Radial Models of Helium and Atomic Hydrogen in Two and Three Dimensions

Abstract We give a uniform radial formulation of atomic hydrogen and helium in three and two dimensions, solving the 3D S -wave ( $$l=0$$ l = 0 ) and 2D $$m=0$$ m = 0 sectors with a single Laguerre-basis method. A one-parameter transcription, $$L=0$$ L = 0 in 3D and $$L=-\tfrac{1}{2}$$ L = - 1 2 in 2D, reduces the dimensional change to one basis parameter, so the same validated code produces both dimensions. For helium, each electron is restricted to the matched radial sector ( $$l_1=l_2=0$$ l 1 = l 2 = 0 in 3D, $$m_1=m_2=0$$ m 1 = m 2 = 0 in 2D), which separates the effects of radial measure, residual inverse-square term, and interaction law. A central observation is that dimension-matched electron–electron interactions project to pure outer-radius kernels. In 3D this gives the standard S -wave Coulomb kernel $$1/r_{12}\rightarrow 1/r_>$$ 1 / r 12 → 1 / r > ; in intrinsic 2D the corresponding logarithmic interaction gives the less commonly noted kernel $$-2\ln (r_{12}/r_0)\rightarrow -2\ln (r_>/r_0)$$ - 2 ln ( r 12 / r 0 ) → - 2 ln ( r > / r 0 ) . Each is the Coulomb Green function of its own dimension after angular projection. The planar 2D model, which keeps the 3D $$1/r_{12}$$ 1 / r 12 law while restricting the electrons to a plane, is therefore mixed-dimensional and yields an elliptic-integral kernel instead. Analytic log–Coulomb crossover and Rytova–Keldysh potentials are also explored as conceptually intermediate one-electron cases between the pure 3D and planar 2D Coulomb models. The method is validated against exact planar 2D hydrogen energies, published logarithmic spectra, and helium benchmarks, with the restricted radial projection recovering the full correlated energies to within about