Undergrad Can the Helium atom be solved by other methods?

[physwiz222]

TL;DR

I hear the helium atom schrodinger equation is unsolvable because the electron repulsion term makes it inseparable thus you cant use separation of variables. Can it be solved by other methods besides separation of variables if it can’t why not.

The Schrödinger equation for helium is
(−ℏ^2/2me(∇21+∇22)+V(r1)+V(r2)+V(r12))ψ=Eψ
V(r12)=1/(r12-r1) which makes the equation inseparable. Can other methods be used to solve it.

 

[hilbert2]

You can approach the solution with numerical methods, as accurately as you want, just like you can calculate the value of $\sin 1.2$ using the power series $\sin x = x - \frac{1}{6}x^3 + \dots$. But if you attempt to find some exact mathematical expression for the energy eigenvalues and eigenfunctions of helium, it is likely to require some special functions that haven't even been named yet. And there's no guarantee that those same new functions are of any use when writing the solution for the lithium atom, for instance.

 

[physwiz222]

You can approach the solution with numerical methods, as accurately as you want, just like you can calculate the value of $\sin 1.2$ using the power series $\sin x = x - \frac{1}{6}x^3 + \dots$. But if you attempt to find some exact mathematical expression for the energy eigenvalues and eigenfunctions of helium, it is likely to require some special functions that haven't even been named yet. And there's no guarantee that those same new functions are of any use when writing the solution for the lithium atom, for instance.

How about other Analytical methods like Fourier, Laplace transform, etc. can those solve it

 

[hilbert2]

Probably not in the way you'd call a "solution".

 

[physwiz222]

Probably not in the way you'd call a "solution".

So basically even if u use those methods the wavefunction would not be extractable and the math would not simplify is that what u are saying

 

[PeroK]

So basically even if u use those methods the wavefunction would not be extractable and the math would not simplify is that what u are saying

I wouldn't be surprised if mathematicians have proved that no solution exists in terms of elementary functions. You could research that online.

In general exact solutions in terms of elementary functions are rare in physics. They are very much the exception.

 

[vanhees71]

Well, even the classical problem is not integrable. I'd not expect to find analytical solutions for the quantum problem either, but for sure He can be described numerically and ab initio including relativistic effects.