🔢 mathematics

The wave equation and two-electron atomic wavefunctions

This paper demonstrates that axisymmetric solutions to the classical wave equation in three spatial dimensions are in a one-to-one correspondence with functions that annihilate the two-electron kinetic energy operator in a fixed-nucleus coordinate system.

Original author: John Lekner

Published 2026-11-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original author: John Lekner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the heart of every atom lies a delicate balancing act. A heavy, positively charged nucleus sits at the center, while two lighter, negatively charged electrons zip around it. To understand how these particles behave, scientists must solve a complex mathematical puzzle known as the wave equation. This equation describes the motion of the electrons, but it is notoriously difficult to crack because the electrons repel each other while being pulled toward the nucleus. The difficulty arises from the kinetic energy of the system—the energy associated with the motion of the particles. If scientists can find a way to describe the motion of these electrons that ignores the forces pushing and pulling them, they can isolate the pure kinetic energy. This simplified version of the problem acts as a foundation, a clean slate upon which the messy reality of electric forces can be built. For decades, researchers have sought specific mathematical shapes, or functions, that vanish when the kinetic energy operator is applied to them. Finding these shapes is like finding the skeleton of the atom before adding the flesh of its interactions.

A physicist named John Lekner has now revealed a surprising connection between this quantum puzzle and a much older, classical problem. He demonstrates that the mathematical functions which describe the motion of two electrons with zero kinetic energy are in a perfect, one-to-one match with solutions to the classical wave equation. This is the same equation that describes how ripples spread across a pond or how sound travels through the air, but applied here to a four-dimensional space-time framework. Lekner's work shows that the complex geometry of two electrons moving around a nucleus can be translated directly into the geometry of a wave moving through ordinary space and time. By using a specific set of coordinates that treat the distances between the particles as if they were dimensions in a higher space, he proves that any solution to the classical wave equation can be instantly converted into a valid solution for the two-electron kinetic energy problem.

The paper does not merely suggest this link; it provides a rigorous proof using a coordinate system that maps the positions of the two electrons onto a four-dimensional space. In this mapping, the distance between the electrons and their distance from the nucleus transform into new variables that behave exactly like the radius, height, and time in a standard wave equation. When the wave equation is solved for a specific shape, that shape can be translated back into the language of the electrons, yielding a function that automatically cancels out the kinetic energy. Lekner shows that this correspondence is not a coincidence but a fundamental structural identity. He uses this bridge to generate new solutions for the electron problem by taking known solutions from classical wave physics. For instance, he takes a well-known solution to the wave equation, which describes a static ripple, and translates it into a new function for the electrons that satisfies the condition of zero kinetic energy.

This approach allows for the creation of a vast library of solutions that were previously difficult to find or verify. The paper highlights that many of these solutions, which had been discovered by other researchers through trial and error or complex expansions, are actually just translations of simple, classical wave patterns. By recognizing this, Lekner unifies two seemingly different areas of physics: the quantum mechanics of atomic structure and the classical mechanics of wave propagation. The work confirms that the mathematical tools developed over a century ago to describe sound and light are directly applicable to the behavior of electrons in an atom. This does not solve the entire problem of the two-electron atom, as the electric repulsion between the electrons still needs to be accounted for, but it provides a powerful new method for generating the essential building blocks needed to do so. The result is a clearer path forward for understanding the motion of electrons, turning a century-old problem into one that can be approached with the clarity of classical physics.

Technical Summary: The Wave Equation and Two-Electron Atomic Wavefunctions

Problem Statement
The paper addresses the mathematical challenge of solving the Schrödinger equation for a two-electron atomic system. The Hamiltonian for such a system includes a kinetic energy operator, $L = \nabla_1^2 + \nabla_2^2$ , and potential terms involving the nucleus-electron and electron-electron interactions. A specific focus is placed on finding functions that annihilate the kinetic energy operator ( $L\psi = 0$ ). Such functions are valuable for augmenting Fock expansion functions to satisfy physical boundary conditions, particularly near singularities where electrons approach the nucleus or each other. The author notes that while solutions to $L\psi=0$ have been identified (e.g., by Gottschalk and Maslen), the underlying mathematical structure connecting these solutions to classical physics remains to be explicitly detailed.

