The quantum state of a system of two particles is represented by a vector in the tensor product of their individual sate spaces.This is made up of linear combinations of products of state vectors (including integrals as well as finite sums), and if the individual states are represented in terms of (say) position observables by functions \psi_1(x_1) and \psi_2(x_2) , then a general state for the composite system is a function \psi(x_1,x_2) where the special case in which \psi(x_1,x_2)=\psi_1(x_1)\psi_2(x_2)
The quantum state of a system of two particles is represented by a vector in the tensor product of their individual sate spaces.
This is made up of linear combinations of products of state vectors (including integrals as well as finite sums), and if the individual states are represented in terms of (say) position observables by functions \psi_1(x_1) and \psi_2(x_2) , then a general state for the composite system is a function \psi(x_1,x_2) where the special case in which \psi(x_1,x_2)=\psi_1(x_1)\psi_2(x_2) corresponds to two separately identifiable particles.
If there is no interaction between the particles then this product form of a “pure tensor” is preserved by the time evolution of the system, but if there is an interaction term in the Hamiltonian then the evolution may carry such a pure tensor into a linear combination (or integral superposition) in which \psi(x_1,x_2) is more conveniently represented in a form like \psi(x_1,x_2)=\psi_{cm}(x_{cm})\psi_{rel}(x_{rel}) where x_{cm} and x_{rel} are the coordinates of the centre of mass and of the internal coordinates describing the relative positions of the two particles.
\psi_{cm}(x_{cm}) is then the wave function of the new composite particle and \psi_{rel}(x_{rel}) is the internal wave function which determines its possible energy levels etc.