⚛️ quantum physics

A solvable quantum many-body problem

This paper demonstrates that for a system of N particles, a many-body wavefunction exponential in the root-mean-square interparticle distance R solves the Schrödinger equation for a potential proportional to 1/R, yielding a set of translationally invariant rotational eigenstates of angular momentum.

Original authors: John Lekner

Published 2026-10-06

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: 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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of quantum physics, scientists often turn to systems of many particles to understand how matter behaves when countless tiny constituents interact. A central challenge in this field is solving the Schrödinger equation, the fundamental rulebook that predicts how a group of particles will move and arrange themselves. While the rules for a single particle or a pair are well understood, the mathematics becomes overwhelmingly complex as more particles are added, usually forcing researchers to rely on approximations. One of the few exceptions where an exact solution exists is a system where particles attract each other through a specific type of spring-like force that grows stronger as they get closer. This "harmonic" interaction allows physicists to calculate the exact energy and shape of the system for any number of particles, providing a rare, clear window into the quantum world. However, most real-world substances, such as liquid helium, do not behave like springs; their particles interact through short-range forces that are far more difficult to model mathematically.

A researcher at Victoria University of Wellington has now identified a new, solvable system that offers a different kind of mathematical clarity. The study focuses on a group of identical particles, specifically bosons, which are a type of subatomic particle that can occupy the same space and move in unison. The author demonstrates that if these particles are subject to a very specific, unusual force, their collective behavior can be described with a simple, exact formula. Unlike the familiar spring-like forces or the standard pairwise interactions seen in most gases, this new system relies on a potential energy that depends on the overall spread of the entire group. Specifically, the force acting on the particles is proportional to the inverse of the root-mean-square distance between all pairs in the system. In simpler terms, the strength of the interaction is determined by a single value representing the average separation of every particle from every other particle, rather than by looking at each pair individually.

The core discovery is that for this specific type of interaction, the wavefunction—the mathematical description of the system's state—takes on a remarkably simple form: an exponential decay based on that single measure of overall separation. This means that for any number of particles, from two up to a large group, the system has a precise, calculable ground state energy. The paper provides the exact formulas for the energy and the strength of the required force for systems with two, three, four, and any number of particles in one, two, or three dimensions. For instance, in a three-dimensional system of three particles, the energy and the necessary force strength are derived exactly, showing a clear pattern that holds true as the number of particles increases. The author also extends this finding to excited states, which are higher energy configurations where the particles possess angular momentum. By constructing specific mathematical forms that include rotational components, the study identifies a set of rotational excited states that are also exact solutions. These states carry specific amounts of angular momentum, increasing as the number of particles grows, while their energy levels follow a predictable relationship to the ground state.

The implications of this work lie in its potential to model complex fluids, such as the microscopic droplets of liquid helium found in nature. While the specific force required to make this system solvable does not perfectly match the physical interactions between helium atoms, the mathematical structure offers a new tool for approximation. The author notes that for a large number of particles, the potential energy in this model scales in a way that could help simplify the description of liquid droplets, replacing a messy collection of individual pairwise interactions with a single, unified many-body force. However, the paper also points out a physical limitation: the model is dominated by the distances between particles that are far apart, which is not entirely realistic for helium atoms that interact primarily when they are close together. Despite this caveat, the existence of these exact solutions provides a valuable benchmark. It gives physicists a concrete, solvable example of a many-body system that is not based on simple pairwise springs, offering a fresh perspective on how to think about the collective behavior of quantum matter. The work stands as a precise mathematical contribution, showing that even in the complex world of many interacting particles, there are still hidden patterns that can be solved exactly.

Technical Summary of "A solvable quantum many-body problem" by John Lekner

Problem Statement
The paper addresses the challenge of finding exact analytical solutions for the Schrödinger equation in quantum many-body systems involving $N$ identical bosons. While the system of pairwise harmonic interactions is a known solvable case, the author seeks to identify a distinct class of solvable potentials that are not strictly pairwise (except for the two-body limit) but depend on the collective geometry of the particle system. The specific focus is on translationally invariant zero-momentum states in one, two, and three dimensions.

