Symmetric and antisymmetric wavefunctions

Wavefunctions must be either symmetric (delete the minus sign from Equation 1.12) or antisymmetric in order to be consistent with the Born interpretation electrons being indistinguishable, W2 must be invariant with respect to an interchange of any pair of electrons, because the probability of finding e, in a volume element around the coordinates qej and ey around qe. must be the same when the labels / and j are exchanged. Both symmetric and antisymmetric wavefunctions would satisfy this condition, but the Pauli principle allows only antisymmetric wavefunctions. 

Next, let us consider the Hartree wavefunction in Eq. (2.5). The symmetric and antisymmetric wavefunctions are given by... 

Hence, it may be seen that Eg < 3 and that Eg > 0. Putting these values back into Equation 3.10, we find that Ci = C2 for Eg and that C = C2 for Eg. We can now construct the symmetric and antisymmetric wavefunctions. 

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