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Quantization
• Matter is not continuous – it is quantized.
– Idea dates back to Democritus (Greece, 450 BC).
– First scientific arguments appear in 18th – 19th century
• Eg, Avogadro’s claim that 22.4 liter volume of a gas
contains the same number of molecules at standard
conditions.
• Towards the end of 19th century, first hints that electric
charge is also quantized.
– Later, found that light is also quantized.
• These discoveries led to the development of quantum
mechanics.
– Very different understanding of the natural laws.
Quantization of Charge
• Early hints:
– Faraday (mid 1800’s) found that he always needed the
same amount of charge to decompose equal amount of
material in electrolysis: Q = NA e
– Later estimates of NA led to the first measurements of e.
– Zeeman (1896): splitting of spectral lines emitted by
atoms in strong magnetic field.
• Thomson (1897): discovered electron using cathode ray
tubes.
Fm = qvB
Thomson’s
Experiment
Fe = qE
v = E/B
• Two parts:
• First pass the cathode rays
through a crossed field.
– Magnetic and electric
force cancel, giving the
speed of the rays.
• Then put the ray in another
magnetic field.
+q
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qvB = mv 2 / R
q
v
=
m BR
Direct
measurement
of the chargeto-mass ratio.
Thomson’s Experiment
• Repeated with different gases in the tube, different
metals for the cathode etc.
• Always the same answer:
e
11
= 0.7 × 10 C/kg
m
• Conclude that cathode rays consist of particles, common
to all metals.
• But still don’t know the charge.
Millikan’s Oil Drop Experiment
• Spray oil into a space between
two metal plates.
– Interaction with the nozzle
charges the oil drops.
• Change electric field until a drop
floats in the air.
qE = mg
• Estimate m from the size of the
drop.
• Found that each drop’s charge
was a multiple of an elementary
unit e, which he measured:
– e = 1.601 x 10-19 C
Blackbody Radiation
• When light falls on a body, it is partly reflected and partly
absorbed.
– Light colored objects reflect more.
• Absorbed light increases the internal energy of the body.
– Atoms move faster, electrons accelerate, and
therefore produce electromagnetic radiation – light.
– This process reduces the temperature (and internal
energy) of the body.
• When the absorption and emission rates are equal, the
body is in equilibrium with the environment.
• So: all bodies continually emit radiation, whose intensity
is dependent of the body temperature.
Stefan-Boltzmann Law
• Stefan empirically found that the
power per unit area emitted by a
black body is:
R = σT
4
σ = 5.67 x 10-8 W/m2/K4
(Stefan’s constant).
• Boltzmann later proved it in
statistical physics.
Visible part of the spectrum is
• Rate only depends on
only populated if T>600 °C
temperature – no other property of
the body matters.
Wien’s Displacement Law
• The spectrum (frequency content)
also only depends on
λmaxT = 2.9 × 10−3 mK
temperature.
Example
• A star’s spectrum is observed to peak at 1 µm. The star
emits 100 times the power of the Sun. How large is the
star?
Blackbody Energy
Density Spectrum
• Best blackbody is a cavity with a
small opening.
Blackbody Spectrum
• Rayleigh-Jeans:
8πkT
u (λ ) =
λ4
• Planck:
u (λ ) =
8πhc
1
λ5 e hc / λkT − 1
e hc / λkT ≈ 1 + hc / λkT
• Large λ:
8πhc λkT 8πkT
= 4
u (λ ) ≈ 5
λ hc
λ
• Small λ:
u (λ ) ≈
8πhc
λ
5
e
− hc / λkT
→0
Recover
Rayleigh-Jeans
at large wavelengths
Blackbody Spectrum
• Planck’s constant, measured to be h = 6.626 x 10-34 Js
• Recover Wien’s Law
• Recover Stefan-Boltzmann Law
Example
• What should be the temperature of the Earth?
– Greenhouse effect and global warming.
/m2
(1-a)
Blackbody Spectrum: Sun
Tsun = 5778 K
Emits mostly in the visible
and infrared parts of the
spectrum.
Atmosphere is transparent in
the visible band.
Blackbody Spectrum: Earth
TE = 20 C
Emits predominantly in
the infrared part of the
spectrum
Atmosphere partly
absorbs in this band.
Greenhouse Effect
• Simple model
– Earth with albedo
(reflection) in visible
– Atmosphere with some
absorption in infrared and
transparent in visible.
• Leads to an increase in the
Earth’s temperature.
• What causes the absorption in
the atmosphere?
– Gases: greenhouse effect.
– Mostly water, but also CO2,
CH4 and others.
"Absorption spectrum of liquid water" by Kebes at English Wikipedia.
