Ambiguity appearing on the Hamiltonian formulation of Quantum Mechanics

TL;DR

This paper investigates how different Hamiltonian formulations for the same classical system can lead to divergent quantum behaviors, highlighting ambiguities in the Hamiltonian approach to quantum mechanics.

Contribution

It demonstrates that multiple Hamiltonians representing identical classical dynamics can produce different quantum dynamics, revealing ambiguities in the Hamiltonian formulation.

Findings

Different Hamiltonians yield distinct quantum behaviors for the same classical system.

The study emphasizes the importance of Hamiltonian choice in quantum modeling.

Quantum dynamics are sensitive to Hamiltonian representation, even when classical dynamics are identical.

Abstract

For the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D, two different Hamiltonian are given describing the same classical dynamics; however, their quantum dynamics behavior are different.