Quaternions Multivariate Vectors

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A level Physics Hidden Depths



Peter Rowlands

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The structure of this presentation
The presentation will be in four main parts 1
Kinematics and kinetic theory 2 Gravity,
photons and electron spin 3 What is the speed
of light? 4 The origins of quantum theory The
idea will be show that A-level incorporates
profound ideas about these things, which a
semi-historical analysis will help to uncover.
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Part One
Kinematics and kinetic theory
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Where does physics begin?
Where does physics, from our point of view, begin?
Merton College in the fourteenth century.
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Merton mean speed theorem
when any mobile body is uniformly accelerated
from rest to some given degree of velocity, it
will in that time traverse one-half the distance
that it would traverse if, in that same time, it
were moved uniformly at the degree of velocity
terminating that latitude. William Heytesbury,
c 1334
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Merton mean speed theorem
Merton mean speed theorem
Add the definition of uniform acceleration
to give the kinematic equations of motion
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Merton mean speed theorem
Combining these equations gives us results
like v2 u2 2as The 2 in
the formula is immensely profound. We will see it
again in many unexpected places. One way of
getting it is by using triangles and rectangles
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Merton mean speed theorem
s ½ vt
s vt
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Two fundamental equations
But it also comes in more general contexts. For
example, there are effectively two ways of
expressing conservation of energy
kinetic energy potential energy
changing conditions steady
state action action
reaction
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Two fundamental equations
escape velocity fixed orbit
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Two fundamental equations
The relation between the two equations looks
trivial, but it isnt. It expresses the
3-dimensionality of space. It is only valid for
inverse-square or constant forces, and these are
characteristic of 3-D space. Immanuel Kant
showed the case for inverse-square forces in the
eighteenth century, and we can show that other
force laws lead to unstable orbits.
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Virial theorem
The more general case is the virial theorem. For
a force proportional to power n of distance or
for potential energies inversely proportional to
power (n 1), the time-averaged kinetic and
potential energies are related by
Only for n 2 (inverse-square force) or n 0
(constant force) is the potential energy
(numerically) twice the kinetic.
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Kinetic theory of gases
The virial theorem is actually used in A-level
physics in the kinetic theory of gases. In fact
from the mathematical point of view, this should
really be called the potential theory. Of
course, Brownian motion demonstrated the truth of
the kinetic theory. But the derivation of
Boyles law does not.
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Kinetic theory of gases
We derive Boyles law by assuming that the system
is constant on a time average. In principle,
this is equivalent to assuming that the gas
molecules are stationary. And we derive a
potential energy relation, not a kinetic one.
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Kinetic theory of gases
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Kinetic theory of gases
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Kinetic theory of gases
We assume that a molecule reflected from the
container wall has change of momentum mv (mv)
2mv. Then, for a molecule travelling twice the
length of the container (2a) between collisions,
we derive a time interval 2a / v, and reaction
force 2mv2 / 2a mv2 / a. Extending this to n
molecules in 3 dimensions with rms speed c, we
find an average force on each wall mnc2 / 3a,
and, for a cubical container of side, an average
pressure P mnc2 / 3a3
Mc2 / 3V rc2 / 3
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Kinetic theory of gases
Let us look at a quite different
alternative. Newton, Principia, Book II,
Proposition 23 If a fluid be composed of
particles fleeing from each other, and the
density be as the compression, the centrifugal
force of the particles will be inversely
proportional to the distances of their centres.
And, conversely, particles fleeing from each
other, with forces that are inversely
proportional to the distances of their centres,
compose an elastic fluid, whose density is as the
compression.
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Kinetic theory of gases
Newton creates an abstract mathematical model in
which the molecules of gases are subject to
repulsive forces between themselves which are
inversely proportional to their separation and
shows that this means P ? r. In fact, if F ? 1
/ rn, in this model, then P ? r(n 2)/3.
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Kinetic theory of gases
At first sight, this looks completely different
to the kinetic model, but, in fact, it is
mathematically the same. An inverse
proportionality between force and distance
between molecules is exactly the same as an
inverse proportionality between force and length
of container.
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Kinetic theory of gases
What has happened in our kinetic model is that
the use of a doubling of momentum by reflection
in a steady state system has taken away our
source of kinetic information. We dont know
anything directly about the kinetic energy
because we have chosen to include both action and
reaction in a system which shows no overall
change. The steady state pressure P gives us
only the potential energy PV, and this is
independent of the constitution of the gas.
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Kinetic theory of gases
We imagine that the fact that our model, by
giving the correct result, is somehow shown to be
true in itself. But Newton knew better. He knew
that his model only had a mathematical
justification But whether elastic fluids
really do consist of particles so repelling each
other, is a physical question. We have here
demonstrated mathematically the property of
fluids consisting of particles of this kind, that
philosophers may take occasion to discuss that
question.
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Kinetic theory of gases
Of course, if we assume kinetic theory to be
true, or base our justification on Brownian
motion (discovered in 1828), and assume that an
observed constant pressure is equivalent to a
constant force for the gas as a whole (not the
