Chapter 6: The old quantum mechanics

Synopsis

Quantum mechanics began with the study of the electromagnetic radiation emitted by hot bodies. Isaac Newton used a simple glass prism to show that white sunlight comprises a spectrum of colours. In the nineteenth century physicists invented spectroscopes to measure spectra precisely. In 1860 Gustav Kirchoff showed that hot black bodies must have a characteristic spectrum depending on their temperature. In 1900 Max Planck found an mathematical representation of this spectrum. He introduced a new fundamental constant, the quantum of action, relating the frequency of radiation to its energy. The problem then became to explain the structures in the atom that gave rise to these spectra. Progress was slow but laboratories continued to generate data and it eventually became clear that atomic structures embody stationary superpositions in a complex vector space representing the electronic structure of the atom.

6.1: The end of the classical era

6.2: Kirchoff, black bodies and Planck

6.3: The quantum of action is a fundamental fixed point in the Universe

6.4: Wave or particle? Einstein and the photon

6.5: The Bohr model of the hydrogen atom

6.6: Louis de Broglie and the wave nature of the electron

6.7: The Old Quantum Mechanics

6.1: The end of the classical era

Isaac Newton’s work laid the foundation for classical mechanics. He was aware of magnetism and static electricity but as he writes at the conclusion of his General Scholium he was in no position to study them experimentally but thought they would be important in the future:

And now we might add something concerning a certain most subtle Spirit, which pervades and lies hid in all gross bodies; by the force and action of which Spirit, the particles of bodies mutually attract one another at near distances, and cohere, if contiguous; and electric bodies operate to greater distances, as well repelling as attracting the neighbouring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explain’d in few words, nor are we furnish’d with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic spirit operates. Isaac Newton (1713): The General Scholium to the Principia Mathematica

Seventy five years after Newton’s death, Alessandro Volta opened a whole new world of electrical phenomena. At the beginning of the nineteenth century he discovered a simple chemical battery that produced continuous electrical current, making possible the development of classical electrodynamics. Many people contributed to this work. Michael Faraday laid the foundations for electric motors, generators, transformers, and many other aspects of electrical power engineering. James Clerk Maxwell summed up the laws of classical electrodynamics in a set of mathematical equations that suggested that light is an electromagnetic phenomenon. Heinrich Hertz showed that radio waves are indeed electromagnetic. Meanwhile spectroscopists had been observing and measuring the spectra of the electromagnetic radiation of hot matter, and the scene was set for quantum mechanics to appear. The spectroscopists produced measurements of atomic spectra with ever increasing precision as their methods and instrumentation improved. Alessandro Volta - Wikipedia, Michael Faraday - Wikipedia, James Clerk Maxwell - Wikipedia, Heinrich Hertz - Wikipedia

6.2: Kirchoff, black bodies and Planck

The journey toward quantum mechanics started in about 1860 when Gustav Kirchoff postulated that the law that spectral radiance, (that is the amount of energy carried by each frequency) is a universal function, one and the same for all black bodies, depending only on wavelength and temperature. This postulate set off first of all a search for a perfect black body, then efforts to measure the spectral radiance of such a body, and finally a search for the universal function. Different authors developed candidates for the function but all failed to reproduce the data until 1900 when Planck (in desperation he said) produced an equation based on the assumption that radiation is emitted in discrete packets each carrying energy proportional to its frequency. This proportionality is expressed by the fundamental equation of quantum theory, E = hf where h is a new universal constant, and f is the frequency expressed in cycles per second. Gustav Kirchoff - Wikipedia, Planck's Law - Wikipedia, Planck constant - Wikipedia

6.3: The quantum of action is a fundamental fixed point in the Universe

The quantum of action is exceedingly small and sets the fundamental scale of activity in the Universe. Interactions between fundamental particles, the basic processes in the Universe, are measured in quanta of action. In Minkowski space Planck’s constant has the same dimensions as angular momentum: ML²T^-1. Since kinetic energy, mv², has the dimension ML²T^-2 and frequency has the dimension of inverse time T^-1, we can see that Planck’s quantum definition of energy is consistent with his interpretation of h as the constant of proportionality between energy and frequency.

