Eigenfunction

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In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has

[\mathcal A f = \lambda f]

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

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

[\mathcal A = \frac - \frac,]

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

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

[i \hbar \frac \psi = \mathcal H \psi]

has solutions of the form

[\psi(t) = \sum_k e^ \phi_k,]

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

Eigenfunction

See also