This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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.)...
View Full Document
- Fall '10
- Physics, Eigenvalue, eigenvector and eigenspace, Eigenfunction, hermitean, Hermitean Operators