[Paper Review] Exact Solution of a N-body Problem in One Dimension

Avinash Khare|arXiv (Cornell University)|Oct 15, 1995

Quantum chaos and dynamical systems4 citations

TL;DR

This paper presents an exact solution for a one-dimensional quantum N-body system with inverse-square and long-range power-law interactions. Using algebraic techniques and symmetry analysis, it derives the complete energy spectrum and proves that scattering processes reverse momenta: p′_i = p_{N+1−i}, establishing a unique scattering behavior in integrable systems.

ABSTRACT

Complete energy spectrum is obtained for the quantum mechanical problem of N one dimensional equal mass particles interacting via potential $$V(x_1,x_2,...,x_N) = g\sum^N_{i < j}{1\over (x_i-x_j)^2} - {α\over \sqrt{\sum_{i < j} (x_i-x_j)^2}}$$ Further, it is shown that scattering configuration, characterized by initial momenta $p_i (i=1,2,...,N)$ goes over into a final configuration characterized uniquely by the final momenta $p'_i$ with $p'_i=p_{N+1-i}$.

Motivation & Objective

• To solve exactly a quantum mechanical N-body problem in one dimension with a specific combination of inverse-square and long-range two-body interactions.
• To determine the complete energy spectrum of the system, which is essential for understanding its quantum behavior.
• To analyze the scattering properties of the system and characterize the final state in terms of initial momenta.
• To establish the integrability of the model through exact solvability and symmetry-based analysis.
• To demonstrate that the scattering process leads to a unique momentum reversal pattern, a hallmark of integrable systems.

Proposed method

• The system is defined by a Hamiltonian with two-body interactions: V = g∑_{i<j} 1/(x_i−x_j)^2 − α/√(∑_{i<j} (x_i−x_j)^2), combining inverse-square and long-range terms.
• The authors use algebraic techniques and symmetry considerations to analyze the system's spectrum and scattering behavior.
• The energy spectrum is derived exactly, showing discrete, quantized energy levels for the N-body system.
• The scattering process is analyzed via momentum conservation and symmetry under particle exchange, leading to the momentum reversal rule.
• The solution relies on the integrability of the system, confirmed by the existence of conserved quantities and exact solvability.
• The analysis corrects a sign error in the original formulation of the first term in the potential, ensuring consistency in the final results.

Experimental results

Research questions

• RQ1 What is the complete energy spectrum of an N-body quantum system in one dimension with both inverse-square and long-range interactions?
• RQ2 How do the scattering processes behave in this system, and what is the relationship between initial and final momenta?
• RQ3 Can the system be exactly solved, and what symmetries or algebraic structures underlie its integrability?
• RQ4 Does the scattering process lead to a unique final state configuration determined solely by the initial momenta?
• RQ5 What role does the long-range potential term play in modifying the dynamics compared to the pure Calogero-type model?

Key findings

• The complete energy spectrum of the N-body system is derived exactly, showing discrete, quantized energy levels.
• The scattering process results in a unique final momentum configuration where p′_i = p_{N+1−i}, indicating a momentum reversal symmetry.
• The system is exactly solvable, confirming its integrability through the existence of conserved quantities and exact spectral analysis.
• The sign error in the first term of the potential (originally written as +g) was corrected to −g, which is essential for consistency with known models.
• The long-range interaction term, proportional to 1/√(∑(x_i−x_j)^2), introduces a non-trivial many-body potential that modifies the dynamics beyond standard Calogero models.
• The solution is valid for N equal-mass particles and provides a new class of exactly solvable one-dimensional quantum many-body systems.