A wave-equation route to building two-electron atomic functions

Describing an atom with two electrons means accounting for three competing interactions: each electron is attracted to the nucleus, while the electrons repel one another. John Lekner’s preprint, submitted on November 9, 2026, explores a mathematical connection that could help construct functions used in this problem. It does not present a complete solution for helium or report new atomic energy predictions.

The central claim is a correspondence between axisymmetric solutions of the classical wave equation and functions that become zero when acted on by the two-electron kinetic-energy operator. Understanding that qualification is essential: these are potential ingredients for atomic calculations, not automatically physically acceptable electron states.

Separating one part of the atomic problem

The paper starts with a nucleus held fixed, neglecting its motion. Its Hamiltonian—the operator representing the system’s total energy—contains electron kinetic energy, electron–nucleus attraction and electron–electron repulsion.

Lekner focuses on the differential operator underlying the kinetic-energy terms. It combines a Laplacian for each electron; a Laplacian measures how a function curves across spatial coordinates. The task is to find functions for which this combined operator gives zero.

That condition is narrower than solving the full atomic Schrödinger equation. The attractive and repulsive potential-energy terms still matter. Nor should the mathematical condition be read as evidence that bound electrons physically possess zero kinetic energy: the functions under discussion need not individually satisfy the requirements of a normalizable atomic state.

Earlier work cited in the preprint used such functions to supplement Fock expansions, mathematical representations of atomic wavefunctions, so that boundary conditions could be satisfied. Lekner’s proposal supplies another route to finding these supplementary functions.

Turning electron geometry into wave coordinates

The derivation describes an electron configuration through three distances: each electron’s distance from the nucleus and the distance between the electrons. These capture the triangle’s shape and size without specifying its overall orientation.

Lekner then introduces hyperspherical coordinates. One variable combines the two electron–nucleus distances into an overall size; two angles describe their relative distances and their angular separation at the nucleus.

A further transformation constructs four coordinates from this geometry. In those variables, the zero condition for the kinetic-energy operator becomes a four-dimensional Laplace equation. Replacing one coordinate by an imaginary multiple of a time-like variable gives the familiar mathematical form of the classical wave equation in three spatial dimensions.

The relevant wave solutions are axisymmetric: they do not change when rotated around a chosen axis. Their radial, axial and time-like arguments can then be replaced by expressions involving the electron distances.

This is a correspondence between equations and functions, not a claim that electrons follow classical wave trajectories. The imaginary time-like substitution is a mathematical device, not the atom’s physical clock. Exchanging the two electrons reverses the sign of the corresponding coordinate, so the paper includes both signs when discussing how to construct electron-exchange symmetry.

Borrowing existing solutions

To illustrate the connection, Lekner transcribes an integral representation associated with Harry Bateman’s early work on the wave equation. He also translates a more recent family of closed-form wave solutions into candidate functions annihilated by the atomic kinetic-energy operator.

The paper works in the opposite direction as well. Previously known logarithmic atomic functions become static wave-equation solutions—functions with no time dependence. These examples show how expressions from the two mathematical settings can be related explicitly.

The possible benefit is access to an existing collection of wave-equation solutions rather than having to construct every atomic building block directly.

What remains unresolved

This is a theoretical preprint, not an experimental result. It provides transformations and examples, but no numerical benchmark showing that they improve atomic energies, accuracy or computational cost.

Physical usefulness also requires more than satisfying a differential equation locally. Singularities, boundary conditions and behavior at large distances must be examined; the logarithmic examples make clear that regularity cannot simply be assumed. Because the transformation uses complex arguments, the domains on which a chosen wave solution is defined also matter.

Lekner expresses hope that the connection will help solve the interacting two-electron problem. The demonstrated contribution is more specific: a way to translate between classical wave mathematics and a class of functions relevant to atomic wavefunction construction. Whether those functions make practical calculations better remains to be shown.