# Solving 1 electron atom Schrodinger Equation in rectangular coordinates?

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.

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Dr Transport
Gold Member
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 Schrodinger 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.

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.