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eigenfunction |
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eigenfunction [′ī·gən‚fəŋk·shən] (mathematics) Also known as characteristic function. An eigenvector for a linear operator on a vector space whose vectors are functions. Also known as proper function. A solution to the Sturm-Liouville partial differential equation. Eigenfunction One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the eigenfunctions (sometimes eigenvectors). In older usage the terms characteristic equation and characteristic values (functions) are common. Eigenvalue equations appear in many contexts, including the solution of systems of linear algebraic equations (matrix equations), differential or partial differential equations, and integral equations. The importance of eigenfunctions and eigenvalues in applied mathematics results from the widespread applicability of linear equations as exact or approximate descriptions of physical systems. However, the most fundamental application of these concepts is in quantum mechanics where they enter into the definition and physical interpretation of the theory. Only linear eigenvalue equations will be discussed. See Eigenvalue (quantum mechanics), Energy level (quantum mechanics), Quantum mechanics How to thank TFD for its existence? Tell a friend about us, add a link to this page, add the site to iGoogle, or visit webmaster's page for free fun content. |
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| Ordinarily, eigenfunctions of this type are obtained via perturbation theory; this method is not applicable to our case, as we deal with states belonging to the continuum whose energies are infinitely close to, and also coincident with, the energies of the discrete states. In the quantum world, these normal modes are known as eigenfunctions, and Sridhar has developed a way to map the eigenfunctions of microwaves bouncing around inside cavities of different shapes. P] is a single particle matrix element of the P-odd interaction, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the eigenfunctions of the nuclear T-invariant Hamiltonian with the appropriate boundary conditions [2,3]. |
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