Eigenfunction
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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
- <math>
\mathcal A f = \lambda f </math>
for some scalar, λ, the corresponding eigenvalue. The existence of eigenvectors is typically a great help in analysing A.
For example, <math>f_k(x) = e^{kx}</math> is an eigenfunction for the differential operator
<math> \mathcal A = \frac{d^2}{dx^2} - \frac{d}{dx}, </math>
for any value of <math>k</math>, with a corresponding eigenvalue <math>\lambda = k^2 - k</math>.
Eigenfunctions play an important role in quantum mechanics, where the Schrödinger equation
- <math>
i \hbar \frac{\partial}{\partial t} \psi = \mathcal H \psi </math>
has solutions of the form
- <math>
\psi(t) = \sum_k e^{-i E_k t/\hbar} \phi_k, </math>
where <math>\phi_k</math> are eigenfunctions of the operator <math>\mathcal H</math> with eigenvalues <math>E_k</math>. Due to the nature of the Hamiltonian operator <math>\mathcal H</math>, its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example <math>A</math> mentioned above).