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:
Equation 1 |
Where n=-∞…-5,-4,-3,-2,-1,0,1,2,3,4,5,…∞
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.
Equation 2 |
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.
Figure 1
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:
Equation 3 |
Where n=1,2,3,4,5…∞
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:
Equation 4 |
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:
Equation 5 |
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 n_{1}^{th} energy state to the n_{2}^{th} energy state, or conversely the amount of energy absorbed if the transition is in the other direction. By letting n_{2} = ꝏ we obtain the energy associated with a transition to or from the maximum possible energy state and its energy in the n^{th} energy state, that is we obtain the energy potential of the atom in the n^{th} energy state. Doing so leads to the Rydberg Series
Equation 6 |
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.
ΔE_{n} represents the difference between the energy of the electron in the n^{th} 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
Equation 7 |
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 n^{th} energy state must be
Equation 8 |
Where v_{n} is the orbital velocity in the n^{th} energy state.
And it is the difference between the energy ceiling and the energy in the n^{th} energy state that is expressed in the Rydberg series. Combining Equation 6, Equation 7 and Equation 8 to calculate the energy potential in the n^{th} energy state gives
Equation 9 |
Equation 9 can be simplified to give
Equation 10 |
The term on the left hand side of Equation 10 will be recognised from Equation 2 as the Lorentz factor Gamma (γ) and hence
Equation 11 |
Since 1/α = 137.03 we can rewrite Equation 11 as
Equation 12 |
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
Equation 13 |
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 …
]]>The quantisation of matter and of electric charge are simple concepts to grasp since they involve merely the absence or presence of a integer number of discrete particles. Particles, like grains of sand, can simply be counted to give the total amount of matter in any given volume. Electric charge is only a little more complicated since it involves the arithmetic sum of particles which can contain unit charge which can be either positive or negative.
The discrete energy levels of the hydrogen atom on the other hand are a completely different matter. Here it is the energy carried by the particle which is somehow constrained to only take on certain discrete values. There is no particle of energy which can simply be counted. Energy is a compound value involving the interactions of several variables. There must be some sort of interplay between the various quantities involved which serves to constrain the overall energy in this particular way. In the past it was deemed that this was because angular momentum was somehow quantized and can only occur in discrete chunks or quanta. However there is no particle of angular momentum and angular momentum is itself a compound value dependent on three variables. No explanation has ever been proposed or found as to how these three variables might interact with one another to produce this quantization effect.
It is not sufficient to simply declare that this or that variable is quantized without any proof or justification. Neither is it sufficient to use this declaration as the basis for justifying the discrete energy levels of the atom. What is necessary is to show that there is some sort of mechanism or process which can cause a variable to be quantized and which in turn leads to the discrete energy levels of the atom.
The Rydberg formula[i] was developed in the late 19^{th} century based on observations of the absorption spectrum of hydrogen[1] and on earlier work by Balmer. It was not based on any theoretical model, but derived empirically from observations of the emissions and absorption of the hydrogen atom. As such it can be regarded as a sort of gold standard against which any theoretical model for the hydrogen atom must be judged. 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 tells us the wavelengths of the photons emitted or absorbed by a hydrogenic atom and in particular the hydrogen atom. The formula for hydrogen is most often quoted as
Equation 1 |
Where n_{1} and n_{2} are the respective energy states for a particular energy transition and R_{H} is a constant, now known as the Rydberg constant, in this case for hydrogen.
In this form the formula tells us little of what is happening within the atom, largely because it is expressed in terms of 1/λ, the wavenumber, which has little direct physical significance. However if we multiply both sides of the formula first by c, the velocity of light, to convert the wavenumber into frequency and then by h, Planck’s constant, to turn this frequency into energy, we get an expression for the energy associated with each type of transition.
Equation 2 |
Based on his model for the hydrogen atom, Niels Bohr was able to determine an analytical expression for the value of R_{H}[ii]
Equation 3 |
Substituting this into Equation 2 and recognising that
Equation 4 |
Equation 5 |
And
Equation 6 |
We obtain the much more useful form
Equation 7 |
Where α is the Fine Structure Constant of which Richard Feynman once said[iii]
“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!”.
The Rydberg formula tells us the amount of energy released when the electron orbiting the hydrogen nucleus makes a transition from the n_{1}^{th} energy state to the n_{2}^{th} energy state, or conversely the amount of energy absorbed if the transition is in the other direction. By letting n_{2} = ꝏ we obtain the energy associated with a transition to or from the maximum possible energy state and its energy in the n^{th} energy state, that is we obtain the energy potential of the atom in the n^{th} energy state. Doing so leads to the Rydberg Series
Equation 8 |
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.
ΔE_{n }represents the difference between the energy of the electron in the n^{th} 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
Equation 9 |
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. Hence the energy of the electron in the n^{th} energy state must be
Equation 10 |
Where v_{n} is the orbital velocity in the n^{th} energy state.
Combining Equation 8, Equation 9 and Equation 10 to calculate the energy potential in the n^{th} energy state gives
Equation 11 |
Equation 11 can be simplified to give
Equation 12 |
And further simplified to give
Equation 13 |
The term on the left of Equation 13 will be recognised as the Lorentz factor Gamma (γ) and hence
Equation 14 |
Equation 13 can be solved for v in the base energy state where n=1. Doing so reveals it to have a value of 0.999973371c. This means that the dynamic range of v is very small, ranging from 99.9973371% of c to 100% of c.
The angular momentum of the orbiting electron is equal to Planck’s constant. The electron is seen to orbit at more or less constant radius given by
Equation 15 |
Because R is the same for all energy states and both m and c are constants it can be seen that the angular momentum is the same in all energy states.
From Equation 14 it is evident that, in an atom where the electron is considered to be an objectively real particle orbiting at near light speed, the variable of quantization is γ and not angular momentum as in the Bohr model and other subsequent models based on the Bohr model.
There is however one important difference to note. In these earlier models angular momentum is taken to be quantum arithmetic value, that is it can only ever take on discrete values which are an integer multiple of Planck’s constant. Here the situation is somewhat different. It is quite evident that γ is a continuous variable, ranging from unity to a theoretical upper limit of infinity. There are numerous examples of circumstances where γ has a value which is not related to α in any way whatsoever. It must therefore be the case that there is something about the dynamics of the atom that cause this otherwise continuous variable to only be capable of taking on one of a series of discrete values. In other words there has to be a mechanism or process which drives the quantization in the context of the dynamics of the hydrogen atom. Equation 14 shows that relativity has a role to play in the dynamics of the atom and it will be shown here that it is indeed instrumental in causing the atom to take on its discrete energy levels.
The year 1905 was an eventful one for Albert Einstein. In that year, he not only published his paper on the discrete nature of the photon for which he later received the Nobel Prize but he also published two further seminal works as well as submitting his PhD thesis. The most famous of these other papers concerned the dynamics of moving bodies^{[iv]}. This is the paper whose later editions contained the equation e=mc^{2}. The paper was based on a thought experiment and concerned the perception of time, distance and mass as experienced by two observers, one a stationary observer and one moving relative to the stationary observer at speeds approaching that of light.
What Einstein showed was that time elapsed more slowly for a moving observer, that distances measured in the direction of travel by a moving observer were foreshortened relative to those same distances measured by a stationary observer and that a stationary observer’s perception of the mass of a moving object was that it had increased. All three effects occur to the same extent and are governed by a factor γ (Gamma). The time between two events observed by a stationary observer as time t is seen by a moving observer as time T=t/γ. Similarly the distance between two point measured by a stationary observer as distance d is seen by a moving observer as distance D=d/γ. And as far as a stationary observer is concerned the mass of the moving object is seen to increase by this same factor γ.
Gamma is referred to as the Lorentz factor and is given by the formula
Equation 16 |
Both observers agree on their relative velocity, but go about calculating it in different ways. For the stationary observer the velocity of the moving observer is the distance travelled divided by the time taken as measured in his stationary domain. For the stationary observer the velocity is
Equation 17 |
For the moving observer the distance as measured in his or her own domain is foreshortened by the factor Gamma, but the time taken to cover that distance reduced by the same factor Gamma hence
Equation 18 |
A slightly different way to view the effect of relativity on distance, rather than imagine that the distance between points changes, is to imagine instead that the scale on which distance is measured changes. It is as if the measurements are made with a tape measure made of elastic, the faster one travels; the more the elastic tape measure is stretched and the further apart the scale markings appear, but only when making measurements in the direction of travel.
There is a great deal of experimental evidence to support Einstein’s Special Theory. One of the more convincing experiments was carried out at CERN in 1977 and involved measuring the lifetimes of particles called muons in an apparatus called the muon storage ring^{[v]}. The muon is an atomic particle which carries an electric charge, much like an electron, only more massive. It has a short lifetime of around 2.2 microseconds before it decays into an electron and two neutrinos.
In the experiment muons are injected into a 14m diameter ring at a speed of 99.94% of the speed of light. At this speed Gamma has a value of 29.33. The muons, which should normally live for 2.2 microseconds, were seen to have an average lifetime of 64.5 microseconds; that is the lifetime of the muon was extended by the factor Gamma. This comes about because the processes which take place inside the muon and which eventually lead to its decay are taking place in an environment which is moving relative to us at 99.94% of the speed of light and in which time, relative to us, is running 29.33 times slower. Hence the muon, in its own domain, still has a lifetime of 2.2 microseconds, it’s just that to us, who are not moving, this appears as 64.5 microseconds.
Travelling at almost the speed of light a muon would normally be expected to cover a distance of 660 metres or roughly 7.5 times around the CERN ring during its 2.2 microsecond lifetime, but in fact the muons travelled almost 20,000 metres or 220 times around the ring. This is because distance in the domain of the muon is compressed so what we stationary observers see as being 20,000 metres the muon sees as being just 660 metres.
The muon ring experiment demonstrates two further important characteristics associated with orbiting objects traveling at near light speed.
The orbital radius is measured at right angles to the direction of travel of the muon and is therefore unaffected by relativity. This means that the angular displacement perceived by the muon must differ from that perceived by the stationary observer. For the stationary observer the muon travels a total of 20,000 m at a radius of 7 m, a total angle of 2857 radians. For the muon however the distance traveled is only 660 m but the radius is still 7 m and so the muon’s perception of the angular displacement is 94 just radians.
Both parties agree that during its lifetime the muon completes some 220 turns around the ring. We stationary observers see this as having taken place in some 64.5 microseconds, corresponding to an orbital frequency of 3.41MHz, while the muon sees these 220 turns as having been completed in just 2.2 microseconds corresponding to an orbital frequency of 100MHz. Hence for the muon orbital frequency is multiplied by a factor γ relative to that seen by a stationary observer.
