Non-degenerate eigenvalues

Notiee that the eigenvalue )ik = 3 appears twiee we say that the eigenvalue )ik = 3 is doubly degenerate. )ik = 2 is a non-degenerate eigenvalue. 

For a continuous spectrum of eigenkets with non-degenerate eigenvalues, it is more convenient to write the eigenvalue equation (3.45) in the form... 

The non-degenerate eigenvalue E for the perturbed system to second order is obtained by substituting equations (9.24) and (9.34) into (9.20) to give... 

The perturbation method presented in Section 9.3 applies only to non-degenerate eigenvalues eX of the unperturbed system. When E is degenerate, the denominators vanish for those terms in equations (9.40) and (9.41) in which Ef is equal to E making the perturbations to E and ipn indeterminate. In... 

Theorem. If A is Hermitian, its eigenvalues are real and its eigenvectors are orthogonal to each other provided they correspond to non-degenerate eigenvalues. 

Hartree-Fock Scheme.- For the sake of simplicity, we will consider the eigenvlue relations (2.3) for a non-degenerate eigenvalue X, for which 0. If Ca and Da are two approximate solutions - characterized by the index "a"- the best approximations are obtained by using the bi-variational principle (B.3.6). In the Hartree-Fock scheme for N fermions, the approximate solutions are represented by two Slater determinants ... 

One obtains one two-fold degenerate eigenvalue Ku and two non-degenerate eigenvalues ... 

Moreover we assume that is a non-degenerate eigenvalue. As a matter of fact, this is never the case, since, because of the indistin-guishability of the electrons the resonance degeneracy, discovered by... 

Finally, a measurement on the final state 0) is performed. The measurement will be done according to a Hermitian matrix with the initial state ip) as an eigenvector with non-degenerate eigenvalue. If this special eigenvalue is measured we can be certain that the computer is in its initial state after the measurement. 

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