Light and Matter Tim Freegarde School of Physics & Astronomy University of Southampton Wave mechanics

2 response to action of neighbour delayed reaction waves are bulk motions, in which the displacement is a delayed response to the neighbouring displacements e.g.

3 Gravitational waves delay may be due to propagation speed of force (retarded potentials) vertical component of force

4 Wave mechanics waves result when the motion at a given position is a delayed response to the motion at neighbouring points derivatives with respect to time and position are related by the physics of the system, which lets us write differential equations in certain circumstances, a wave may propagate without distortion: e.g. the solutions depend upon whether the system shows linearity or dispersion a surface of constant phase,, is known as a wavefront, and propagates with the phase velocity,

5 Linearity and superpositions if the system is linear, then the wave equation may be split into separate equations for superposed components; i.e., if y 1 and y 2 are wave solutions, then so is any superposition of them if sinusoidal solutions are allowed, then the wave shape at any time may be written as a superposition of sinusoidal components Fourier analysis complex coefficients allow waves which are complex exponentials:

6 Dispersion linear systems may show dispersion – that is, the wave speed varies with frequency if sinusoidal solutions are allowed, then the wave shape may still be written as a superposition of sinusoidal components dispersion causes the components to drift in phase as the wave propagates the wave may no longer be written as

7 Dispersion 2 sinusoidal components: 10 sinusoidal components: spreading of wavepacket this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation

8 Plane wave solutions to wave equations linear, non-dispersive linear, dispersive non-linear solitons:

9 Alternative solutions show that spherical waves of the form are valid solutions to the Schrödinger equation of a free particle

10 Wave mechanical operators an operator is a recipe for determining an observable from a wave function e.g. an operator could yield the parameter from the wave for convenience, to avoid the observable depending upon the magnitude of the wavefunction, we instead define the general operator i.e. the square brackets are commonly omitted

11 Messenger Lecture Richard P. Feynman ( ) Nobel prize 1965 Messenger series of lectures, Cornell University, 1964 Lecture 6: ‘Probability and Uncertainty – the quantum mechanical view of nature’ see the later series of Douglas Robb memorial lectures (1979) online at