James Maxwell and those of his time laid the foundation for Einstein and those who would develop modern physics. The inquiry would explore the aspects of one of the most fundamental physical phenomena: light. There were many problems with the existing field of ‘Classical electromagnetism’ founded by Maxwell, as the experiments became more advanced, they began to find limit cases and experimental disagreement with some of his theories. The primary one was that of continuity. Maxwell argued that electromagnetic energy should be distributed continuously over space when analyzing an emission. Max Planck found that this was not precisely true – his studies into ‘black-body radiation’ had established that the energy was being emitted in discrete parts, with energy being radiated in ‘countable’ integers of an elementary value, never having a continous disribution. To make the physical models of each of them consistent, Albert Einstein studied the photoelectric effect of black-body radiators and applied the concept of Quantum Mechanics (QM) to light. Many other experiments would be conducted in this field of science to eventually form the field of modern physics as we know it today. The groundbreaking theory of relativity, created by Einstein, pivoted on the historical revolution in QM that redefined the scientific model of light and how it exists in nature – not a wave or particle, but somehow both. With special and later, general relativity developed for their toolbox, along with the myriad of technologies developed with electricity, the scientists that came after Einstein and Planck were able to advance their studies in all fields by leaps and bounds.
The study of electricity and magnetism has puzzled humanity for millennia. Since the beginning of our time on this planet, the power of electricity in the form of lightning has puzzled us for ages, with studies of this mysterious force ranging from Greeks rubbing amber on fur to the modern particle experiments of today’s physicists. For civilization to develop a thorough understanding of this awe-inspiring aspect of nature, it would take centuries for the state of technology to advance to a level that would allow for a working understanding of the mechanisms, and we are still struggling to fully comprehend it today. The classical laws of electromagnetics form the foundation of our understanding of how this strange force behaves in nature, postulated by James Maxwell in the late 1800s. He did not come to this conclusion alone, as it took the efforts of many others over hundreds of years to build up the study of electricity and magnetism before he could synthesize the laws that the phenomena followed, known as Maxwell’s Equations. In their original form, they were twenty equations that described the mathematical models that were used to understand electrical and magnetic fields, along with how the two interacted while changing with time.1
A few years later, Oliver Heaviside condensed the twenty equations down into the modern form of four condensed differential equations that elegantly describe the laws of electromagnetism. The complex “electromagnetic potential field” was a difficult analytical tool to use and Heaviside wished to simplify them to make them more accessible. The first of these in the modern form is Gauss’s Law, which basically states the relationship between electrical charges and the fields they emit. The second equation states the nonexistence of isolated magnetic charge, but sometimes is known as Gauss’s law for magnetic fields – in a practical sense, it states that no closed surface will contain a “magnetic charge” that emits a magnetic field in the same manner as electric charges emit electric fields. For this, the use of Ampere’s circuit law is necessary to understand how magnetic fields are created – it states that the magnetic fields are a direct result of electric current or the displacement of charges. The last equation, Faraday’s law, describes how a changing magnetic field will create an electrical field, or “induce” a voltage on a conductor. With these four laws of electromagnetism, the power of electricity was now in humanity’s grasp.
The journey of discovery that Maxwell undertook to reach these conclusions was long and arduous, and in the end Maxwell was still left with a few unanswered questions. The Michaelson-Morley experiment had failed to prove the existence of a ‘luminiferous aether’ – a medium for the passage of light, but Maxwell hadn’t entirely dispensed with the idea during his research.2 His many experiments and theories on ‘electromagnetic radiation’ had yielded an interesting result: the speed of electromagnetic radiation in vacuum is equal to the speed of light. It wouldn’t be confirmed for decades, but this discovery had actually illustrated the relationship between electric and magnetic fields, and one of the most mysterious phenomena in science: light.
