A new Approach to the N-particle Problem in QM

TL;DR

This paper introduces an analytical approach to simplify the N-particle quantum problem, enabling the use of modern numerical methods by approximating the Hamiltonian with finite-dimensional operators.

Contribution

It presents a novel method that reduces the complex N-particle Hamiltonian to a sum of finite-dimensional operators using the Cook-Schroeck formalism and occupation-number representation.

Findings

The N-particle Hamiltonian is determined by a 2-particle Hamiltonian.

An approximation yields a block-diagonal form suitable for numerical analysis.

The method offers a classification of the resulting matrices.

Abstract

In this paper the old problem of determining the discrete spectrum of a multi-particle Hamiltonian is reconsidered. The aim is to bring a fermionic Hamiltonian for large numbers N of particles by analytical means into a shape such that modern numerical methods can successfully be applied. For this purpose the Cook-Schroeck Formalism is taken as starting point. This includes the use of the occupation-number representation. It is shown that the N-particle Hamiltonian is determined in a canonical way by a \fictional 2-particle Hamiltonian. A special approximation of this 2-particle operator delivers an approximation of the N-particle Hamiltonian, which is the orthogonal sum of finite dimensional operators. A complete classification of the matrices of these operators is given. Finally the method presented here is formulated as a work program for practical applications. The connection with other methods for solving the same problem is discussed.