Solving 1 electron atom Schrodinger Equation in rectangular coordinates?

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Discussion Overview

The discussion revolves around the solvability of the Schrödinger equation for a one-electron atom in rectangular coordinates versus spherical coordinates. Participants explore the implications of the Coulomb potential and the geometry of atomic systems, focusing on the appropriateness of coordinate systems for solving the equation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that spherical coordinates are preferred due to the spherical nature of atoms and the central potential represented by the Coulomb interaction.
  • Others argue that the mathematical complexity in Cartesian coordinates is a significant barrier, as the Coulomb potential is not separable in those coordinates.
  • A participant notes that while spherical coordinates allow for a fully separable Schrödinger equation, the situation changes with multiple electron atoms due to additional electron-electron repulsion terms, complicating the potential landscape.
  • One participant emphasizes that the final solution's form is less important than the fact that the Coulomb potential is inherently spherical, making spherical coordinates more convenient.

Areas of Agreement / Disagreement

Participants generally agree on the challenges of using rectangular coordinates for this problem, but there is no consensus on the necessity of using spherical coordinates or the implications of the potential's geometry.

Contextual Notes

Some limitations include the unresolved nature of why spherical coordinates are deemed necessary, the dependence on the central potential assumption, and the complexities introduced by multiple electron interactions that prevent analytical solutions.

Who May Find This Useful

This discussion may be useful for undergraduate physics students, particularly those studying atomic physics and quantum mechanics, as well as educators seeking to understand common student confusions regarding coordinate systems in quantum problems.

VulpineNinja
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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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The math is so horrendous in Cartesian coordinates, besides the fact that the Coulomb potential is not separable.
 
VulpineNinja said:
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.
 
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.
 
I've found the answer already, but not the elaboration.
So, thank you!
 

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