The man who would bridge the gaps in understanding to allow for a more complete view of electromagnetic radiation was German physicist Heinrich Hertz. A man with strong aptitudes in sciences and engineering, Hertz earned his PhD in 1880 in the University of Berlin and continued his scientific work under Hermann von Helmholtz. Helmholtz suggested to Hertz that proving Maxwell’s theory would be a good topic for his PhD dissertation. With this landmark discovery he proved the existence of a self-propagating electromagnetic field first envisioned by Maxwell.3 The astonishing caveat to this discovery was that the speed of this radiation was equal to that of light in vacuum – which would later prove that “visible light” was actually just a form of electromagnetic radiation of a specific frequency. This frequency, bearing an inextricable relationship with the wavelength of the emission, was theorized years before in the field of optics, and while many questions such as the existence of the aether had been solved (in this specific case, disproven), still more remained – why were astronomers still limited to the solar system, unable to look any further with reliable accuracy? While the “wave” aspect of light had been proven with the works of these great scientists, the question of how a wave could pass through empty space, or vacuum, still puzzled the scientific community.
These questions would be unraveled and more startling aspects of the physical nature of the universe were revealed, and still more questions appeared as science’s understanding of the fundamental phenomena in existence became deeper. Albert Einstein brought great strides to the scientific community with his theories of special and general relativity, along with the foundation of quantum mechanics that colours our comprehension of reality to this very day. He and many others including Schrodinger, de Haas, Bohr and Rutherford crafted several theories to aid in our understanding of some of the most difficult physics of their time. Their discoveries were used to contemplate some of the greater mysteries that were unsolved since the time of Newton – by proving that space was no longer an absolute, further inquiries into the secrets of the universe experienced an evolution into what we know today as modern physics. This fundamental advancement of the organization of thought would lead to a new era of science – an era from the quantum perspective, accepting that nature could seemingly obey contradictory principles simultaneously. The idea that things at an imperceptibly small level could somehow act as both particles and waves, along with many other surreal ideas, would be explored at length in this field.
The field of QM had been established before the time of Einstein with the works of Boltzmann, Planck, and Bohr, developing many underlying theories that were necessary for science to deepen its understanding of light, including the discovery of the photoelectric effect by Hertz in 1887. The effect observed by physicists was that light shining on metal would appear to release some electrical energy in this process – “knocked free” by the incoming light. With the publication of this paper in March 1905, Einstein synthesized the underlying theories into a mechanism which could reconcile Maxwell’s theories of electromagnetism with the modern particle experiments and theories of Planck and Boltzmann that seemed to verify the presence of the ‘quantized’ aspect that energy emissions in nature appeared to have. To quote Einstein’s paper directly: “…phenomena involving the emission or conversion of light can be better understood on the assumption that the energy of light is distributed discontinuously in space.”4 In essence, Einstein stated that light appears to exist as discrete, quantizable ‘pieces’ rather than a continuous wave of energy.
It may seem like this idea is fundamentally incompatible with the Maxwell’s theory of an electromagnetic wave that was ultimately confirmed years later by Hertz, but applying the idea to most light sources in nature had proven to be practical. If a source emits light, the energy is not emitted continuously through the light’s path. It may appear to be the case, but when considering a much slower timescale than humans experience, and a over a much smaller space than we can actually see, scientific experiments can prove the existence of this phenomenon. Counter-intuitive, but the data appeared to show that it was in fact the case, and following the example set by Newton and Gallileo, Einstein attempted to craft a theory that would reconcile the existing theories with the data provided by the many physicists working in the field of Quantum Mechanics. Particularly, working at the problem of ‘black-body’ radiation that posed an interesting problem for the physicists of the time.
