A level Physics Hidden Depths Peter Rowlands
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.
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 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 energypotential energy changing conditions steady state action action + reaction
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: 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 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
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 Pr. In fact, if F 1 / rn, in this model, then Pr(n + 2)/3.
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 = rc2 / 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.
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 1772 Laplace 1796
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
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 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.
Gravitational light bending hyperbolic orbit 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 and Unfortunately, Soldner didn’t because he used d instead of 2d.
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 w0 to w in a magnetic field B, where ww0, is m (w2 – w02) = m (w + w0) (w – w0) = ewrB , with frequency change
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.
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.