Linearly independent eigenfunctions

If the unperturbed system is degenerate, so that several linearly independent eigenfunctions correspond to the same energy value, then a more complicated procedure must be followed. There can always be found a set of eigenfunctions (the zeroth order eigenfunctions) such that for each the perturbation energy is given by equation 9 and the perturbation theory provides the... 

In this interpretation Q is the number of linearly independent eigenfunctions of the unperturbed Hamiltonian in the interval AE. From (36) the microcanonical average of an observable represented by the operator A in an arbitrary basis, is... 

In general, we cannot prefer the description 1/ 2 5 3 the description 0i, 02 5 03 of the entire collection of possibilities, we are required f or completeness to choose any three linearly independent eigenfunctions. Beyond that, it is a matter only of convenience. As we shall see later, in systems with certain symmetry properties, some combinations are more convenient than others. 

As in Case (i) the ntoleeular orbital problem is solved completely by symmetry for the doubly degenerate rep. The expansions are similar to Case (ii) except that in this case the linearly independent eigenfunctions belong to the same eignevalue and do not require sorting out by the secular equation. As an example we will consider cyclobutadiene. We will assume that the symmetry is />4 ( a subgroup of the probable... 

Fig. C.4. Each energy level corresponds to an irreducible representation of the symmetry group rf the Hamiltonian. Its linearly independent eigenfunctions that correspond to a given level form a basis of the irreducible Fepiesenlation, or in other words, transform according to this representation. The number of the basis functions is equal to the degeneracy of the leveL...

The methods that we consider fall into two types synthetic (see e.g. Kotani et ai, 1955), in which the full set of linearly independent eigenfunctions of given 5 and M is built up by some systematic procedure and analytic (Ldwdin, 1955, 1956a), in which a spin eigenfunction with the required values of 5 and M is extracted from an arbitrary function (i.e. a mixture of spin eigenfunctions) by means of a suitable projection operator. There are many possible procedures of both types, all exhaustively treated in the book by Pauncz (1979). 

In the last section it was noted that a full set of linearly independent eigenfunctions of the spin operators and S, provides a basis... 

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