Correlated Wave Function

Question

Imagine two correlated charged particles which have opposite charges. How can we write a total correlated wave function that describes these two particles?

I know that simplest or the most intuitive way to treat such problem is product wave function which has been multiplied by a correlation factor (or Geminal), $$ \Psi= \psi_1 \psi_2 *G$$ where $\psi_1$ stands for wave function of particle 1 and $\psi_2$ for particle 2. But I mean a more creative way to including correlation other than product form.

Answer 1

The most general wave function describing two distinguishable particles is $$\Psi(x_1,x_2) = \sum_{i=1}^N\lambda_i \psi_i(x_1) \phi_i(x_2),$$ where $\psi_i$ and $\phi_i$ are normalised wave functions for each particle with coordinates $x_1$ and $x_2$, and $\lambda_i$ are the corresponding probability amplitudes, i.e. they satisfy $\sum_i|\lambda_i|^2 = 1$. The particles are correlated (specifically, entangled) if and only if it is not possible to express $\Psi$ in the above form with $N=1$. Thus, the wave function you have written is not correlated unless $G$ is a non-trivial function of both $x_1$ and $x_2$. Assuming that this is what you have in mind, then the form you have written is actually an extremely specific (and therefore, arguably, quite creative) way of writing a correlated state.