a level physics hidden depths

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



tvu

s2

t

uva

Add the definition of uniform acceleration

to give the kinematic equations of motion


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:


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


escape velocity fixed orbit


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.


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.


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


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.’


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


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.


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.


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.’


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.


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 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?


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.


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 …


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.


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.


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.


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.


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.


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 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.


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.


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 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.


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.


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?


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:


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


The predictions are easy to see from diagrams. Wave theory:


Or in a more simplified form:


Corpuscular theory:


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


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.


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.


Louis de Broglie reconciled the theories with his wave-particle duality in 1924:

hp


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


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.


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.


Anomalous dispersion can reverse a spectrum if the material has a refractive index < 1 for the wavelengths used.


In fact, anomalous dispersion is the more regular phenomenon over the full wavelength range. It’s ‘normal’ dispersion that’s anomalous.


One particularly unusual case of anomalous dispersion:

is only visible on The Dark Side of the Moon!

http://images.google.com/imgres?imgurl=www.ilbaluardo.com/Cover/Audio/P%2520-%2520Q%2520-%2520R/PINK%2520FLOYD%2520-%2520Dark%2520side%2520of%2520the%2520moon%2520-%2520Back.jpg&imgrefurl=http://www.ilbaluardo.com/coverzone-P.htm&h=1656&w=2120&sz=273&tbnid=148DpSqy6Y0J:&tbnh=116&tbnw=149&start=3&prev=/images%3Fq%3Ddark%2Bside%2Bmoon%26hl%3Den%26lr%3D%26ie%3DUTF-8%26sa%3DN

http://images.google.com/imgres?imgurl=www.ilbaluardo.com/Cover/Audio/P%2520-%2520Q%2520-%2520R/PINK%2520FLOYD%2520-%2520Dark%2520side%2520of%2520the%2520moon%2520-%2520Back.jpg&imgrefurl=http://www.ilbaluardo.com/coverzone-P.htm&h=1656&w=2120&sz=273&tbnid=148DpSqy6Y0J:&tbnh=116&tbnw=149&start=3&prev=/images%3Fq%3Ddark%2Bside%2Bmoon%26hl%3Den%26lr%3D%26ie%3DUTF-8%26sa%3DN

http://images.google.com/imgres?imgurl=otterpop.sbay.org/album/darkside.jpg&imgrefurl=http://tabacaria.blogspot.com/2002_09_01_tabacaria_archive.html&h=694&w=700&sz=24&tbnid=AneIAriVUzQJ:&tbnh=136&tbnw=137&start=17&prev=/images%3Fq%3Ddark%2Bside%2Bmoon%26hl%3Den%26lr%3D%26ie%3DUTF-8


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.


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.


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.


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?


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


Quantum theory arrived, historically, with Planck’s theory of black body radiation. Was this the only way it could have happened?

http://academic.brooklyn.cuny.edu/history/virtual/portrait/planck.jpg


Planck introduced the quantum, but Einstein converted Planck’s idea into the particle-like photon in explaining the photoelectric effect, etc.


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


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.


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).


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).


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 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


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?


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.


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.


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.


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


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.


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.


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.


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

a level physics hidden depths

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