molecules), then we can apply the virial
theorem. In fact, we have to do this to derive
the average kinetic energy of a molecule. At this
point, we introduce the virial factor ½ , and
assume that temperature is a measure of kinetic
energy, but there is no derivation.
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Kinetic theory of gases
Perhaps the lack of real connection between the
model and the results derived from it may explain
why the kinetic theory was twice rejected before
being finally accepted. Herapath 1813 ignored
as the work of an eccentric Waterston
1845 rejected by the Royal Society Several
authors took it up around 1858, partly influenced
by Waterstons abstract.
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Daltons atomic theory
Interestingly, at least one major piece of work
resulted directly from a misreading of Newtons
Book II, Proposition 23, along with a double
misreading of Newtons views about atoms! This
was John Daltons atomic theory (? nucleon
number).
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Daltons atomic theory
Dalton was a meteorologist with no apparent
interest in chemistry. He collected data about
rainfall throughout his entire life, and his last
recorded act was to write down the weather for
that day in a shaky hand. The big question for
him was, if air was composed of several gases of
different densities, why didnt it separate out
into layers?
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Daltons atomic theory
He hit upon Newtons model of gases being
composed of particles repelling each other
mutually, and then decided that each type of gas
only repelled molecules of its own type. He saw
atmospheric gases as solvents for each
other. This theory of mixed gases went down like
a lead balloon. But his friend William Henry had
shown that gases dissolved in inverse proportion
to their density.
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Daltons atomic theory
So he decided to use Henrys law to shore up his
theory. But he needed data on the relative
masses of the gas particles. To interpret the
chemical data to justify his theory of gases he
had to assume that elementary chemical substances
were composed of unbreakable atoms, each having
characteristic weights
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Daltons atomic theory
His justification for these assumptions were 2
(misinterpreted) paragraphs from Newtons
Opticks it seems probable to me that God in
the beginning formed matter in solid, massy,
hard, impenetrable, moveable particles and that
these primitive particles being solids, are
incomparably harder than any porous bodies
compounded of them it may be also allowed
that God is able to create particles of matter of
several sizes and figures and perhaps of
different densities and forces, and thereby
make worlds of several sorts in several parts of
the Universe.
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Part Two
Gravity, photons and electron spin
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Photon gases
One interesting consequence of gas theory emerges
if we replace the material gas particles with
photons, as Einstein did. Amazingly, the photon
gas behaves in exactly the same way as the
material gas, generating a pressure proportional
to density via an entirely analogous formula
P rc2 / 3
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Photon gases
This may seem strange, because photons are
relativistic particles with energy E mc2, while
gas molecules are classical with kinetic energy ½
mv2. All kinds of explanations have been put
forward involving doubling and halving, but the
simple fact is that it really demonstrates that
the Boyles law relation is nothing to do with
kinetic energy.
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Photon gases
It also demonstrates something equally
fascinating, that E mc2, in the case of
photons, is equivalent to potential energy, and
has exactly the same form that a photon would
have if it were a classical particle of mass m
travelling at speed c. Despite its connection
with Einsteins theory of relativity, E mc2
emerges as an integration constant, which is
introduced specifically to preserve classical
conservation laws.
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Kinetic energy and photons
Studies of the historical record show that the
classical corpuscular theory of light used
terms equivalent E mc2 in this way. But what
about kinetic energy? Does it ever make sense to
write down a term like ½ mc2 for a
photon? Surprisingly, it seems it does, but only
under special conditions.
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Kinetic energy and photons
Photons in a medium, such as plasma, can slow
down and acquire an effective rest mass. However,
a more direct slowing down occurs under the
action of gravity. Of course, general relativity
preserves the unchangeability of c at the expense
of curving space-time, but many calculations can
be done by assuming that classical conditions
apply. The trick is to use the fact that the
mass and energy of photons are defined to
preserve classical energy conservation.
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Black holes
The most obvious example is the calculation of
the radius for a black hole. Until the 1960s it
was assumed that this concept originated with
General Relativity (Schwarzschild
radius). However, it was subsequently discovered
that there were at least two calculations from
the eighteenth century Michell 1772 Laplac
e 1796
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Black holes
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Black holes
Of course, the calculation itself is relatively
simple. All we have to do is to write down the
kinetic energy equation for changing conditions
(escape)
from which
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Black holes
Laplaces calculation led to another significant
consequence
Pierre Simon Laplace
Johann von Soldner
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Gravitational light bending
Johann von Soldner used Laplaces black hole
calculation in 1801 to estimate the gravitational
deflection of a light ray grazing the sun. In
1919 Arthur Eddington used an eclipse expedition
to measure the deflection, and found that it was
twice this value, according to the new
predictions of General Relativity. Soldners
calculation wasnt rediscovered until 1921. It
has been misunderstood ever since.
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Gravitational light bending
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Gravitational light bending
Modern authors have claimed that Soldner assumed
that a light ray travelling at c would have a
hyperbolic orbit with eccentricity e much
greater than 1 and deflection d 1 / e.
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Gravitational light bending
hyperbolic orbit
with total deflection (in and out of the Sun)