The idea that the discrete energy levels of the hydrogen atom are associated with a harmonic series was first proposed by the French physicist Louis de Broglie. He suggested that the electron had an associated wavelength that was equal to Planck’s constant divided by its linear momentum, effectively this is a restatement of Bohr’s earlier assumption that angular momentum is quantized.
Here we see a different situation, Equation 14 tells us that each energy state is associated with a value of γ that is an integer multiple of that of the base energy state and as we have seen, orbital frequency for a moving object is multiplied by γ relative to that seen by a stationary observer. So while we stationary observers see the orbital frequency as being more or less constant, the orbiting electron sees the orbital frequency as being one of a series of frequencies which form a harmonic series.
For the stationary observer the orbital frequency is approximately
Equation 19 |
For the moving electron however the orbital frequency is seen as
Equation 20 |
Forming a harmonic series in which the orbital frequency in the n^{th} energy state is the n^{th} multiple of that of the base energy state.
Wherever we see a harmonic series in nature there must always be a corresponding sampling process. This becomes evident if we consider the Fourier representation of a harmonic series. Such a Fourier representation comprises a series of spikes equally spaced along the frequency axis. For a real function these are disposed equally on both the positive and negative frequency axes. These spikes are referred to as Dirac functions and such a collection of equally spaced Dirac functions is referred to as a Dirac comb.
The inverse Fourier transform of a Dirac comb in the frequency domain is itself another Dirac comb in the time domain[vi]. Such a Dirac comb in the time domain can be regarded as a sampling function, since if it is multiplied by any other signal it effectively takes a sample at regular intervals in time. All of this points to the idea that somewhere within the dynamics of the atom we can expect to find a sampling process. It means that there is something within the atom that happens or can happen only once per orbit of the orbiting electron.
Sampling Theory is the branch of mathematics which deals with continuous variables and discrete solutions. It was developed in the 1930’s and 1940’s at Bell labs in order to deal with capacity problems on the telephone network.
At that time telecommunications engineers were concerned to increase the capacity of the telephone network. One of the ideas that surfaced was called Time Division Multiplexing. In this each of a number of incoming telephone lines is sampled by means of a switch, the resulting samples are sent over a trunk line and are decoded by a similar switch at the receiving end before being sent on their way. This allowed the trunk line to carry more telephone traffic without the expense of increasing the number of cables or individual lines. The question facing the engineers at the time was to determine the minimum frequency at which the incoming lines needed to be sampled in order that the telephone signal can be correctly reconstructed at the receiving end.
The solution to this problem was arrived at independently by a number of investigators, but is now largely credited to two engineers. The so called Nyquist-Shannon sampling theorem is named after Harry Nyquist[vii] and Claude Shannon[viii] who were both working at Bell Labs at the time. The theorem states that in order to reproduce a signal with no loss of information, then the sampling frequency must be at least twice the highest frequency of interest in the signal itself. The theorem forms the basis of modern information theory and its range of applications extends well beyond transmission of analog telephone calls, it underpins much of the digital revolution that has taken place in recent years.
What concerned Shannon and Nyquist was to sample a signal and then to be able to reproduce that signal at some remote location without any distortion, but a corollary to their work is to ask what happens if the frequency of interest extends beyond this Shannon limit? In this condition, sometimes called ‘under sampling’, there are frequency components in the sampled signal that extend beyond the Shannon limit and maybe even beyond the sampling frequency itself.
The following example serves to illustrate the phenomenon. Suppose there is a cannon on top of a hill, some distance away an observer is equipped with a stopwatch. The job of the observer is to calculate the distance from his current location to the cannon. Sound travels in air at roughly 340 m/s. So it is simply a matter of the observer looking for the flash as the cannon fires and timing the interval until he hears the bang. Multiplying the result by 340 gives the distance D to the cannon in metres.
This is fine if the cannon just fires a single shot, but suppose the cannon is rigged to fire at regular intervals, T seconds apart. For the sake of argument and to simplify things, we can make T equal to 1. If the observer knows he is less than 340 m from the cannon there is no problem. He makes the measurement and calculates the distance D as before. If on the other hand he is free to move anywhere with no restriction placed on his distance to the cannon then there is a problem. There is no way that the observer can know which bang is associated with which flash, so he might be located at any one of a number of different discrete distances from the cannon. Not just any old distance will do however. The observer must be at a distance of D or D + 340 or D + 680 and so on, in general D + 340n. The distance calculated as a result of measuring the time interval between bang and flash is ambiguous. In fact there are an infinite number of discrete distances which could be the result of any particular measured value. This phenomenon is known as aliasing. The term comes about because each possible distance is an alias for the measured distance.
Restricting the observer to be within 340 m of the cannon is simply a way of imposing Shannon’s sampling limit and by removing this restriction we open up the possibility of ambiguity in determining the position of the observer due to aliasing.
Turning the problem around slightly; instead of measuring the distance to the cannon the position of the observer is fixed. Once again, to make things simpler, we can choose a distance of 340m. This time however we are able to adjust the rate of fire of the cannon until the observer hears the bang and sees the flash as occurring simultaneously. If the rate of fire is one shot per second then the time taken for the slower bang to reach the observer exactly matches the interval between shots and so the two events, the bang and the flash are seen as being synchronous. Notice that the bang relates, not to the current flash, but to the previous flash.
If the rate of fire is increased then at first, for a small increment, the bang and the flash are no longer in sync. However they come back into sync again when the rate of fire is exactly two shots per second, and again when the rate is three shots per second. If we had a fast enough machine gun this sequence would extend to infinity but only for a rate of fire which is an integer number of shots per second. Notice that now the bang no longer relates to the previous flash, but to a previous flash. The fact that there are intermediate bangs and flashes is irrelevant. If we look at any arbitrary flash then there will be a synchronous bang provided the rate of fire is an integer number of shots per second.
It is interesting to note also that if the rate of fire is reduced from once per second then the observer will never hear and see the bang and the flash in sync with one another and so once per second represents the minimum rate of fire which will lead to a synchronous bang and flash. In fact what we have is a system that has as its solutions a base frequency and an infinite set of harmonic frequencies.
Here is a system which can cause a variable, in this case the rate of fire of the gun, to take on a series of discrete values even though, in theory at least, the rate of fire can vary continuously. Equally important is that if the system is capable of syncing to the lowest such frequency then all the multiples of this frequency are also solutions, in other words if the base frequency is a solution then so are harmonics of the base frequency. It is suggested here that this is precisely the type of mechanism that occurs inside the atom and leads to its discrete energy levels.
We saw in Equation 18 that velocity is generally taken to be invariant with respect to relativity. Indeed this idea is axiomatic in the derivation of special relativity. Hence for the moving observer both the distance and the time are scaled by the same factor γ relative to those seen by the stationary observer and these cancel such that the velocity is the same for both observers.
In order to measure the speed of an object moving at close to the speed of light in real time it is necessary for a stationary observer to use two clocks, at least conceptually. One clock must be set up at the point of departure and another at the point of arrival. The two clocks must then be synchronized before the measurement can begin. The time that the moving object leaves the point of departure is noted on the departure clock and the time of its arrival is noted on the arrival clock. At least one of these measurements must then be transmitted to the other location before the difference can be taken and the speed calculated. Any attempt to measure such a velocity in real time is thwarted by the fact that the clock would have to move with the moving object and so would itself be slowed down due to the effects of relativity.
There is however one circumstance where this is not the case and that is when the moving object is in orbit. Under this circumstance the object returns to its point of origin once per orbit and so it is possible conceptually at least, to measure its orbital velocity in real time using a single clock. Such measurement is only possible when the object returns to its point of departure that is once per complete orbit. This then is the sampling process of which I spoke earlier. The orbital velocity of the electron is such that it can only be determined once per orbit.
It is thus possible to define a velocity term which couples the two domains, that of the stationary observer and that of the moving electron. Such a velocity is calculated as the distance as measured by the moving object divided by the time as measured by the stationary observer, this latter can only meaningfully be measured for one or more complete orbits. For obvious reasons I have called this type of velocity Relativistic Velocity as opposed to the Actual Velocity and propose that it is this Relativistic Velocity that applies to phenomena associated with objects in orbit, specifically to centrifugal and centripetal force and acceleration and to angular momentum. Relativistic Velocity has the important characteristic that it gets smaller as the actual velocity approaches the speed of light.
Equation 21 |
Returning to Equation 15 which gives us the radius of the atom (reproduced here)
Equation 15 |
In this form the equation fails to take account of relativity. For a stationary observer located at the atomic nucleus electron is seen to be travelling at near light speed and so the mass term should be multiplied by γ. However it is argued here that the velocity term should be considered to be affected by relativity meaning that this should use the term for Relativistic Velocity, which would then mean that it should be divided by γ and hence
Equation 22 |
Although in strict mathematical terms the two γ terms could cancel to return to Equation 15, it is important to note that there is a subtle difference between these two equations. In Equation 22 the value of the radius is actively driven to have the value ħ/mc and so it is perhaps more meaningful to state that R is identically equal to ħ/mc, rather than simply being equal to it.
Equation 23 |
This provides an explanation as to why the orbiting electron does not decay due to the emission of synchrotron radiation. Rather than being driven in any conventional manner to adopt a circular orbit, here the atom is constrained by the combined effects of relativity and Planck’s constant to always have a constant value. It is as if the electron is orbiting on a hard surface, one which it cannot penetrate and from which it cannot depart. This is more akin to the way in which we view general relativity, where objects move in straight lines on a curved space.
For the hydrogen atom to be stable it is necessary that the forces acting on the electron be in balance. The electrical force tending to pull the electron towards the nucleus must balance the centrifugal force tending to throw it off.
The electron is orbiting at near light speed where the effects of relativity must be taken into account. The mass term is affected by relativity and once again it is argued here that the velocity term should be based on the relativistic velocity and so for the base energy state
Equation 24 |
This can be combined with Equation 23 and simplified to give
Equation 25 |
The term on the LHS of Equation 25 is recognised as α, the Somerfield fine structure constant and so in the base energy state
Equation 26 |
The forces are in balance because the Relativistic Velocity term causes the centrifugal force to decrease as the Actual Velocity increases eventually reaching the point where it exactly matches the attractive electrical force. At this point γ has a value of approximately 137 and the orbital path length has been reduced by a factor γ=1/α to 2πRα =1.77076*10^{-14} while the orbital period is 2πR/c = 8.093*10^{-21} and the Relativistic Velocity is cα .
