Hermitean operators

# Hermitean operators - A list of important properties of the...

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Hermitean operators Hermitean operators are those where the outcome of any measurement is always real, as they should be (complex position?). This means that both its eigenvalues are real, and that the average outcome of any experiment is real. The mathematical definition of a Hermitean operator can be given as (8.4) show that and (in 1 dimension) are Hermitean. Eigenvalues of Hermitean operators Eigenvalues and eigen vectors of Hermitean operators are defined as for matrices, i.e., where there is a matrix-vector product we get an operator acting on a function, and the eigenvalue/function equation becomes (8.5) where is a number (the ``eigenvalue'' )and is the ``eigenfunction''.

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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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Hermitean operators - A list of important properties of the...

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