Suppose we have a quantum system with one degree of freedom, ##x##, and a heat bath (possibly modeled by a large amount of coupled oscillators) with many degrees of freedom, ##q_1 , q_2 , \dots , q_n##. If the system and bath are coupled by some interaction, this is a simple model of a system interacting with a measuring apparatus. If the wavefuntion of the total system (system+bath) is of product form, ##\psi (x) \phi (q_1 , q_2 , \dots , q_n )##, it seems natural to say that the wavefunction of system only is ##\psi (x)## and the wavefunction of bath only is ##\phi (q_1 , q_2 , \dots , q_n )##. However, when the system and bath are interacting, entanglement between them rapidly occurs and the wavefunction is no longer of product form. (isn't this what is called "decoherence" and what happens in a measurement) How would we assign a "wavefunction" to one small part of a larger system in the case when there is entanglement between the small part and the rest of the system? Of course there has to be some way to do this, as otherwise we would have to consider the wavefunction of whole universe every time we do QM... Do we need density matrices to do this?