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If I have a composite system, like a two particle system, for exemple, I can construct my Hilbert space as the tensor product of the hilbert spaces of these particles, and, if ##\{|A;m \rangle \}## and ##\{|B;n \rangle \}## are basis in these hilbert spaces, a basis in the total hilbert space is ##\{|A;m \rangle \otimes |B;n \rangle \}##. But, to it make sense, shuoldn't the hilbert spaces of the subsystems represent non-interacting systems? For exemple, one of the most know exemples of composite system is the Hydrogen atom, where the hamiltonian is $$H = \frac{p_{p}^{2}}{2m_{p}} + \frac{p_{e}^{2}}{2m_{e}} + V(r),$$where ##r## is the relative position of the proton and the electron and ##V## is the Coulomb potential. However, we can write this in the rest of C.M. $$H = \frac{P^{2}}{2M} + \frac{p^{2}}{2\mu} + V(r).$$This hamiltonian is equivalent to a system with a free particle and a particle subject to an external potential, but these particles do not interact with each other. So, in this case, I can write a basis of the total system using the tensor products ##|P \rangle \otimes |n, l, m \rangle##.