# Spacetime curvature around Gaussian wave packets

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.

• Wave function is just an abstract quantity and should not be viewed as some real physical wave evolving in space time
– KP99
Aug 27 '21 at 3:53
• @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 Aug 27 '21 at 6:22
• @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.
– KP99
Aug 27 '21 at 9:12
• @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 Aug 29 '21 at 21:37
• @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.
– KP99
Aug 30 '21 at 4:52