The problem with the study of ‘black-body radiation’ was that at higher frequencies of radiation, the classical model suggested that the energy of such an emission would approach infinity. This posed a fundamental problem with the model, known as the ‘ultraviolet catastrophe’. Experimentally and fundamentally, this was proven not to be true, as the energy radiated from a ‘black-body radiator’ was never, in practice, observable with an infinite range of frequency emission. Max Planck verified this with his experiments on “Elementary Quanta” – energy appeared to be released and transferred in quantizable, discrete “packets” of energy, not a continuous distribution as one might intuitively believe from physical experience.5
Einstein’s March 1905 paper on the photoelectric effect applies Planck’s theory of QM to light, and in doing so proves the existence of the “light energy quanta” that would later come to be known as a “photon”. The work of Newton had proven that for physics to work, it must be consistent with all possible scenarios, and it must be consistent within itself. Basically, what Einstein had done was take the existing experiments and data of the time and reconcile all of the other proven theories with this anomalous data. The goal of physics here was basically to make it so that the relations that hold for physical reality can be expressed in a mathematical (and therefore measurable and understandable) relationship. When problems occur in the existing model, new theories need to be constructed to reconcile the data with the existing model, to create a new version that is consistent within itself and also covers every physical possibility – in mathematician’s terms, a ‘general’ solution to understand the underlying mechanism.
Reconciling the notion of quantizable light energy ‘particles’ was difficult when the presiding theory was that light existed as a form of wave. But to reconcile the theories of quantum mechanics along with the experimental results that the black-body radiation experiments produced, a notion of ‘duality’ was considered by Einstein. What if the photons could be both – particles and waves somehow? It seemed like this idea is the one that held up to experiment, as reality shows that the behaviour of light can follow the rules of both, depending on the situation. The idea was that the photons could be just little bursts of energy that ripple through our world, and over a period of time which we experience, we perceive them as an aggregate of the tiny particles which can be understood as a wave.6 But Einstein and his compatriots’ ideas were still compatible with the classical physics that had been developed by Maxwell and those who came before. Somehow, the universe allowed light to follow the physical laws of both – particlesand waves – a dual natured physical phenomenon. It appeared that QM had to be considered when analyzing the atomic world – nature just seemed to work that way at the smallest order of magnitude.
At this point, the ‘subatomic’ level was being considered by numerous physicists, perhaps foremost among them Niels Bohr, Ernest Rutherford, and J. J. Thomson. Thomson was an English physicist that conducted several experiments with cathode rays, and in 1897 he had proven the existence of the electron and in doing so, had discovered the first subatomic particle.7 His model for atoms was flawed, but Rutherford, working as his student at the time, discovered that the atoms (or as Thomson called them, corpuscles) weren’t just made up of negatively charged electrons as Thomson thought – with the gold foil experiment, Rutherford had proven that the positive charge is concentrated in the center of the atom. This would take a lot of time, and the presence of the nucleus which contained the positive charge of atoms wouldn’t be verified until his formulation of the famed Rutherford model of the atom in 1911 – that instead of Thomson’s ‘plum pudding’ model that theorized the positive charge was evenly distributed throughout the atom, it was tightly concentrated in the center.8
Niels Bohr built on the ‘raisin bun’ model that Rutherford had proposed, theorizing a model instead that was consistent with Max Planck’s studies in Quantum Mechanics – leading to the creation of the Bohr model in July 1913 with the publication of his paper “On the Constitution of Atoms and Molecules”.9 The model formed the basis for the ‘old quantum model’ which basically stated that all of the ‘quanta’ that exist in nature appear to follow the laws of classical physics, but for a variety of reasons, the exchange of energy is not of a ‘continuous’ nature – it is discrete. In addition, the formal foundation of the ‘correspondence principle’ by Bohr in 1920 with the publication of the work of the same name had made a startling conclusion – with the numbers of a quantum system being sufficiently large, the classical equations of mechanics must agree.
