Chapter 7
Equations
n  d sin 
Warning!
“What I am going to tell you is what we teach
our physics students in the third or fourth
year of graduate school - and you think I
am going to explain it to you so you can
understand it? No, your not going to be
able to understand it….you see my physics
students don’t understand it either, that’s
because I don’t understand it. Nobody
does.”
Richard P. Feynman - QED
Feynman Lectures

http://vega.org.uk/video/subseries/8
If its so difficult why do we use it?
I would again like to impress you with the vast range
of phenomena that the theory of quantum
electrodynamics describes; It’s easier to say it
backwards: the theory describes all the phenomena
of the physical world except the gravitational
effect,..,and radioactive phenomena, which involves
nuclei shifting in their energy levels.
So if we leave out gravity and radioactivity (more
properly, nuclear physics), what have we got left?
Gasoline burning in automobiles, foams and
bubbles, the hardness of salt or copper, the
stiffness of steel. In fact, biologists are trying to
interpret as much as they can about life in terms of
chemistry, and as I already explained, the theory
behind chemistry is quantum electrodynamics.’
“Richard P. Feynman - QED”
Q.E.D
The most accurate scientific theory ever
developed
 For example it predicts the value of a
particular constant to be

 Predicted value
 Measured value

1.00115965221±4
1.00115965265±20
This is the same as measuring the
distance from London to New York with
an error equal to the thickness of human
hair
Photography with photons
A frustratingly unpredictable process – except on average!
making a photo using
a 3 bit greyscale CCD
the CCD
Old expectations
New reality
Smooth arrival of energy
Photons arriving
Photons arrive
randomly in space
and time
more
exposure
Only averaging
over a large
number of arrivals
is predictable
This does not happen!
This happens
Evidence for the graininess of light
Light-emitting diodes
LEDs are engineered to drop each
electron by a fixed p.d. and to emit
a photon of a definite colour. A range
of LEDs, of different colours, illustrate
the relationship between energy and
frequency for photons.
Increase the p.d. until the LED just
glows. This is the striking p.d.
energy transferred
to each electron
= e  V
energy
transferred to
each photon
= e  V
V
V
True as long as the LED
does not warm up.
Striking p.d. fixes
energy
E/J
E = qV
E = e Vblue
+
E = e  Vgreen
E = e  Vred
+
+
fred light
fgreen light
f
f/Hz
blue light
Constant slope, E/f. The number of joules per hertz is uniform for all
electromagnetic radiation.
h, the gradient, is 6.634  10–34 J Hz–1. More often written as
h = 6.634  10–34 J s
Spectral lines and energy levels
Particular colours of light are associated with certain
energies.
The same pattern extends beyond the
visible, to all parts of the
electromagnetic spectrum.
E/J
atom fixes
energy
Constant slope, E/f. The number of joules per
hertz is uniform for all radiation.
h, the gradient, is 6.634  10–34 J Hz–1
More often written as h = 6.634  10–34 J s
E/J
Eblue light
+
+
Egreen light
Ered light
That there are sharp spectral lines
means some rungs of an energy
ladder exist – a clue about the
structure of atoms.
+
fred light
f/Hz
fgreen light fblue light
Frequency determines colour. Frequency = speed/wavelength.
Photoelectric effect
The ejection of electrons from metals by photons was important in establishing the photon description.
The energy from a single photon
is transferred to a single electron.
energy to just climb
potential hill = e V
A
potential difference
just stops electrons
energy transferred by each
photon = e  V + 
Stopping p.d.
measures electron E/J
energy E = qV
Constant slope, E/f. The number of joules per
hertz is uniform for all electromagnetic radiation.
E = e  Vblue
E = e  Vgreen
h, the gradient, is 6.634  10–34 J Hz–1
More often written as h = 6.634  10–34 J s
+
E/J
+
metal that gives
up electrons more
easily
original
metal
f/Hz
fred light f0 fgreen light
too low a frequency to
provide the energy to
eject electron
V
fblue light
 = hf0 energy needed
to eject one electron from
the metal
f/Hz
How a path is explored

‘waypoints’ define paths
for the photons to explore
an arrow spins at
the frequency of
the photon as the
path is explored
arrow moves at
the speed of the
photon
D
‘path’ is one of
the many routes
that the photon
must explore to
calculate the
fraction of the
emitted photons
found at the
detector
S
‘source’ is
where the
photons come
from
An arrow is the
output from this
process

