LECTURE 14

WAVEPACKETS

THE UNCERTAINTY PRINCIPLE

PHYS 420-SPRING 2006

Dennis Papadopoulos

2/

...3,2,1

h

nnvrme

an integer number of wavelengths fits into the circular orbit

rn 2

where

p

h

is the de Broglie wavelength

Particle Value of

Electrons of kinetic energy

Protons of kinetic energy

Thermal neutrons (300K)

Neutrons of kinetic energy (14 MeV)

He atoms at 300K

1 eV 100 eV 10000 eV

12.2 A 1.2 A 0.12A

1 keV 1 MeV 1 GeV

0.009 A 28.6 F 0.73 F

1.5 A

9.0 F

0.75 A

A10cm10 F 1 cm;10A1 5-138

me, walking to the student union for lunch at 2 miles per hour

m1054.2 34

Fig. 5-6, p. 157

Fig. 5-4, p. 156

d=2.15 A

Fig. 5-5, p. 156

sin(50)=1.65 A

Fig. 5-7, p. 157

50 keV electronsIn 400 A alloy film

Fig. 5-8, p. 158

Neutrons inNaCl crystal

Electrons have a wavelength that is much shorter than visible light. The smallest detail that can be resolved is equal to one wavelength.

Human hair

Red Blood Cells

Table Salt

bu

lk

(th

ick)

foil

(th

in)

Specimen interactions involved in forming an image

Backscattered electrons can be used to identify elements in the material.

Auger electrons also can give compositional information.

Secondary electrons are low in energy and thus can’t escape from the interior of the material. They mostly give information about the surface topography.

Secondary electrons are low in energy and thus can’t escape from the interior of the material. They mostly give information about the surface topography.

X-rays are produced by de-excited atoms.

Unscattered electrons are those which are transmitted through the material. Since the probability of transmission is proportional to thickness, it can give a reading of thickness variations.Elastically scattered Bragg electrons give info about atomic spacing, crystal orientations, etc.

Loss of energy by electrons is characteristic of bulk composition.

Fig. 5-18, p. 165

2 cos( )2

tA

Beats

http://www.school-for-champions.com/science/soundbeat.htm

http://library.thinkquest.org/19537/java/Beats.html

Fig. 5-19, p. 166

(1/ 2) k x

How does this wave behave at a boundary?

at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave

at a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change)

When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary

In this animation, the density of the thick string is four times that of the thin string …

Fig. 5-20, p. 168

The word “particle” in the phrase “wave-particle duality” suggests that this wave is somewhat localized.

How do we describe this mathematically?

…or this

…or this

Fig. 5-17, p. 164

FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite

number of harmonic waves

FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite

number of harmonic waves

dkekaxf ikx)(

2

1)(

spatially localized wave group

amplitude of wave with wavenumber k=2/

adding varying amounts of an infinite number of

wavessinusoidal expression

for harmonics

Adding several waves of different wavelengths

together will produce an interference pattern which

begins to localize the wave.

To form a pulse that is zero everywhere outside of a finite spatial range x requires adding together

an infinite number of waves with continuously varying wavelengths and

amplitudes.

Fig. 5-21, p. 171

Fig. 5-22, p. 171

sin( )2( )

( )o

o

Tg T

T

t TT

Fig. 5-23, p. 172

Fig. 5-24, p. 173

4k x

hh

xpx

sin2sin

2

anglescattering

/)sin(momentum electron /)sin( hxh )sin2/( x

Uncertainty on optics:

Remember our sine wave that went on “forever”?

We knew its momentum very precisely, because the momentum is a function of the frequency, and the frequency was very well defined.

But what is the frequency of our localized wave packet? We had to add a bunch of waves of different frequencies to produce it.

Consequence: The more localized the wave packet, the less precisely defined the momentum.

Consequence: The more localized the wave packet, the less precisely defined the momentum.

kp

E