An Eigenvalue Problem for a Fermi System and Lie Algebras

TL;DR

This paper investigates the eigenvalue problem of a non-commuting Fermi Hamiltonian, explores its algebraic structure, and discusses entanglement, providing new insights into the algebraic and quantum properties of such systems.

Contribution

It introduces a detailed analysis of the eigenvalue problem for a specific Fermi Hamiltonian and classifies the Lie algebras generated by its terms and the number operator.

Findings

Eigenvalues and eigenstates of the Hamiltonian are explicitly solved.

The Lie algebra generated by the Hamiltonian terms is characterized.

The algebraic structure related to the number operator is classified.

Abstract

We study a Fermi Hamilton operator $\hat K$ which does not commute with the number operator $\hat N$. The eigenvalue problem and the Schr\"odinger equation is solved. Entanglement is also discussed. Furthermore the Lie algebra generated by the two terms of the Hamilton operator is derived and the Lie algebra generated by the Hamilton operator and the number operator is also classified.