Water Absorption Spectrum:
4.2 micron
15 micron
"Absorption spectrum of liquid water" by Kebes at English Wikipedia.
Pulling things together:
- Significant part of the Earth’s emission spectrum is absorbed by the
atmospheric gases, leading to the greenhouse effect.
- Changing the composition of the atmosphere can modify the
magnitude of the greenhouse effect: global warming
Evidence for Global Warming
http://www.ipcc.ch/report/ar5/
From the latest (fifth) assessment report of
the Intergovernmental Panel on Climate
Change (IPCC).
- Founded by the World Meteorological
Organization and the UN Environment
Programme.
Evidence for Global Warming
http://www.ipcc.ch/report/ar5/
Drivers of Global Warming
http://www.ipcc.ch/report/ar5/
Models require
anthropogenic
contributions to explain
the observed data.
Natural effects
(volcanic eruptions,
solar cycle effects)
alone cannot explain
the data.
Strong evidence that
CO2 emission
dominates the changes
in the greenhouse
effect.
Photoelectric Effect
• Light kicks out electrons from
a material.
• First hint: Hertz experiments
with spark gaps (1887).
– Observed that if light is
present, it is easier for
sparks to jump a gap (i.e.
they can jump a larger
gap).
• Lenard (1900) conducted a
series of experiments with a
cathode ray tube.
Photoelectric Effect
• Increasing voltage increases
collection of cathode rays
(electrons), until they are all
collected.
• Negative voltage repels
electrons. No signal if
1 2
mv = eV0
2
Stopping
Potential
• Increasing light intensity
increases the maximum signal,
but it does not change V0.
– Changing the light frequency
does!
Photoelectric Effect
• Einstein solved the problem in 1905, using Planck’s
quantization of energy.
• Light consists of quanta (photon) each of energy hν.
• A photon enters the cathode and gives its energy (and
momentum) to an electron.
– If it takes energy φ (work function) to remove the
electron from the material, the electron will have
energy hν-φ.
1 2
• The stopping potential is then: eV0 = mvmax
= hν − φ
2
– It does not depend on the light intensity (rate of
photons), but linearly depends on the light frequency.
Photoelectric Effect
• Millikan later measured the
stopping potential as a
function of frequency.
– The slope yields h/e, in
agreement with Planck’s
value.
– V0 = 0 gives the
measurement of the
work potential, for the
material used.
• Typically a few eV.
h
φ
V0 = ν −
e
e
Demos
• 7A10.10 - Discharging Aluminum Plate
• 7A10.12 - Photoelectric Charging
• 7A10.30 - Stopping Potential
Example
• A light source at wavelength 555 nm radiates at 100 W. If
the human eye can register 10 photons/sec, how far
could the source be placed (and still be seen by eye).
X-rays
• Roentgen also worked with cathode ray tubes (1895+).
• He found that cathode rays generate new “rays” when
they interact with the glass tube.
– These rays typically pass through materials opaque to
light, and can activate photographic film (or
fluorescent screen).
• Don’t bend in magnetic field, don’t seem to do
interference, called them X-rays.
• Produced by accelerated charges (electrons), so they
are a form of light.
– Generally, radiation formed in this way is called
bremsstrahlung (German for breaking radiation).
X-rays
• Generated by electrons with V
~ 10 kV.
• So, electron’s (and X-ray’s)
energy is ~10 keV.
• Or, X-ray wavelength is of
order 0.1 nm = 10-10 m = 1 Å.
• This is of the same scale as
spacing between atoms in a
crystalline lattice.
• So, use different planes of
atoms in the crystal to do
interferometry.
2d sin θ = mλ
Bragg Condition
By changing the incident
angle, can measure the
spectral content of the
incoming wave.
• Found that the spectrum is
material dependent!
• Continuous component,
expected from
bremsstrahlung.
• Sharp lines (characteristic
spectrum) dependent on
the cathode target material.
– Quantum atomic
structure.
• Continuum stops at λm.
– Independent of material.
– Dependent on the CRT
voltage.
– Corresponds to highest
energy electrons
produced in CRT:
V↑, Ek↑, νm↑, λm↓
X-Rays
Bragg
Spectrometer
Demo
• 7A60.50 - Microwave Bragg Diffraction
Compton Effect
• Scattering of photons off of electrons
– Electron absorbs part of the incoming energy and
momentum.
– Have to do it relativisticaly.
Compton Effect
• Compton confirmed this experimentally.
• Used CRT with molybdenum to generate monochromatic
beam of photons of wavelength 0.0711 nm.
• Smashed them into graphite target, producing scattered
photons.
• Measured the content of scattered photons with a Bragg
spectrometer.