quantum mechanics - What is meant by the term "single particle state"

In a lot of quantum mechanics lecture notes I've read the author introduces the notion of a so-called single-particle state when discussing non-interacting (or weakly interacting) particles, but none that I have read so far give an explicit explanation as to what is exactly meant by this term.

Is it meant that, in principle, each individual state constituting a multi-particle system can be occupied by a single particle (contrary to an entangled state, where is impossible to "separate" the particles), such that the state as a whole can be de-constructed into a set of sub-states containing only one particle each, and each being described by its own Hamiltonian?

Sorry to ramble, I'm a bit confused on the subject, in particular as I know that more than one particle can occupy a single-particle state (is the point here that the particles can still be attributed their own individual wave functions, and it just so happens that these individual wave functions describe the same state?).

Answer

Many particle wavefunctions are generally appallingly complicated objects. One way to get a handle on them is to break them down into simpler parts, understand those parts and then put them back together again. We do this by constructing the space of many particle wavefunctions as either a tensor product space or a Fock space.

An obvious way break down a many particle system is to try to consider what each particle is doing individually. Obviously there will be emergent effects in the many body system due to entanglement that were not present when only considering one particle and for strongly interacting systems this breakdown may not be possible, but often it is the only method we have.

So the single particle states are those states which on their own describe a single particle and from which we construct the full space as a tensor product space (i.e. the tensor product of single particle states and linear combinations thereof) or Fock space.