The quantum of action is now a precisely defined natural constant which is used as a foundation for units of classical physics. The unit of time is the second. It was once defined as 1 / 86 400 of a day, since there are 86 400 seconds in a day of 24 hours x 60 minutes x 60 seconds. Since day lengths vary, the second is now defined as 9 192 631 770 periods of the frequency of a certain electronic transformation in caesium. Second - Wikipedia

The metre is was originally defined as one ten millionth of the distance from the equator to the North Pole along a great circle. Later it was defined more conveniently as the length of a metal bar, and now it is defined in terms of the speed of light as the distance travelled by a light beam in 1 / 299 792 458 of the second defined by the caesium frequency. Metre - Wikipedia

The kilogram was was first defined as the mass of a a certain volume of water measured in terms of the metre. In 1889 this was converted into a mass of metal known as the International Prototype Kilogram. Now, since 2019, it has been defined in terms of the Planck constant, the speed of light and the caesium frequency, effectively defining the kilogram in terms of the second and the metre. Planck’s constant has now been given a fixed value, as has the speed of light. Planck’s constant is now defined to be 6.626 070 15 ×10⁻³⁴ kg.m².s⁻¹. David Brynn Hibbert (2019): The way we define kilograms, metres and seconds changes today

Many consider Planck’s constant to be a measure of uncertainty, but this is a fictitious interpretation which causes much trouble in physics, as we shall see as we go on.

6.4: Wave or particle? Einstein and the photon

It was commonly believed that Maxwell and Hertz had shown beyond reasonable doubt that light is an electromagnetic wave. Since it was hard to believe that a wave could exist without something to carry it, as water carries the waves in the sea, it was widely believed that the Universe must be permeated with ether, a term coined by analogy to the aether that the ancients believed to be the fifth element that occupied the heavens.

On the other hand, Planck had found the Kirchoff’s universal function by assuming that radiation was emitted in packets or quanta whose energy depended on their frequency. This conflicted somewhat with the idea that radiation is a wave. To avoid this contradiction Planck’s idea was cast as a mathematical trick, or perhaps as a property of the radiating body rather than of the energy radiated.

Einstein was not happy with this and produced a statistical argument that radiation could be considered as a gas of particles (later called photons). Thus began the quantum conundrum of wave particle duality. This was solved by Louis de Broglie, who explained that one cannot have particles without waves. First, however, we must stop to consider the fourth major breakthrough on the road the quantum mechanics (the first three having been achieved by Kirchoff, Planck and Einstein). Albert Einstein (1905c): On a heuristic point of view concerning the production and transformation of light

6.5: The Bohr model of the hydrogen atom

At the turn of the century the existence of atoms had not been settled but experimenters had become quite familiar with electrons. These particles had been discovered by Joseph J Thompson in 1897 while studying cathode rays. Thompson had proposed a plum pudding model of the atom, the electrons being the plums, as some believed that they are constituents of atoms.

In 1911 Ernest Rutherford had his a students Geiger and Marsden bombard gold foil with alpha particles emitted by radium. Alphas particles are helium nuclei about 7000 times heavier than an electron. If gold foil was made of plum pudding, the alpha particles could be expected to go straight through. This did not always happen. Some of the alpha particles bounced straight back, leading Rutherford to believe that that the gold atoms had a heavy nucleus in the middle. The new model of the atom became a nucleus surrounded by electrons. Rutherford model - Wikipedia

By this time a lot of spectroscopic data had been collected from atoms and people were looking for meaning in the forest of wavelengths. In 1885 Johannes Balmer found a simple formula that connected four lines in the hydrogen spectrum. This showed there was some order in the chaos. Meanwhile Niels Bohr in Copenhagen had begun to imagine that the electrons in an atom circulated around the nucleus like a little solar system. Balmer series - Wikipedia

Classical electrodynamics said electrons moving in a curved orbit should radiate electromagnetic waves, lose energy and spiral into the nucleus. Bohr, without knowing why, decided that this didn’t happen. Instead an electron which absorbed a photon would gain energy and jump to a higher orbit. An electron that radiated a photon would lose energy and fall to a lower orbit. Using the data he had, he calculated that this process could be the source of the Balmer series. This was a first definitive step in the study of the electronic structure of atoms. This was a great start, but it only worked for hydrogen, an atoms with one electron. Bohr model - Wikipedia

6.6: Louis de Broglie and the wave nature of the electron

Wave particle duality seemed to be an impossible contradiction until people began to think of particles as harmonies, fixed points in moving waves, the nodes of standing waves. This idea occurred to Louis de Broglie. He explained his train of thought in his Nobel lecture:

The necessity of assuming for light two contradictory theories - that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics. Now a purely corpuscular theory does not contain any element permitting the definition of frequency. This also renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity.