Figure 1 Graph of force vs Gamma
The situation is shown in Figure 1 which shows the centrifugal force derived from the relativistic velocity plotted against Gamma. It also shows the electrostatic force acting between the electron and the proton (in red) which is independent of Gamma and therefore constant. The two curves intersect when Gamma is equal to 1/α that is when it equals approximately 137.03.
Using the analogy of the elastic tape measure and taking a tape measure of natural length 2πRα the tape measure has been stretched as the Actual Velocity increases. Stability is achieved when this tape measure is stretched sufficiently to encircle the Actual orbit of the electron exactly once. See figure 2.
Figure 2 Orbital path length for a stable orbit in the base state
Here also we see the true nature of the Somerfield Fine Structure Constant. It is seen as the extent to which the orbital path length must be foreshortened due to the effects of relativity in order to produce a stable atom. Conversely it can be seen as the extent to which our elastic tape measure must be stretched under relativity to describe one Actual orbit.
It is important to note however that while 2πRα is the distance as measured around the orbital circumference by the electron, it is not the only possible distance. Every integer multiple of this distance is an alias for this distance, so the distance could be interpreted as 4πRα or as 6πRα or in general as 2πRαn where n = 1,2,3,4... As far as the electron is concerned all of these possible distances are indistinguishable from one another and any one of these distances could be the distance travelled since it was overhead its point of departure on the orbital circumference. In the base state however none of these other aliased distances correspond to the atom being in a stable state, in much the same way as the point at which the cannon described earlier comes first into sync at 1 shot per second.
Figure 3 Centrifugal force showing aliased values
Figure 3 shows how each of these aliases for the centrifugal force varies with Gamma, that of the base energy state is shown highlighted.
For an atom in the base energy state a small increase in orbital velocity the forces are no longer in balance and the atom would be unstable. They next come into balance when γ=2/α or approximately 274. At this Actual velocity the Actual orbital period remains substantially unaltered at 2πR/c. The relativistic path length as seen by the electron is however halved over that of the base state, although again using the analogy of the elastic tape measure which is now stretched by a factor of 274 the electron is has completed exactly one orbit during this period. When the Actual orbital velocity is such that distance travelled by the electron is 4πR/γ = 2πRα which in turn means that the centrifugal force exactly matches the electrical force. The situation is shown in Figure 4 where the second alias for the centrifugal force is this time shown highlighted.
Figure 4 Graph of centrifugal force based on aliased velocities
This situation repeats again in an exactly similar manner when n=3 only this time the alias that results in a stable atom is the one corresponding to three orbits around the nucleus. The situation repeats for every integer value of n.
In effect therefore the distance around the orbital path in the n^{th} energy state is the n^{th} alias of the distance around a single orbit as perceived by the electron or 2πRαn/n for integer n. The period in the domain of the observer is always roughly the same at 2πR/c and therefore the relativistic velocity is always the same cαn/n for integer n.
In general therefore each successive stable state occurs as γ equals an integer multiple of 1/α, so
Equation 27 |
Just as with the cannon, if the base frequency is a solution, then so are all the harmonics a solution. And just as with the cannon where multiple bangs and flashes occurring between the ones of interest here we see that the electron perceives itself as having completed more than one orbit in order to achieve stability which leads directly to the idea of frequency multiplication and the stable states of the atom corresponding to a harmonic series.
From this we can calculate the actual orbital velocity in the nth energy state as
Equation 28 |
And from this we can calculate the various energy levels and their differences, which exactly match those of the Rydberg Formula.
The energy levels for the hydrogen atom predicted by the model exactly match those of the Rydberg formula. The electron orbits at a constant radius and at a velocity very close to the speed of light hence changes in energy level are accomplished by a change in orbital velocity with no change in orbital radius.
The model for the hydrogen atom described here effectively unifies classical mechanics with quantum mechanics. It does so by showing the mechanism which causes the electron orbiting the hydrogen nucleus to do so only at certain very particular velocities, each associated with its respective energy level. The model is based on a single postulate, that certain orbital velocity terms should be considered as themselves being affected by relativity. In doing so the model takes full account of the effects of relativity on the various components that make up the atom. Indeed relativity is seen as being at the very heart of the model, not just an adjunct to be added later.
The discrete energy levels of the atom are associated with a series of harmonic frequencies which are experienced by the moving electron, but are not directly experienced by external stationary observers. These harmonics are in turn associated with the quantization of the variable Gamma, which is constrained to only take on certain values each of which is an integer multiple of the reciprocal of the Fine Structure Constant. It is important to understand that Gamma is not itself inherently quantized. The model shows that Gamma is quantized only in the context of the dynamics of the atom and that this comes about because orbital velocity as it affects centrifugal and centripetal force and acceleration and angular momentum is itself affected by relativity causing the effective velocity to reduce by the factor Gamma. In other contexts Gamma is free to take on any value over the dynamic range of 1 to infinity.
The electron is seen as a particle in the classical sense, a point particle of almost infinitesimal size and having deterministic position and velocity. This is not to say that the uncertainty principle does not exist, but rather that uncertainty is not an inherent property of the particle. It is instead a practical difficulty of measurement which occurs when the object being measured and the tools used to measure it are of the same order of magnitude – the so called Observer Effect.
The electron orbits at a constant radius irrespective of energy level. It should be noted that this is a necessary condition for the electron to be considered an objectively real particle, since anything else implies the existence of the physically unrealizable quantum leap or its latter day equivalents. Changes in energy level are then associated with changes in orbital velocity with no change in orbital radius. There is therefore no change in potential energy with change in energy level, merely a change in kinetic energy. Hence the morphology of the atom does not vary with energy level and so it is evident that such an atom would have the same physical and chemical properties irrespective of energy state which is what we observe in practice.
The constant orbital radius of the electron is driven by the combined effects of relativity on both the mass of the electron and its orbital velocity. Rather than simply cancelling one another out, these effectively constrain the orbital radius to have a constant value and it is this that explains why the orbiting electron does not emit synchrotron radiation.
The electron has wave like characteristics which derive directly from its orbital motion. We stationary observers, viewing the atom from an external viewpoint, see the frequency of this wavelike motion as being more or less constant. Viewed from the electron’s point of view however, where time is slowed due to the effects of relativity, the orbital frequency of each successive energy state is an integer multiple of that seen by the electron in the base energy state forming a harmonic series with successive harmonics each being associated with a discrete energy level.
Louis de Broglie was the first to propose that the electron has a dual nature. In it the electron, is seen as being both a particle and a wave at the same time. De Broglie struggled to validate this idea and spent much of the last 40 years of his life trying and failing to do so. In de Broglie’s duality the nature of the particle is seen as being split between that of a wave on the one hand and that of a particle on the other. De Broglie identified the wavelength of the particle with Planck’s constant divided by its linear momentum and in doing so devised a set of wavelike properties for the electron which are not capable of physical realization and in fact amount to little more than a euphemism for the equally unrealizable quantum leap of the earlier Bohr model.
In developing his ideas about the wave particle duality, De Broglie made two key observations. First he proposed that the discrete energy levels of the atom were in some way associated with a harmonic sequence and secondly his proposed waves were at a frequency higher than that of the orbiting electron, implying that some sort of frequency multiplication process is taking place within the atom.
Here we find that both of these conditions are met but not quite in the way that de Broglie envisaged. An object orbiting at near light speed experiences time as passing at a slower rate than does a stationary observer. However the number of orbits in any given period is the same for both observers meaning that the moving observer sees the orbital frequency as being higher than does the stationary observer, the same number of cycles having occurred in a shorter period for the moving observer. Hence the moving observer sees the orbital frequency as having been multiplied by γ.
De Broglie understood that the dynamics of the hydrogen atom required some sort of dual solution and chose to identify these separately with the wavelike properties of the electron and its particle like properties. For de Broglie therefore the duality existed between the wave and the particle.
Here the situation is somewhat different. The electron still has wavelike properties but these derive directly from its orbital motion as an objectively real discrete particle in the classical sense having both deterministic position and deterministic velocity in much the same way as any object in orbit will display wavelike properties of wavelength, amplitude frequency and phase to an external observer. The relationship between the wave like properties of the particle and its orbital motion is unique in exactly the same way as it is on any other scale.
The duality applies separately, but not independently, to both the particle like properties and the wavelike properties. It stems from the fact that relativity means that both time and distance as far as the electron is concerned each have two different values depending on the perspective of the observer and on the velocity of the electron. For the particle the length of the orbital path seen by an external, stationary observer is Gamma times longer than that seen by the moving electron. Hence the orbital path length is considered to be 2πR by an external observer, but is perceived as being 2πR/γ when viewed from the perspective of the moving electron. This then is the dual nature of the particle.
Similarly for the wavelike characteristics, the orbital frequency is seen as having one value as far as an external stationary observer is concerned, but having Gamma times this value when viewed from the perspective of the moving electron. This is because time for the moving electron is slowed by the factor Gamma, but the number of orbits remains the same for both observers and hence the same number of orbits is completed in a shorter interval for the moving electron than for the stationary observer. This is the dual nature of the wavelike properties of the electron.
It is therefore not the case that the electron is either a particle or a wave. It always has both particle like and wave like properties, the former because it is a discrete point particle in the classical sense and the latter because it is following a circular orbit which subtends a wave to any external observer. It is the properties of both the particle and of the wave that each display a dual nature and that these are brought about by the effects of relativity. For the stationary observer where v≈c the frequency . For the moving electron frequency is multiplied by Gamma due to the effects of relativity and so ω_{e}=cn/Rα. For the stationary observer the orbital velocity of the electron is close to the speed of light and substantially constant for all energy levels, while for the moving electron the Relativistic Velocity is cα/n.
It is therefore appropriate to describe the wave/particle relationship not as the wave particle duality but as the wave particle identity, and to describe the particle as having a dual nature and to describe the wave as having a dual nature.
The model provides a simple physical interpretation of the physical nature of the Somerfield Fine Structure Constant. This constant is a pure number and therefore must be derived from the ratio of two quantities with similar Dimensions or units. Here it is seen as the ratio of the orbital velocity as experienced by the stationary observer to that experienced by the moving electron observer in relation to its orbital or Relativistic velocity. Since these share the same orbital period it can also be seen as the ratio of the distance around the orbital circumference foreshortened due to relativity to the actual distance around the orbital circumference as seen by a stationary observer.
