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EigenfunctionIn 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, is an eigenfunction for the differential operator for any value of , with a corresponding eigenvalue . Eigenfunctions play an important role in quantum mechanics, where the Schrdinger 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 are eigenfunctions of the operator with eigenvalues . Due to the nature of the hamiltonian operator , its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example mentioned above).
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