Eigenfunction

This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace's equation on a disk. The Vibrations of an idealized circular Drum, essentially an elastic Membrane of uniform thickness attached to a rigid circular frame are solutions of the In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function More precisely, one has

$\mathcal A f = \lambda f$

for some scalar, λ, the corresponding eigenvalue. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes 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

$\mathcal A = \frac{d^2}{dx^2} - \frac{d}{dx}$

for any value of $k$ , with a corresponding eigenvalue $λ = k 2 - k$ . In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator 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

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

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 $E k$ . Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system 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. In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system In Mathematics, two functions f and gare called orthogonal if their Inner product \langle fg\rangle is zero 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 linearly independent.

Dictionary

-noun

(mathematics) 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.
(quantum mechanics) Any eigenfunction of the Hamiltonian operator, representing a quantum state whose energy level is given by the corresponding eigenvalue.

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Applications

See also

Dictionary

eigenfunction

-noun