How far is it around the earth?
More specifically, if I was to set off from here in Wokingham following a great circle route, how far would I have to travel before I found myself back here in Wokingham?
Well, if you look it up on Google the circumference of the earth is near enough to 40,000 km. So it would seem to be that the distance from Wokingham to Wokingham is 40,000km.
But this is only part of the answer. I did not ask for the shortest distance from Wokingham to Wokingham, which would after all be zero, and there is no rule that says we have to complete just one orbit. After two orbits and the corresponding 80,000km we would find ourselves back in Wokingham. But why stop there? We could opt for three or more orbits, in fact there is no limit as to how many orbits we could count before we deem our task to be complete.
This means that in effect there are an infinite number of discrete distances around the earth all of which lead back to our point of departure. Equally we could go in the opposite direction, in which case we can regard the distance as being negative. Again there are an infinite number of such distances. We can write a simple formula to calculate them:
Even this falls short of a complete answer because in our imaginary orbiter we can travel as fast or as slow as we like. The distances we have measured so far are measured at a low speed where the effects of relativity are negligible. But if we were to travel much faster, at close to the speed of light, then the distance we perceive is reduced or foreshortened.
It was Einstein who gave us our present understanding of how relativity affects distance. He did so initially for objects travelling at constant speed in what is now called Special Relativity, Special because it deals with the special case of things moving at constant speed. Later on he was to deal with objects that are accelerating or decelerating in what has come to be known as General Relativity. Here we need only concern ourselves with the special case since our orbiter is assumed to be traveling at constant speed, that is it has constant tangential velocity.
What Einstein showed was that distances measured in the direction of travel are foreshortened or compressed, those at right angles to the direction of travel are unaffected. The extent of this foreshortening is governed by a factor called the Lorentz factor. The Lorentz factor is usually referred to as Gamma (γ) given by a simple formula and tells us the extent of foreshortening for a given speed.
Where c is the speed of light.
If we plot the value of Gamma against speed we see that for very low speeds it has a value of 1, but that it diverges rapidly to infinity as we approach the speed of light.
So for example if we are traveling at 86.6% of the speed of light, where Gamma has a value of 2, then the distance that we see from our moving perspective is half that seen by a stationary observer. So from our orbiter the earth would seem to be only 20,000km around. Of course as before it is also 40,000km and 60,000km and so on depending on how many orbits we decide to complete before we arrive back at our departure point. By choosing the right speed and number of orbits we can arrange to make the distance around the earth anything we care to choose.
So what about the distance around the earth? Just how far do I have to travel to get from Wokingham to Wokingham? Well the short answer to the question; How far is it around the earth? Has to be: How far do you want it to be?
Let’s say we want to always travel from Wokingham to Wokingham covering a distance of 400km, how many different ways can we find to achieve this?
We might achieve this by completing one orbit at a speed where Gamma has a value of 100, but we could equally well complete two orbits where Gamma equals 200 or three orbits where Gamma equals 300. Once again we can write a simple formula which describes all the possible cases:
There are in fact an infinite number of ways in which the distance around the earth can be arranged to be 400km and Equation 3 represents the complete set. Each successive strategy involves an integer multiple of the value of Gamma in the first or base strategy. We can regard these solutions as being associated with a quantisation of the value of Gamma in increments of the base value. This is despite the fact that Gamma is in all other respects a continuous variable.
Relativity not only affects the observer’s perception of the distance travelled but also the time taken to travel it. For such a moving observer time is dilated or slowed down. The extent to which it is slowed is the same factor Gamma as affects the perception of distance. For an orbiter travelling at 99.995%c, a speed where Gamma equals 100, a stationary observer would measure the time of a single orbit as being roughly 133 msecs. For the observer travelling in the orbiter the time taken to complete each orbit is slowed down by the factor Gamma, effectively divided by Gamma and so would appear to be 1.33msecs. This change in the perception of time has a knock on effect on the perception of frequency. Orbital frequency is the reciprocal of the orbital period, so the stationary observer will see the orbital frequency as 7.5Hz, whereas the moving observer will see his orbital frequency as 750Hz for the case where Gamma equals 100. In the case where Gamma equals 200 the moving observer would see the frequency as 1500Hz and so on.
