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https://en.wikipedia.org/wiki/Wave_packet#/media/File:Wavepacket-a2k4-en.gif

From quantum mechanics we know how to describe, statistically, an unbound particle floating in space. Treat it as a dispersing, normalizable, Gaussian wave packet. We know how the wave packet evolves, and can give "what would we measure" descriptions of it's properties by sandwiching the right operator representing some/any physical quantity.

Question 1: why can't we just work with the expectation values of properties of a massive spread-out wave function ( < x >, < p > ), write up some statistical, or averaged version of a stress-energy-momentum tensor, plug it into the Einstein field equations, tidy out issues, and voila: wave-function sourced probabilistic description of the spacetime metric? I'm certain there's some problems with this "program", but what?

Question 2: put two apples into space with a zero relative velocity. Due to gravity, they will eventually move towards each other. Now lets say it's not two apples, but two Gaussian wave packets of neutral particles. What has to be plugged into this two-particle $\Psi$ that would evolve it in a way where the two maxima of the probability density would move closer to each other over time at the right rate?

Especially Q2 seems to be such a simple setup that smart people must have written something up that works, at least approximately.

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  • $\begingroup$ Wave function is just an abstract quantity and should not be viewed as some real physical wave evolving in space time $\endgroup$
    – KP99
    Aug 27 at 3:53
  • $\begingroup$ @KP99 I find that a misleading assertion. Matter is made of wavefunctions, regardless of how that wavefunction manifests observationally (rate density of scatterings) it is by definition a "physical wave", in any sense where that word has meaning and can be applied. To the best of our knowledge, the "physical waves" you are thinking off are made themselves of complex hierarchies of "abstract" wavefunctions $\endgroup$
    – lurscher
    Aug 27 at 6:22
  • $\begingroup$ @lurscher I meant that wavefunctions only give a probability distribution of matter(say electron) and not the actual profile in which this electron is distributed/dispersed in space. Wave function can only give information about various states of an electron. Of course subatomic particles show non-local behaviors. but that is not well understood other than what our QM postulates says. Even wave function is just an approximation of quantum fields. I will apologize if I made another misleading assertion, I am saying this from my undergraduate knowledge only. $\endgroup$
    – KP99
    Aug 27 at 9:12
  • $\begingroup$ @KP99 : "abstract quantity and should not be viewed as some real physical wave evolving in space time" show me why. The idea of only-probabilistically existing physical objects works very well $\endgroup$
    – JohnDeeDoe
    Aug 29 at 21:37
  • $\begingroup$ @JohnDeeDoe Here I am particularly talking about wave function. So if I haven't misinterpreted your original question, you are replacing the idea of free particle with wave packet (a scalar field $\Psi(x)$) as a real entity, evolving in space time. Is this the same wave function as in Schrödinger's equation? None of the postulates of QM suggest that we can do so (atleast within Copenhagen interpretation). The first question is independent of this assertion though. $\endgroup$
    – KP99
    Aug 30 at 4:52
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Regarding the first part of the question, the problem with this approach is that you are combining general relativity with wave packets, which evolve according to Schrodinger equation, and so do not behave well under Lorentz transformations. You would get some non-Lorentz-covariant result.

Before you can combine quantum particles and general relativity, you need to find a special relativistic description of quantum particles. In doing so, you will find that to avoid paradoxes and inconsistencies it is necessary to abandon the concept of single particle wavefunction and wave packet. A special relativistic description of quantum mechanics requires fileds and it is called quantum field theory.

In order to fit gravity in a quantum field theory (an make it interact with other fields) you need to make it a quantum field. And that is the challenge we haven't been able to overcome.

Regarding the second part of the question, if we assume Newtonian gravity, and that the two particles are neutral and without spin, then this is mathematically equivalent to the problem of the hydrogen atom.

There are two particles that attract each other with an inverse square law, just like a proton and an electron. You can just write down the hamiltonian and solve it numerically with standard tools of quantum mechanics.

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