This conclusion, along with the work of Einstein, aided in the reconciliation of quantum mechanics with the classical laws of physics. It appeared that the classical laws held for most systems of practical scale, but the laws of quantum mechanics dominated at the subatomic level. In 1924, a French physicist named Louis de Broglie introduced the ‘wave theory of matter’ to help understand the problems with the current models of electrons. He built on Einstein’s idea of the duality of light, and applied it to matter at the quantum level in the form of the electrons discovered by Thomson and modeled more accurately by Bohr and Rutherford.10
The developments of the wave-particle duality of light that appeared to extend to all particles at the quantum level, as formulated by de Broglie in his 1924 thesis, were carried on by the work of Erwin Schrodinger in 1926 when he postulated his famed ‘Schrodinger Equation’ which was a wave equation that only utilized quantum mechanics – in other words it was completely independent of the classical physical equations of Newton and Maxwell that described. This caused a change in the old quantum model, and with the work of Heisenberg and Pauli during this period onward, the modern theory of Quantum Mechanics began to take shape. The two of them contributed fundamental principles to the field that advanced the scientific community’s understanding with historical experiments, leading the way into yet another era of science from the ‘old quantum’ perspective into the modern one that we still use to this very day.
The two foundational principles that Pauli and Heisenberg introduced to Quantum Mechanics were those of ‘exclusion’ and ‘uncertainty’. These two ideas helped scientists understand the statistical properties predicted by Schrodinger’s equations of quantum waveforms. It was difficult for physicists looking into the structure of atoms to understand how it could align with the matter-wave theory of de Broglie along with the correspondence principle established by Bohr. How could some quanta be seemingly obeying the classical laws of physics with more effect than others? The ‘Pauli Exclusion Principle’ states that quanta are restricted to a certain ‘allowable’ number of ‘states’ – in the case of electrons in atoms, their position in the orbit surrounding the nucleus is restricted depending on a set number of parameters known as ‘quantum numbers’, and it is impossible for the electrons of an atom bearing many electrons for any two of them to have the same set of quantum numbers. The numbers themselves correspond to the position in the orbit, angular momentum, magnetic quantum number, and the spin quantum number – in combination they essentially predict the amount of ‘energy’ that the individual electron has in its orbit.11
To illustrate with a hydrogen atom, the electrons in ‘lower’ shells have lower energy than if the atom is energized and the electron becomes excited to a higher orbit above the nucleus. Many physicists have used this practice, known as ‘spectroscopy’ for the spectrum of light emitted by hydrogen and other gas atoms when the electrons are excited and allowed to relax, releasing this emission as the electron ‘falls’ in orbit and gets closer to the nucleus. In fact, this was how Niels Bohr proved the existence of his Bohr Model in 1913 in his paper “On the Constitution of Atoms and Molecules”. In part one of the three-part paper, he describes how the Balmer series of line spectra (along with many other series of ‘spectral lines’ provided by astronomers) was fundamentally related to the change in the orbits of electrons in hydrogen atoms.12 This idea, along with the ‘conservation of energy’ had informed science’s view of the atom and the way that the electrons could move in different ways around the nucleus, and when an electron loses energy and ‘falls down’ orbital levels, an emission of some kind of energy is the result. Whether its light in the form of photons, heat, vibrations, or something else. Bohr had proven the fundamental mechanism for the emission of light, and with the help of thermodynamics it was made intuitive – the energy cannot be “created” or “destroyed”, it will be passed to something else in the universe – in this case, emitted as a photon.
The study of the physics of atoms and the quantum world is at it’s heart, a question about the behaviour of how reality actually functions. Science since the time of Newton has been a method for the discovery of truth in a world as chaotic and turbulent as ours. With the tools of mathematics and experiment, Scientists are able to form theories and model the behaviour of physical objects smaller than they can even see. The accuracy and gravity of their discoveries changed the way that humanity understands physics on a fundamental level, bringing us forward into a new era of understanding. The main problem with the theory of Quantum Mechanics and the reason it was incompatible with the Classical Mechanics developed by Newton is that the two of them used very different kinematic descriptions, or in other words, the equations used to represent the motion of objects were extremely different, to the level that they could not be reconciled. It was Bohr’s view, in his later writings and after committing years of thought to the physics, that with the correspondence principle, the effects of quantum mechanics over a macroscopically large enough system, classical mechanics could be used as the analysis requires the knowledge of the ‘initial’ state of the subject of analysis.13 When analyzing systems that are sufficiently small, in essence, if the number of ‘Planck Constants’ or elementary particles involved in the actions of the particles is small enough, then classical mechanics will not be adequate for theory to match experiment. The strangeness of this idea is one that has puzzled scientists since the beginning of the study of quantum mechanics – how could reality follow one set of rules on one scale, and then seemingly contradict those rules at a fundamentally small enough level?