‘detector’ is where
we look to find out
the fraction of the
emitted photons
arriving
The spinning
arrow freezes
when it arrives
at the detector
to give an arrow
One arrow by itself means nothing - you
need to sum arrows from all possible paths
I am describing to you
how nature works you
won’t understand why
nature works that way.
But you see nobody
understands that
“Richard P. Feynman
QED”
Calculating probabilities from arrows
D
Each path explored
delivers one arrow
Add these nose to tail to
give the amplitude
Square the amplitude to
give a number
proportional to the
probability that a photon
is detected
2
0.7 = 0.49
Exploring three paths to calculate an amplitude
S
D
S
D
S
D
Intensity
a
b c
d
e f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y z
31
30
29
28
27
26
25
way point
arows lining up
arows curling up
31
30
29
28
27
26
25
Reflection - explorations over a surface
S
D
S
D
S
D
length = 0.4
chance = 0.42
= 0.16
from end of mirror
Place the source,
detector and mirror.
Fix the frequency of
the photon and
define a set of paths
for each photon to
explore by flagging
waypoints. All
chosen paths go via
the mirror.
Explore each path by
moving a phasor
along that path. Start
with a fresh phasor
each time and record
the final arrow.
Record these arrows
in order.
Place all these
arrows nose to tail in
order. The sum of
these arrows is the
amplitude. Square
the amplitude to find
the chance that a
photon ends up at
this detector.
Reflection occurs quantum mechanics
says mirrors should
work. Most of the
final amplitude
comes from paths
with waypoints on
the middle of the
mirror.
Exploring more paths
gives more arrows,
which increases the
precision of the
calculation.
The pattern is clear.
Almost all the
amplitude comes
from the centre of the
mirror, only a little
from the ends.
The intensity, equal
to the number of
photons per second,
does not change
much if the ends of
the mirror are cut off.
from middle of mirror
These
phasors all
come from a
narrow slice
at the middle
of the mirror
Use a restricted set
of paths (with only
one waypoint each)
to keep things
simple.
from end of mirror
Making a grating from a mirror
We aim to make the ends count
S
D
concentrate on this
piece and see how
to get the arrows to
line up
Now remove the middle one
Take out the middle one
by eliminating that path
best detector to have a chance of
finding red photons
best detector to have a chance of
finding green photons
S
best detector to have a chance of
finding blue photons
This is a reflection grating - useful for analysing spectra
Source
Detector
separation of source and detector (y)
A=
2
z +x
2
perpendicular distance
detector to mirror (z)
B=
x
2
2
z +(y–x)
mirror
time =
A+B
speed
time
position
starting with a plane mirror
S
D
set up
detector
where we
would like
to get a
focus
not much chance
of getting
photons here
start bending the mirror to get the arrows to line up
S
keep bending until
the arrows line up
D
up a little here
?
up a little more here
down a little here
Photon trip time for refraction
separation of source and detector, y
height of source
above surface, h
A =h2+ x2
y –x
water
depth of
detector below
surface, d
x
B = d2+ (y – x)2
time A =
h2+x2
speed in air
d2+(y–x)2
time B =
speed in water
trip time = time A+ time B
position of impact
on surface
Refraction – explorations through a surface
S
Place the source, detector
and surface.
Light appears to travel more
slowly below the surface, so
we reduce the speed of the
exploring phasor. The
frequency is unchanged.
Choose a photon frequency
and define a characteristic
set of paths going via the
surface.
The trip time is calculated in
two parts: above and below
the surface. The phasor spins
at the same frequency. The
time taken determines the
angle through which it has
turned.
D
S
D
S
Explore each path by moving
a phasor along the path.
Start with a fresh phasor each
time and record the final
arrow.
Record these arrows in order.
D
near least
time path
far from
least time
path
Obtain and square the
amplitude to find the chance
that a photon ends up at this
detector.
Refraction occurs – quantum
mechanics says that there is
a large chance that the photon
be found at the detector.
Most of the final amplitude
comes from paths just to the
right of the straight line path;
paths close to the path of least
time.
Explore more paths to get
more arrows, a clearer
picture and greater accuracy.
The pattern is clear. Most of
the amplitude comes from the
paths close to the path that
takes least time, only a little
from those far out.
High kinetic energy
E
f= k
h
Small differences in trip
time are enough to allow
arrows to curl up
WP
WP
WP
D
S
At high
frequencies
3 chosen
waypoints
give arrows
that curl up
Removing these paths does
not have much effect on the
probabilitiy of finding a
particle at the detector
Low kinetic energy
E
f= k
h
Small differences in trip
time mean arrows line up
WP
WP
WP
D
S
At low
frequencies
the same 3
waypoints
give arrows
that line up
Removing these paths
significantly affects the
probabilitiy of finding a
particle at the detector
Note: At higher kinetic energies we need only consider paths closer to the straight line
Trying to pin down photons
Very wide slit
x
The photon has lots of space to
explore between x and y: as a result
its likely arrival places are not much
spread out.
S
D
Only near the straight through path
do the phasor arrows make a large
resultant.
y
barrier to restrict
paths explored
scan detector to predict
brightness on a screen
chance the photon ends
up at each place
Wide slit
x
As the photon passes xy it has
only a few paths to explore. Path
differences are small.
S
D
Phasor arrows add to a large
resultant at a wide spread of
places.
y
barrier to restrict
paths explored
scan detector to predict
brightness on a screen
chance the photon
ends up at each place
Very narrow slit
x
As photon passes xy it has only one
path to explore: an infinitely thin slit!
Now it could go anywhere!
S
D
The narrower slit the wider the
spread.
y
barrier to restrict
paths explored
scan detector to predict
brigtness on a screen
chance the photon ends
up at each place