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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.

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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.

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  • $\begingroup$ Thanks a lot, but the wave function that you have suggested has product form again, namely it is product of wave function of each particle. Yes factor G in my wave function is a function of both particles. Anyway I look for a correlated wave function that has not a product form. $\endgroup$ – Wisdom Apr 1 '18 at 18:59
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    $\begingroup$ @NahidSR What I wrote is the most general possible wave function for any two-particle quantum mechanical system. It is actually not a product, unless $N=1$. Any wave function can be brought into the above form, allowing for $N$ to be possibly infinite. So it's not clear what other kind of answer you could be looking for. It might help you to try to think of an example that cannot be written in the above form. $\endgroup$ – Mark Mitchison Apr 1 '18 at 21:25
  • $\begingroup$ Please tell me where is the correlation hidden? in λ? and what does show N exactly? In fact I look for a total wave function which can include the correlation in its heart, not as a product of wave functions of two particles. I'll be so grateful if you introduce me a textbook which discuss about this subject. $\endgroup$ – Wisdom Apr 2 '18 at 4:41
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    $\begingroup$ @NahidSR The correlation enters by the fact that $\Psi(x_1,x_2)$ cannot be written as a product $\Psi = \psi\phi$, but rather as a sum over such products. The minimum number of terms $N$ in this sum is called the Schmidt number, and it (or, rather, $N-1$) is a measure of entanglement (correlation). If you want to learn about quantum correlations then a quantum information textbook might be useful for you, e.g. Nielsen & Chuang. $\endgroup$ – Mark Mitchison Apr 2 '18 at 9:42
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    $\begingroup$ @NahidSR I don't think you do understand. Every two-particle wave function is of the form I wrote. There are no other possibilities. $\endgroup$ – Mark Mitchison Apr 2 '18 at 19:28

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