Uncertainty relations for the Hohenberg-Kohn theorem

TL;DR

This paper explores the physical mechanism behind the Hohenberg-Kohn theorem, showing how charge density constrains many-body wavefunctions through effective operators, and provides initial energy functional results aligning with Quantum Monte Carlo data.

Contribution

It introduces effective canonical operators related to electric field and particle momentum that explain the theorem's physical basis in many-body systems.

Findings

Derived the functional form of total energy for interacting systems.

Found agreement with Quantum Monte Carlo simulations in the uniform density limit.

Provided a physical interpretation of charge density constraints in quantum systems.

Abstract

How does charge density constrain many-body wavefunctions in nature? The Hohenberg-Kohn theorem for non-relativistic, interacting many-body Schr\"odinger systems is well-known and was proved using \emph{reductio-ad-absurdum}; however, the physical mechanism or principle which enables this theorem in nature has not been understood. Here, we obtain effective canonical operators in the interacting many-body problem -- (i) the local electric field, which mediates interaction between particles, and contributes to the potential energy; and (ii) the particle momenta, which contribute to the kinetic energy. The commutation of these operators results in the charge density distribution. Thus, quantum fluctuations of interacting many-particle systems are constrained by charge density, providing a mechanism by which an external potential, by coupling to the charge density, tunes the quantum-mechanical many-body wavefunction. As an initial test, we obtain the functional form for total energy of interacting many-particle systems, and in the uniform density limit, find promising agreement with Quantum Monte Carlo simulations.