Lie Groups and Quantum Mechanics

TL;DR

This paper examines inconsistencies in quantum mechanical models of the harmonic oscillator, demonstrating how Lie symmetries and Jacobi Last Multipliers can resolve issues in quantization, especially for dissipative systems.

Contribution

It introduces a novel approach using Lie symmetries and Jacobi Last Multipliers to address quantization inconsistencies in dissipative systems.

Findings

Identifies inconsistencies in standard Hamiltonian formulations.

Shows how Lie symmetries lead to consistent Lagrangians.

Proposes a resolution to the debate on quantizing dissipative systems.

Abstract

Mathematical modeling should present a consistent description of physical phenomena. We illustrate an inconsistency with two Hamiltonians -- the standard Hamiltonian and an example found in Goldstein -- for the simple harmonic oscillator and its quantisation. Both descriptions are rich in Lie point symmetries and so one can calculate many Jacobi Last Multipliers and therefore Lagrangians. The Last Multiplier provides the route to the resolution of this problem and indicates that the great debate about the quantisation of dissipative systems should never have occurred.