This is what Soldner got, and its only half the
true value.
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Gravitational light bending
However, this not what Soldner did. The potential
energy equation applies to an orbit already in
existence. But he assumed that the orbit still
had to be formed (the reverse of gravitational
escape) and so used kinetic energy. Stanley
Jaki, who republished Soldners paper in 1978,
complained about him using the wrong equation,
but actually he used the right one.
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Gravitational light bending
Using kinetic energy, we get the right answer,
because
and
Unfortunately, Soldner didnt because he used d
instead of 2d.
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The spin of the electron
A very similar case occurs with electron spin,
which is supposedly one of the most mysterious
aspects of quantum physics. It is possible to
derive this quantity in a wide variety of ways,
one, at least, of which looks rather
simple. However, this simple derivation is in
many ways the most profound.
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The spin of the electron
According to classical reasoning, we are told,
the energy acquired by an electron changing its
angular frequency from w0 to w in a magnetic
field B, where w ? w0, is m (w 2 w02)
m (w w0) (w w0) ewrB , with frequency
change
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The spin of the electron
The frequency change we actually observe is twice
this value
The only way round this is to suppose that the
electron has to spin round 2 revolutions to
complete a cycle. In quantum terms it has spin ½.
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The spin of the electron
Many kinds of reasoning have been used to derive
this strange result of spin ½, and they are all
actually true. Uhlenbeck and Goudsmit were so
embarrassed about putting forward the original
hypothesis in 1925 that they tried to withdraw
their paper. However, a relativistic (reference
frame) effect, was invoked by L. H. Thomas in
February 1926 (the Thomas precession) and this
made everyone happy, though still puzzled.
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The spin of the electron
Dirac then produced his relativistic quantum
theory of the electron (1927), and derived the
spin from first principles. Funnily enough, it
wasnt the relativistic aspect of the Dirac
equation that produced the doubling or halving
effect, but the anticommuting property of the
momentum operator. But we dont need either
quantum theory or relativity to derive spin ½,
only A-level physics.
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The spin of the electron
The equation we really need to get the correct
value is m (w 2 w02) m (w w0) (w
w0) 2ewrB (1) This is a kinetic energy
equation (applied at the moment of applying the
field B) and the frequency change becomes
A-level physics? Well, we might recognise (1) as
v2 u2 2as
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Newtons third law of motion
Why is this simple derivation particularly
profound? It is because it ultimately derives
from one of the deepest laws in the whole of
physics Newtons third law of motion. It is
often said that this law is simple to state, but
difficult to apply but, at the deepest level, it
is also difficult to state.
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Newtons third law of motion
We say that the law describes the mutual
interactions of two bodies on each other, but
really this is only an approximation. The real
situation is that each of the two bodies acts on
the rest of the universe.
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Newtons third law of motion
In quantum physics, the rest of the universe is
called vacuum. For the complete picture (action
and reaction), we need both electron and
vacuum. When we consider the electron only, we
are considering action only, and so we should
expect to use kinetic energy equations, and so
obtain spin ½.
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General Relativity
Of course, we havent really derived spin ½ from
first principles, as we do with quantum
mechanics, but we have gained a new insight into
what this strange property means. It also
suggests a new meaning for the parallel doubling
effect in gravitational light deflection, for
General Relativity, which requires it, is a
vacuum theory. Curvature of space can be seen as
another way of expressing the vacuum effect.
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Critical density
It is interesting that years after GR had been
used to predict the 3 possible universe outcomes,
Milne and McCrea showed that the same could have
been done using the much simpler Newtonian theory.
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Critical density
In this context we note that the critical density
equation for the expanding universe is only a
rearranged version of
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One last twist
In historical terms, Eddingtons announcement of
a double gravitational light deflection,
predicted by Einstein, clinched the acceptance of
General Relativity in 1919. Einstein was mainly
a theorist, but he did one notable experiment,
with de Haas, in February 1915, on the magnetic
moment of the electron. They found a value in
agreement with the then prediction, but 2 the
(correct) value found by Barnett in October.
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One last twist
It is interesting to quote Einsteins own words
on this experiment. How tricky nature is, when
one tries to approach it experimentally!
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Part Three
What is the speed of light?
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What is the speed of light?
The law of refraction says that the refractive
index of a medium is the ratio of the speed of
light in vacuum to the speed of light in a
medium, or that between media 1 and 2
But what do mean by speed of light in this
context? Can it be ever true that, as some people
once believed
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What is the speed of light?
The story is immensely complicated, both
physically and historically, though it used to be
presented as a classic case in which a decisive
experiment produced a definitive answer. In
principle, wave theory said that
while the old (discarded) corpuscular theory said
that
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What is the speed of light?
The predictions are easy to see from diagrams.
Wave theory
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What is the speed of light?
Or in a more simplified form
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What is the speed of light?
Corpuscular theory
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What is the speed of light?
Huygens Newton
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Prehistory of light (1)

