November 26, 2022November 27, 2023 by David D. Nolte

A Short History of Quantum Entanglement

Despite the many apparent paradoxes posed in physics—the twin and ladder paradoxes of relativity theory, Olber’s paradox of the bright night sky, Loschmitt’s paradox of irreversible statistical fluctuations—these are resolved by a deeper look at the underlying assumptions—the twin paradox is resolved by considering shifts in reference frames, the ladder paradox is resolved by the loss of simultaneity, Olber’s paradox is resolved by a finite age to the universe, and Loschmitt’s paradox is resolved by fluctuation theorems. In each case, no physical principle is violated, and each paradox is fully explained.

However, there is at least one “true” paradox in physics that defies consistent explanation—quantum entanglement. Quantum entanglement was first described by Einstein with colleagues Podolsky and Rosen in the famous EPR paper of 1935 as an argument against the completeness of quantum mechanics, and it was given its name by Schrödinger the same year in the paper where he introduced his “cat” as a burlesque consequence of entanglement.

Here is a short history of quantum entanglement [1], from its beginnings in 1935 to the recent 2022 Nobel prize in Physics awarded to John Clauser, Alain Aspect and Anton Zeilinger.

The EPR Papers of 1935

Einstein can be considered as the father of quantum mechanics, even over Planck, because of his 1905 derivation of the existence of the photon as a discrete carrier of a quantum of energy (see Einstein versus Planck). Even so, as Heisenberg and Bohr advanced quantum mechanics in the mid 1920’s, emphasizing the underlying non-deterministic outcomes of measurements, and in particular the notion of instantaneous wavefunction collapse, they pushed the theory in directions that Einstein found increasingly disturbing and unacceptable.

This feature is an excerpt from an upcoming book, Interference: The History of Optical Interferometry and the Scientists Who Tamed Light (Oxford University Press, July 2023), by David D. Nolte.

At the invitation-only Solvay Congresses of 1927 and 1930, where all the top physicists met to debate the latest advances, Einstein and Bohr began a running debate that was epic in the history of physics as the two top minds went head-to-head as the onlookers looked on in awe. Ultimately, Einstein was on the losing end. Although he was convinced that something was missing in quantum theory, he could not counter all of Bohr’s rejoinders, even as Einstein’s assaults became ever more sophisticated, and he left the field of battle beaten but not convinced. Several years later he launched his last and ultimate salvo.

At the Institute for Advanced Study in Princeton, New Jersey, in the 1930’s Einstein was working with Nathan Rosen and Boris Podolsky when he envisioned a fundamental paradox in quantum theory that occurred when two widely-separated quantum particles were required to share specific physical properties because of simple conservation theorems like energy and momentum. Even Bohr and Heisenberg could not deny the principle of conservation of energy and momentum, and Einstein devised a two-particle system for which these conservation principles led to an apparent violation of Heisenberg’s own uncertainty principle. He left the details to his colleagues, with Podolsky writing up the main arguments. They published the paper in the Physical Review in March of 1935 with the title “Can Quantum-Mechanical Description of Physical Reality be Considered Complete” [2]. Because of the three names on the paper (Einstein, Podolsky, Rosen), it became known as the EPR paper, and the paradox they presented became known as the EPR paradox.

When Bohr read the paper, he was initially stumped and aghast. He felt that EPR had shaken the very foundations of the quantum theory that he and his institute had fought so hard to establish. He also suspected that EPR had made a mistake in their arguments, and he halted all work at his institute in Copenhagen until they could construct a definitive answer. A few months later, Bohr published a paper in the Physical Review in July of 1935, using the identical title that EPR had used, in which he refuted the EPR paradox [3]. There is not a single equation or figure in the paper, but he used his “awful incantation terminology” to maximum effect, showing that one of the EPR assumptions on the assessment of uncertainties to position and momentum was in error, and he was right.

Einstein was disgusted. He had hoped that this ultimate argument against the completeness of quantum mechanics would stand the test of time, but Bohr had shot it down within mere months. Einstein was particularly disappointed with Podolsky, because Podolsky had tried too hard to make the argument specific to position and momentum, leaving a loophole for Bohr to wiggle through, where Einstein had wanted the argument to rest on deeper and more general principles.

