Solving 1 electron atom Schrodinger Equation in rectangular coordinates?

[VulpineNinja]

Hi, I'm a physics undergraduate, and I'm having trouble understanding Atomic Physics right now.

At first I thought it (the question in the title) is possible, but then I think I got confused with particle in a box. So I refer to textbooks and look for answers in the internet, and confirmed that we can only solve it by using spherical coordinates. But I couldn't find the reason why we should only use spherical coordinates, but not rectangular coordinates. My only guess is it's because atoms are said to be spherical. Am I correct?

Whether I'm right or not, can anyone elaborate this? Thank you in advance.

 

[Dr Transport]

The math is so horrendous in Cartesian coordinates, besides the fact that the Coulomb potential is not separable.

 

[SpectraCat]

Hi, I'm a physics undergraduate, and I'm having trouble understanding Atomic Physics right now.

At first I thought it (the question in the title) is possible, but then I think I got confused with particle in a box. So I refer to textbooks and look for answers in the internet, and confirmed that we can only solve it by using spherical coordinates. But I couldn't find the reason why we should only use spherical coordinates, but not rectangular coordinates. My only guess is it's because atoms are said to be spherical. Am I correct?

Whether I'm right or not, can anyone elaborate this? Thank you in advance.

Perhaps a bit more precise way to say it is that the coulomb potential between the nucleus and electrons represents a central potential in 3-D, which means that spherical coordinates are a natural representation for the single-electron problem, and the Schrödinger equation is fully separable in those coordinates. Note that for multiple electron atoms, the full potential is no longer a central potential, since there are electron-electron repulsion terms that depend inversely on the electron-electron displacement. In fact, there is no analytical solution for the energies and wavefunctions of multiple electron atoms. However, the problem can be solved approximately by expansion in a basis of 1-electron, H-atom-like states, so that representation is still useful.

 

[0xDEADBEEF]

It is not so important what the final solution will look like, but the Coulomb potential is spherical. So it looks much easier in spherical coordinates.

 

[VulpineNinja]

I've found the answer already, but not the elaboration.
So, thank you!