Methodology
The author employs a direct ansatz approach, proposing a many-body wavefunction of the form $\Psi = e^{-KR}$ , where $R$ is defined as the square root of the sum of the squares of the interparticle distances:
$R^2 = \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N r_{ij}^2 = \sum_{j>i} r_{ij}^2$
The study proceeds by substituting this wavefunction into the Schrödinger equation ( $H\Psi = E\Psi$ ) for a Hamiltonian containing a kinetic energy term and an unknown potential $V(\{r_{ij}\})$ . The potential is required to be proportional to $R^{-1}$ .

To verify the solution, the author calculates the action of the Laplacian operator on the wavefunction. For $N=3$ and $N=4$ , this is performed using relative coordinates ( $r_{ij}$ ) and specific forms of the relative-motion Laplacian ( $\Lambda_N$ ) found in existing literature. For the general case, the derivation extends to arbitrary $N$ in one, two, and three dimensions. The author also constructs rotational excited states by multiplying the ground state wavefunction by polynomial factors of the complex relative coordinates $(x_{ij} + i y_{ij})$ and adjusting the decay constant $K$ to maintain the same potential form.

Key Contributions and Results
The paper establishes that the wavefunction $\Psi = e^{-KR}$ is an exact eigenstate of the Schrödinger equation for a potential proportional to $R^{-1}$ . The specific results are:

Ground State Solutions:
- Energy: The ground state energy is given by $E_N = -N \frac{\hbar^2 K^2}{2M}$ , where $M$ is the particle mass.
- Potential: The interaction potential required to sustain this state is $V_N = -C_D \frac{\hbar^2 K}{2M R}$ $V_{N} = - C_{D} \frac{ℏ ^{2} K}{2 M R}$ , where the coefficient $C_D$ $C_{D}$ depends on the number of particles $N$ $N$ and the spatial dimension $D$ $D$ :
  - 1D ( $N>2$ ): $C_1 = N(N-2)$
  - 2D: $C_2 = N(2N-3)$
  - 3D: $C_3 = N(3N-4)$
- General Form: The potential can be summarized as $V_N = -N(DN - D - 1) \frac{\hbar^2 K}{2M R}$ for $D$ dimensions (with a noted exception for the 1D two-body delta function case).
Rotational Excited States:
- The author derives a set of translationally invariant rotational eigenstates with non-zero angular momentum. For 3D, an excited state is constructed as $\Psi_N^e = \prod_{j>i} (x_{ij} + i y_{ij}) e^{-K_e R}$ .
- These states possess an orbital angular momentum eigenvalue of $L_z = \frac{1}{2}N(N-1)\hbar$ .
- To maintain the same $R^{-1}$ potential form as the ground state, the decay constant must be adjusted to $K_e = \frac{3N-4}{N^2+2N-4}K$ .
- The energy of these excited states scales as $E_N^e = \left(\frac{3N-4}{N^2+2N-4}\right)^2 E_N^0$ .

Significance and Claims
The paper claims to present a new "solvable many-body problem" distinct from the harmonic oscillator model. The primary significance lies in the existence of exact solutions for a potential that scales with the inverse of the root-mean-square interparticle distance, a collective variable rather than a sum of pairwise terms.

The author notes that while these solutions are mathematically exact, their physical application to systems like liquid helium droplets is limited. The potential scales as $N^2 R^{-1}$ for large $N$ , and the variable $R$ is dominated by pairs with large separations, which does not align well with the short-range nature of helium interactions. Consequently, the paper presents these results as a theoretical model for exact solvability and a source of rotational eigenstates, rather than a direct physical model for specific experimental systems. The work is framed as a contribution to the catalog of exactly solvable quantum mechanical models.

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