3. Two bosons in a well. Two particles of mass $m$ are in the spherical potential
well
$V(r) = \begin{cases} 0 & r \le r_0 \\ \infty & r > r_0 \end{cases}$
where $r_0$ is a positive constant. Assume the two particles do not interact, so
that the Hamiltonian is
$H = \frac{p_a^2}{2m} + \frac{p_b^2}{2m} + V(r_a) + V(r_b).$
Suppose the two particles are identical spin-1 bosons. The symmetry condition
in this case is that the total wavefunction is symmetric under interchange of all
the quantum numbers belonging to the two particles.
In terms of the eigenfunctions $\psi_n(r)$ for a single particle in a square well, (a)
find the allowed values of the total spin of the two spin 1 bosons in the ground
state, and write down the ground state wave function. (Extra credit: write
down the wave function in terms of the actual square well wave functions). (b)
Find the allowed values of the total spin in the first excited state.
