Time-Independent Degenerate Perturbation Theory

The non-degenerate perturbation expressions that have been developed in the previous section will result in zero denominators for degenerate systems. As a result, the approach used to obtain the expressions for the various orders of corrections for a degenerate system must be modified from the approach used in non-degenerate systems. 

Consider an unperturbed system where a given energy level n has an r-fold degeneracy. This means that there are r wavefunctions that will result in the same energy,, when applied to the Hamiltonian,. In this notation scheme, the n refers to the various degenerate states (1 n r ). 

Now suppose this degenerate system experiences a perturbation. The 

Hamiltonian for this perturbed system is //, and the wavefunctions for the perturbed system are. The wavefunctions i// may be non-degenerate, have a fraction of the degeneracy, or in some cases no change in the degeneracy relative to the unperturbed system. The change in degeneracy in 

For the unperturbed system, any normalized linear combinations of the r unperturbed wavefunctions are acceptable solutions however for the perturbed system, oidy certain normahzed linear combinations form the correct zero-order perturbed (unperturbed) wavefunctions.  

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Let Ho be the Hamiltonian of the independent electron atom. We use the formalism of time-independent, degenerate perturbation theory to describe the problem, the variation being that, in the present case, the states which are degenerate in energy belong to the continuum on one hand and to the discrete spectrum on the other. This is a very interesting complication it is fundamental to quantum mechanics that discrete energy levels appear in what would otherwise be a fully continuous spectrum. Autoionisation is a mechanism which couples bound states of one channel to continuous states of another. 

These are standard results of the time-independent nondegenerate perturbation theory which are presented in every quantum naechanical textbook. The term nondegenerate refers to the zeroth-order spectrum where degenerate levels would lead to singularities in the perturbation expressions, cf. Eqs. (12.10). 

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