Physics of Oscillators

The oscillator is an essential concept in physics: matter is often modeled by a collection of oscillators (harmonic or not) interacting with each other and with the external environment. The latter acts on the matter through a wave, such as an acoustic or electromagnetic wave. This makes it possible to lay the theoretical foundations of the problems of radiation-matter interaction and thus to build one of the fundamental tools for the study of matter (in the broad sense): spectroscopy.

Spectroscopy is indeed the basic tool for the study of the physical properties of objects that surround us, such as a molecule, a crystal, a star, a galaxy. These properties are deduced either from their spontaneous emission or from their response to an external excitation. For example we measure the properties of absorption, reflection, transmission of an applied electromagnetic radiation (visible, infra-red, X, neutrons, ...). The response to this radiation is then a way to discover which are the various types of oscillators constituting the studied medium.

 In short, the study of the physical media that surround us requires the use of two fundamental theoretical tools: oscillators and waves, which are the subject of this course.  

The principle adopted here is a step-by-step progression from the harmonic oscillator, then from coupled oscillators, to waves treated in the framework of discrete systems: infinite and then finite coupled oscillators with different edge conditions.

Acquire the basic knowledge necessary to interpret certain spectroscopy experiments. Know how to perform mode calculations and construct general solutions by modal superpositions. Basic knowledge of waves and dispersion relations.

Basic Newtonian dynamics: point mechanics: elementary oscillators.  

Mastery of basic mathematics (L1): analysis and algebra.

Recommended prerequisites*: Any in-depth knowledge of point mechanics.

CT 100%.

1 - Oscillators: The harmonic oscillator (definition, solution, phase space, energy). The potential method (reminders, potential, nonlinear oscillations, linear limit). Forced and damped oscillator (equation of motion, solution, power consumption).

2 - Coupled oscillators: Two oscillators (equations, natural frequencies, natural modes). Coupled particles (diatomic molecule, eigenmodes of CO2).


3 - Waves in molecular chains: Coupled particle system (potential energy, equations of motion, modes, dispersion relation, transverse and longitudinal traveling waves, band edge wave, evanescent wave). Finite system of coupled particles (edge conditions, standing waves, fixed, free and periodic ends). Forced end (monochromatic forcing, signal propagation, low pass filter, resonantforcing, transmitted power, attenuation). Diatomic chain (dispersion relation, gap opening, acoustic branch and optical branch). Coupled oscillator chain, band gap (dispersion relation, travelling and standing waves, evanescent wave, wave packet, phase and group velocities, bandwidth, Schrödinger equation, wave packet spreading).