Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
Light and optics
RAYS
• straight propagation paths
• least time (Fermat’s principle)
focus
• reflection, refraction, lenses, telescopes, microscopes
directrix
WAVES
• Huygens’ description of propagation, reflection, refraction
• polarization, colour (wavelength, frequency)
• diffraction, interference, beats, interferometers
• Maxwell’s electromagnetism, Einstein’s relativity
PHOTONS
• Planck, Compton, Einstein
2
Radiation pressure
• light carries both energy and momentum
(Maxwell’s electromagnetism)
• intensity (energy per unit time per unit area)
I   0c E 2
• pressure (momentum per unit time per unit area)
I
p
c
© Malcolm Ellis
Comet Hale-Bopp, 1997
• torque (angular momentum per unit
time per unit area)
T 
I
2
sunshine
sun
3
Radiation pressure
• light carries both energy and momentum
(Maxwell’s electromagnetism)
• intensity (energy per unit time per unit area)
I   0c E 2
• pressure (momentum per unit time per unit area)
I
p
c
© Malcolm Ellis
Comet Hale-Bopp, 1997
Crookes radiometer
www.a3bs.com
• torque (angular momentum per unit
time per unit area)
T 
I
2
sunshine
R A Beth, Phys Rev 50
115 (1936)
sun
4
RAYLEIGH-JEANS DISTRIBUTION
• consider modes of given volume of space
intensity
Blackbody radiation
RayleighJeans
• assume equipartition: average energy kT
per mode
2
• mode density
• hence
8V
N   d 
d
3
c
8 kT
   d  4 d
wavelength

5
intensity
Blackbody radiation
RayleighJeans
observed
spectrum

wavelength
BLACK BODY
• perfect absorber, hence ‘ideal’ emitter
• no spectral features beyond Planck curve
ULTRAVIOLET CATASTROPHE
• classical thermodynamics predicts monotonic increase with
frequency
• quantization of radiation field supplies required correction
6
intensity
Blackbody radiation
PLANCK’S DERIVATION
• energy quantized in units of
• modify equipartition:
• hence
   d 

8 hc

5
0
e 0
0
kT
e
hc kT
observed
spectrum
1
1
1
RayleighJeans
d
wavelength
BLACK BODY
• perfect absorber, hence ‘ideal’ emitter
• no spectral features beyond Planck curve
ULTRAVIOLET CATASTROPHE
• classical thermodynamics predicts monotonic increase with
frequency
• quantization of radiation field supplies required correction
7
WORK FUNCTION
• threshold for photocurrent
• no current above threshold
wavelength regardless of intensity
A
photocurrent
Photoelectric effect
increasing
intensity
• applied voltage
BIAS VOLTAGE
• applied voltage changes threshold
• threshold voltage proportional to
optical frequency
optical frequency
h    eV
voltage
electron charge
work function
optical frequency
Planck’s constant
8
Light and optics
RAYS
• straight propagation paths
• least time (Fermat’s principle)
focus
• reflection, refraction, lenses, telescopes, microscopes
directrix
WAVES
• Huygens’ description of propagation, reflection, refraction
• polarization, colour (wavelength, frequency)
• diffraction, interference, beats, interferometers
• Maxwell’s electromagnetism, Einstein’s relativity
PHOTONS
• energy quantized in units of
h (h = Planck’s constant)

h
• momentum quantized in units of h  k 
c

h
• angular momentum quantized in units of

2
9
wavelength shift
Compton scattering
GRAPHITE
TARGET

0.711 Å
X-RAYS
0
45
90
135
angle
A H Compton, Phys Rev 22 409 (1923)
• photon momentum
h
p



10
Davisson-Germer experiment
NICKEL
TARGET
C Davisson & L H Germer, Phys Rev 30 705 (1927)
ELECTRON DIFFRACTION
• electrons behave like waves
• electron wavelength
h

p
11
Diffracting molecules
S Gerlich et al, Nature Physics 3 711 (2007)
MOLECULE DIFFRACTION
• molecules behave like waves
h
• molecule wavelength  
p
12
Quantum theory
PHOTONS
• energy quantized in units of
(h = Planck’s constant)
h
• blackbody radiation
• photoelectric effect

h
 k 
c

h
• angular momentum quantized in units of

2
• momentum quantized in units of
h
• Compton scattering
PARTICLES
• frequency determined by energy
• de Broglie wavelength determined
by momentum
E  h
p  k 
• angular momentum quantized in units of
h

h

2
• electron diffraction
• atomic theory
13
Bohr model of the hydrogen atom
BOHR MODEL
• circular orbits
e2
mv2

2
40 r
r
+
• quantized angular momentum
mvr  n
• de Broglie wavelength

• quantized energy levels
h
p
E  h
• Hydrogen energy level
measurements and calculations
agree to 15 figures
f1S 2 S  2 466 061413187 074 34 Hz
R  10 973 731.568 527 73 m1
14
Bohr model of the hydrogen atom
E
me4 1
 3 2 2
hc
8h  0 c n
R Rydberg
  2 constant
n
energy
• allowed energies
me4
1
E
2
2
240   n
n=
0
n=3
hcR
4
n=2
 hcR
n=1

• emission wavelengths
 1 1
E Ei  E j 


  R  2  2 
n n 
 hc
hc
j 
 i
1
R  10 973 731.568 527 73 m1
15
Atomic line spectra
E
me4 1
 3 2 2
hc
8h  0 c n
R Rydberg
  2 constant
n
energy
• allowed energies
me4
1
E
2
2
240   n
n=
0
n=3
hcR
4
n=2
 hcR
n=1

• emission wavelengths
 1 1
E Ei  E j 


  R  2  2 
n n 
 hc
hc
j 
 i
1
R  10 973 731.568 527 73 m1
16
energy
Atomic line spectra
n=
0
n=3
Paschen

n=2
hcR
4
Balmer
universe-review.ca
scope.pari.edu
 hcR
n=1
Lyman
R  10 973 731.568 527 73 m1
17
Hydrogenic atoms
E
me4 Z 2
 3 2 2
hc
8h  0 c n
R 2 Rydberg
  2 Z constant
n
energy
• allowed energies
Z 2 me4 1
E
2
2
240   n
n=
0
n=3
hcR
4
n=2
 hcR
n=1

• emission wavelengths
 1 1
E Ei  E j 
2


  Z R  2  2 
n n 
 hc
hc
j 
 i
1
R  10 973 731.568 527 73 m1
18
Franck-Hertz experiment
J Franck & G Hertz, Verh. Dtsch. Phys. Ges. 16 457 (1914)
• accelerate electrons through atomic vapour
• periodic modulation of measured current
• inelastic collisions when electron energy equals
atomic transition energy
singlet
triplet
Hg
G Rapior et al., Am J Phys 74 423 (2006)
19