To reiterate, the correspondence principle founded by Bohr is a very useful principle when considering the world of particle physics – it may seem strange that due to the nature of quantum mechanics and the uncertainty of the physical state of things at this level illuminated by the Heisenberg uncertainty principle and the Schrodinger Wave equation seemed to make the science redeemable with reality.14 After all, the basis for all of these experiments and theories is the scientific method, so in general they should be acceptable as true. The difficulty is accepting that the data might be accurate, and understanding the what the implications might be of the experiment, something that most scientists have to suspend in their search for the truth, as the senses are not reliable and only data can be relied upon to be impartial. By selecting specific problems and working away at them, physicists are able to theorize mechanisms that explain the results of their experiments – and once proven to be consistent they can slowly be accepted as true, which takes time for the general population and even the scientific community. Even still, there are some questions that can be difficult to address, and remain unanswered to this day because there has not been a perfectly consistent physical model of the universe created to this day, that agrees with experiment.
One of the primary difficulties with Maxwell’s electromagnetism model was that it seemed to require the notion of ‘action-at-a-distance’, which appears to be incompatible with Newtonian physics. The idea that an object can be moved by a force that does not have a mass associated with it basically shatters the physical reality that we experience – something can only move if force acts on it, and according to Newton all force requires mass. The ‘zeroth’ law of motion is a bit of an assumption made when making analysis of systems with classical mechanics – there cannot be motion without force. It can be supposed that Newton began his study of the celestial bodies by this very principle – what is making the planets move and stay in orbit, if not some kind of force? The study of gravity operates with the assumption of ‘action-at-a-distance’ as well, but the work of scientists for ages has been to refine the models and make them closer to the actual truth in any study of nature. Newton knew that his models were imperfect, but he also knew that they were more consistent than anything developed up until that point. Many noticed the imperfections and attempted to rectify them over time – Maxwell and his colleagues with the study of ‘unseen force’ and then Einstein and the other pioneers of Quantum Mechanics that have formed the functional models that we use to this very day.
The remaining unanswered question that the current model discussed here bears is related to the atomic structure that make up the particles that surround us. The electromagnetic forces between the electrons and the nucleus – made up of protons and neutrons in most atoms with the exception of hydrogen, which has only a proton, according to the known laws it the two particles should accelerate towards one another. Yet, as Bohr proved with his experiments and by the fact we continue to exist, the electron has a certain amount of energy that dictates how far from the nucleus it will orbit the positively charged mass, and Pauli would later formulate his exclusion principle which stated that no two electrons can bear the same electron state. Schrodinger, along with de Broglie’s theory for electron waves, helped illuminate why the electrons appear to behave that way according to the laws of quantum mechanics – Schrodinger’s quantum wave model allowed for the analysis of waves with the use of quantum mechanics. Applying the Schrodinger wave model to de Broglie’s electron waves and bearing the Pauli exclusion principle in mind can be used to engineer devices that can energize particles and produce miraculous effects by evaluating the energy levels of existing materials in nature.
Annotated Bibliography
Maxwell, James. A Treatise on Electricity and Magnetism, Cambridge Press, 1873.
This is the primary source for Maxwell’s equations, and functions as one of the first textbooks on electricity and magnetism, synthesizing many of the developments in the field up until that point.
Michaelson, Albert A., On the Relative Motion of the Earth and the Luminiferous Ether, American Journal of Science, 1887.
This paper outlines the Michaelson-Morley experiment proving false the notion of the existence of this ‘luminiferous ether’ envisioned by Maxwell.
Planck, Max, On the Law of Energy Distribution in the Normal Spectrum, 1901, Annalen der Physik.
