About ambiguities appearing on the study of classical and quantum harmonic oscillator

Abstract

A family of nonequivalents Lagrangians and Hamiltonians are given for the one-dimensional harmonic oscillator. These Lagrangians are deduced using the constant of motion approach. The study is focused on one of these Lagrangians and Hamiltonians to analyze their implications on the quantization of the one-dimensional harmonic oscillator. The main feature is the incomnpatibilities of the units in the usual quantization approaches. Using the velocity quantization approach it is possible to get rid of this incompatibility units problem.