Finally the model extends the laws of physics down to the scale of the atom and most likely beyond. It does however demand a subtle change to those laws which would apply equally on any scale, notably that certain orbital velocity terms should be taken as being affected by relativity.
[1] In fact the Rydberg formula works for any so called hydrogenic atom, that is an atom which has been ionized to the extent that it has only one orbiting electron. The value of R is then specific to each type of atom.
[i] Heilbron, J. L. (1981), Historical Studies in the Theory of Atomic Structure, Arno Press
[ii] Niels Bohr (1913). On the Constitution of Atoms and Molecules, Part I. Philosophical Magazine 26: 1–24.
[iii] Feynman, Richard P., (1985) QED: The Strange Theory of Light and Matter, Princeton University Press, p. 129.
[iv] Einstein, Albert (1905), Zur Elektrodynamik bewegter Körper (On the Electrodynamics of Moving Bodies), Annalen der Physik 17 (10): 891–921,
[v] Bailey, H.; Borer, K.; Combley F.; Drumm H.; Krienen F.; Lange F.; Picasso E.; Ruden W. von; Farley F. J. M. ; Field J. H.; Flegel W. & Hattersley P. M. (1977). Measurements of relativistic time dilatation for positive and negative muons in a circular orbit. Nature 268 (5618): 301–305
[vi] Bracewell, R.N. (1986), The Fourier Transform and Its Applications (revised ed.), McGraw-Hill; 1st ed. 1965, 2nd ed. 1978
[vii] H. Nyquist, Certain Topics in Telegraph Transmission Theory, Trans. AIEE, vol 47, pp 617-644, Apr 1928
[viii] C. E Shannon, Communication in the Presence of Noise, Proc. Institute of Radio Engineers, vol 37 no. 1 pp 10-21, Jan 1949.
]]>At this point the cognoscenti will sigh and condescendingly explain that the universe does not have a centre, that it is expanding everywhere in all directions from everywhere. They will typically explain that the universe is not expanding into space but that space itself is expanding. If pressed this usually leads to an explanation expressed in terms of an analogy based on a balloon.
Picture a balloon, they will say, and imagine that its surface is covered in dots. The dots represent the stars and galaxies and the surface of the balloon represents a two dimensional analog of a three dimensional space. As the balloon is inflated its surface stretches in all directions and the dots move away from one another. If we consider just the surface of the balloon, there is no centre; all of the dots move way from one another.
Of course the balloon does have a centre, but that only exists if we add a third dimension and consider the balloon in a three dimensional space. Here we are constrained to exist only on the surface of the balloon.
At this point most laymen will give up and concede defeat, accepting that science has found the answer, that the universe must be expanding and desperately trying to comprehend what a balloon would look like in a four dimensional space.
There is however a fundamental problem with the balloon analogy and by implication therefore with the theory that it supports. The problem lies in the fact that the balloon’s surface is curved and not completely flat. The surface of the balloon has what is known as positive curvature.
In a positively curved space the angles of a triangle add up to more than 180° whereas in negatively curved space they add up to less than 180° and in a flat space they add up to exactly 180°. We can see this if we consider a traveler setting off from the North Pole and heading due south along the Greenwich meridian. When our traveler reaches the equator he turns right through 90° heading west. Somewhere off the coast of Ecuador at 90°W he turns right again through 90° and heads north, back to the pole, where he arrives at 90° to his original heading. Our intrepid traveler has completed a triangular course, but in doing so has turned through 270° and not the 180° we would expect if the earth was flat.
The balloon analogy only works because, no matter how big the balloon gets, and how close it approximates a flat space, its surface has positive curvature. The problem is that as far as we can tell the universe is flat. It shows none of the characteristics of a curved space. The angles of triangles all add up to 180°.
We can visualise how this affects the balloon analogy by imagining the balloon pressed against a flat sheet of glass. The region where the balloon’s surface comes into contact with the piece of glass is flat. As before if the balloon is inflated the dots in that region all move apart, but there is one important difference. This flat region of the balloon has a boundary. Indeed it has to have a boundary, the line where the balloon membrane leaves the surface of the glass. Such a boundary has to be a closed contour and therefore it has to have what can be regarded as a centre, that is a centroid or a centre of gravity.
It is simply not possible to create a flat membrane in a flat space that does not have a boundary. Exactly the same holds true in a three dimensional space, if space is flat, that is if the angles of triangles add up to 180°, then if it is expanding it must have a boundary. In this case the boundary is a bounding surface, not a contour. Such a bounded region must also have a centre of some description.
So the balloon analogy is invalid and it is perfectly reasonable to ask the question where the centre in an expanding universe is. If the universe is expanding and has no centre then space viewed on a large scale has to be curved, which it is not. If space is flat and expanding it has to have a centre, which it does not. There is of course a third alternative and that is that the universe is not expanding at all. That it is static and infinite. But I will deal with that in a later post.
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Rydberg’s formula is the more general case of a formula developed by Balmer to describe the emission spectra of hydrogen. Where Balmer only dealt with one set of spectral lines, Rydberg dealt with all the emission lines of the hydrogen atom.
Balmer’s formula dealt with the wavelength of the emitted light, whereas Rydberg found it more convenient to deal with the reciprocal of the wavelength or wavenumber and developed the formula:
Equation 1 |
R_{Ꝏ }is known as the Rydberg constant. Rydberg originally used a value based on Balmer’s formula, which was derived empirically, but Niels Bohr was able to calculate a value analytically for this constant based on his model for the hydrogen atom.
Equation 2 |
Bohr had assumed that the proton around which the electron orbits was much more massive than the electron itself. In such a case the electron orbits around the centre of gravity of the proton. In fact the electron orbits around the combined centre of gravity of the proton and the electron, which is somewhat closer to the electron than the centre of gravity of the proton alone. In 1917 Arnold Somerfield provided a correction called the reduced mass:
Equation 3 |
Where m_{p} is the rest mass of the proton and m_{e} is the rest mass of the electron and m_{e}/m_{p}=1/1847.
The use of wavenumber in Rydberg’s formula provides very little insight into what is actually going on and so it is useful to express the formula in terms of frequency or energy multiplying both sides by either by c for frequency or by 2πħc for energy. Expressing the Rydberg formula in terms of energy is particularly useful:
Equation 4 |
If we ignore the slight error due to the effect of the reduced mass then:
Equation 5 |
Recognizing that K, the Coulomb constant or electrostatic force constant, is given by:
Equation 6 |
That
Equation 7 |
And that Alpha, the fine structure constant is given by:
Equation 8 |
Equation 5 can be simplified to:
Equation 9 |
This in itself is interesting because ½mc^{2} is the kinetic energy that an electron would have if it were travelling at the speed of light and cα is the Bohr velocity, the velocity at which the electron supposedly travels in the Bohr model.
By setting n_{2} = ∞ and allowing n_{1} to take on successive integer values we obtain the difference between the energy level in any particular energy state and the maximum possible energy that the atom can contain. This forms a series of values called the Rydberg series.
Equation 10 |
The Rydberg series is useful because it allows us to calculate the energy, frequency or wavelength associated with any energy transition within the atom.
By setting n =1 in the Rydberg series we obtain the maximum possible energy that the hydrogen atom can absorb or emit, this is called energy potential [1]of the atom, which for the base energy state is
Equation 11 |
All of this is consistent with the Bohr model, however the Bohr model is fraught with difficulties which are also present in other later models. Most notable in the Bohr model is the quantum leap; the need for the electron to move instantly from one place to another without occupying anywhere in between. Other, later models seek to avoid this by asserting that the state of the electron is inherently uncertain, until it is subject to some sort of observing process, at which time its wave function (whatever that is) collapses (however that works) to reveal either its position or velocity. Of course the collapsing wave function is merely a euphemism for the quantum leap and is equally unrealizable and unrealistic.
Since nothing can travel faster than light, the maximum kinetic energy that the orbiting electron in the hydrogen atom could ever possibly achieve is given by:
Equation 12 |
If we are to avoid the problem of the quantum leap, then the electron must orbit at the same orbital radius in all energy states, since anything else implies that the quantum leap exists. If the orbital radius remains constant for all energy states then there can be no change in potential energy between energy states and all of the change in energy must therefore be kinetic in nature. In which case the energy in the base state must be given by
Equation 13 |
Where v is the orbital velocity in the base energy state.
But the difference between these two values (Equation 12 and Equation 13) is the energy potential of the atom and so we can write:
Equation 14 |
Simplifying and rearranging this gives:
Equation 15 |
Readers will be familiar with the LHS of this equation as the Lorentz factor Gamma (γ) associated with the effects of special relativity. Hence
Equation 16 |
And from this we can calculate the value of v as 0.999973371c.
What this reveals is an atom in which the electron in the base energy state is orbiting at 99.9973371%c and in the maximum possible energy state is orbiting at velocity c. There is no substantial change in orbital radius across this dynamic range and hence no need to introduce the idea of the quantum leap or its latter day equivalents – collapsing wave functions. The angular momentum too remains substantially the same for all energy levels and if we equate that to Planck’s constant we find that the orbital radius is
Equation 17 |
We stationary observers then see the orbital frequency as
Equation 18 |
However the orbiting electron is in an environment where time is slowed down due to the effects fo relativity by a factor Gamma and therefore where frequency is multiplied by the same factor Gamma and hence, as far as the orbiting electron is concerned, its orbital frequency is
Equation 19 |
We can repeat this exercise for the values in the Rydberg series and not just the base energy state, in which case we will obtain the orbital velocities and corresponding orbital frequencies for each respective energy level.
Equation 20 |
The orbital frequency seen by us stationary observers remains substantially unaltered as in Equation 18, however that for the orbiting electron is different for each energy level and is given by
Equation 21 |
From Equation 21 it can be seen that the frequencies experienced by the orbiting electron form a harmonic series, starting with a base frequency experienced in the base energy state an rising in integer multiples of Gamma with each succeeding energy state. We stationary observers, on the other hand, always see the orbital frequency as being substantially constant. This shows that at the heart of the discrete energy levels of the atom lies a harmonic series, much as de Broglie suggested, only rather than appearing directly in our observing domain, it appears instead in the domain of the moving electron.
Equation 20 shows us that the variable of quantisation within the atom is not angular momentum as postulated by Niels Bohr and incorporated into subsequent models, but instead is Gamma, the Lorentz factor. In the base energy state Gamma has a value of 137.03, in the second energy state this rises to 2*137.03 and so on. It is important to note that Gamma is not inherently quantised, that is it can vary continuously, rather that the dynamics of the atom are such that the atom is only stable when it takes on values which are an integer multiple of 1/α.
