# The Harmonic Oscillator in the Framework of Scale Relativity

## TL;DR

This paper explores the harmonic oscillator within the scale relativity framework, demonstrating that its solutions can be derived from the Schrödinger equation, thus linking scale relativity to quantum mechanics.

## Contribution

It introduces a method to solve the harmonic oscillator problem in scale relativity using known quantum solutions, bridging the two theories.

## Key findings

- Solutions of the scale relativity harmonic oscillator are expressed via Schrödinger solutions.
- The approach reduces the dynamical law to a Riccati equation.
- The method provides a new perspective on quantum systems in scale relativity.

## Abstract

The dynamical law obeyed by the one-dimensional physical systems in the scale relativity approach is reduced to a Riccati nonlinear differential equation. Applied to the harmonic oscillator potential, we show that such an approach permits the calculation of the solutions of the scale relativity problem in terms of the well known solutions of the Schroedinger equation for the harmonic oscillator.