A Unified Theoretical Framework for Mapping Models for the Multi-State Hamiltonian

TL;DR

This paper introduces a unified theoretical framework for representing multi-state Hamiltonians and mapping them onto phase space, enabling new quantum-classical correspondence models.

Contribution

It presents a comprehensive framework for Hamiltonian mapping, including derivation of relations and examples, advancing the understanding of quantum-classical mappings.

Findings

Six mapping models demonstrated as examples

Derived commutation and anti-commutation relations

Unified approach to Hamiltonian representations

Abstract

We propose a new unified theoretical framework to construct equivalent representations of the multi-state Hamiltonian operator and present several approaches for the mapping onto the Cartesian phase space. After mapping an F-dimensional Hamiltonian onto an F+1- dimensional space, creation and annihilation operators are defined such that the F+1 dimensional space is complete for any combined excitations. Commutation and anti-commutation relations are then naturally derived, which show that the underlying degrees of freedom are neither bosons nor fermions. This sets the scene for developing equivalent expressions of the Hamiltonian operator in quantum mechanics and their classical/semiclassical counterparts. Six mapping models are presented as examples. The framework also offers a novel way to derive such as the well-known Meyer-Miller model.