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 For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...

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. In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

For example, f_k(x) = e^kx is an eigenfunction for the differential operator In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

for any value of k, with a corresponding eigenvalue λ = k² − 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 .

Eigenfunctions play an important role in many branches of physics. An important example is quantum mechanics, where the Schrödinger equation Fig. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ...

has solutions of the form

where φ_k are eigenfunctions of the operator 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 defining 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 , 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, have the property that The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In mathematics, two functions and are orthogonal if their inner product is zero. ...

whenever , in which case the set is said to be linearly independent.

See also

• Eigenvector