Here we are dealing with speeds where the orbital velocity is very close to the speed of light and while there is a difference in the orbital period between successive choices of Gamma. The speed we need to be travelling for Gamma to equal 100 is 99.995% c, so the dynamic range is very small. To all intents and purposes it is travelling at c, which means that the orbital frequency seen by the stationary observer remains more or less constant for all of our various strategies.
This is not the case for the orbital frequency seen by the moving observer. He or she is moving at close to the speed of light where time is slowed by the factor Gamma and so sees the orbital frequency as increasing directly with the value of Gamma or the number of orbits completed in our strategy. Taken overall then, these frequencies form a harmonic series.
If we turn this on its head and look at the wavelength of the waves whose frequency this represents. When n equals 1 and with Gamma equal to 100, the distance travelled during one orbit of our orbiter is 400km. In the next state, where n equals 2 and Gamma is equal to 200, the distance it takes two orbits to complete the 400km target distance, so the distance around a single orbit, the wavelength, is 200km. With n equal to 3 the distance around a single orbit is 133.3km and so on for n equals 4 it is 100km, n equals 5 it is 80km.
We have seen that relativity affects the perception of distance travelled, of the time taken to travel that distance and, in the case of an orbtining object, it also affects the perception of orbital frequency. There is one further effect of relativity on orbiting objects which has some importance and that is its effect on the perception of angular displacement. For the stationary observer the orbiter circumnavigates the earth every 133msecs and in doing so its angular displacement is 360 degrees or 2π radians for each orbit. Hence the total angular displacement for each of the strategies we have described is 2πn radians. This is not the case for an observer on board the orbiter. For such an observer the orbital radius is at right angles to the direction of travel and so is unaffected by relativity. The strategy chosen is such that each orbit is some 400km or one hundredth of the actual circumference of the earth and so the angular displacement seen by the observer on the orbiter is 2π/100 for all of the strategies described. Putting this another way it is 2π/100n for each orbit.
Now let’s take a look at the hydrogen atom.
During the 18th and 19th century it was discovered that when shining white light through a gas the resulting spectrum contained dark lines. These were located at wavelengths which were specific to the type of gas and later formed the basis of spectroscopy. Work by a Swiss mathematician and numerologist, Balmer, led to a formula that linked six of the various wavelengths for hydrogen. Using this Balmer was able to predict a seventh spectral line, which was later found by the German physicist Fraunhofer. However Balmer’s formula did not predict all of the spectral lines of hydrogen. The Swedish physicist Johannes Rydberg was able to generalise Balmer’s formula in such a way that his new formula was able to predict all the spectral lines of hydrogen. The atom is seen as occupying one of a number of discrete energy states, that energy being carried by the orbiting electron. Transitions between a high energy state and a low energy state result in the release of energy in the form of a photon. Those from a low energy state to a high energy state are the result of energy being absorbed from an incident photon.
The Rydberg formula is most often written as:
It is important to understand that Rydberg’s formula is based on the results of experiment and observation. It does not seek to explain the spectral lines, rather it seeks to describe them and it is complete, that is it describes objectively all of the spectral lines for hydrogen. As such it is a sort of gold standard which any successful model for the hydrogen atom must satisfy in order to be valid.
The first such model was described by Niels Bohr around 1912. It simply balances the electrical force of attraction between the hydrogen nucleus and the orbiting electron with the centrifugal force tending to separate them. Bohr needed a second equation in order to solve for the two unknown quantities of orbital velocity and orbital radius. He found one in the work of a colleague, John W Nicholson. Nicholson had observed that Planck’s constant had units or dimensions which were the same as those of angular momentum and so suggested that the angular momentum of the orbiting electron was equal to Planck’s constant. He went one step further and argued that angular momentum could only ever take on values which were an integer multiple of Planck’s constant. In other words he argued that angular momentum was quantised. Armed with this assumption, Bohr was able to solve his equations in such a way that the differences in energy between the various energy levels exactly matched those of the Rydberg formula.
