = 0}} and {{math|1=sin(''ψ'') = 0}}, so the phase angles {{math|1=''φ'' = ''ψ'' = 0}}, and
<math display="block" alt="sine of omega divided by c quantity equals 0">\sin\left(\frac{\omega L}{c}\right) = 0.</math>

This last boundary condition constrains {{mvar|ω}} to take a value {{math|1=''ω<sub>n</sub>'' = {{sfrac|''ncπ''|''L''}}}}, where {{mvar|n}} is any integer. Thus, the clamped string supports a family of standing waves of the form
<math display="block">h(x,t) = \sin\left(\frac{n\pi x}{L} \right) \sin(\omega_n t).</math>

In the example of a string instrument, the frequency {{math|''ω<sub>n</sub>''}} is the frequency of the {{mvar|n}}-th [[harmonic]], which is called the {{math|(''n'' − 1)}}-th [[overtone]].

===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.

===Signals and systems===
In the study of [[LTI system theory|signals and systems]], an eigenfunction of a system is a signal {{math|''f''(''t'')}} that, when input into the system, produces a response {{math|1=''y''(''t'') = ''λf''(''t'')}}, where {{mvar|λ}} is a complex scalar eigenvalue.{{sfn|Girod|Rabenstein|Stenger|2001|p=49}}