32 - Physics 6210/Spring 2007/Lecture 32 Lecture 32...

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Physics 6210/Spring 2007/Lecture 32 Lecture 32 Relevant sections in text: § 5.2 Degenerate Perturbation Theory (cont.) Degenerate perturbation theory leads to the following conclusions (see the text for details of the derivation). To compute the first-order corrections to the energies and eigen- vectors when there is degeneracy in the unperturbed eigenvalue one proceeds as follows. Step 1 Consider the restriction, ˜ V of the perturbation V to the d -dimensional degenerate subspace D . ˜ V is defined as follows. The action of V on a vector from D is some other vector in the Hilbert space. Take the component of this vector along D , i.e., project this vector back into D . This process defines a Hermitian linear mapping ˜ V from D to itself. In practice, the most convenient way to compute ˜ V is to pick a basis for D . Compute the matrix elements of V in this basis. One now has a d × d Hermitian matrix representing ˜ V . Step 2 Find the eigenvectors and eigenvalues of ˜ V . Again, it is most convenient to do this using the matrix representation of ˜ V . Step 3 The first-order energy corrections to E (0) n are the eigenvalues of ˜ V . The zeroth order limit of the true eigenvectors that correspond to the true eigenvalues are the eigenvectors of ˜ V that correspond to the first-order eigenvalue corrections.
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32 - Physics 6210/Spring 2007/Lecture 32 Lecture 32...

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