Methodology
The paper employs a coordinate transformation approach to establish a correspondence between the two-electron kinetic energy operator and the classical wave equation.

Hyperspherical Coordinates: The author introduces hyperspherical coordinates ( $r, \alpha, \theta$ ) defined by the distances of the electrons from the nucleus ( $r_1, r_2$ ) and their separation ( $r_{12}$ ).
Fock Coordinates Transformation: The author defines a set of four variables ( $X, Y, Z, T$ $X, Y, Z, T$ ) based on the electron coordinates:
- $X = 2r_1r_2 \sin\theta \cos\phi$
- $Y = 2r_1r_2 \sin\theta \sin\phi$
- $Z = 2r_1r_2 \cos\theta$
- $T = r_1^2 - r_2^2$
Operator Mapping: By substituting these variables, the two-electron Laplacian $L$ is shown to transform into a four-dimensional Laplacian: $\partial_X^2 + \partial_Y^2 + \partial_Z^2 + \partial_T^2 = 0$ .
Wave Equation Analogy: The author identifies that if the variable $T$ is treated as imaginary time ( $T = i c t$ ), the four-dimensional Laplacian becomes the axisymmetric wave equation in three spatial dimensions ( $\rho, z, ct$ ), where $\rho = \sqrt{X^2+Y^2}$ and $z=Z$ .

Key Contributions and Results
The primary contribution of the paper is the demonstration of a one-to-one correspondence between axisymmetric solutions of the classical wave equation and functions that annihilate the two-electron kinetic energy operator.

General Correspondence: Any solution $\Psi(\rho, z, ct)$ to the axisymmetric wave equation $\{\partial_\rho^2 + \frac{1}{\rho}\partial_\rho + \partial_z^2 - \partial_{ct}^2\}\Psi = 0$ maps directly to a solution $\psi(r_1, r_2, r_{12})$ of $L\psi=0$ via the coordinate transformations defined in Section 2.
Generation of Solutions: The paper utilizes known solutions of the wave equation to generate new solutions for the two-electron problem:
- Bateman Solutions: Using Bateman's integral solution for the wave equation, the author derives a general integral form for $\psi(r_1, r_2, r_{12})$ involving an arbitrary function $F$ .
- Closed-Form Solutions: The paper applies recent closed-form solutions of the wave equation (parametrized by a length $a$ ) to generate new families of zero-kinetic-energy functions. These are expressed in terms of complex lengths $R$ and coordinates $z$ .
Verification of Existing Solutions: The paper demonstrates that previously known solutions by Gottschalk and Maslen (such as logarithmic functions of electron distances) are specific instances of static solutions to the wave equation transcribed into electron coordinates. For example, the logarithmic solution $\ln(\frac{1+\cos\theta}{1-\cos\theta})$ corresponds to a static wave equation solution.

Significance and Claims
The paper modestly claims to provide a theoretical link between classical wave physics and the quantum mechanical description of two-electron atoms.

Unification: It establishes that the mathematical tools used to solve the classical wave equation (dating back to Bateman and extended in modern electrodynamics) can be directly transcribed to solve the kinetic energy component of the two-electron Schrödinger equation.
Utility: The author suggests that this correspondence offers a systematic method for generating functions that satisfy the boundary conditions required for analytic expansions of atomic wavefunctions.
Scope: The paper does not propose new experimental applications or solve the full Schrödinger equation (including potential terms). Instead, it focuses strictly on the kinetic energy operator, noting that because the kinetic energy dominates the wavefunction's generation, these solutions are relevant for non-Coulombic potentials and other multi-body systems as well.

In summary, the paper posits that the century-old problem of constructing two-electron wavefunctions can be illuminated by recognizing that the condition $L\psi=0$ is mathematically equivalent to the classical wave equation in a transformed coordinate space.

More like this