Schrodinger equation permutational symmetry

Second, suitable solutions to the Schrodinger equation for electrons must have appropriate permutational symmetry. That is, an interchange of the space-spin coordinates Xj and Xj must not alter the probability density For this to be true, the interchange operator has to have the effect, = P. For... 

For a system of either bosons or fermions, the wavefunction must have the correct properties of symmetry and antisymmetry. Particles with half-integral spin, such as electrons, are fermions and require antisymmetric wavefunctions. Particles with integral spin, such as photons, are bosons and require symmetric wavefunctions. The complete space-spin wavefunction of a system of two or more electrons must be antisymmetric to the permutation of any two electrons. Except in the simplest cases, the wavefunction for a system of n fermions is positive and negative in different regions of the 3 -dimensional space of the fermions. The regions are separated by one or more (3 - 1 )-dimensional hypersurfaces that cannot be specified except by solution of the Schrodinger equation. 

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