Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
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
• discrete energy levels for bound particles
h

h

2
• electron diffraction
• atomic theory
• Stern-Gerlach
• atomic theory
2
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!
3
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
4
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
5
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
6
Atomic clock
energy
2 e i0t
0
1

0
2
• Cs atom
1
•  = 9.1926 GHz
• electron density depends upon relative
phase of superposition components
7
Atomic clock
1
x/a0
• atomic wavefunction
2
x/a0
  ae i  E  t
• electron density depends upon relative
phase of superposition components
8
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
9
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
10
Messenger Lecture
• Richard P. Feynman (1918-1988)
Nobel prize 1965
• Messenger series of lectures, Cornell University, 1964
• Lecture 6: ‘Probability and Uncertainty – the quantum mechanical view
of nature’
• The Character of Physical Law - Penguin
• see the later series of Douglas Robb memorial lectures (1979) online
at http://www.vega.org.uk/series/lectures/feynman/
11
The experiment with the two holes
y
p0 

h


0

a
0
s
x

• fringe maxima when
a
• equivalent to change in illumination angle and hence
 by   p y p0
a sin   m0
 fringe spacing
  0 a
• smallest visible feature size 
 illumination wavelength   a
 illumination momentum p y  h a
a
12
Single slit diffraction

amplitude
y

a
x
intensity

s
13
Uncertainty
HEISENBERG’S UNCERTAINTY PRINCIPLE
• certain pairs of parameters may not simultaneously be exactly determined
• {position, momentum}
• {position, wavelength}
• {time, energy}
• {time, frequency}
• {orientation, angular momentum}
• {intensity, phase}
• {x, y}, {x, z}, {y, z} components of angular momentum
QUANTUM MEASUREMENT
• measurement changes observed system so that parameter measured
is subsequently definite
• conjugate parameters cannot be simultaneously definite
• process measure A, measure B not the same as measure B, measure A
• measure A, measure B are not commutative / do not commute
• commutator [measure A, measure B]  0
14
Uncertainty
BEATING OF TWO DIFFERENT FREQUENCIES
cos 1t  cos 2t  cos
1  2
2
t cos
1  2
2
t
15
Terminology
UNCERTAINTY IN MEASUREMENT
• repeated experiment yields range of results
• expectation value = mean
• uncertainty = standard deviation
1
a   an
n n
2
1
2
2
a    a   an  a
n n


• before measurement, system was in a superposition
• probability of given result
a
given by
 a 
2
16