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why we cannot solve many body Schrodinger equation like this in finite different method? Could you give me some details?

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Qmechanic
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王庆轩
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    You can. But Schrödinger's equation is an equation in configuration space, which means that you will have 3n dimensions for n particles, and most of that will be useless space. If you split each dimension into N equal widths, then you will have to keep track of 3nN positions. For all but the smallest systems, this is just too much computational time and storage usage. – naturallyInconsistent May 25 '23 at 05:59
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    @naturallyInconsistent That should be an answer! – John Rennie May 25 '23 at 06:18
  • @JohnRennie the site rules on this makes me confused to no end. Anyway, posted. – naturallyInconsistent May 25 '23 at 06:22
  • There are more problems than just that. The Schroedinger equation in itself doesn't give us any physics. The physics lies in determining the spectrum of the system, which finite difference methods can't do (at least not all by themselves). – FlatterMann May 25 '23 at 06:22
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    Does this answer your question? What are the primary obstacles to solve the many-body problem in quantum mechanics? – Tobias Fünke May 25 '23 at 06:27
  • @FlatterMann "The Schroedinger equation in itself doesn't give us any physics", could you elaborate on this please? – Marius Ladegård Meyer May 25 '23 at 07:08
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    We strongly discourage posting images of text or mathematics on the site. Please use Mathjax to display mathematics. – StephenG - Help Ukraine May 25 '23 at 09:02
  • @MariusLadegårdMeyer We aren't interested in solving the time dependent SE for a concrete set of localized wave functions. That's not how we make measurements in e.g. atomic physics. We make measurements by performing spectroscopy, i.e. we measure scattering coefficients. In mathematical terms that is equivalent to finding the spectrum. The OP is asking about a numerical method that can, to the best of my knowledge, not calculate a spectrum. – FlatterMann May 25 '23 at 13:29

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You can. But Schrödinger's equation is an equation in configuration space, which means that you will have $3n$ dimensions for $n$ particles, and most of that will be useless space. If you split each dimension into $N$ equal widths, then you will have to keep track of $N^{3n}$ positions. For all but the smallest systems, this is just too much computational time and storage usage.

nanoman
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naturallyInconsistent
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