Wave packets: The solution to the particle-wave dilemma One point in space Wave packet: Can be squished to a point and stretched to a wave. Wave: Everywhere in space Ch. 13.1

Interference between two waves with different wavelength Waves add up Waves cancel x

Forming a wave packet Combine waves with different wavelengths. Waves synchronized here Waves getting out of synch

Spatial spread x of a wave packet

x, p in the particle and wave limit Particle Wave x  0 x  ∞ p  ∞ p  0 , p=h/ not defined , p=h/ well-defined

The general case: A wave packet There is a tradeoff between particle and wave character. They have to add up to 100% . An electron can be 100% particle: x=0, p=∞ or it can turn into 100% wave: x=∞, p=0 In between it is a wave packet: x0, p0 The uncertainty relation quantifies the tradeoff.

The uncertainty relation x · p  h/4 x = Position uncertainty p = Momentum uncertainty (Werner Heisenberg 1927)

Comparison with diffraction (Lect. 9, Slides 11,12) Real space (x) versus reciprocal space (p): Large objects in real space (large x) Small objects in reciprocal space (small p) 

Generalization of the uncertainty relation to space-time Real space-time Space x Time t Reciprocal space-time Momentum p Energy E x · p  h/4 t · E  h/4 The variables (x,t) are incompatible with (p,E). They are also fundamental variables in relativity.

Removing h from the uncertainty relation (optional) t · E  h/4 t · f  1/4 Divide by h Dividing the uncertainty relation by h and using E = h f gives an uncertainty relation between time and frequency. Planck’s constant of quantum physics h has disappeared. The uncertainty relation is due to the use of wave packets !

Connection to mathematics (optional) The mathematical operation that transforms space and time to momentum and energy is a Fourier transform. A Fourier transform describes a wave packet as a sum of plane waves. The Fourier transforms is used widely in sig- nal processing and data compression . The JPEG (MPEG) compression of images (movies) is based on a Fourier transform. The image is compressed by omitting short wavelengths.