• 1640 Hobbes (c1 gt c2) Descartes (c2 gt c1)
• Fermat least time
• 1670 Huygens wave theory Newton corpuscular
  theory
• 1676 Roemer measures c using eclipses of
  Jupiters satellites
• 1740 Maupertuis least action

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Prehistory of light (2)

• Young interference
• Fresnel full wave theory
• 1850 Foucault measures c on Earth (c1 gt c2 for
  air / water)
• Maxwell electromagnetic (wave) theory
• Planck quantum theory (black body radiation)
• Einstein photon
• 1924 de Broglie wave particle duality

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What is the speed of light?
The outline history leaves out many of the most
interesting developments. The wave theorists
used least time, proportional to 1 /speed. The
corpuscular theorists used least action (mvs), a
quantity proportional to speed. They each
converted each others results by inverting
speeds.
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What is the speed of light?
A classic case was the work of Hamilton, who, in
the 1830s, worked out a theory of dynamics by
analogy with optics. This was based on the
corpuscular theory of light, which was closer to
classical mechanics, but in about 1837 he found
he could switch entirely to the wave theory by
inverting velocities. His system was later used
by Schrödinger as the basis for wave mechanics.
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What is the speed of light?
Louis de Broglie reconciled the theories with his
wave-particle duality in 1924
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What is the speed of light?
How does this explain Hamilton, etc?

• If p mu, then u is equivalent to a
  corpuscular velocity, while l v / f, where v
  is a wave (phase) velocity.
• So, inverting one will always produce the other.