The model proposed here therefore gives a physical significance to α, the Fine Structure Constant. In the past the precise nature of what this constant represents has been missing. This has led to some wild speculation as to its true nature and significance. Here it is simply seen as the ratio of the orbital circumference as traversed at relativistic speeds and so foreshortened, to that traversed at non relativistic speeds.
The stability of such an atom depends on their being a force balance between the orbiting electron and the nuclear proton. It must be stable in each of the various energy states.
We also know that wherever we see a harmonic series in the frequency domain there must be a corresponding sampling process in the time domain. This comes about because a harmonic sequence in the Fourier space appears as a series of impulses or spikes, equally spaced along the frequency axis, forming what is known as a Dirac comb. The inverse Fourier transform of such a Dirac comb in the time domain is itself a Dirac comb formed of a series of impulses or spikes, only this time in the time domain and spaced T seconds apart where T=1/F and where F is the fundamental frequency of the harmonic series. Such a Dirac comb in the time domain represents a sampling signal and so we can expect to find some sort of sampling process taking place in the time domain when we examine the electro dynamics of the atom.
Such an analysis is beyond the scope of this post, however if you want to understand the role of the sampling process and the internal dynamics of the hydrogen atom then this is covered in much more detail in Sampling the Hydrogen Atom.
[1] Note that energy potential should not be confused with potential energy. Energy potential is the difference between the energy contained within the atom and the maximum possible energy that the atom can absorb, while potential energy is energy associated with the difference in orbital radius between a particular energy state and the base energy state in the presence of an electric field and is analogous to gravitational potential energy associated with the height of an object above some datum level.
]]>In 1905 Einstein had shown that the hitherto wavelike nature of light concealed an underlying particle-like nature. In 1923 de Broglie was struck with the idea that maybe this situation could be reversed, that perhaps a particle could be described in terms of wave. De Broglie discovered that if he assigned a wavelength and a frequency to an electron he could explain the location of the atoms in the Bohr model of the atom. He found that the orbiting electrons could only occupy orbits which contained a whole number of such wavelengths.
De Broglie’s idea hinges around the notion of standing waves. A standing wave occurs for example in a taught string, anchored at both ends, that is plucked. The fundamental frequency occurs when the whole length of the string vibrates. Other modes of vibration are also possible, for example where the centre of the string remains stationary and the two halves of the string each vibrate at what is referred to as a second harmonic frequency. This can also happen at the third, fourth and other higher harmonics. Here then was a possible explanation as to why the electron orbits could only take on whole number multiples of a base value.
Figure 1
De Broglie supposed that the electron had a natural frequency which existed as standing wave and which described the orbit of the electron in its base energy state. The standing wave was at a frequency which equated to the orbital frequency of the electron and the wavelength was equal to the orbital path length.
At higher energy levels he found that the standing wave was at a frequency which was an integer multiple of the electron’s orbital frequency and that it had a wavelength which was an integer fraction of the orbital path length. He could therefore describe the orbit in terms of a fundamental frequency in the base state and a series of harmonic frequencies in the higher energy states.
De Broglie developed a formula relating the wavelength of the electron to its mass and velocity.
Equation 1 |
Where p is the linear momentum of the particle: the product of its mass and its velocity.
Substituting the values for velocity and radius of the Bohr model in this equation:
Equation 2 |
And
Equation 3 |
So
Equation 4 |
But
Equation 5 |
Which means that
Equation 6 |
In the base state, where n = 1, the wavelength equivalent of the particle is simply 2π times the radius, that is equal to the circumference of its orbit. In the second energy state the wavelength equivalent is equal to one half the circumference of the now larger orbit and so on.
De Broglie went on to argue that it was not just the electron that had this wave equivalence but that it was true for all particles. All particles could be regarded as a wave having a wavelength based on this same formula.
Equation 7 |
The formula links the fundamental Planck’s constant to the momentum of the particle. Momentum is however the product of mass and velocity, so the de Broglie wavelength is a function of both mass and velocity.
In his calculations de Broglie chose to identify the wavelength of the particle with Planck’s constant, defining it as Planck’s constant divided by the linear momentum of the particle.
Equation 8 |
However this is an artificial and arbitrary device. On any other scale, from that of a star orbiting a galaxy, through a planet orbiting a star, right the way down through a conker whirled on a piece of string, down to and including the base energy state of the atom we identify the wavelength of an orbiting object with its angular momentum divided by linear momentum. De Broglie chooses not to and instead invents a new type of wave, one for which there is no real physical interpretation, based on dividing Planck’s constant by the linear momentum. He does so in the full knowledge of Bohr’s adopted assumption that angular momentum is an integer multiple of Planck’s constant. De Broglie’s standing waves are an inevitable consequence of this invention.
Objects which are in circular motion can always be associated with a wave. The wave simply describes the motion of the object in one axis, normally the axis which along which the observer views the orbiting object. The diameter of the orbit is then the amplitude of the wave, the circumference is the wavelength and the tangential velocity of the orbiting object is identified with the velocity of propagation of the wave.
The radius of the orbit of such an orbiting object can then simply be calculated by dividing the angular momentum of the object by its linear momentum; a fact which derives directly from the definitions of these two quantities. This holds true for any object in orbit, so for example the orbital radii of the moons of Jupiter can be calculated by dividing their angular momentum by their linear momentum.
Equation 9 |
The wavelength of such wave is then simply the radius multiplied by 2π
Equation 10 |
In Bohr’s model the angular momentum is an integer multiple of Planck’s constant.
Equation 11 |
Equation 12 |
So the wavelength of the particle is no longer the radius of the generating circle multiplied by 2π, it is the radius times 2π divided by n.
Equation 13 |
De Broglie’s harmonics are therefore a direct result of him applying Bohr’s adopted assumption that the angular momentum of the orbiting electron can only take on values which are integer multiples of Plank’s constant while at the same time identifying the wavelength of the particle, not with its angular momentum, but with Planck’s constant. Viewed in this light de Broglie’s contribution amounts to nothing more than a restatement of the Bohr model using different language; the language of waves and frequency rather than that of particles. In effect de Broglie’s harmonics are as artificial as Bohr’s and Nicholson’s assumption and can be seen as a contrivance rather than an insight into the mechanics of the atom.
De Broglie published his results as his PhD Thesis in 1924 and was subsequently awarded the Nobel Prize for physics in 1929.
De Broglie was less than satisfied with his own ideas on the wave nature of matter and in particular was concerned to establish the link between his new wave mechanics and classical mechanics. His early attempts were abandoned in 1927 due to the adherence of physicists to the probabilistic interpretation of Born, Bohr, and Heisenberg. He resumed this search in 1951 and spent much of the rest of his life trying to find such a link. What de Broglie was looking for was a causal relationship between the mechanics of his waves and classical mechanics. In effect this is the same as looking for a mechanism in the classical domain which leads to quantisation of angular momentum. Needless to say de Broglie was not successful in this search. He does however provide us with a number of useful insights.
Firstly de Broglie recognises that the discrete energy levels of the atom are in some way associated with a harmonic sequence. That is with a series of frequencies which are an integer multiple of a base or fundamental frequency. A harmonic sequence appears in the frequency domain as a series of impulses equally spaced equally along the frequency axis F Hz apart and forming what is commonly referred to as a Dirac comb. The inverse Fourier transform of such a Dirac comb into the time domain is itself a Dirac comb, or series of impulses equally spaced along the time axis T seconds apart where T=1/F. Such a series of impulses in the time domain corresponds to a sampling process. The uniqueness of the Fourier transform means that if ever we encounter a harmonic series in the frequency domain there must be a corresponding sampling process taking place in the time domain. In this case the sampling frequency is equal to the orbital period of the orbiting electron. That is there must be something that happens within the atom which occurs or can occur only once per orbit of the electron.
De Broglie’s second insight comes in recognising that there is some sort of frequency multiplication process taking place within the atom. In the case of de Broglie, this is confined to the domain of the orbiting electron in the form of his harmonic waves where in the n^{th} energy state he cites a frequency of n times that of the orbiting electron. The problem with this is that there is no physical way to interpret such waves, the fastest moving object in the atom is the orbiting electron itself and it does so at the fundamental frequency. Nothing about the electron is oscillating at any higher frequency than this, at least nothing that can be said to be real. There is however one circumstance where such frequency multiplication does occur naturally; and that is under conditions of relativity.
Some of the most compelling evidence for the effects of relativity comes in an experiment carried out at CERN in 1977 called the Muon Ring Experiment. Muons are charged particles much like an electron, only more massive. When observed at low, non-relativistic, speeds they have an average lifetime of 2.2μsecs after which time they decay into an electron and two neutrinos. In the experiment muons are injected into the 14m diameter ring at 99.94% of the speed of light where Gamma has a value of almost 30. The muons now appear to have lifetime of some 65μsecs or Gamma times that which they have at non-relativistic speeds. This is because the processes which take place inside the muon and eventually lead to its decay are doing so in an environment where time is running slower by a factor of 30. Hence as far as the muon is concerned, it still has a lifetime of 2.2μsecs, but as far as us stationary observers are concerned it has a lifetime of 65μsecs. Travelling for 2.2 microseconds at close to the speed of light you would expect the muon to cover some 660m or roughly 14 times around the ring, but in fact it is observed to travel some 20,000m or 440 times around the track. This comes about because for the muon, travelling at near light speed, distance is foreshortened by a factor Gamma. So as far as the muon is concerned, it has travelled 660m, but to us stationary observers, where distance is not foreshortened, this appears as 20,000m.
Both parties agree that the muon passes a fixed reference point on the ring some 440 times during its lifetime. For us stationary observers that is 440 times in 65μsecs or roughly 6.8mHz. For the muon however the 440 turns around the ring are completed in just 2.2μsecs, corresponding to a frequency of 200mHz. For the muon, indeed for any object in orbit under relativistic conditions, orbital frequency is increased by a factor Gamma. We see the muon as travelling some 20,000m during 440 cycles around the ring or .022 revolutions/m. The muon, on the other hand sees itself as travelling some 660m in 440 turns or .667 revolutions/m. Hence for objects travelling at near light speed both temporal and spatial frequencies are multiplied by Gamma. This really is quite a remarkable result, there is no other phenomenon where both temporal and spatial frequency is multiplied in this way.
