Eigenfunction


Eigenfunction

[
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 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^{kx} 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 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{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 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 eq j, in which case the set {f_i ,|, i in I} is said to be orthogonal. Also, it is linearly independent.

ee also

* Eigenvalue, eigenvector and eigenspace
* Hilbert-Schmidt theorem
* Spectral theory of ordinary differential equations


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