Quantum mechanics, symmetrization postulate

Special attention must be paid in systems of identical particles, where we have to take into account the symmetry postulate of quantum mechanics. This means that the space of states for fermions is the antisymmetric subspace of while the symmetric subspace dK+N refers to bosons. 

According to the symmetrization postulate of quantum mechanics, the spin-space state function of a system of N nondifferentiable nuclei must be invariant under any of the A /2 even permutations IV W performed simultaneously on the space (7 ) and spin (S) particles coordinates. Under odd permutations, the state function of N fermions changes sign while that of N bosons remains invariant. 

Summarizing, when the matrix elements of the correct quantum mechanical Hamiltonian of the system in question are calculated in a special basis conforming to the symmetrization postulate then, upon integrating over the spatial variables, the following effective spin Hamiltonian is obtained... 

We have shown that there are two possible cases for the wave function of a system of identical particles, the symmetric and the antisymmetric cases. Experimental evidence (such as the periodic table of the elements to be discussed later) shows that for electrons only the antisymmetric case occurs. Thus we have an additional postulate of quantum mechanics, which states that the wave function of a system of electrons must be antisymmetric with respect to interchange of any two electrons. Was important postulate is called the Pauli principle, after the physicist Wolfgang Pauli. 

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