Parametric Resonance

Michael Fowler

Introduction

(Following Landau para 27)

A one-dimensional simple harmonic oscillator, a mass on a spring,

has two parameters,  and  For some systems, the parameters can be changed externally (an example being the length of a pendulum if at the top end the string goes over a pulley).

We are interested here in the system's response to some externally imposed periodic variation of its parameters, and in particular we'll be looking at resonant response, meaning large response to a small imposed variation.

Note first that imposed variation in the mass term is easily dealt with, by simply redefining the time variable to , meaning 

Then

... Read more »

Dynamics of a One-Dimensional Crystal

Michael Fowler

The Model

Notation! In this lecture, I usefor the spring constant (is a wave number) and  for frequency ( is a root of unity).

A good classical model for a crystal is to represent the atoms by balls held in place by light springs, representing valence bonds, between nearest neighbors. The simplest such crystal that has some realistic features is a single chain of connected identical atoms. To make the math easy, we'll connect the ends of the chain to make it a circle. This is called "imposing periodic boundary conditions". It is common practice in condensed matter theory, and makes little difference to the physics for a large system.

We'll take the rest positions of the atoms to be uniformly spaced,  apart, with the first atom at  ... Read more »

Driven Oscillator

Michael Fowler (closely following Landau para 22)

Consider a one-dimensional simple harmonic oscillator with a variable external force acting, so the equation of motion is

which would come from the Lagrangian

(Landau "derives" this as the leading order non-constant term in a time-dependent external potential.)

The general solution of the differential equation is  , where  , the solution of the homogeneous equation, and  is some particular integral of the inhomogeneous equation.

An important case is that of a periodic driving force  A trial solution  ... Read more »

Small Oscillations

Michael Fowler

Particle in a Well

We begin with the one-dimensional case of a particle oscillating about a local minimum of the potential energy  We'll assume that near the minimum, call it , the potential is well described by the leading second-order term, , so we're taking the zero of potential at  assuming that the second derivative , and (for now) neglecting higher order terms.

To simplify the equations, we'll also move theorigin to , so

... Read more »

Elastic Scattering

Michael Fowler

Billiard Balls

"Elastic" means no internal energy modes of the scatterer or of the scatteree are excited -- so total kinetic energy is conserved. As a simple first exercise, think of two billiard balls colliding. The best way to see it is in the center of mass frame of reference. If they're equal mass, they come in from opposite directions, scatter, then move off in opposite directions. In the early days of particle accelerators (before colliders) a beam of particles was directed at a stationary target. So, the frame in which one particle is initially at rest is called the lab frame. What happens if we shoot one billiard ball at another which is initially at rest? (We'll ignore possible internal energies, including spinning.) The answer is that they come off at right angles. This follows trivially from conservation of energy and momentum (in an obvious notation)

and Pythagoras' theorem.

Discovery of the Nucleus

The first significant use of scattering to learn about the internal structure of matter was Rutherford's use of  ... Read more »