Consider a composite system of several identical particles. Describe how the multiparticle state is constructed from single-particle states. For the case of two identical particles, describe how considering the interchange symmetry leads to the definition of bosons and fermions.

Consider two non-interacting, identical particles, each with spin 1 . The singleparticle, spin-independent Hamiltonian $H\left(\hat{\mathbf{x}}_{i}, \hat{\mathbf{p}}_{i}\right)$ has non-degenerate eigenvalues $E_{n}$ and wavefunctions $\psi_{n}\left(\mathbf{x}_{i}\right)$ where $i=1,2$ labels the particle and $n=0,1,2,3, \ldots$ In terms of these single-particle wavefunctions and single-particle spin states $|1\rangle,|0\rangle$ and $|-1\rangle$, write down all of the two-particle states and energies for:

(i) the ground state;

(ii) the first excited state.

Assume now that $E_{n}$ is a linear function of $n$. Find the degeneracy of the $N^{\text {th }}$energy level of the two-particle system for:

(iii) $N$ even;

(iv) $N$ odd.