# Solving the Helium Eigenvalue Problem—where to start?

**URL:** <https://discourse.julialang.org/t/solving-the-helium-eigenvalue-problem-where-to-start/137783>\
**Category:** Specific Domains\
**Tags:** question, quantum, physics, eigenvalues\
**Created:** [June 24, 2026, 5:37pm UTC](https://discourse.julialang.org/t/solving-the-helium-eigenvalue-problem-where-to-start/137783 "2026-06-24T17:37:36Z")\
**Posts on this page:** 1\
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**Author:** ![ducksoverip](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ducksoverip/32/31967_2.png) [@ducksoverip](https://discourse.julialang.org/u/ducksoverip)\
**Post date:** [June 27, 2026, 9:44pm UTC](https://discourse.julialang.org/t/solving-the-helium-eigenvalue-problem-where-to-start/137783/7 "2026-06-27T21:44:22Z")

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So, thinking about the problem a bit more and doing some more research, I’ve determined the following:

- Yes, the method used in the [paper](https://www.academia.edu/download/95215177/fulltext.pdf) is a pseudospectral collocation method (per [Wikipedia](https://en.wikipedia.org/wiki/Pseudo-spectral_method), apparently pseudospectral is a synonym for discrete variable representation, which is indeed what the Fortran code uses. Seems I’m one of today’s [lucky ten thousand](https://xkcd.com/1053/).)
- I inadvertently left a detail out of my description; after expanding \psi and 1/|\vec{r\_1} - \vec{r\_2}| in terms of coupled spherical harmonics, the entire equation is integrated over both pairs of angular coordinates, which (because of the orthogonality properties of the coupled harmonics) yields the following, where V\_{F0}(i,j) is the expanded and integrated form of 1/|\vec{r\_1} - \vec{r\_2}|, and i and j index bundles of the integer partial wave parameters L, l\_1, and l\_2:

-\frac{1}{2}\frac{\partial^2}{\partial r\_1^2} -\frac{1}{2}\frac{\partial^2}{\partial r\_2^2} + \frac{l\_1(l\_1 + 1)^2}{2r\_1^2} + \frac{l\_2(l\_2 + 1)^2}{2r\_2^2} - \frac{2}{r\_1} - \frac{2}{r\_2} \\+ \sum\_j V\_{F0}(i,j) = i \frac{\partial}{\partial t} \psi\_i(r\_1, r\_2)

- The above coupled PDEs are what are properly discretized on a 2D grid of Legendre polynomial roots and solved via reverse imaginary time propagation.
- If I’m purely interested in the ground-state eigenvalues and eigenvectors of this problem, then if I can construct the operators as a matrix, I can use KrylovKit.jl or ArnoldiMethod.jl to solve the eigenvalue problem for a given set of partial waves, making sure that I use enough partial waves to converge on the correct ground-state energy.

So, with this in mind, I think I can ask some more specific questions, namely:

1. Is pseudospectral collocation (what the paper calls finite element-discrete variable representation) the only viable approach, or is it worth trying a different method?
2. Based on the answer to the above, what Julia packages will help me specify the grid and construct the operators for this problem? I’m not really sure what the operators will look like if I’m not doing the split-operator + reverse imaginary time method.

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