Transcript Wave packets

Wave packets:
The
solution to the particle-wave dilemma
Particle:
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
x
Waves add up
Waves cancel
Forming a wave packet
Combine waves with different wavelengths.
2 waves
3 waves
5 waves
Waves synchronized here
Waves getting out of synch
Spatial spread x of a wave packet
8
4
0
-4
x
-8
-15
-10
-5
0
J
5
10
15
x, p in the particle and wave limit
x
x
x
Particle
Wave
x  0
x  ∞
∞
p  0
p 
, 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
Reciprocal space-time
Space x
x · p  h/4
Momentum p
Time t
t · E  h/4
Energy E
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
Divide by h
t · f  1/4
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 signal 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.