Two identical spin 1/2 particles

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  • Thread starter Lebnm
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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?
 

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DrClaude
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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.
 
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