Despite Bohr’s victory, Einstein had been correct in his initial formulation of the EPR paradox that showed quantum mechanics did not jibe with common notions of reality. He and Schrödinger exchanged letters commiserating with each other and encouraging each other in their counter beliefs against Bohr and Heisenberg. In November of 1935, Schrödinger published a broad, mostly philosophical, paper in Naturwissenschaften [4] in which he amplified the EPR paradox with the use of an absurd—what he called burlesque—consequence of wavefunction collapse that became known as Schrödinger’s Cat. He also gave the central property of the EPR paradox its name: entanglement.

Ironically, both Einstein’s entanglement paradox and Schrödinger’s Cat, which were formulated originally to be arguments against the validity of quantum theory, have become established quantum tools. Today, entangled particles are the core workhorses of quantum information systems, and physicists are building larger and larger versions of Schrödinger’s Cat that may eventually merge with the physics of the macroscopic world.

Bohm and Ahronov Tackle EPR

The physicist David Bohm was a rare political exile from the United States. He was born in the heart of Pennsylvania in the town of Wilkes-Barre, attended Penn State and then the University of California at Berkeley, where he joined Robert Oppenheimer’s research group. While there, he became deeply involved in the fight for unions and socialism, activities for which he was called before McCarthy’s Committee on Un-American Activities. He invoked his right to the fifth amendment for which he was arrested. Although he was later acquitted, Princeton University fired him from his faculty position, and fearing another arrest, he fled to Brazil where his US passport was confiscated by American authorities. He had become a physicist without a country.

Despite his personal trials, Bohm remained scientifically productive. He published his influential textbook on quantum mechanics in the midst of his Senate hearings, and after a particularly stimulating discussion with Einstein shortly before he fled the US, he developed and published an alternative version of quantum theory in 1952 that was fully deterministic—removing Einstein’s “God playing dice”—by creating a hidden-variable theory [5].

Hidden-variable theories of quantum mechanics seek to remove the randomness of quantum measurement by assuming that some deeper element of quantum phenomena—a hidden variable—explains each outcome. But it is also assumed that these hidden variables are not directly accessible to experiment. In this sense, the quantum theory of Bohr and Heisenberg was “correct” but not “complete”, because there were things that the theory could not predict or explain.

Bohm’s hidden variable theory, based on a quantum potential, was able to reproduce all the known results of standard quantum theory without invoking the random experimental outcomes that Einstein abhorred. However, it still contained one crucial element that could not sweep away the EPR paradox—it was nonlocal.

Nonlocality lies at the heart of quantum theory. In its simplest form, the nonlocal nature of quantum phenomenon says that quantum states span spacetime with space-like separations, meaning that parts of the wavefunction are non-causally connected to other parts of the wavefunction. Because Einstein was fundamentally committed to causality, the nonlocality of quantum theory was what he found most objectionable, and Bohm’s elegant hidden-variable theory, that removed Einstein’s dreaded randomness, could not remove that last objection of non-causality.

After working in Brazil for several years, Bohm moved to the Technion University in Israel where he began a fruitful collaboration with Yakir Ahronov. In addition to proposing the Ahronov-Bohm effect, in 1957 they reformulated Podolsky’s version of the EPR paradox that relied on continuous values of position and momentum and replaced it with a much simpler model based on the Stern-Gerlach effect on spins and further to the case of positronium decay into two photons with correlated polarizations. Bohm and Ahronov reassessed experimental results of positronium decay that had been made by Madame Wu in 1950 at Columbia University and found it in full agreement with standard quantum theory.

John Bell’s Inequalities

John Stuart Bell had an unusual start for a physicist. His family was too poor to give him an education appropriate to his skills, so he enrolled in vocational school where he took practical classes that included brick laying. Working later as a technician in a university lab, he caught the attention of his professors who sponsored him to attend the university. With a degree in physics, he began working at CERN as an accelerator designer when he again caught the attention of his supervisors who sponsored him to attend graduate school. He graduated with a PhD and returned to CERN as a card-carrying physicist with all the rights and privileges that entailed.

During his university days, he had been fascinated by the EPR paradox, and he continued thinking about the fundamentals of quantum theory. On a sabbatical to the Stanford accelerator in 1960 he began putting mathematics to the EPR paradox to see whether any local hidden variable theory could be compatible with quantum mechanics. His analysis was fully general, so that it could rule out as-yet-unthought-of hidden-variable theories. The result of this work was a set of inequalities that must be obeyed by any local hidden-variable theory. Then he made a simple check using the known results of quantum measurement and showed that his inequalities are violated by quantum systems. This ruled out the possibility of any local hidden variable theory (but not Bohm’s nonlocal hidden-variable theory). Bell published his analysis in 1964 [6] in an obscure journal that almost no one read…except for a curious graduate student at Columbia University who began digging into the fundamental underpinnings of quantum theory against his supervisor’s advice.

