1. 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

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

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

4. Spatial spread x of a wave packet

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

6. 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.

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

8. 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) 

9. 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.

10. 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 !

11. 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.