De Broglie failed to find a way to justify the wave mechanics for which he is now credited. In doing so however he provides use with useful insights into the inner workings of the atom. The discrete energy levels of the atom are associated with a harmonic sequence. This is inevitably associated with some sort of sampling process taking place within the atom. There is also some sort of frequency multiplication taking place and this is most likely associated with the effects of relativity. The electron orbiting the hydrogen nucleus must therefore be doing so at near light speed, where these effects become significant and not at the much lower Bohr velocity that current models suggest.
These ideas are explored more fully in Sampling the Hydrogen Atom.
]]>Erwin Schrödinger was intrigued by de Broglie’s idea of the particle as a wave and set about developing an equation to describe de Broglie’s waves. There are many ways to derive Schrödinger’s wave equation, but by far the simplest is to substitute de Broglie’s wavelength into the canonical form of an undamped second order differential equation, in other words into the standard equation for a wave.
Schrödinger’s time independent wave equation can be written:
Equation 1 |
Where
Equation 2 |
de Broglie identifies a new type of wave defined by dividing Planck’s constant by the linear momentum. The wavelength is then given by:-
Equation 3 |
So we can identify the equivalent orbital radius associated with such a wave as
Equation 4 |
The canonical form of a second order differential equation which is associated with a continuous wave described by a body orbtiting at radius r is
Equation 5 |
Substituting for r gives
Equation 6 |
The energy of the electron is made up of its kinetic energy E combined with its potential energy V
Equation 7 |
And so
Equation 8 |
Extending this to a three dimensional space instead of this simpler one dimensional case gives
Equation 9 |
Because the potential energy V is a function of the distance this is sometimes written:
Equation 10 |
Hence Bohr’s original model, de Broglie’s waves and Schrödinger’s wave equation all owe their existence to Bohr’s adopted assumption that angular momentum is quantised.
If you ask almost any physicist today to prove that angular momentum is quantised, they will invariably tell you that if you apply 360 degree rotational symmetry to the solutions of Schrödinger’s equation you will obtain quantum angular momentum. Or that the Eigenvalues of the Hamiltonian lead to a similar conclusion. This of course is total nonsense. Since Schrödinger’s wave equation is derived directly from the assumption that angular momentum is quantised, it cannot then be used to prove that it is quantised. Such a proof is almost the very definition of an ontological one. By the way the Hamiltonian referred to here is simply a way to express Schrödinger’s equations in terms of a state space using matrices, and the Eigenvalues represent the solutions to the equation when expressed in this form.
The fact of the matter is that the idea that angular momentum is quantised has never been proven. No mechanism has ever been found which could link orbital radius with orbital velocity and take account of relativity acting on the mass term to produce such quantisation. No lesser person that Louis de Broglie spent the better part of 20 years trying and never succeeded. If it ever could be proved this would represent the missing link between classical and quantum physics. And if it ever had been proved you certainly have heard about it; the person who did so would for sure have received the Nobel Prize.
In Sampling the Hydrogen Atom a model is derived for the hydrogen atom based on the idea that it is not angular momentum that is quantised, but Gamma, the Lorentz factor and in which the orbital radius is fixed at
Equation 11 |
Since the electron is orbiting at a fixed radius, the potential energy term in the Schrödinger wave equation makes no contribution to the atomic spectra and is therefore zero. The electron is orbiting at near light speed and so the kinetic energy of the electron is
Equation 12 |
Substituting these into the Schrödinger wave equation gives
Equation 13 |
Which simplifies to
Equation 14 |
Or
Equation 15 |
Which is the equation of a particle in circular orbit at radius R.
Quantum theory suggests that when a particle is observed its wave front (whatever that is) collapses to reveal either the location or the velocity of the particle. In a sense we can think of the above as the wave equation itself collapsing to reveal the particle as being objectively real.
In Sampling the Hydrogen Atom changes in energy level are accompanied by a change in orbital velocity, not by changes in orbital radius. Unlike the Bohr model, which requires the introduction of the mysterious quantum leap, or the Schrödinger model which requires that energy level is encoded in some mysterious representation of a probability, such changes as are proposed in Sampling the Hydrogen Atom are perfectly possible within the realm of conventional mechanics. They happen every time you change gear in a car.
Particles in such a model are objectively real. All of the exotic paraphernalia of quantum theory, with its wave/particle duality, inherent uncertainty, particles which are neither here nor there, representations of particles as probabilities are unnecessary and can be eliminated to yield a simple mechanical model for the atom. Particles are objectively real point particles having deterministic position and deterministic velocity exactly as Einstein would have understood them.
The debate between Bohr and Einstein, which ended prematurely and unresolved with Einstein’s death in 1955, is decided firmly in Einstein’s favour. Bohr and his cohort were simply wrong – they attempted to build on top of the Bohr model for hydrogen by carrying on with the supposition that angular momentum is quantised, a model which was fundamentally flawed, and as a consequence were forced down a path which led to such fanciful ideas as intrinsic uncertainty, wave particle duality and subjective reality. If they had recognised that it was the Bohr model itself which was at fault and sought to find a viable alternative, they would not have needed to follow this path. Einstein is not entirely blameless. While his instincts told him that Bohr and his colleagues were wrong, he was so engrossed in his work on grand unified field theory that he spent little time countering Bohr. In particular he only ever bothered to criticise Bohr’s proposals rather that to conduct his own investigation to discover the root of the problem and possibly propose an alternative theory
[1] It would have been far more conventional to suggest that the wavelength was angular momentum divided by linear momentum, which is what happens on any other scale. Indeed this marks the precise point where quantum theory takes on the idea that the laws of physics are different on the scale of the atom.
]]>The Stern Gerlach experiment was first carried out by German physicists Otto Stern(1888-1969) and Walther Gerlach(1889-1979) in 1922. It involved firing silver atoms through an inhomogeneous magnetic field at a target. The atoms are electrically neutral (not ionized) which avoids large deflections due to the orbit of charged particles moving through a magnetic field and allows spin effects to dominate.
The spinning electron can be regarded as a dipole and the non-uniform nature of the magnetic field means that the forces acting on one end of this dipole differ slightly from the forces acting on the other end. As a result the atoms are deflected by an amount which varies according to the angular momentum associated with the spinning electron. If the spin of the electron was to be a continuous variable then the extent of the deflection would vary continuously, but it does not. The electrons are seen to cluster into one of two regions, indicating that the spin on the electron is not continuous. Physicists argue that this is because angular momentum is quantised and can only take on certain discrete values and that these two regions of deflection correspond in some way to that quantisation.
Indeed it does show that the spin on the electron can only take on one of two discrete values, but this does not mean that angular momentum is quantised, far from it. What it does indicated is that the spin on the electron is always at some maximum value, but that it can take on one of two types, we can think of these as being clockwise or anticlockwise with respect to some datum within the atom. It means that the electron is never not spinning, it must always spin, but when it does so, it is always at some maximum value in one of only two possible orientations. The electron has very little mass and correspondingly low moment of inertia and so it comes as no surprise that almost any stimulus could cause it to spin. The fact that there is a limit to the rate of spin should also come as no surprise; if we think of the electron as a solid sphere, the limit to its spin must occur when the equatorial speed equals that of light.
So why does this not indicate that angular momentum is quantised everywhere?
The answer is simple and has to do with the size of the quantum of angular momentum postulated by physicists in the so called Standard Model. The Standard model holds that angular momentum is quantised and that the size of a quantum of angular momentum is equal to Planck’s constant. The trouble is that the Stern Gerlach experiment shows that the angular momentum of the spinning electron is some 10^{6} times smaller than this. If angular momentum were quantised in units of Planck’s constant then the spin on the electron could only take on values which are equal to an integer multiple of Planck’s constant, the smallest such being unity. The fact that the spin on the electron is 10^{-6} times this blows this idea out of the water. If on the other hand the fundamental unit of angular momentum is associated with that of the spin on the electron, it makes a nonsense of the idea that angular momentum can only take on values which are an integer multiple of Planck’s constant necessary for the Bohr model and its descendants. On the scale of the atom angular momentum would, to all intents and purposes be continuously variable.
]]>Our view of science is as a discipline where everything can be reduced to a fundamental set of axioms each of which has a logically consistent proof or is capable of irrefutable experimental verification. The mechanics of motion, for example, can be reduced to a set of basic equations. These in turn are derived from other more basic equations or from direct observation. In science we are asked to believe only what can be proved from first principles.
Religion on the other hand is a discipline where everything can be reduced to a set of propositions each of which must be accepted as an article of faith. In religion we are asked to believe what other people (I use the word advisedly) tell us are the fundamental truths.
How can it be that these two, seemingly opposite philosophical approaches, be said to be convergent?
Is the universe founded on fundamental principles that can only be divined by soothsayers and prophets?
Have we given up on the scientific method as a means of obtaining an understanding of the universe?
Or is the scientific method subject to some sort of limitation which ultimately forces us to rely on articles of faith to divine the true nature of the universe?
Has religion taken the characteristics of a scientific discipline?
Or has science taken on the characteristics of a religion?
It is certainly not the case that religion has changed – all of the major religions remain steadfast in the idea that we as humans cannot know everything and that in the final analysis we must accept certain ideas as articles of faith. The fact that these are somewhat of a moving target does not seem to matter too much. So for example up until the 17^{th} century Christians were supposed to accept that the earth was the centre of the universe, then that the sun was the centre, and now that the universe has no centre. Religion, in other words, has been forced to bow to the pressure of scientific discovery and knowledge, but remains at its heart unaltered.
Science has undergone many transitions as successive discoveries have been made and theories proven or disproven. Science admits to not knowing all the answers but seeks to search them out, one by one. Science is therefore dynamic in a way that religion is not. So what has happened to bring about the current state of affairs? What has changed to lead us to believe that science and religion have anything at all in common? It would appear that it is science that has changed and not religion.
We must first ask ourselves since when this change came about. Since when have science and religion been seen to be on convergent paths? Up until the beginning of the 20^{th} Century science and religion were clearly on divergent paths. The works of the astronomers such as Copernicus, Galileo, Newton, Hooke and generations of other scientists have gradually driven a wedge between science and religion culminating in the works of Darwin. This is despite the fact that many of these individual scientists held on to their religious sensibilities.
It is only over the last 100 or so years that such comparisons have been meaningfully drawn. I would go further and suggest that in fact it is only since the advent of quantum theory that science has taken on the characteristics of a religion.
Of course there are references to God by scientists,
“Black holes are where God divided by zero.”
― Albert Einstein
“Coincidence is God’s way of remaining anonymous.”
― Albert Einstein, The World As I See It
“God does not play dice with the universe.”
