Download CHAPTER 3: The Experimental Basis of Quantum

CHAPTER 3
Prelude to Quantum Theory
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Discovery of the X-Ray and the Electron
Determination of Electron Charge
Line Spectra
Quantization
Blackbody Radiation
Photoelectric Effect
X-Ray Production
Compton Effect
Pair Production and Annihilation
Max Karl Ernst Ludwig Planck
(1858-1947)
Problems due Wednesday Sept. 23rd: Chapter 3: problems: 7, 9, 14, 15,
16, 18, 19, 24, 25, 26
X-Ray Production: Theory
An energetic electron
passing through matter will
radiate photons and lose kinetic
energy, called bremsstrahlung.
Since momentum is conserved, the
nucleus absorbs very little energy,
and it can be ignored. The final
energy of the electron is determined
from the conservation of energy to be:
E f  Ei  h
Ei
Ef
h
Photons also have momentum!
Use our expression for the relativistic
energy to find the momentum of a photon,
which has no mass:
E  (mc )  p c
2
2 2
Alternatively:
When
radiation
pressure is
important:
2 2

E h h
p 

c
c 
h 2
p
 k
2 
Comet tails (other forces are small)
Viking spacecraft (would've missed Mars by 15,000 km)
Stellar interiors (resists gravity)
X-Ray
Production:
Experiment
Current passing through a filament produces copious numbers of
electrons by thermionic emission. If one focuses these electrons by
a cathode structure into a beam and accelerates them by potential
differences of thousands of volts until they impinge on a metal
anode surface, they produce x rays by bremsstrahlung as they stop
in the anode material.
Compton Effect
Ep, pp
Photons have energy and momentum:
E  hc / 
p h/
When a photon enters matter, it can
interact with one of the electrons. The
laws of conservation of energy and
momentum apply, as in any elastic
collision between two particles.
This yields the change in
wavelength of the scattered
photon, known as the
Compton effect:
Ee, pe
Ep’ , pp ’
Read: section 3.4 Kane
Pair Production and Annihilation
In 1932, C. D. Anderson observed a
positively charged electron (e+) in
cosmic radiation. This particle, called a
positron, had been predicted to exist
several years earlier by P. A. M. Dirac.
A photon’s energy can be converted
entirely into an electron and a positron
in a process called pair production:
Paul Dirac
(1902 - 1984)
Pair Production
in Empty Space
h
Conservation of energy for pair
production in empty space is:
E
E+
h  E  E
The total energy for a particle is:
So:
E  p c
This yields a lower limit on the photon energy:
Momentum conservation yields:
h  p c  p c
h  pc cos( )  p c cos( )
This yields an upper limit on the photon energy:
h  p c  p c
A contradiction! And hence the conversion of energy and momentum
for pair production in empty space is impossible!
Pair Production
in Matter
In the presence of matter, the
nucleus absorbs some energy
and momentum.
The photon energy required for
pair production in the presence
of matter is:
h  E  E  K .E.(nucleus)
h  2me c 2  1.022 MeV
Pair Annihilation
A positron passing through matter
will likely annihilate with an
electron. The electron and positron
can form an atom-like configuration
first, called positronium.
Pair annihilation in empty space
produces two photons to conserve
momentum. Annihilation near a
nucleus can result in a single
photon.
hv
hv
Pair Annihilation
Conservation of energy:
2me c 2  hv1  hv2
Conservation of momentum:
hv1 hv2

0
c
c
So the two photons will have the
same frequency:
v1  v2  v
The two photons from positronium
annihilation will move in opposite
directions with an energy:
hv  me c 2  0.511 MeV
hv
hv
What is a photon?
Photons move at the speed of light, just like an electromagnetic wave
They have zero rest mass and rest energy
They carry energy and momentum E=h and p=h/
They can be created and destroyed when radiation is emitted or
absorbed
They can have particle-like collisions with other particles such as
electrons
Probability to observe photons|electric field amplitude|2
Indeed, photographs taken in dimmer light look grainier.
Very very dim
Bright
Very dim
Very bright
Dim
Very very bright
When we detect very weak light, we find that it’s made up of
particles. We call them photons.
CHAPTER 4
Wave Properties of Particles
4.1
4.2
4.3
4.4
4.6
4.8
4.7
X-Ray Scattering
De Broglie Waves
Electron Scattering
Wave Motion
Uncertainty Principle
Particle in a Box
Probability, Wave Functions, and
the Copenhagen Interpretation
4.5 Waves or Particles?
Louis de Broglie
(1892-1987)
I thus arrived at the overall concept which guided my studies: for both
matter and radiations, light in particular, it is necessary to introduce the
corpuscle concept and the wave concept at the same time.
- Louis de Broglie, 1929
Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu
X-Ray Scattering (Diffraction)
In 1912, Max von Laue suggested that, since x-rays were a form of
electromagnetic radiation, they should diffract.
Crystals have interatomic separations similar to x-ray wavelengths
and so act as three-dimensional diffraction gratings, diffracting x-rays.
Bragg’s Law
William Lawrence Bragg
showed that x-ray diffraction
acted like reflection from
planes of atoms in the
crystal.
There are two conditions for
constructive interference of
the scattered x-rays:
1) The angle of incidence must equal
the angle of reflection.
2) The difference in path lengths must
be an integral number of
wavelengths.
Bragg’s Law: n = 2d sin (n = integer)
The Bragg Spectrometer
A Bragg spectrometer diffracts x-rays
from a crystal.
It measures the intensity of the diffracted
beam vs. angle.
When x-rays pass through a powdered
crystal, the dots become a series of
rings.
De Broglie Waves
In his thesis in 1923, Prince Louis V.
de Broglie suggested that mass
particles should have wave properties
similar to electromagnetic radiation.
The energy can be written as:
h = pc

If a light-wave could also
act like a particle, why
shouldn’t matter-particles
also act like waves?
h = p
The wavelength of a matter wave is
called the de Broglie wavelength:
 h/ p
So do experiments confirm this idea?
Louis V. de Broglie
(1892-1987)
Electron Scattering
George P. Thomson (1892–1975),
son of J. J. Thomson, saw electron
diffraction from celluloid, gold,
aluminum, and platinum.
A randomly oriented
polycrystalline sample of SnO2
produces rings.
In 1925, Davisson and
Germer observed
electrons diffracting
(much like x-rays) from
nickel crystals.
Beautiful Proof That Electrons are
Waves: Imaging Using Them
Imaging using
light waves is
well known.
But optical
microscopes’
resolution is
only /2 ~
200nm.
Electron micrograph of pollen grains with ~0.1nm
resolution
Electrons have
much smaller
wavelengths,
and electron
microscopes
can achieve
resolutions of
~0.05nm.
Recall that waves diffract through slits.
Fraunhofer diffraction patterns
One slit
Two slits
In 1803, Thomas Young saw the two-slit pattern for light,
confirming the wave nature of light. But particles are also
waves. So they should exhibit similar patterns when passing
through slits, especially pairs of slits.
Electron Double-Slit Experiment
C. Jönsson of Tübingen,
Germany, succeeded in 1961
in showing double-slit
interference effects for
electrons by constructing very
narrow slits and using
relatively large distances
between the slits and the
observation screen.
This experiment demonstrated
that precisely the same
behavior occurs for both light
(waves) and electrons
(particles).