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What is the speed of light?
Q. But didnt Foucaults experiment decide the
issue? A. No, because it didnt measure either
corpuscular velocity or wave velocity. It
measured group velocity.
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What is the speed of light?
Finding the ratio of signal velocities in two
media doesnt predict the refractive index. For
air / water at the wavelengths used by Foucault
it happens to be about the same (1.5
discrepancy), but in many other cases it is
widely different. A standard optical medium of
the time, carbon disulphide, exhibits an 8
discrepancy. And there are very much bigger
discrepancies, even infinite ones.
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What is the speed of light?
Anomalous dispersion can reverse a spectrum if
the material has a refractive index lt 1 for the
wavelengths used.
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What is the speed of light?
In fact, anomalous dispersion is the more regular
phenomenon over the full wavelength range. Its
normal dispersion thats anomalous.
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What is the speed of light?
One particularly unusual case of anomalous
dispersion
is only visible on The Dark Side of the Moon!
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What is the speed of light?
Anomalous dispersion was discovered in Foucaults
time. Jamin 1847 metallic reflection Leroux
1860 iodine vapour Christiansen 1870 rosanil
ine Metallic mirrors, microwave waveguides and
reflection of radiowaves from the ionosphere
would be impossible without it needing total
internal reflection in the less dense
medium. X-rays in glass show almost entirely
corpuscular properties.
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What is the speed of light?
Group velocity was also known in Foucaults time
(Hamilton, 1839). Its relationship to wave
(phase) velocity was established by Rayleigh
(1877), who also showed its significance for
Foucaults experiment (1881). A hundred years
later, however, people were still quoting the
Foucault experiment as the classic example of a
decisive one.
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What is the speed of light?
The individual wave (phase) velocity does always
decrease in the ratio c2 / c1, which determines
the refractive index, though not the velocities
which will be measured. But is there any meaning
to saying that a particle or corpuscular
velocity simultaneously increases in the ratio of
c1 / c2? Remarkably, there is. In 1977 R. V.
Jones showed that at a refracting boundary,
photon momentum increases in exactly this way.
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What is the speed of light?
So, everyone was right, after all! Foucault had
shown only that no signal velocity in a medium
could exceed that of light in a vacuum, though
the velocity of an individual wave
could. However, vacuum itself is a medium, so
even this velocity can be exceeded!
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Part Four
The origin of quantum theory
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The development of quantum theory
The general history of the theory of light
suggests that two revolutions occurred, one
when the wave theory replaced the more dominant
corpuscular (particle theory), and one when
quantum theory was introduced to modify the wave
theory 1670 wave theory versus particle
theory 1820 wave theory (confirmed 1850)
1900 wave theory plus quantum theory This picture
assumes quantum theory ? corpuscular theory. Is
this true?
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The development of quantum theory

• 1900 Planck black body radiation E hf
• Einstein photoelectric effect photon
• 1913 Bohr atomic structure
• 1922 Compton Compton effect
• 1923 de Broglie duality p h / l
• 1925 Heisenberg matrix mechanics
• 1926 Schrödinger wave mechanics

