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Is it possible to map quantum states of different physical (quantum) systems to the same Hilbert space? For example if I consider two different molecules in the ground state, may I represent them as kets in the same Hilbert space? I am not considering to construct a tensor product of their own Hilbert spaces like in the case of composite systems.

If I am not wrong, I recall that in S matrix scattering theory, different colliding particle are mapped to kets in the same Hilbert space.

If it is correct, may I think that a mixture state of a quantum system is a result of a statistical ensemble of states of different (quantum) physical systems?

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  • $\begingroup$ "I am not considering to construct a tensor product of their own Hilbert spaces like in the case of composite systems." Why, uh, not? (If the systems are non-interacting, you won't get entangled states, but it still works) $\endgroup$
    – ACuriousMind
    Apr 8 '15 at 17:54
  • $\begingroup$ No, they are not interacting. I want to know if different systems can be represented by kets in the same Hilbert space. $\endgroup$
    – Caute
    Apr 8 '15 at 17:59
  • $\begingroup$ You didn't answer why you think the tensor product unsuitable for this. $\endgroup$
    – ACuriousMind
    Apr 8 '15 at 18:08
  • $\begingroup$ Sorry, I missed your question. I want to know if in principle a Hilbert space can accomodate quantum states of different systems with similar degrees of freedom. $\endgroup$
    – Caute
    Apr 8 '15 at 18:14
  • $\begingroup$ I added a question mark to the last sentence of the post. Please use correct punctuation; it helps readers understand the question and offer good answers. $\endgroup$
    – DanielSank
    Apr 8 '15 at 18:27