# User Contributed Dictionary

### Etymology

From eigen + “function”### Noun

- A function φ such that, for a generic linear operator D (e.g. differential one), Dφ = λφ where λ is an eigenvalue of the operator.
- Any eigenfunction of the Hamiltonian operator, representing a quantum state whose energy level is given by the corresponding eigenvalue.

#### Synonyms

#### Translations

- Finnish: ominaisfunktio
- Italian: autofunzione
- Japanese: (koyū kansū)
- Russian: собственная функция (sóbstvennaja fúnktsija)
- Swedish: egenfunktion
- ttbc Arabic: (dála χáʂa)
- ttbc Spanish: autofunción , función propia

# Extensive Definition

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 \lambda=\lambda_n (n=1,2,3,...) that admit a
corresponding solution for f=f_n (with each f_n belonging to the
eigenvalue \lambda_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^ is an eigenfunction for
the differential
operator

\mathcal A = \frac - \frac

for any value of k, with a corresponding
eigenvalue \lambda = 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_n^2-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 \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. The fact that only certain
eigenvalues E_k with associated eigenfunctions \phi_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 \ is said
to be linearly independent.

eigenfunction in German: Eigenfunktion

eigenfunction in French: Fonction propre

eigenfunction in Italian: Autofunzione

eigenfunction in Dutch: Eigenfunctie

eigenfunction in Swedish: Egenfunktion

eigenfunction in Ukrainian: Власна функція

eigenfunction in Chinese: 特征函数