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```Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
1
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
2
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



3
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
4
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
5
Bohr model of the hydrogen atom
BOHR MODEL
• circular orbits
e2
mv 2

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  2 466 061 413187 074 34 Hz
R  10 973 731.568 527 73 m 1
6
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
7
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
8
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
9
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
10
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)
11
Quantum theory
PHOTONS
• energy quantized in units of
(h = Planck’s constant)
h

h
• momentum quantized in units of h  k 
c

h
• angular momentum quantized in units of

2
• photoelectric effect
• Compton scattering
PARTICLES
• frequency determined by energy
• de Broglie wavelength determined
by momentum
E  h
p  k 
• angular momentum quantized in units of
• discrete energy levels for bound particles
h

h

2
• electron diffraction
• atomic theory
• Stern-Gerlach
• atomic theory
12
Wave-particle duality
WHAT SORT OF WAVE?
• transverse/longitudinal motion?
+
• density?
QUANTUM WAVEFUNCTION
• amplitude2 describes probability
• phase has no classical analogue
• rate of phase variation defines
frequency and wavelength
transverse
?
+
• amplitude and phase combined to
form complex number
  ae
i
density
• phase matters!
13
Diffracting molecules
S Gerlich et al, Nature Physics 3 711 (2007)
MOLECULE DIFFRACTION
• molecules behave like waves
h
• molecule wavelength  
p
ikx
• molecular wavefunction   ae
14
Ramsauer-Townsend effect
Ar
A
S G Kukolich, Am. J. Phys. 36 701 (1968)
• anomalous dip in scattering probability at low energy
• proves to be interference from front and rear ‘reflections’ from Ar atom
15
Particle interference
MOLECULE DIFFRACTION and RAMSAUER-TOWNSEND
• give particle two or more routes through experiment
• interference depends upon relative phases of contributions
• phase depends upon path difference and wavelength
STATIONARY PARTICLES
• give particle two or more routes through experiment
• interference depends upon relative phases of contributions
• phase depends upon frequency difference and duration
16
Atomic clock
energy
2 e  i 0 t
0
1

0
2
• Cs atom
1
•  = 9.1926 GHz
• electron density depends upon relative
phase of superposition components
17
Atomic clock
1
x/a0
• atomic wavefunction
2
x/a0
  ae i  E  t
• electron density depends upon relative
phase of superposition components
18
Quantum measurement
• allowed energies
energy
THE HYDROGEN ATOM
me4
1
E
2
2
240   n
n=
0
n=3
hcR
4
n=2
 hcR
n=1

QUANTUM MEASUREMENT
1.
measured energy must be one of allowed
values
2.
…but until measurement, any energy possible
3.
after measurement, subsequent measurements
will give same value
19
Quantum mechanics
1.
particles behave like waves, and vice-versa
2.
energies and momenta can be quantized, ie measurements yield
particular results
3.
all information about a particle is contained within a complex
wavefunction, which determines the probabilities of experimental
outcomes
4.
80 years of experiments have found no inconsistency with quantum
theory
5.
explanation of the ‘quantum measurement problem’ – the collapse of the
wavefunction upon measurement – remains an unsolved problem
20
```
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