So I just learned how to solve the Schroedinger Equation for the Hydrogen atom. The set up looked like this: $$\left[\dfrac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\dfrac{\partial}{\partial r}\right)+\frac{1}{r^2\sin \theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta \dfrac{\partial}{\partial \theta}\right) + \frac{1}{(r\sin \theta)^2}\frac{\partial^2}{\partial \phi^2}+\frac{2m}{\hbar^2}\left(E +\frac{\epsilon}{r}\right) \right]\varphi(r,\theta,\phi)=0$$ Where $r$ is the position vector and $\theta$ and $\phi$ are polar angles. $\epsilon=\frac{e^2}{4\pi \epsilon_0}$ and $E$ is the energy spectrum. So, just for practice, I want to solve the shroedinger equation for the helium atom. All I need is the potential energy. What is it?
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$\begingroup$ Use the standard Coulombic electrostatic potential with three terms: between electron 1 and the nucleus; between electron 2 and the nucleus; between electron 1 and electron 2. Expect to encounter difficulties and bitter disappointment, because the resulting 6D PDE isn’t separable. The good news is that there are analytic approximation techniques, which you will probably learn later in your QM course. $\endgroup$– GhosterCommented Sep 21, 2022 at 19:56
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1$\begingroup$ The Helium atom is an example of the Three Body Problem and no general solution method produces a simple solution like the two body hydrogen atom. A suggested practice system would be the two body harmonic potential which has great utility in quantum theory. $\endgroup$– StephenG - Help UkraineCommented Sep 21, 2022 at 23:28
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1$\begingroup$ Google for helium atom Hamiltonian or some similar phrase and you'll find lots of articles. See for example this article. $\endgroup$– John RennieCommented Sep 22, 2022 at 5:36
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