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The development of quantum theory
Quantum theory arrived, historically, with
Plancks theory of black body radiation. Was this
the only way it could have happened?
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The development of quantum theory
Planck introduced the quantum, but Einstein
converted Plancks idea into the particle-like
photon in explaining the photoelectric effect,
etc.
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The development of quantum theory
Einstein got the Nobel Prize for this work, but
not until 1923. Until then, it was completely
rejected.
In 1915 Robert Millikan verified Einsteins
equation hf W ½ mv2
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The development of quantum theory
But he categorically rejected Einsteins theory
as the explanation despite the apparently
complete success of the Einstein equation for the
photoelectric effect the physical theory of which
it was designed to be a symbolic expression is
found to be so untenable that Einstein himself, I
believe, no longer holds to it. Even Max
Planck, who had published the 1905 paper,
rejected Einsteins photon theory, as did Bohr in
1913.
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The development of quantum theory
Eventually, politics decided it. The spectacular
success of General Relativity in 1919 led to a
reassessment of everything that Einstein had
previously done.
Arthur Compton, at the last minute, inserted a
photon explanation of the Compton effect into a
lengthy report (1922).
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The development of quantum theory
Comptons discovery had the effect of a crystal
dropped into a supersaturated solution. Everyone
was now in favour. Einstein was sent de
Broglies thesis by examiners who were unsure of
its value. The American G. N. Lewis, who had
long had his own ideas on the subject, came up
with the name photon (1926). Einstein got the
Nobel Prize (1923).
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The development of quantum theory
Why was Einsteins theory rejected for 18
years? The answer lies in a phrase Einstein used
at least as early as 1909. He was proposing a
Newtonian emission theory. And Millikan
referred to it as semicorpuscular. People were
absolutely convinced that the corpuscular theory
had been dead in the water since 1850, and didnt
want it resurrecting. Were they right?
92
The development of quantum theory
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The development of quantum theory
The Balmer series (1884) is probably the only
successful example of numerology in the history
of science,
Balmers formula
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The development of quantum theory
Bohr is said to have developed his quantum atomic
theory in 1913 immediately upon seeing it. Yet
line spectra were the obvious problem for a fully
wave theory of light from the beginning. David
Brewster said, as early as 1832, that absorption
spectra couldnt explain wave theory. There could
be no basis for such an extraordinary selection
of the undulations which the wave medium stops
or transmits. Was he a diehard reactionary or
proto-modern?
95
The development of quantum theory
One way of deciding the issue is to do a bit of
counter-factual history. The corpuscularians
could have derived p h / l and an approximate
value for h from a classical experiment Newtons
rings.
96
The development of quantum theory
No one, of course, did do this but Newton himself
came close. In his earliest, draft, account of
the experiment he came up with a succession of
hypotheses to explain the phenomenon, some of
which contradicted each other, According to the
modern reasoning the thickness of the film
required to produce a given ring is inversely
proportional to the photon momentum, and directly
proportional to the area of the ring or its
diameter squared.
97
The development of quantum theory
At one point in his manuscript Newton writes if
ye medium twixt ye glasses bee changed ye bignes
of ye circles are also changed. Namely to an eye
held perpendicularly over them, the difference of
their areas (or ye thicknesses of ye interjected
medium belonging to each circle) are reciprocally
as ye subtlity of ye interjected medium or as ye
motions of ye rays in that medium. By motions
he means momentum.
98
The development of quantum theory
Newton, of course, was not a conventional wave
theorist, but he did have a conception of
periodicity, and measured a quantity which he
called the interval of fits which is, in
principle, equivalent to a wavelength, if we
neglect the halving effect due to interference.
As is usual in the experiment, he determined
this quantity from the thicknesses of the film
associated with each circle. They range from 2.0
107 m for violet to 3.2 107 m for red
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The development of quantum theory
To calculate h from this (since, from Roemer, he
had c 2.47 108 ms1) we would need a figure
for the mass of a light corpuscle which he could
reasonably have calculated. We also need his
speculation that similar molecular forces are
involved in refraction and cohesion or capillary
action and have the same (electrical) origin.
100
The development of quantum theory
A late experiment on the capillary action of a
drop of oil of oranges on glass balanced against
gravity gave an inverse proportionality between
force and distance, with a constant equivalent in
modern terms to 2.8 102 Nm1. Newton quoted
the force value at 107 inches in his Opticks,
presumably guessing from his optical experiments
that this was a molecular level distance
scale. At the same time, he was working on a
manuscript calculation of the force of optical
refraction per unit mass as c2 / r, with a
distance which he seems to have pitched at lt 5
107 inches.
101
The development of quantum theory
If we use Newtons own figure for c, and choose r
at, say, 2 107, we can easily calculate the
mass of a light corpuscle at 0.9 1035
kg. This is a calculation Newton certainly could
have done. His eighteenth century editor Horsley
used other Newtonian data to calculate the mass
at 1.6 1037 kg. At any rate, we can now use
the data to estimate h mcl at between 4.5 and
7.2 1034 Js.
102
The development of quantum theory
Obviously the answer must be out by a systematic
factor of 2 because of the neglect of
interference (though this was added later, in the
nineteenth century). In addition, there will be a
fairly wide variation due to the choice of
distance r. The result is only intended to
indicate that, if that duality that was inherent
in the corpuscular theory had been considered
seriously from the beginning, there would have
been a much smoother transition to a quantum
theory, and Plancks constant could have been
determined entirely from optics.
103

       
The End