a level physics hidden depths
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A level Physics
Hidden Depths
Peter Rowlands
The structure of this presentation
The presentation will be in four main parts:
1 Kinematics and kinetic theory2 Gravity, photons and electron spin3 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.
Part One
Kinematics and kinetic theory
Where does physics begin?
Where does physics, from our point of view, begin?
Merton College in the fourteenth century.
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
Merton mean speed theorem
Merton mean speed theorem
tvu
s2
t
uva
Add the definition of uniform acceleration
to give the kinematic equations of motion
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:
Merton mean speed theorem
s = ½ vt s = vt
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
r
GMmmv 2
changing conditions steady state
action action + reaction
r
GMmmv
2
2
Two fundamental equations
escape velocity fixed orbit
Two fundamental equations
The relation between the two equations looks trivial, but it isn’t. 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.
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:
V
nT
2
1
Only for n = 2 (inverse-square force) or n = 0 (constant force) is the potential energy (numerically) twice the kinetic.
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 Boyle’s law does not.
Kinetic theory of gases
We derive Boyle’s 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.
Kinetic theory of gases
Kinetic theory of gases
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 = c2 / 3
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.’
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 .
In fact, if F 1 / rn, in this model, then P (n + 2)/3.
rF
1
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.
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 don’t 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.
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.’
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.
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 Waterston’s abstract.
Dalton’s atomic theory
Interestingly, at least one major piece of work resulted directly from a misreading of Newton’s Book II, Proposition 23, along with a double misreading of Newton’s views about atoms!
This was John Dalton’s atomic theory ( nucleon number).
Dalton’s 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 didn’t it separate out into layers?
Dalton’s atomic theory
He hit upon Newton’s 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.
Dalton’s atomic theory
So he decided to use Henry’s 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 …
Dalton’s atomic theory
His justification for these assumptions were 2 (misinterpreted) paragraphs from Newton’s 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.
Part Two
Gravity, photons and electron spin
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 = c2 / 3
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 ‘Boyle’s law’ relation is nothing to do with kinetic energy.
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 Einstein’s theory of relativity, E = mc2 emerges as an integration constant, which is introduced specifically to preserve classical conservation laws.
F.vdt
dT
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.
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.
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 1772Laplace 1796
Black holes
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):
R
GMmmc
2
2
from which
2
2
c
GMR
Black holes
Laplace’s calculation led to another significant consequence:
Pierre Simon Laplace Johann von Soldner
Gravitational light bending
Johann von Soldner used Laplace’s 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.
Soldner’s calculation wasn’t rediscovered until 1921. It has been misunderstood ever since.
Gravitational light bending
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 = 1 / e.
Gravitational light bending
hyperbolic orbit
12 er
GMmmc
rc
GM
e 2
222
with total deflection (in and out of the Sun)
This is what Soldner got, and it’s only half the true value.
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 Soldner’s paper in 1978, complained about him using the wrong equation, but actually he used the right one.
Gravitational light bending
Using kinetic energy, we get the right answer, because
rc
GM
e 2
442
12
2
er
GMmmc
and
Unfortunately, Soldner didn’t because he used instead of 2.
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.
The spin of the electron
According to ‘classical’ reasoning, we are told, the energy acquired by an electron changing its angular frequency from 0 to in a magnetic field B, where 0, is
m ( 2 – 02) = m ( + 0) ( – 0) = erB ,
with frequency change
mr
eB
2
The spin of the electron
The frequency change we actually observe is twice this value:
mr
eB
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.
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.
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 wasn’t the relativistic aspect of the Dirac equation that produced the doubling or halving effect, but the anticommuting property of the momentum operator.
But we don’t need either quantum theory or relativity to derive spin ½, only A-level physics.
The spin of the electron
The equation we really need to get the ‘correct’ value is
m ( 2 – 02) = m ( + 0) ( – 0) = 2erB (1)
This is a kinetic energy equation (applied at the moment of applying the field B) and the frequency change becomes
mr
eB
A-level physics? Well, we might recognise (1) as
v2 = u2 + 2as
Newton’s 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: Newton’s 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.
Newton’s 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.
Newton’s 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.
General Relativity
Of course, we haven’t 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.
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.
Critical density
In this context we note that the critical density equation for the expanding universe
is only a rearranged version of
r
GMmmv
2
2
G
Hc
8
3 2
One last twist
In historical terms, Eddington’s 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.
One last twist
It is interesting to quote Einstein’s own words on this experiment.
‘How tricky nature is, when one tries to approach it experimentally!’
Part Three
What is the speed of light?