Job done you might think, but there was a problem, in fact there were a number of problems with the Bohr model. The most alarming was that the model required that the electron should be capable of moving from one orbit to another without ever occupying anywhere in between, the so called quantum leap. But this was not the only problem. The model failed to take account of the recently described phenomenon of relativity. It failed to explain why the orbiting electron did not emit a type of radiation called synchrotron radiation, which is characteristic of all other orbiting charges. It failed to explain a phenomenon called Zero Point Energy, which is the residual energy present in each hydrogen atom, even when it is cooled to a temperature of absolute zero where all Brownian motion has ceased. And it predicts that the size of the atom increases as the square of the energy level. Since there is no theoretical limit on the order of the highest energy level this would allow for atoms where the nucleus is in one location and the orbiting electron tens of metres if not hundreds of metres away. This change in physical size of the atom presents another problem: The hydrogen atom has the same physical and chemical properties irrespective of its energy level. It is difficult to imagine that this can be the case when these properties depend on the morphology of the atom and if that morphology can vary over such a large dynamic range.
The idea that angular momentum is quantised in units of Planck’s constant has pervaded physics ever since. It forms an integral part of later work by the French physicist Louis de Broglie in his wave/particle duality and the Austrian physicist Erwin Schrödinger in his eponymous wave equation. And yet there is considerable evidence to suggest that angular momentum cannot be quantised in this way. This in part comes about because it turns out that the spectral lines are not single lines at all, but closely spaced pairs of lines. The explanation for this is that the electron itself is spinning on its axis and that the sense in which it is spinning can be either the same as that of the electron orbit or the opposite. Hence the angular momentum of the spinning electron either adds to that of the orbiting electron or it detracts from it. This alters the energy associated with each energy level, but only by the smallest amount some ten thousand times less than that of the orbit. If angular momentum were only ever to take on values which are integer multiples of Planck’s constant then the angular momentum associated with the electron spin could only ever be equal to Planck’s constant or a multiple of it. That means that the total angular momentum could only ever be Planck’s constant plus Planck’s constant or Planck’s constant minus Planck’s constant. It clearly isn’t and the only sensible explanation is that the angular momentum associated with the spin of the electron is at least ten thousand times less than that of the electron orbit, which is supposedly equal to Planck’s constant. Hence angular momentum cannot be quantised in the way suggested since there exist entities whose angular momentum is less than Planck’s constant.
Back to Rydberg: Rather than use the somewhat obscure wave number (1/λ), the Rydberg formula can be expressed in terms of the energy emitted or absorbed when a transition takes place. This is achieved by multiplying both sides of Equation 4 by c, the velocity of light and by h, Planck’s constant. Gathering terms and substituting the analytical value for RH gives:
Where m is the rest mass of the electron and α is a constant known as the Fine Structure Constant of which Richard Feynman once said:
“It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!”.
Well we are about to find out.
The Rydberg formula tells us the amount of energy released when the electron orbiting the hydrogen nucleus makes a transition from the n1th energy state to the n2th energy state, or conversely the amount of energy absorbed if the transition is in the other direction. By letting n2 = ꝏ we obtain the energy associated with a transition to or from the maximum possible energy state and its energy in the nth energy state, that is we obtain the energy potential of the atom in the nth energy state. Doing so leads to the Rydberg Series
The Rydberg Series is particularly useful because it allows us to easily calculate the energy associated with any energy transition, simply by taking the difference between two values in the series.
ΔEn represents the difference between the energy of the electron in the nth energy state and the most energetic energy state possible, the ꝏ energy state or energy ceiling of the atom. The energy ceiling of the atom represents the maximum energy that an orbiting electron could ever possibly have. Since nothing can ever travel faster than the speed of light, the energy ceiling is limited by the speed of light to be
It is reasoned here that the electron orbiting the atomic nucleus must do so at the constant radius, that is at the same orbital radius for every energy state. Anything other than this would imply the existence of the physically impossible ‘quantum leap’, the ability to move from A to B without occupying anywhere in between. This in turn means that there can be no change in potential energy when the electron transitions from one energy state to another energy state. All changes in energy must therefore be kinetic in nature. Hence the energy of the electron in the nth energy state must be
Where vn is the orbital velocity in the nth energy state.
