Energy of two spin-1 bosons in 1D harmonic oscillator I'm taking a QM class and I'm trying to work out a problem given to us by the professor.
We have two identical, non-interacting, spin-1 bosons that are located in a one dimensional harmonic oscillator potential with a characteristic frequency $\omega$. The problem to solve is to find the three lowest energy levels of this two-particle system, as well as their degeneracies. 
I think that I need to use the Clebsch-Gordan table to combine the single-particle states into a combined state, and then act on this state with the hamiltonian to find the energies.
But in trying to do this I don't get very far. In earlier problems I always know the $m_{1}$ and $m_{2}$ quantum numbers ($m_{i} \in \{-s, -s+1,\dots,s\}$) that I need in order to use the table. This time the values of $m$ is not specified, so I am not sure how to proceed.
I\m thinking it might just be to write out a linear combination of all the combined states that I can get from every permutation of $m$'s, but that doesn't seem to right (plus I'm not getting anywhere when trying to do so). I assume I'm missing something quite elementary in understanding this.
Any help on how I should go about doing this would be great! 
 A: Since the two particles are bosons you need to analyse which combinations of the two spins are consistent with a total wave function that is even under particle exchange.
The total wave function is the product of the spatial wave function from the harmonic oscillator (a product state for non-interacting particles) and the spin wave function. For example, if the two bosons are both in the ground state of the harmonic oscillator their spatial wave function is a product of two Gaussians. This is clearly even under particle exchange, and therefore the total spin wave function should be even as well. This can only be the symmetric states with $S_{tot} = 0,2$ and therefore the total degeneracy of the ground state is 6.
A: In this case the particles are non-interacting and there is no magnetic field. 
Therefore the quantum numbers $m_i$ do not influence the energy of the system. 
You can therefore calculate the energy eigenstates of the system while ignoring the quantum numbers $m_i$. You should however take the quantum numbers $m_i$ into account when you want to determine the degeneracy of the found states.
