Dictionary - Definition of EIGENFUNCTION

Eigenfunction

This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace's equation on a disk.

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 solution of the differential eigenvalue problem also depends upon any boundary conditions required of $f$ . In each case there are only certain eigenvalues $λ = λ n$ ( $n = 1,2,3,...$ ) that admit a corresponding solution for $f = f n$ (with each $f n$ belonging to the eigenvalue $λ n$ ) when combined with the boundary conditions. The existence of eigenvectors is typically the most insightful way to analyze $A$ .

For example, $f k (x) = e k x$ is an eigenfunction for the differential operator

$\mathcal A = \frac{d2}{dx2} - \frac{d}{dx}$

for any value of $k$ , with a corresponding eigenvalue $λ = k 2 - k$ . If boundary conditions are applied to this system (e.g., $f = 0$ at two physical locations in space), then only certain values of $k = k n$ satisfy the boundary conditions, generating corresponding discrete eigenvalues $\lambda_n=k_n2-k_n$ .

Applications

Eigenfunctions play an important role in many branches of physics. An important example is quantum mechanics, where the Schrödinger equation

$i \hbar \frac{\partial}{\partial t} \psi = \mathcal H \psi$

has solutions of the form

$\psi(t) = \sum_k e{-i E_k t/\hbar} \phi_k,$

where $φ k$ are eigenfunctions of the operator $\mathcal H$ with eigenvalues $E k$ . The fact that only certain eigenvalues $E k$ with associated eigenfunctions $φ k$ satisfy Schrödinger's equation leads to a natural basis for quantum mechanics and the periodic table of the elements, with each $E k$ an allowable energy state of the system. The success of this equation in explaining the spectral characteristics of hydrogen is considered one of the great triumphs of 20th century physics.

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). Orthogonal functions $f i$ , $i=1, 2, \dots,$ have the property that

$0 = \int f_i{*} f_j$

where $f_i{*}$ is the complex conjugate of $f i$

whenever $i\neq j$ , in which case the set $\{f_i \,|\, i \in I\}$ is said to be orthogonal. Also, it is linearly independent.

Domain	Definition
Electrical Engineering	An eigenvector for a linear operator on a vector space whose vectors are functions. Source: European Union. (references)
Nuclear Energy & Physics	The solution of an equation compatible with the boundary conditions associated with possible values of a parameter of the equation (the eigenvalue). Source: European Union. (references)
Source: compiled by the editor from various references; see credits.		Top

Topics by Level of Interest: EIGENFUNCTION
Topics sorted by level of Interest	Level (1=low, 600=high)	Topics sorted Alphabetically	Level (1=low, 600=high)
Eigenfunction	8	Eigenfunction	8
Source: the editor, created by/for EVE to gauge likely levels of human interest in linguistically triggered topics (compiled across various sources, such as Wikipedia and specialty expression glosses).

Specialty Definition: EIGENFUNCTION

Extended Definition: Eigenfunction

Eigenfunction

Applications

See also

Topics by Level of Interest: EIGENFUNCTION