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:
2
121 c
cn
1
221 c
cn
But what do mean by speed of light in this context? Can it be ever true that, as some people once believed:
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:
2
121 c
cn
1
221 c
cn
while the old (discarded) corpuscular theory said that
What is the speed of light?
The predictions are easy to see from diagrams. Wave theory:
What is the speed of light?
Or in a more simplified form:
What is the speed of light?
Corpuscular theory:
What is the speed of light?
Huygens Newton
Prehistory of light (1)
1640 Hobbes (c1 > c2) Descartes (c2 > c1)
1650 Fermat – least time
1670 Huygens wave theory Newton corpuscular theory
1676 Roemer measures c using eclipses of Jupiter’s satellites
1740 Maupertuis least action
Prehistory of light (2)
1801 Young interference
1820 Fresnel full wave theory
1850 Foucault measures c on Earth (c1 > c2 for air / water)
1864 Maxwell electromagnetic (wave) theory
1900 Planck quantum theory (black body radiation)
1905 Einstein photon
1924 de Broglie – wave particle duality
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 other’s results by inverting speeds.
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.
What is the speed of light?
Louis de Broglie reconciled the theories with his wave-particle duality in 1924:
hp
What is the speed of light?
How does this explain Hamilton, etc?
hp
If p = mu, then u is equivalent to a ‘corpuscular’ velocity, while = v / f, where v is a wave (phase) velocity.
So, inverting one will always produce the other.v
cu
2
What is the speed of light?
Q. But didn’t Foucault’s experiment decide the issue?A. No, because it didn’t measure either corpuscular velocity or wave velocity. It measured group velocity.
What is the speed of light?
Finding the ratio of signal velocities in two media doesn’t 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.
What is the speed of light?
Anomalous dispersion can reverse a spectrum if the material has a refractive index < 1 for the wavelengths used.
What is the speed of light?
In fact, anomalous dispersion is the more regular phenomenon over the full wavelength range. It’s ‘normal’ dispersion that’s anomalous.
What is the speed of light?
One particularly unusual case of anomalous dispersion:
is only visible on The Dark Side of the Moon!
What is the speed of light?
Anomalous dispersion was discovered in Foucault’s time.
Jamin 1847 metallic reflectionLeroux 1860 iodine vapour Christiansen 1870 rosaniline
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.
What is the speed of light?
Group velocity was also known in Foucault’s time (Hamilton, 1839). Its relationship to wave (phase) velocity was established by Rayleigh (1877), who also showed its significance for Foucault’s experiment (1881).
A hundred years later, however, people were still quoting the Foucault experiment as the classic example of a decisive one.
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.
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!
Part Four
The origin of quantum theory
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?
The development of quantum theory
1900 Planck black body radiation E = hf
1905 Einstein photoelectric effect photon
1913 Bohr atomic structure
1922 Compton Compton effect
1923 de Broglie duality p = h /
1925 Heisenberg matrix mechanics
1926 Schrödinger wave mechanics
The development of quantum theory
Quantum theory arrived, historically, with Planck’s theory of black body radiation. Was this the only way it could have happened?
The development of quantum theory
Planck introduced the quantum, but Einstein converted Planck’s idea into the particle-like photon in explaining the photoelectric effect, etc.
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 Einstein’s equation:
hf = W + ½ mv2
The development of quantum theory
But he categorically rejected Einstein’s 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 Einstein’s photon theory, as did Bohr in 1913.
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).
The development of quantum theory
Compton’s ‘discovery had the effect of a crystal dropped into a supersaturated solution’.
Everyone was now in favour. Einstein was sent de Broglie’s 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).
The development of quantum theory
Why was Einstein’s 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 didn’t want it resurrecting. Were they right?
The development of quantum theory
The development of quantum theory
The Balmer series (1884) is probably the only successful example of numerology in the history of science,
22
2
nm
hm
Balmer’s formula
22
11
mnRf
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 couldn’t 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?
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 / and an approximate value for h from a classical experiment: Newton’s rings.
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.
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.
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 × 10–7 m for violet to 3.2 × 10–7 m for red
The development of quantum theory
To calculate h from this (since, from Roemer, he had c = 2.47 × 108 ms–1) 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.
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 × 10–2 Nm–1. Newton quoted the force value at 10–7 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 < 5 × 10–7 inches.
The development of quantum theory
If we use Newton’s own figure for c, and choose r at, say, 2 × 10–7, we can easily calculate the mass of a light corpuscle at 0.9 × 10–35 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 × 10–37 kg.
At any rate, we can now use the data to estimate h = mc at between 4.5 and 7.2 × 10–34 Js.
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 Planck’s constant could have been determined entirely from optics.
The End
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