Ambiguities on the Hamiltonian formulation of the free falling particle with quadratic dissipation

TL;DR

This paper investigates the Hamiltonian formulation of a free falling particle with quadratic dissipation, revealing ambiguities in quantization and statistical mechanics despite equivalent classical descriptions.

Contribution

It demonstrates that different Hamiltonians for the same classical dissipative system lead to distinct quantum and statistical behaviors.

Findings

Two equivalent classical Hamiltonians yield different quantum systems.

Quantization depends on the chosen Hamiltonian, affecting physical predictions.

Dissipative effects are significant at first order in the dissipation parameter.

Abstract

For a free falling particle moving in a media which has quadratic velocity force effect on the particle, two equivalent constants of motion, with units of energy, two Lagrangians, and two Hamiltonians are deduced. These quantities describe the dynamics of the same classical system. However, their quantization and the associated statistical mechanics (for an ensemble of particles) describe two completely different quantum and statistical systems. This is shown at first order in the dissipative parameter.