Undergrad Two identical spin 1/2 particles

[Lebnm]

I am studying identical particles and I have some doubts. Considerer two identical spin 1/2 particles interacting through a central potential $V$. In the rest of CM, the hamiltonian is $$ H = \frac{\textbf{P}^{2}}{2M} + \frac{\textbf{p}^{2}}{2\mu} + V(r),$$ where $\textbf{P}$ is the momentum of CM, $\textbf{p}$ is the momentum associated with the relative coordinate $\textbf{r}$, $M$ is the total mass and $\mu$ is the reduced mass. The text I am reading write the state of the system as $$| \psi \rangle = | \textbf{P} \rangle \otimes | n,l,m \rangle \otimes | S,M \rangle.$$ Here, $| \textbf{P} \rangle$ is an eigenstate of the operator $\textbf{P}$, $| n,l,m \rangle$ is the solution of the central potential problem and $| S,M \rangle$ are the eigenstates of $\textbf{S}^{2}$ and $S_{z}$, being $\textbf{S}$ the total spin of the system. My questions are: it's not exactly a state, but a bases, isn't it? To construct a state,do I need to take linear combinations of it? In this case, How can I write this?

 

[DrClaude]

The $|\psi\rangle$ you wrote there represents an eigenstate of the Hamiltonian (in addition to being a spin eigenstate).

It is a state. The set of all such eigenstates forms a complete basis, such that any state can be written as a sum of these eigenstates.