9.6: Wavefunction for many spin one-half particles

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The exchange arguments for two-particle systems can be extended to many particle systems: The indistinguishable wavefunction consists of all possible permutations of the product of one electron wavefunctions. For the symmetric case \(\hat{P}_{nm} \Phi = \Phi\), a product of these permutations will suffice. For the antisymmetric case, the correct form turns out to be given by the determinant of a matrix:

\[\Phi = \frac{1}{\sqrt{N!}} \text{det} \begin{pmatrix} \phi_a(1) & \phi_b(1) & ... & \phi_N (1) \\ \phi_a(2) & \phi_b(2) & ... & \phi_N (2) \\ ... & ... & ... & ... \\ \phi_a(N) & \phi_b(N) & ... & \phi_N (N) \end{pmatrix} \nonumber\]

This is called a Slater Determinant. For fermions, where \(\hat{P}_{nm} \Phi = −\Phi\) the Slater Determinant obeys the Pauli exclusion principle: if any two of the one-particle wavefunctions were identical \((\phi_n = \phi_m)\), then the wavefunction would be the determinant of a matrix with two identical rows, i.e. zero.

Note also that \(\langle \Phi |\hat{H}|\Phi \rangle\) has many more exchange terms than direct ones.