Why specifically cant QM handle 2 electrons? Ie what about the

>>8494666
because according to QM, all electrons are completely interchangeable. Therefore, having "two electrons" is logically incoherent within this framework, since all electrons are the same.

>>8494671
so what do they mean when they say the SE cant be solved for 2 electron systems?

i always took it to mean it was too complex, but i guess it implies a more fundamental reason

>>8494673
It's not even a well-posed question, because QM doesn't allow for the existence of two separate electrons

>>8494671
>>8494675
Your pretty good

>>8494675
i dont understand what you mean.

we can approximate helium to 99.9999999999999999999999999999999999999999999999999999999999999999999999% accuracy if we had the computing power, but we can never solve it exactly

is it because we don't have all of the variables?

like we dont have terms for what happens when the electrons interact with each other?

>>8494666
I don't know how satisfying this will be, but... We want to evaluate the amplitude [eqn] \mathcal { A } = \langle x | e^{ -i \hat { H } t } | 0 \rangle = \int d^3 p \langle x | e^{ -i \hat { H } t } | p \rangle \langle p | 0 \rangle \\ = \frac { d^3 p } { (2 \pi )^3 } e^{i p \dot x } e^{ - i E_p t } [/eqn] Doing some integrals we end up with [eqn] \mathcal { A } = \frac { i } { 2 \pi ^2 | x | } e^{-m |x| } \int ^{ \infty } _{ m } dz ~z e^{ - ( z - m) |x| } \sinh \left ( t \sqrt { z^2 - m^2 } \right ) [/eqn]

But the integrand is positive definite, which means that there's a non-zero chance of finding an electron outside it's forward light cone. So that means it's impossible for single particle quantum mechanics and special relativity to be reconciled.

Now [math] why [/math] is that the case, I've no idea. Also these
>>8494675
>>8494671

>>8494673
It's the potential term makes it intractable.

>>8494673
>i always took it to mean it was too complex, but i guess it implies a more fundamental reason
it has no analytical solution thats all. numerical you can solve it of course.

>>8494666
Quantum mechanics itself does not have any conceptual problem with systems with more than one electron and you can in fact just add additional terms for the other electrons. However, since they are treated as indistinguishable, you have to be careful so that your results do not depend on "which" electron is in which state (this is ensured by wave function antisymmetrization).
Most problems in quantum mechanics (such as atoms with more than one electron) do not have analytical solutions just because differential equations are fucky, but that's a completely different issue (the equivalent problem in classical mechanics, the three-body problem, doesn't have an analytical solution either).