Degeneracy, exchange

Exchange Degeneracy.—All of the methods which we shall consider are based on a first approximation in which the interaction of the electrons with each other has either been omitted or been replaced by a centrally symmetric field approximately representing the average effect of all the other electrons on the one under consideration. We may first think of the problem as a perturbation problem. The wave equation for an atom with N electrons and a stationary nucleus is 

If the terms in are omitted, this equation is separable into N three-dimensional equations, one for each electron, just as was found to be the case for helium in Section 236. To this 

1 This chapter can be omitted by readers not interested in atomic spectra and related subjects however, the treatment is closely related to that for molecules given in Chapter XIII. 

Any of the above permutations can be formed from xt, ij, x by successive interchanges of pairs of symbols. This can be done in more than one way, but the number of interchanges necessary is either always even or always odd, regardless of the manner in which it is carried out. A permutation is said to be even if it is equivalent to an even number of interchanges, and odd if it is equivalent to an odd number. We shall find it convenient to use the symbol ( — IE to represent +1 when P is an even permutation and — 1 when P is an odd permutation. 

Multiplication of the operators P and P means that P and P are to be applied successively. The set of all the permutations of N symbols has the property that the product PP of any two of them is equal to some other permutation of the set. A set of operators with this property is called a group, if in addition the set possesses an identity operation and if every operation P possesses an inverse operation P l such that PP-1 is equivalent to the identity operation. There are N permutations of N different symbols. 

The concept of exchange degeneracy is of appreciable import herein (see Section 7.1 below). Baym (Ref. 16, p. 391) defines it by considering the permutations P among a set of identical particles as applied to a Hamiltonian operator opH symmetric in these. Various of the different energy eigenstates of opP opH will be degenerate, with such superpositions not affected by Zeeman interactions.13 

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Since the Hamiltonian is symmetric in space coordinates the time-dependent Schrodinger equation prevents a system of identical particles in a symmetric state from passing into an anti-symmetric state. The symmetry character of the eigenfunctions therefore is a property of the particles themselves. Only one eigenfunction corresponds to each eigenfunction and hence there is no exchange degeneracy. 

Key Words NMR, EPR, Spin-spin coupling, Equivalent nuclides, Spin-Hamiltonian energies and eigenstates, Exchange degeneracy. 

A semiclassical estimate of the integral in (44) can be obtained from the stationary-phase approximation. Exchange degeneracy splitting is clearly a case where the Cl approach results in a very simple, practically useful expression. 

Now consider the removal of the exchange degeneracy. The functions (9.96) with which we began the perturbation treatment have each electron assigned to a definite... 

We cannot say which orbital electron 1 is in for either or 4. This property of the wave functions of systems containing more than one electron results fi-om the indistinguishability of identical particles in quantum mechanics and will be discussed further in Chapter 10. Since the functions and have different energies, the exchange degeneracy is removed when the correct zeroth-order functions are used. 

To determine the asymptotic expression of the interaction for very large R we use the Schrodinger perturbation theory, where Eq. (44) is expanded in powers of l/R. Since exchange degeneracy is not lifted at large distances, it is sufficient to consider the perturbation of for instance... 

Because of the equality of the electrons we have exchange degeneracy. In the case of [tt] and [tt] exchange of the electrons yields nothing new, since these states are symmetric in the coordinates of the electrons. In the case [7r]u[7r]p instead the state is double degenerate without considering the interaction. 

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