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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.

Qmechanic
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Wisdom
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1 Answers1

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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.

Mark Mitchison
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  • 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. – Wisdom Apr 01 '18 at 18:59
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    @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. – Mark Mitchison Apr 01 '18 at 21:25
  • 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. – Wisdom Apr 02 '18 at 04:41
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    @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. – Mark Mitchison Apr 02 '18 at 09:42
  • Thanks. I understood your mean, it is similar to wave function of some Post-HF methods such as Coupled Cluster. Such wave functions have been examined for particles such as electron-proton and it diverges.Thus it needs to a more creative wave function! – Wisdom Apr 02 '18 at 16:30
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    @NahidSR I don't think you do understand. Every two-particle wave function is of the form I wrote. There are no other possibilities. – Mark Mitchison Apr 02 '18 at 19:28
  • Mark, does this most general formulation of the wave function of two indistinguishable particles only hold for "non-interacting" particles, or can they also be interacting, like the electron and proton in a hydrogen atom? – freecharly Apr 29 '18 at 22:54
  • @freecharly This holds for interacting or non-interacting distinguishable particles. Formally speaking, this is a purely kinematical question which is independent of dynamical details such as whether the particles interact or not. However, if you actually want to see correlations between the particles (which means $N>1$), then they need to interact somehow. – Mark Mitchison Apr 30 '18 at 16:17
  • Thank you, Mark! So it would be correct to consider the proton wave function in the hydrogen atom with a bound electron to be perfectly correlated to the electron wave function? – freecharly Apr 30 '18 at 16:24
  • @freecharly I am not sure about "perfectly" correlated. In an infinite-dimensional Hilbert space there is no maximally entangled state with finite energy, I believe. But there are certainly strong correlations. This can be seen since the wavefunction is approximately of the form $\Psi(r_p,r_e) = \phi(r_p)\psi(|r_e-r_p|) \neq \phi(r_p)\psi(r_e)$, where $\phi$ is the centre-of-mass (approximately the proton position $r_p$) wavefunction and $\psi$ is the usual solution of the Schrodinger equation for the relative coordinate in terms of Laguerre polynomials and spherical harmonics. – Mark Mitchison Apr 30 '18 at 16:50
  • Mark, thank you very much for your insights. I am mainly an experimentalist and was wondering whether my derivation of the correlated wave functions of the electron and proton in hydrogen https://physics.stackexchange.com/questions/401853/hydrogen-atom-whats-the-wave-equation-for-the-atoms-nucleus/402086#402086 was OK also in the eyes of an expert – freecharly Apr 30 '18 at 18:53