A unified theoretical framework for mapping models for the multi-state Hamiltonian

Area of Science:

• Quantum mechanics
• Theoretical chemistry
• Mathematical physics

Background:

• The Hamiltonian operator is central to quantum mechanics, describing system evolution.
• Representing multi-state Hamiltonians and their classical/semiclassical counterparts poses theoretical challenges.

Purpose of the Study:

• To develop a unified theoretical framework for constructing equivalent representations of multi-state Hamiltonians.
• To explore mapping approaches onto Cartesian phase space for quantum operators.

Main Methods:

• Mapping an F-dimensional Hamiltonian to an F+1 dimensional space.
• Defining creation and annihilation operators within the extended space.
• Deriving commutation and anti-commutation relations for the new operators.

Main Results:

• The framework provides a unified approach to Hamiltonian representation.
• The derived operators indicate underlying degrees of freedom are neither bosons nor fermions.
• Six example mapping models are presented, including a novel derivation of the Meyer-Miller model.

Conclusions:

• The proposed framework offers a new perspective on quantum mechanical operators and their classical/semiclassical connections.
• This work facilitates the development of equivalent Hamiltonian expressions across different domains of physics.