― Albert Einstein, The Born-Einstein Letters 1916-55
“Stop telling God what to do with his dice.”
― Niels Bohr
These do not necessarily imply that science has any of the properties of a religion, more they are comments made by scientists in the context of a society where religion exists.
There are however comments which attempt to draw direct comparison between science and religion. They seek to impart some of the mystery normally associated with religion into a scientific discipline.
“Those who are not shocked when they first come across quantum theory cannot possibly have understood it.”
― Niels Bohr, Essays 1932-1957 on Atomic Physics and Human Knowledge
“Everything we call real is made of things that cannot be regarded as real.”
― Niels Bohr
“What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school… It is my task to convince you not to turn away because you don’t understand it. You see my physics students don’t understand it… That is because I don’t understand it. Nobody does.”
― Richard P. Feynman, QED: The Strange Theory of Light and Matter
“If you thought that science was certain – well, that is just an error on your part.”
― Richard P. Feynman
Such comments seek to mystify science, to establish the scientist as some sort of wise man or priest whose role is to translate to the poor foolish layman all of the intricacies of a discipline which he is not capable of understanding.
To understand how and why this situation came about we need to look at how physics has evolved over the last century and to identify the points at which it has taken on this mantle of a religion. Over the last century, physics has come to rely, not on a set of axioms to sustain its position, but on a set of beliefs. These are the “articles of faith” that characterize modern quantum theory, the quantum catechism referred to in the title.
The list goes on…
Bit by bit these ideas have grown, each one just a small increment on what had gone before, each one seemingly correct given what had gone before, growing like a mound of sticking plasters on a festering sore.
While at first site each of these ideas appears to be capable of being traced back to first principles, in fact they are not. When we look at these ideas in detail we find that they can each trace their origin back to the day in 1916 when Niels Bohr published his paper on the structure of the hydrogen atom. That the paper was seriously flawed was widely understood at the time, but the assumption which underlay it was accepted and has been accepted as a matter of fact ever since. It is this assumption that is the festering sore that lies behind the slide of physics from a scientific discipline to a quasi-religious system of beliefs.
Hydrogen is the simplest of atoms, comprising a single nuclear proton orbited by a single negatively charged electron. In his model for the structure of the hydrogen atom Bohr first balances the electrical force acting to draw these two particles towards one another against the centrifugal force tending to throw them apart. Bohr needed a second equation in order to solve for the two unknowns of orbital radius and orbital velocity.
He took the idea of another physicist, John W Nicholson, that Planck’s constant was a measure of angular momentum, but also that angular momentum could only take on values which were an integer multiple of this basic value. This is the theory that angular momentum is quantized and which forms the first article of faith in the quantum catechism. When Bohr solved these two equations he found that his model yielded results for the energy levels of the atom which matched those of the empirically derived Rydberg model.
When the model was rejected as being unsatisfactory, nobody sought to question Bohr’s assumption which lay behind it and since then this assumption has been incorporated one way or another into every model that has been proposed. To the point where the assumption that angular momentum is quantized is now never questioned and has taken on the status of an article of faith.
Of even greater concern however is just how its advocates seek to justify this assumption. The assumption that angular momentum is quantized enjoys a ‘proof’, but it is in the nature of that proof that the similarities between religion and science are to be found.
No one can prove a fundamental religious truth, by definition they must be accepted as articles of faith, but that has never stopped people from trying[i]. As a result a number of ‘proofs’ of such things as the existence of god or the existence of miracles or any number of other religious ideas has taken on a common form. Such proofs are described as ontological. An ontological proof is one which postulates that something is true and then goes on to show that therefore it must be true, usually through a tortuous series of arguments whose main purpose is to obfuscate the link between the initial proposition and the final conclusion.
Proofs of the existence of god or of miracles almost invariably take on this form. It is postulated that god exists and then through a series of logical steps it is shown that therefore god does exist. When expressed in these terms it is clear that such so called proofs are nothing of the sort. What they amount to is a restatement of belief.
The proof of the quantization of angular momentum takes on just such a form. From Bohr’s assumption the model of the hydrogen atom is described, the model is dropped but the assumption remains. Louis de Broglie then goes on to express this assumption in a different form, in terms of wavelengths of fictitious waves that deny any physical reality. No matter the waves are what is important here. Schrödinger then develops a set of equations which describe these waves, unwittingly incorporating Bohr’s assumptions into the equations themselves. The equations are expressed in matrix terms, a move which obfuscates the underlying assumption even more. The matrix equations are solved and lo and behold they show that the solutions are consistent with the idea that angular momentum is quantised.
This line of reasoning, this so called proof, is the very definition of an ontological argument and such arguments have no validity. This means that the whole issue of whether angular momentum is quantized remains an open question.
In the case of religion, the argument stops there, either you believe in god or you don’t. If you don’t; then god’s existence defies proof and if you do; it doesn’t need to be proven. Science is a different matter or it should be. It is not sufficient to simply ask whether you believe in the quantization of angular momentum or not. It is not sufficient to accept any proof which begins in the domain of quantum theory, since any such proof is bound to be ontological. When we recognise an ontological argument as scientists we have to question it. One of the first questions to ask is what would constitute a scientifically valid proof?
In the case of the quantization of angular momentum the only valid form of proof is one which does not begin with the assumption that lies at its heart. There can therefore be no proof of the quantization of angular momentum which begins from the assumption that angular momentum is quantized. This ultimately means that it is not possible to prove the quantization of angular momentum based on quantum theory. The only valid proof of the quantization of angular momentum is one which is firmly rooted in classical mechanics.
This is the missing link between quantum theory and classical theory. If it could be proved using classical mechanics, that angular momentum is quantized, we would have the bridge that divides classical theory from quantum theory. Such a proof would remove any scientific doubt, it would restore physics to a scientific discipline and would banish the nonsense of comparing science with religion to the dustbin of history.
Angular momentum is the product of the orbital velocity, the orbiting mass and the orbital radius. We must also take into account the effects of relativity, which at velocities approaching the speed of light affect the mass component of this product.
To prove that angular momentum is quantized we must show a causal link between the change in radius and the change in orbital velocity which results in an increase or decrease in angular momentum by one or more quanta, taking into account the effects of relativity on mass. It is not sufficient to show, as Bohr did, that if we assume angular momentum is quantised, then the energy levels of the atom match those of the empirically derived Rydberg formula. That is a necessary condition for such a proof, but not a sufficient one. There might be any number of variables or combinations of variables that produce such a match besides that of angular momentum being quantized.
In short then, to answer the question as to why physics and religion appear to be on a converging path, we must look to the tenets which underlie modern quantum theory. When we do so we find that quantum theory is not founded on scientific fact, not one which can be traced back to a set of fundamental axioms, but is instead founded on a belief, namely that angular momentum can only take on values which are an integer multiple of Planck’s constant.
Perhaps it is not surprising that I am of the opinion that no such proof is possible. That angular momentum is not quantized and that it is therefore not possible to prove that it is quantized. The reasons for suspecting that this is the case stem from the fact that in the classical domain, angular momentum is the product of three separate and unconnected variables, the orbital radius, the orbital velocity and the mass. At least one of these, the mass term, is definitely affected by relativity and at least one of them, the orbital radius, is definitely not affected by relativity. Relativity introduces a continuously variable element into the value of the mass term and hence the value of the angular momentum. This is inconsistent with the idea of quantization.
Even if we ignore the effects of relativity which, by the way, is what Niels Bohr chose to do, then for angular momentum to be quantized its three constituent variables would have somehow to collaborate with one another in ways which we cannot imagine, which we cannot describe and which we never experience anywhere else in the universe.
Set all of this against a much simple idea, that orbital velocity is subject to the effects of relativity and quantization is seen to be the function of a single variable; the orbital velocity. Furthermore we find that this leads to a simple mechanism which causes the energy levels to be quantized. The electron is seen to be a point particle in the classical sense. The wave properties of the electron derive directly from its orbital motion and we get a simple explanation for the Fine Structure constant. These ideas are explored more fully in https://quantum-reality.net/sampling-the-hydrogen-atom/
In effect we have only to make one slight adjustment to the classical laws of mechanics; to take account of the effects of relativity on orbiting objects and the whole edifice of quantum theory collapses. Physics is restored to being a scientific discipline and not a system of beliefs. We can tear up the quantum catechism.
The reason why it has been meaningful to compare physics with religion is that over the last 100 years physics has adopted a set of theories which are based on belief rather than being related to fundamental axioms. It has sought to prove these beliefs based on the flawed logic of an ontological proof. It is only by recognizing that this is the case and finding the root cause that physics will be restored to a true scientific discipline.
[i] One of the first Ontological arguments was proposed by Anslem of Canterbury in 1078 to prove the existence of god.
]]>The paradox argues that an infinite, eternal and static universe would have a night sky which was bright and not the dark sky that is known to exist. The essence of the argument is that any ray traced from an observer on earth out into such a space would ultimately end on a star and hence no matter what the direction an observer looked he would see a point of light.
At first sight it might be argued that this is an inevitable consequence of the inverse square law. The further away a star is from us here on earth, the less of its light reaches us. So if you look far enough into space, you will reach a point where there is virtually no light reaching us from that distance. The problem however with this argument is that while the intensity of light from each star falls off as the inverse square of the distance to the star, the number of stars at any given distance increases in as the square of the distance, in other words in exact proportion to the attenuation due to the inverse square law. The result is that we should expect the night sky to be bright. All of the gaps that appear due to the inverse square law are filled by more stars the further away we look.
Another argument that has been put forward is that dust and other particles in the intervening space between us and distant stars would absorb some of the light, leading eventually to a dark night sky. Once again there is a flaw in this argument. The problem here is that the intervening dust would absorb energy from the light and that would raise its temperature, causing it to glow. If all of the energy along a particular path were absorbed, sufficient to render that point a dark point in space, then the intervening path would have absorbed exactly that amount of energy such that it would glow as brightly as the distant star.
The currently held view is that the dark sky happens because the universe has a finite age. The further one looks into space, the further back one looks in time. Eventually the point is reached where one is looking back into a time from before the Big Bang. Unless there happens to be an intervening star, then any ray traced from here on earth will not appear as a bright spot, but will be dark. The population of stars in the universe is sufficiently sparse that the night sky then appears predominantly dark.
