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Unformatted text preview: A list of important properties of the eigenvalue-eigenfunction pairs for Hermitean operators are: 1. The eigenvalues of an Hermitean operator are all real. 2. The eigenfunctions for different eigenvalues are orthogonal. 3. The set of all eigenfunction is complete. Ad 1. Let be an eigenfunction of . Use (8.6) Ad 2. Let and be eigenfunctions of . Use (8.7) This leads to (8.8) and if , which is the definition of two orthogonal functions. Ad 3. This is more complex, and no proof will be given. It means that any function can be written as a sum of eigenfunctions of , (8.9) (A good example of such a sum is the Fourier series.)...
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- Fall '10