This paper forms the basis for Max Plancks observations on black-body radiators, and details his experiments and models for the foundation of Quantum Mechanics.
Bohr, Niels, On the Constitution of Atoms and Molecules, Philosophy Magazine 26, 1913.
This revolutionary work outline’s Bohr’s new atomic model, and details his experiments with spectroscopy and energized gases. It’s in three parts, but the first part is primarily considered here.
Bohr, Niels, Niels Bohr, Collected Works, Volume 3: The Correspondence Principle, 1918-1923, trans. 1976, Amsterdam, North Holland
This is the work in which Bohr discusses his correspondence principle and elucidates on how the models of classical physics and modern physics should be eventually reach agreement.
D’Agostino, Salvo., Hertz’s Researches on Electromagnetic Waves, 1975, Historical Studies in the Physical Sciences, Vol. 6, pp 261-323
This background source provides academic validation of Hertz’s discoveries. Heinrich Hertz’s career was tragically short, so the material he produced was slightly limited, but his discoveries were earthshattering.
Thomson, J. J., Cathode Rays, 1897, Philosophy Magazine, 44, 293
Thomson presents his research and discovery of electrons in this paper, and begins the study of subatomic particles in this paper.
Rutherford, Ernest., The Scattering of α and β Particles by Matter and the Structure of the Atom, 1911, Philosophical Magazine, Series 6, vol. 21, p. 669-688
In this paper Rutherford proves the existence of the nucleus and disproves Thomson’s “plum pudding” model.
Einstein, Albert. On a Heuristic Point of View about the Creation and Conversion of Light, 1905, Annalen der Physik. 17(6): 132–148
This is the famous paper describing the mechanism of the photoelectric effect, and contributed to the field of QM by introducing the concept that photons could be discrete entities with wave-particle duality.
de Broglie, Louis, On the Theory of Quanta, Annalen der Physik, 10(3), 1925
Louis de Broglie outlines the idea for ‘matter-waves’ in regards to the theory of electrons, to great effect for the future research of semiconductors.
Schrödinger, Erwin. Collected Papers on Wave Mechanics, University of Berlin, 1928 (Translation: Blackie & Son Ltd.)
In this collection, the quantum mechanical wave equation was formed, providing a statistical insight for the physicists working at the quantum level.
APS News, January 1925: Wolfgang Pauli announces the exclusion principle, January 2007 (https://www.aps.org/publications/apsnews/200701/history.cfm )
This article just serves for some supplementary information regarding the Pauli principle. Apparently, Pauli didn’t publish with great frequency.
B.J. Hunt .The Maxwellians, Cornell University Press (1991)
This work contains some background information on Oliver Heaviside and some of the others that worked with Maxwell during the period. It provides a bit of extra information regarding how the original Maxwell’s equations were simplified (much to the appreciation of students of science everywhere).
1Maxwell, A Treatise on Electricity and Magnetism, p. xiv
2Michaelson, The Relative Motion of the Earth and the Luminiferous Ether, American Journal of Science, p. 22
3D’Agostino, Hertz’s Researches on Electromagnetic Waves, Historical Studies in the Physical Sciences, Vol. 6 p. 261
4Einstein, On a Heuristic Point of View about the Creation and Conversion of Light, p. 92
5Planck, On the Law of the Energy Distribution in the Normal Spectrum, p. 1
6Einstein, Creation and Conversion of Light, p. 102
7Thomson, Cathode Rays, p. 1
8Rutherford, The Scattering of Alpha and Beta Particles by Matter and the Structure of the Atom, p. 688
9Bohr, II: On the constitution of atoms and molecules, p. 2
10Louis de Broglie, On the Theory of Quanta, p. 1
11APS News, January 2007 – This Month in Physics History, January 1925: Wolfgang Pauli announces his exclusion principle
12Bohr, I. On the constitution of atoms and molecules, p 8-9
13Bohr, The Correspondence Principle, p.6
14Schrodinger, Collected Papers on Wave Mechanics, p. ix
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