On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too. Louis de Broglie (1929): Nobel Lecture: The Wave Nature of the Electron

Now we can see how the complex numbers (described in Chapter 5) are a bridge between the classical and quantum worlds. Classical physics assumes that the world is continuous. The old saying natura non facit saltum, nature does not jump, is wrong. Nature is quantized. Complex numbers are continuous. They rotate continuously on the complex plane like the second hand of a stopwatch. But they are also quantized. Each complete turn is a unit, a quantum of time or action.

6.7: The Old Quantum Mechanics

The first era of quantum mechanics, from 1900 to about the middle of the twenties is known as the old quantum mechanics. Observation was paramount, many new features were discovered by spectroscopists. In general explanation trailed experiment until Werner Heisenberg revealed the basic problem and its solution with a mathematical construction that came to be called matrix mechanics. Not long afterwards Erwin Schrödinger, following Louis de Broglie, produced wave mechanics. John von Neumann showed that Heisenberg and his contemporaries Born, Jordan and Schrodinger needed a new space to describe the world. Von Neumann showed that our natural home is Hilbert space, an abstract invisible musical space that handles all the computational work necessary to make the Universe go. It is the soul or mind of cognitive cosmogenesis. Stern-Gerlach experiment - Wikipedia, John von Neumann (2014): Mathematical Foundations of Quantum Mechanics

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Notes and references

Links

Albert Einstein (1905c), On a heuristic point of view concerning the production and transformation of light, ' The wave theory of light, which operates with continuous spatial functions, has proved itself splendidly in describing purely optical phenomena and will probably never be replaced by another theory. One should keep in mind, however, that optical observations apply to time averages and not to momentary values, and it is conceivable that despite the complete confirmation of the theories of diffraction, reflection, refraction, dispersion, etc., by experiment, the theory of light, which operates with continuous spatial functions, may lead to contradictions with experience when it is applied to the phenomena of production and transformation of light. Indeed, it seems to me that the observations regarding "black-body" light, and other groups of phenomena associated with the production or conversion of light can be understood better if one assumes that the energy of light is discontinuously distributed in space.' back

Alessandro Volta - Wikipedia, Alessandro Volta - Wikipedia, the free encyclopedia, 'Alessandro Giuseppe Antonio Anastasio Volta (18 February 1745 – 5 March 1827) was an Italian physicist and chemist who was a pioneer of electricity and power who is credited as the inventor of the electric battery and the discoverer of methane. He invented the voltaic pile in 1799, and reported the results of his experiments in 1800 in a two-part letter to the president of the Royal Society.' back

Balmer series - Wikipedia, Balmer series - Wikipedia, the free encyclopedia, ' The Balmer series is characterized by the electron transitioning from n ≥ 3 to n = 2, where n refers to the radial quantum number or principal quantum number of the electron. The transitions are named sequentially by Greek letter: n = 3 to n = 2 is called H-α, 4 to 2 is H-β, 5 to 2 is H-γ, and 6 to 2 is H-δ. As the first spectral lines associated with this series are located in the visible part of the electromagnetic spectrum, these lines are historically referred to as "H-alpha", "H-beta", "H-gamma", and so on, where H is the element hydrogen.' back

Bohr model - Wikipedia, Bohr model - Wikipedia, the free encyclopedia, 'In atomic physics, the Rutherford–Bohr model or Bohr model, introduced by Niels Bohr in 1913, depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus—similar in structure to the solar system, but with attraction provided by electrostatic forces rather than gravity.' back

David Brynn Hibbert (2019), The way we define kilograms, metres and seconds changes today, ' From today the kilogram is defined using the Planck constant, something that doesn’t change from quantum physics. The challenge now though is to explain these new definitions to people – especially non-scientists – so they understand. Comparing a kilogram to a metal block is easy. Technically a kilogram (kg) is now defined: […] by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10–34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and ΔνCs.' back

Gustav Kirchoff - Wikipedia, Gustav Kirchoff - Wikipedia, the free encyclopedia, ' Gustav Robert Kirchhoff (12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coined the term "black body" radiation in 1862, and two different sets of concepts (one in circuit theory, and one in spectroscopy) are named "Kirchhoff's laws" after him; there is also a Kirchhoff's Law in thermochemistry. The Bunsen–Kirchhoff Award for spectroscopy is named after him and his colleague, Robert Bunsen.' back

Heinrich Hertz - Wikipedia, Heinrich Hertz - Wikipedia, the free encyclopedia, ' Heinrich Rudolf Hertz (22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The unit of frequency, cycle per second, was named the "hertz" in his honor.' back

