# Eigenfunction

In mathematics, an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies

[itex]

\mathcal A f = \lambda f [itex]

for some scalar λ, the corresponding eigenvalue. The existence of eigenvectors is typically a great help in analysing A.

For example, [itex]f_k(x) = e^{kx}[itex] is an eigenfunction for the differential operator

[itex] \mathcal A = \frac{d^2}{dx^2} - \frac{d}{dx}, [itex]

for any value of [itex]k[itex], with a corresponding eigenvalue [itex]\lambda = k^2 - k[itex].

Eigenfunctions play an important role in quantum mechanics, where the Schrödinger equation

[itex]

i \hbar \frac{\partial}{\partial t} \psi = \mathcal H \psi [itex]

has solutions of the form

[itex]

\psi(t) = \sum_k e^{-i E_k t/\hbar} \phi_k, [itex]

where [itex]\phi_k[itex] are eigenfunctions of the operator [itex]\mathcal H[itex] with eigenvalues [itex]E_k[itex]. Due to the nature of the hamiltonian operator [itex]\mathcal H[itex], its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example [itex]A[itex] mentioned above).de:Eigenfunktion

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