And it is the difference between the energy ceiling and the energy in the nth energy state that is expressed in the Rydberg series. Combining Equation 6, Equation 7 and Equation 8 to calculate the energy potential in the nth energy state gives
Equation 9 can be simplified to give
The term on the left hand side of Equation 10 will be recognised from Equation 2 as the Lorentz factor Gamma (γ) and hence
Since 1/α = 137.03 we can rewrite Equation 11 as
Where n=1,2,3,4,5…∞ We can calculate the orbital velocity in the base energy state using Equation 2 and 137.03 as the value of Gamma. This gives us an orbital velocity close to the speed of light at 99.997331% of c. This means that the dynamic range of the orbital velocity between the lowest or base energy state and the energy ceiling of the atom is extremely small.
A comparison between Equation 3 and Equation 12 is obvious and shows them to be identical in form. Indeed had we chosen to circumnavigate the earth in 291.9km instead of 400km they would have been identical. This is suggests that the mechanism that underpins the dynamics of the hydrogen atom is centred on the effects of relativity and not on the arbitrary quantisation of angular momentum as in the current standard model.
We also see that Planck’s constant takes on a special significance. If the orbital angular momentum of the electron is taken to be Planck’s constant then it is seen to be substantially unaltered between the various energy levels. It has a value equal to the product of mass, orbital radius and orbital velocity and this latter varies over an extremely small dynamic range, close to c, for all energy levels. Turning this on its heads reveals that the orbital radius is indeed constant for all energy levels and has a value of
The orbital angular momentum does not change as the energy level changes, so rather than being quantised in the way that the Standard model suggests it has the same value for all energy levels.
Louis de Broglie was first to suggest that the orbiting electron could be regarded as a wave and this lead eventually to the idea of the wave particle duality. De Broglie struggled to reconcile the wave solution, ultimately expressed in terms of the Schrödinger equation, and the particle solution, represented by the Bohr model. He recognised that there were two solutions but believed that these could be expressed as one solution in the wave domain and effectively a different solution in the particulate domain, hence the expression the wave particle duality.
Here we see that the wave characteristics of the particle derive directly from its orbital motion, the wavelength being the orbital circumference, the amplitude is the orbital diameter and the frequency is the orbital frequency. This is true both in the domain of the stationary observer and that of the orbiting electron. However the properties of these two views of the same wave are different and the difference is brought about by the effects of relativity. The stationary observer sees the orbital frequency as being more or less constant while the orbiting electron sees the frequency as being dependent on the energy level and forming a harmonic sequence.
It is a similar story with the properties of the particle. Here the orbital path length is seen by the stationary observer as being more or less constant while that of the orbiting electron is seen as being a fraction of that and dependent on the energy level.
Hence there is not so much a wave particle duality, indeed this can be more accurately described as a wave particle identity, since the relationship between wave and particle is consistent across each domain. There is however a wave duality and a separate but related particle duality.
The picture that emerges of the hydrogen atom is one in which the electron orbits at near light speed and at a constant radius, irrespective of energy level. This is consistent with the atom having the same physical and chemical properties for all energy levels.
Thus far we have obtained a possible description of the hydrogen atom, one which is far more rational than that proposed by Bohr or in the standard model. We have also provided an explanation for the mysterious Fine Structure Constant, done away with the quantum leap, restored the status of the electron to that of a discrete particle in the classical sense, explained the dual nature of the wave like behaviour of the electron and the dual nature of its particle like behaviour and explained the phenomenon of Zero point energy.
However this still falls short of what is necessary. If this model is to be deemed correct, it must further explain precisely how and why the value of Gamma is quantised in the way that we have seen. To do so it is necessary to describe the mechanism that underpins this quantisation and that is what I plan to cover in the next post. It is this lack of a mechanism to describe the quantisation of angular momentum in the Standard model that is its Achilles Heel.
TO BE CONTINUED …