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https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.118.240402

It is known that the Schrödinger equation can not be solved for many body system, but in this article cited above, the wave function for the Bose-Einstein condensate was found experimentally. Does this mean that mathematics is fundamentally limited in describing nature, or that modern mathematics is not sufficiently developed to solve such equations, or that quantum systems are deterministic?

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Firstly, the many-body wave function of systems can actually be found (two entangled photons is a many-body system and we can describe it's wave function). The issue is, as Willy Billy Williams' comment eludes to, that the problem of finding it scales non-polynomially (NP). So it's not that we "can't" find it, it's more that it would take (with our current methods) EXTREMELY long to find the wave function of a large many body system. This is because as you add systems together in quantum mechanics the dimensionality of the state space grows exponentially (which grows faster then any polynomial). However, often the reason why we would like to know the wave function of a many body system is so that we can calculate some interesting quantities. This paper shows how you can find some of these quantities without ever knowing the wave function (by using a BEC).

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  • $\begingroup$ Yep. In particular, for added explanation, each added particle adds 3 more dimensions to the wave function. And thus if you divide each axis into $N$ "cells" as for a numerical differential equations solver, then there are $N^D$ "hyper voxels" in the wave function input space to sample, where $D$ is the dimension, equal to 3 times the number of particles. This is obviously an exponential function in the dimension $D$ and thus the number of particles. So it is not only NP, it is exponential time AND space. $\endgroup$ – The_Sympathizer Jun 23 '17 at 4:16
  • $\begingroup$ For example, for 3 particles and 1024 cells, it will take $1024^{3 \cdot 3} = 1024^9$ cells. If you need 8 bytes per cell -- e.g. two single precision IEEE 754 floats, one for the real and one for the imaginary part of the wave function, it will take 8192 Yobibytes to store all this data. For comparison Google's whole server farm is about 13 Exbibytes according to one figure I could dig up provided it was interpreted correctly (and Google keeps the true figure secret so this is a guesstimate), at least as of 2016. So it would take several hundred million Google server farms just to store ... $\endgroup$ – The_Sympathizer Jun 23 '17 at 4:22
  • $\begingroup$ ... all this wavefunction. $\endgroup$ – The_Sympathizer Jun 23 '17 at 4:22
  • $\begingroup$ Of course I'm sure there are ways to "fake" and "cheat" for certain kinds of problems, e.g. I suspect you might be able to expand say the wave function of atoms, like Helium, to an infinite series of 6, 9, etc. dimensional generalized spherical harmonics or something like that, thus making a compact, if not strictly "closed form" formula, but I haven't tried that so don't take my word for it; it's just a guess and could be total crap. But if you're talking generally 3 particles quantuming about in whatever way they want, 8192 Yobibytes or more is where it's at. $\endgroup$ – The_Sympathizer Jun 23 '17 at 4:25
  • $\begingroup$ tl;dr ENTANGLEMENTS result in EXPLOSIONS of data. Even "Big Data" p-sses its pants. :) $\endgroup$ – The_Sympathizer Jun 23 '17 at 4:28

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