The Big Bang theory however introduces a new paradox. If looking further into space involves looking back in time, then eventually the observer will see back sufficiently far to see the Big Bang itself, which was by definition a very hot, bright event. The paradox is explained by invoking the expansion of the universe itself. Such an expansion is accompanied by a cooling effect, similar to that which happens when gas is expelled from an aerosol can. It is argued that the resulting cooling of the remnants of the Big Bang is now seen as the microwave background radiation. The problem here is that light from 13.8 billion years ago originated in the big bang and so we should not be looking at the way the universe now is, cool and radiating microwave energy, but hot, exactly as it was at the time of the big bang. The way that the big bang theory gets around this slight inconvenience is to argue that there was a period of rapid inflation when the universe grew at enormous speed, pushing the constituents of the early universe apart far faster than the speed of light. That way we are seeing it as it was after this rapid expansion.
While the Big Bang may be consistent with the idea of a dark night sky, this is by no means the only explanation, indeed it is not even a very good explanation of the phenomenon since it requires that we overturn Einstein’s postulate that nothing can travel faster than the speed of light.
It is well understood that the further out into space we peer, the further back in time we are looking, this is the essence of the big bang explanation for Olber’s paradox. So when we look at the sun for example we are not seeing it as it is now, but as it was 8 minutes or so ago. This is the time it takes for light to travel from the sun to reach us here on earth. Similarly when we look at a star which is say 50 light years away, we are seeing it as it was 50 years ago. This is one reason why measuring cosmological distances using units of time, light years, makes sense.
However there is another phenomenon which is closely related and which is often overlooked. Light from distant objects is shifted towards the red end of the spectrum. With this so called Red Shift the further away the object is, the more the light from that object is shifted down the spectrum. Looking at this from a slightly different perspective we can see that this is equivalent to saying that the further out into space we look the further up the spectrum we are looking.
A question then arises over the bandwidth of the photon which has to be limited in some way, otherwise we could expect photons with infinite frequency and infinite energy. Infinite energy means enough energy in a single photon to destroy a star, enough energy in a single photon to destroy a galaxy, indeed it means enough energy to destroy all of the galaxies. Clearly this is not the case and so there has to be a limit on the maximum energy that a single photon can carry. In Shedding Some Light on the Nature of the Photon, a model is developed for the photon in which the bandwidth is seen to be limited by its inherent structure.
With photons having limited bandwidth and the further out into space we peer the further up the spectrum we are looking. There must therefore come a point where we are looking so far out into space that we have reached the upper frequency limit of the photon and all we can see from beyond that point is black. Hence the dark night sky.
Looking at it another way, photons from this critical distance and beyond are red shifted to such an extent that they are no longer visible but are to be found in the part of the spectrum below the infra-red. The critical distance at which this occurs then forms a visible event horizon from beyond which we cannot receive any visible light. There are other similar event horizons for different frequencies, so for example there is a distance from beyond which we can never receive any X Rays and one for Ultra Violet. In effect we are sitting at the centre of a series of such event horizons which form a set of concentric spheres. We can only ever probe what lies beyond these event horizons by looking at lower and lower frequencies where, paradoxically, the resolution we can obtain gets less and less the further we look.
So here is a very simple rational explanation for the dark knight sky, one which does not require that we disbelieve Einstein’s theory of special relativity, one which does not requite that the universe started with a big bang, nor indeed that it is even expanding. All that is necessary to fully explain Olber’s paradox is that there is a red shift, whatever its cause, and that the bandwidth of the photon has an upper limit.
]]>Attempts at unifying the forces of nature are called Grand Unification Theories (GUTs) and a theory which unifies all of the forces of nature is called a Theory of Everything (ToE).
One of the first such attempts was by William Gilbert (1544-1603). It was Gilbert who first suggested that the earth had an iron core and acted as a magnet. He tried unsuccessfully to show that gravity and magnetism were both manifestations of a single more fundamental force. Ultimately Gilbert’s attempts were unsuccessful and we now know that gravity and magnetism are two distinct forces, although they do have some aspects in common, for example they both obey the inverse square law. Despite this early failure physicists are still trying to find a common link between the various forces of nature.
The first really successful attempt at unification was the unification of magnetism and electricity in the 19^{th} century. Until then it was thought that these were two independent forces. It was Hans Christian Ørsted (1777-1851) who first discovered that an electric current could affect a compass needle. He did so initially while conducting a demonstration of the heating effects of an electric current and when he noticed that a nearby compass needle was deflected whenever he turned on the current. Michael Faraday went on to show that a changing magnetic field could induce an electric current.
In 1864 James Clerk Maxwell published his famous paper on the electromagnetic field. This was the first example of a theory that combined what had previously been thought to be separate fields into a single unified field theory and in so doing showed that electromagnetic waves propagate at the speed of light. Einstein built on the idea that the velocity of electromagnetic waves is constant to combine our notions of space and time into the unified concept of space time. He then used this to describe a curved geometry of four dimensional space time to encompass gravity.
During the 20^{th} Century there have been various attempts at unification, the Holy Grail being to unify all the forces of nature into a single force, in effect to combine the wave/particle model of the atom with that of General Relativity into a single coherent theory. All such attempts so far have ended in failure.
In the current Standard Model it is held that there are four fundamental forces of nature:
Thus far it has been possible to unify the electromagnetic, strong and weak nuclear forces but only under conditions of high energy, which leaves gravity out on its own as a separate force.
All of these attempts at unification focus on force as the variable of unification. It seems that modern physicists are fixated on the idea that it is forces that can be combined into a single force. Here it is argued that the seat of unification does not lie with force, but with energy and that all forms of energy can be ultimately resolved to be mechanical in nature. In doing so it can be shown that there are just two fundamental forces in nature, not the four of the Standard Model. Based on two simple postulates that are described in more detail elsewhere on this blog, it can be shown that there are two fundamental forces, gravity and electromagnetism and that both the strong and weak nuclear forces are really just forms of electromagnetism and/or gravity and hence can be thought of as being unified.
In Sampling the Hydrogen Atom it was shown that the forces which bind the orbiting electron to the atomic nucleus are electrostatic. They are the same forces that we encounter when considering electricity on any other scale. The energy which is associated with binding the electron to the atomic nucleus is shown to be kinetic in nature, deriving directly from the mechanics of the system. This theory rests on the idea that certain orbital velocity terms are affected by relativity. That when it comes to objects in orbit the orbital velocity can be treated as the distance compressed by the effects of relativity divided by the orbital period as measured by the stationary observer. Using this hybrid velocity term the centrifugal force acting on the orbiting electron balances the electrostatic force at a series of velocities close to the speed of light and which correspond to the energy levels of the atom.
If we apply this same postulate to the atomic nucleus we find that it too is governed by conventional mechanics and that the strong nuclear force is in reality just another manifestation of the electromagnetic force and gravity.
The problem with the dynamics of the atomic nucleus is that it is made up of electrically charged particles, protons, which repel one another. These forces of repulsion are governed by the inverse square law and so on the scale of the atomic nucleus are very strong indeed. The solution that the Standard Model adopts to resolve this problem is to argue that forces are mediated by particles and then to invent a particle, called the Gluon, which has whatever properties are necessary to overcome these forces and bind the nuclear protons together.
All of this subterfuge becomes unnecessary if we adopt the postulate that orbital velocity is affected by relativity. If we examine what happens to a pair of protons which are in mutual orbit under such circumstances we find that as the orbital velocity approaches the speed of light, the centrifugal force acting on the protons becomes vanishingly small. At the same time the angular momentum reaches its lower limit of Planck’s constant and so the radius of the orbit is constrained by this to be:
Where m_{p} is the mass of the proton
Hence the orbital radius of the protons within the nucleus is 2.10308783323*10^{-16}m.
To a stationary observer located at the centre of the atomic nucleus, the effect of relativity acting on the two protons is to increase their mass by the factor Gamma. At sufficiently high speeds the mass of the protons is such that the gravitational force acting between the protons matches the electrical force tending to drive them apart and the system becomes stable and hence:
From this we can calculate the value of Gamma as
And from this we can calculate the orbital velocity of the proton as:
99.9999999999999999999999999999999999% of c.
To all intents and purposes the velocity of the protons around their respective orbits is equal to the speed of light. The binding energy, which is the energy associated with the stable orbit of the protons is then equal to the combined kinetic energy of the two particles and has a value of mc^{2} and this energy is seen to be mechanical in nature.
In Shedding Some Light on the Nature of the Photon a second postulate is introduced: that gravitational mass is bipolar in nature, i.e. that it can take on both positive and negative values and that these add arithmetically. Negative gravitational mass is then associated with antimatter, while positive gravitational mass is associated with matter.
Based on this assumption it is shown that radiation energy is fundamentally kinetic in nature. The energy of the photon is the binding energy associated with the forces that maintain the structure of the photon as a binary system comprising a particle and its antiparticle equivalent, specifically an electron and a positron, locked in mutual orbit.
Although the forces associated with gravitational mass on the atomic scale are very small indeed, it is postulated that with gravitational mass like poles are attracted to one another and unlike poles are repulsive to one another. This is in contrast to the electrical force where like polarities are repulsive and unlike polarities are attractive.
Thus the two fundamental forces, gravity and electricity form a fully subscribed set of attraction and repulsion that can be formed from bipolar forces. In other words there cannot be any other forces that obey the inverse square law and which offer combinations of attraction or repulsion based on polarity since these would simply add to or detract from one of these two forces.
The situation is summarised in the tables below
+ |
+ |
Repulsive |
+ |
– |
Attractive |
– |
+ |
Attractive |
– |
– |
Repulsive |
Table 1 Electrostatic Force
+ |
+ |
Attractive |
+ |
– |
Repulsive |
– |
+ |
Repulsive |
– |
– |
Attractive |
Table 2 Gravitational Force
We have seen, in Shedding Some Light on the Nature of the Photon, that by postulating that gravitational mass is bipolar and that orbital velocity is relativistic, that radiation energy is mechanical or kinetic in nature. The energy of the photon is carried, rather like a flywheel, by the two constituent particles that are locked in mutual orbit.
The Kinetic theory of heat was proposed by Daniel Bernoulli and later his ideas were consolidated by Maxwell and Boltzmann. It shows that temperature and pressure are phenomena associated with the motion of atoms. In other words heat energy is ultimately kinetic in nature.
Chemical energy is similarly mechanical in nature being the energy associated with the bonds between atoms.
Hence with just two simple and plausible postulates it can be shown that all forms of energy are ultimately mechanical in nature. All the various kinds of energy can be unified into a single sort of energy – mechanical or kinetic energy. This means that in the realm of force, there are just two fundamental forces, not the four prescribed by the Standard Model. These are gravity and electromagnetism and these represent a fully subscribed set of attractive and repulsive forces.
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