Isaac Newton (1713), The General Scholium to the Principia Mathematica, ' Published for the first time as an appendix to the 2nd (1713) edition of the Principia, the General Scholium reappeared in the 3rd (1726) edition with some amendments and additions. As well as countering the natural philosophy of Leibniz and the Cartesians, the General Scholium contains an excursion into natural theology and theology proper. In this short text, Newton articulates the design argument (which he fervently believed was furthered by the contents of his Principia), but also includes an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity, which Newton saw as a post-biblical corruption. The English translation here is that of Andrew Motte (1729). Italics and orthography as in original.' back

James Clerk Maxwell - Wikipedia, James Clerk Maxwell - Wikipedia, the free encyclopedia, 'James Clerk Maxwell FRS FRSE (13 June 1831 – 5 November 1879) was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.' back

John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back

Louis de Broglie (1929), Nobel Lecture: The Wave Nature of the Electron, ' Nevertheless, it was still necessary to adopt the wave theory to account for interference and diffraction phenomena and no way whatsoever of reconciling the wave theory with the existence of light corpuscles could be visualized. The necessity of assuming for light two contradictory theories-that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics.Now a purely corpuscular theory does not contain any element permitting the definition of frequency. This also renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity. On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr's expression, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. . . .. They showed clearly that it was possible to establish a correspondence between waves and corpuscles such that the laws of mechanics correspond to the laws of geometrical optics. . . .. This prompted the thought that classical mechanics is also only an approximation relative to a vaster wave mechanics. I stated as much almost at the outset of my studies, i.e. "A new mechanics must be developed which is to classical mechanics what wave optics is to geometrical optics". This new mechanics has since been developed, thanks mainly to the fine work done by Schrödinger. . . .. I cannot attempt even briefly to sum up here the development of the new mechanics. I merely wish to say that on examination it proved to be identical with a mechanics independently developed, first by Heisenberg, then by Born, Jordan, Pauli, Dirac, etc quantum mechanics. The two mechanics, wave and quantum, are equivalent from the mathematical point of view. . . .. Since the wavelength of the electron waves is of the order of that of X-rays, it must be expected that crystals can cause diffraction of these waves completely analogous to the Laue phenomenon. . . . Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predict a whole group of phenomena without their actually having been discovered. And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.' back

Metre - Wikipedia, Metre - Wikipedia, the free encyclopedia, 'The metre (British spelling), or meter (American spelling), (SI unit symbol: m), is the fundamental unit of length (SI dimension symbol: L) in the International System of Units (SI), which is maintained by the BIPM. [The International Burea of Weights and Measures] Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level), its definition has been periodically refined to reflect growing knowledge of metrology. Since 1983, it has been defined as "the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second." ' back

Michael Faraday - Wikipedia, Michael Faraday - Wikipedia, the free encyclopedia, ' Michael Faraday FRS September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis. Although Faraday received little formal education, he was one of the most influential scientists in history. It was by his research on the magnetic field around a conductor carrying a direct current that Faraday established the concept of the electromagnetic field in physics. . . . His inventions of electromagnetic rotary devices formed the foundation of electric motor technology, and it was largely due to his efforts that electricity became practical for use in technology.' back

Planck constant - Wikipedia, Planck constant - Wikipedia, the free encyclopedia, ' Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram (IPK), that had defined the kilogram since 1889. . . . For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.626 070 150 × 10-34 J⋅s exactly. ' back

Planck's Law - Wikipedia, Planck's Law - Wikipedia, the free encyclopedia, 'In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature T. As a function of frequency ν. back

Rutherford model - Wikipedia, Rutherford model - Wikipedia, the free encyclopedia, ' The Rutherford model was devised by the New Zealand-born physicist Ernest Rutherford to describe an atom. Rutherford directed the Geiger–Marsden experiment in 1909, which suggested, upon Rutherford's 1911 analysis, that J. J. Thomson's plum pudding model of the atom was incorrect. Rutherford's new model for the atom, based on the experimental results, contained new features of a relatively high central charge concentrated into a very small volume in comparison to the rest of the atom and with this central volume also containing the bulk of the atomic mass of the atom. This region would be known as the "nucleus" of the atom.' back

Second - Wikipedia, Second - Wikipedia, the free encyclopedia, ' The current and formal definition in the International System of Units (SI) is more precise: The second [...] is defined by taking the fixed numerical value of the caesium frequency, Δν_Cs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s⁻¹. back

Stern-Gerlach experiment - Wikipedia, Stern-Gerlach experiment - Wikipedia, the free encyclopedia, 'The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. It demonstrated that atomic-scale systems have intrinsically quantum properties, and that measurement in quantum mechanics affects the system being measured.' back