Invariance with respect to permutation of identical particles fermions and bosons

4 INVARIANCE WITH RESPECT TO PERMUTATION OF IDENTICAL PARTICLES (FERMIONS AND BOSONS) 

The Hamiltonian has also permutational symmetry. This means that if someone exchanged labels numbering the identical particles, independently of how it was done, they would always obtain the identical mathematical expression for the Hamiltonian. This implies that any wave function has to be symmetric (for bosons) or antisymmetric (fermions) with respect to the exchange of labels between two identical particles (cf. p. 33). 

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Invariance with Respect to Permutation of Identical Particles (Fermions and Bosons)... 

Invariance with respect to permutations of identical particles (fermions and bosons)... 

J, quantizes its component along the z axis, and II = 1 represents the parity with respect to the inversion. As to the invariance with respect to permutations of identical particles an acceptable wave function has to be antisymmetric with respect to the exchange of identical fermions, whereas it has to be s)mmetric when exchanging bosons. 

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