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When we solve the Schrodinger equation for the Helium atom, we usually resort to approximation methods citing that it cannot be exactly solved analytically.

When it is said that it "cannot" be solved does that translate to "an analytical solution does not exist" or "nobody has yet found a way to solve it exactly analytically"?

Qmechanic
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IAmOneWithTheScientist
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1 Answers1

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Solving Helium atom means solving a three-body problem - moreover, quantum three body problem, which is unlikely to be easier than a classical one:

Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

Furthermore:

A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.

This last passage contains reference to Griffiths' Introduction to Quantum Mechanics. The passage that I found in Griffiths' book, preceding equation (7.16) is:

It is curious that such a simple and important problem has no known exact solution 3. The trouble comes from the electron-electron repulsion, [...].

Footnote 3 reproduces what has been also mentioned in Wikipedia - that the problem is solvable for some special kinds of potential, but not the Coulomb one.

Roger V.
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    Does it? You can make a very good approximation (1 part per thousand in energy) by saying the nucleus is fixed at the center of the coordinate system, then you have two dynamical bodies in a fixed potential. Is that a three body problem? Note none of this detracts from the original statement - I'm pretty sure it is widely expected that no analytic solution to the helium Schrodinger equation exists. – AXensen Apr 06 '23 at 14:12
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    @AndrewChristensen If we believe Griffiths than no such solution exists (see the material that I added to the answer.) – Roger V. Apr 06 '23 at 14:14
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    Yes. Exactly what @Andrew said. It is mentioned in the book I am reading "Quantum Chemistry" by McQuarrie that even after the approximation of nucleus being stationary the two - body Schrodinger equation can still not be solved exactly. – IAmOneWithTheScientist Apr 06 '23 at 14:16
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    @IAmOneWithTheScientist I am not sure that fixing origin really makes into a two-body problem, sinc enow we have two bodies in an external potential. – Roger V. Apr 06 '23 at 14:18
  • @RogerVadim Can you comment on whether "solution does not exist" is the same as "no closed form solution exists" ? Wikipedia mentions that a closed form solution is one that can be expressed with finite number of "standard" operations/functions. 'Standard' is subject to change right? It may happen that mathematicians find a class of functions that can solve the system exactly right? – IAmOneWithTheScientist Apr 06 '23 at 14:20
  • I think going in the com makes it a two body problem just like a two body problem reduces into a one body problem in com frame. There also the single mass experiences an external potential (due to the 2nd body) no? – IAmOneWithTheScientist Apr 06 '23 at 14:23
  • @IAmOneWithTheScientist regarding what is exactly solvable see https://physics.stackexchange.com/a/559754/247642 and https://physics.stackexchange.com/a/587485/247642 – Roger V. Apr 06 '23 at 14:25
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    @IAmOneWithTheScientist moving to COM makes reduces the number of variables, but doesn't make it any less of a three-body problem (because real two-body problem is one-body in a potential in COM.) This is however not what at_ AndrewChristensen suggests - their approach is approximate: since the nucleus is much heavier than electrons, we can try to assume that COM and nucleus coincide. – Roger V. Apr 06 '23 at 14:28
  • Ah crap good point - you can always go to the center of mass frame, reducing the number of dynamical variables. So just because the center of mass is one particular body in this case doesn't probably make the system any more solvable. – AXensen Apr 06 '23 at 15:07
  • @Andrew Christensen In the present case, the COM is approximated to be located at the nucleus itself because of which it then is essentially a two-body problem. Even after this approximation the equation is not solvable – IAmOneWithTheScientist Apr 06 '23 at 15:48
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    One should add that even in classical mechanics only a handful (about a dozen) integrable Hamiltonians exist. All the other systems are "unsolvable". This doesn't mean that we can't give closed form solutions for special cases and some initial conditions. Hundreds, if not thousands of those can be found in the literature. One simply can't construct the entire solution for those systems. If I understand correctly, most of the research in many-particle quantum systems is therefor focused on predicting the ground state energy and first excitations correctly, rather than the entire spectrum. – FlatterMann Apr 06 '23 at 16:47