Chemistry - Eigenfunction

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In mathematics, an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies

$\mathcal A f = \lambda f$

for some scalar λ, the corresponding eigenvalue. The existence of eigenvectors is typically a great help in analysing A.

For example, $f k (x) = e k x$ 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 $λ = k 2 - k$ .

Eigenfunctions play an important role in 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 $E k$ . 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).

Categories: Functional analysis