General aim:

The general aim of this course is three fold:

to develop and put on a firm footing the mathematical techniques needed to discuss both qualitatively and quantitatively the physics of waves and vibrations both classical and quantum,

to look in detail at standing and travelling wave solutions to the wave equation and give an introduction to Fourier analysis,

to introduce the ideas of matter waves, standing waves as a representation of a particle in a box and the time independent Schrodinger wave equation.

Objectives

On completion of this course you should be able to:

• manipulate and evaluate partial derivatives of functions of two variables;
• write down the 1-D wave equation;
• solve the wave equation for harmonic travelling and standing waves;
• solve the wave equation for waves on a string and apply boundary conditions;
• apply the principle of superposition and express standing waves sum of travelling waves;
• calculate phase and group velocity;
• explain the meaning of normal modes;
• calculate the allowed vibration modes of a string as a function of the boundary conditions and explain the generalisation to modes in a box;
• use a Fourier series to express an arbitrary vibration of a string of fixed length as a superposition of normal modes;
• calculate the de Broglie wavelength of a particle given its momentum;
• calculate the allowed energies of a particle in a 1D box from the normal modes;
• estimate optical excitation energies of atoms and molecules by treating the atom as an electron in a 1D well;
• write down the 1-D Schrodinger equation for a particle and appreciate its significance.