Eigenfunction

\mathcal A f = \lambda f

for some Scalar , λ, the corresponding Eigenvalue . The existence of eigenvectors is typically a great help in analysing ''A''.

For example,

f_k(x) = e^{kx}

is an eigenfunction for the Differential Operator

\mathcal A = rac{d^2}{dx^2} - rac{d}{dx},

for any value of

k

, with a corresponding eigenvalue

\lambda = k^2 - k

.

Eigenfunctions play an important role in Quantum Mechanics , where the Schrödinger Equation

:

i \hbar rac{\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

. 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).

SEE ALSO

	CATEGORIES ABOUT EIGENFUNCTION

	functional analysis

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