For a system of two noninteracting particles of spins $j_1$ and $j_2$, the joint Hilbert space $\mathcal{V}$ is the tensor product of the individual Hilbert spaces $\mathcal{V}_1$ and $\mathcal{V}_2$. Notationally, $\mathcal{V}=\mathcal{V}_1\otimes\mathcal{V}_2$ which is spanned by the $(2j_1+1)(2j_2+1)$ product states or their linear combinations.
- What is the Hilbert space when there is an interaction between the particles? Do the product states still serve as a basis which spans the space?