 Solution

For distinguishable particles with one-particle Hilbert spaces ${\mathcal{H}}_1,\dots ,{\mathcal{H}}_N$, the multiparticle quantum state belongs toA product of one-particle states gives a product vector, and arbitrary linear combinations give entangled states. A noninteracting Hamiltonian is the sum of one-particle Hamiltonians acting on their respective tensor factors, so product-state energies add.

For two identical particles, the particle exchange operator $P_{12}$ swaps every degree of freedom. Since $P_{12}^2=I$, its eigenvalues are $\pm 1$. Particles whose allowed total states satisfyare bosons, while those satisfyingare fermions. Thus bosonic states are symmetric and fermionic states antisymmetric under interchange. The same condition applies to every transposition in a system of more than two identical particles.