Problems (N.B. Neglect spin in the following questions.)

1. Consider a system consisting of two non-interacting particles, and three one-particle states, $\psi_a(x)$, $\psi_b(x)$, and $\psi_c(x)$. How many different two-particle states can be constructed if the particles are (a) distinguishable, (b) indistinguishable bosons, or (c) indistinguishable fermions? [modified from Griffiths]

2. Consider two non-interacting particles, each of mass $m$, in a one-dimensional harmonic oscillator potential of classical oscillation frequency $\omega$. If one particle is in the ground-state, and the other in the first excited state, calculate $\langle (x_1-x_2)^2\rangle$ assuming that the particles are (a) distinguishable, (b) indistinguishable bosons, or (c) indistinguishable fermions. [from Gasiorowicz]

3. Two non-interacting particles, with the same mass $m$, are in a one-dimensional box of length $a$. What are the four lowest energies of the system? What are the degeneracies of these energies if the two particles are (a) distinguishable, (b) indistinguishable bosons, or (c) indistingishable fermions? [modified from Squires]

4. Two particles in a one-dimensional box of length $a$ occupy the $n=4$ and $n'=3$ states. Write the properly normalized wave-functions if the particles are (a) distinguishable, (b) indistinguishable bosons, or (c) indistinguishable fermions. [modified from Harris]