# Quantum Ornstein-Zernike Equation

## TL;DR

This paper formulates a quantum version of the Ornstein-Zernike equation by representing non-commutativity as an effective potential, enabling classical statistical mechanics techniques to analyze quantum systems.

## Contribution

It introduces a novel quantum Ornstein-Zernike equation with a series expansion of many-body terms and provides methods to solve the resulting non-linear PDEs.

## Key findings

- Explicit linear solution valid at high/intermediate temperatures
- Algorithm for solving the full non-linear problem
- Effective pair potentials incorporate quantum effects

## Abstract

The non-commutativity of the position and momentum operators is formulated as an effective potential in classical phase space and expanded as a series of successive many-body terms, with the pair term being dominant. A non-linear partial differential equation in temperature and space is given for this. The linear solution is obtained explicitly, which is valid at high and intermediate temperatures, or at low densities. An algorithm for solving the full non-linear problem is given. Symmetrization effects accounting for particle statistics are also written as a series of effective many-body potentials, of which the pair term is dominant at terrestrial densities. Casting these quantum functions as pair-wise additive, temperature-dependent, effective potentials enables the established techniques of classical statistical mechanics to be applied to quantum systems. The quantum Ornstein-Zernike equation is given as an example.