Fig. 4 Polarization measurements on entangled photons violate Bell’s inequality.

John Clauser’s Tenacious Pursuit

As a graduate student in astrophysics at Columbia University, John Clauser was supposed to be doing astrophysics. Instead, he spent his time musing over the fundamentals of quantum theory. In 1967 Clauser stumbled across Bell’s paper while he was in the library. The paper caught his imagination, but he also recognized that the inequalities were not experimentally testable, because they required measurements that depended directly on hidden variables, which are not accessible. He began thinking of ways to construct similar inequalities that could be put to an experimental test, and he wrote about his ideas to Bell, who responded with encouragement. Clauser wrote up his ideas in an abstract for an upcoming meeting of the American Physical Society, where one of the abstract reviewers was Abner Shimony of Boston University. Clauser was surprised weeks later when he received a telephone call from Shimony. Shimony and his graduate student Micheal Horne had been thinking along similar lines, and Shimony proposed to Clauser that they join forces. They met in Boston where they were met Richard Holt, a graudate student at Harvard who was working on experimental tests of quantum mechanics. Collectively, they devised a new type of Bell inequality that could be put to experimental test [7]. The result has become known as the CHSH Bell inequality (after Clauser, Horne, Shimony and Holt).

When Clauser took a post-doc position in Berkeley, he began searching for a way to do the experiments to test the CHSH inequality, even though Holt had a head start at Harvard. Clauser enlisted the help of Charles Townes, who convinced one of the Berkeley faculty to loan Clauser his graduate student, Stuart Freedman, to help. Clauser and Freedman performed the experiments, using a two-photon optical decay of calcium ions and found a violation of the CHSH inequality by 5 standard deviations, publishing their result in 1972 [8].

Fig. 6 CHSH inequality violated by entangled photons.

Alain Aspect’s Non-locality

Just as Clauser’s life was changed when he stumbled on Bell’s obscure paper in 1967, the paper had the same effect on the life of French physicist Alain Aspect who stumbled on it in 1975. Like Clauser, he also sought out Bell for his opinion, meeting with him in Geneva, and Aspect similarly received Bell’s encouragement, this time with the hope to build upon Clauser’s work.

In some respects, the conceptual breakthrough achieved by Clauser had been the CHSH inequality that could be tested experimentally. The subsequent Clauser Freedman experiments were not a conclusion, but were just the beginning, opening the door to deeper tests. For instance, in the Clauser-Freedman experiments, the polarizers were static, and the detectors were not widely separated, which allowed the measurements to be time-like separated in spacetime. Therefore, the fundamental non-local nature of quantum physics had not been tested.

Aspect began a thorough and systematic program, that would take him nearly a decade to complete, to test the CHSH inequality under conditions of non-locality. He began with a much brighter source of photons produced using laser excitation of the calcium ions. This allowed him to perform the experiment in 100’s of seconds instead of the hundreds of hours by Clauser. With such a high data rate, Aspect was able to verify violation of the Bell inequality to 10 standard deviations, published in 1981 [9].

However, the real goal was to change the orientations of the polarizers while the photons were in flight to widely separated detectors [10]. This experiment would allow the detection to be space-like separated in spacetime. The experiments were performed using fast-switching acoustic-optic modulators, and the Bell inequality was violated to 5 standard deviations [11]. This was the most stringent test yet performed and the first to fully demonstrate the non-local nature of quantum physics.

Anton Zeilinger: Master of Entanglement

If there is one physicist today whose work encompasses the broadest range of entangled phenomena, it would be the Austrian physicist, Anton Zeilinger. He began his career in neutron interferometery, but when he was bitten by the entanglement bug in 1976, he switched to quantum photonics because of the superior control that can be exercised using optics over sources and receivers and all the optical manipulations in between.

Working with Daniel Greenberger and Micheal Horne, they took the essential next step past the Bohm two-particle entanglement to consider a 3-particle entangled state that had surprising properties. While the violation of locality by the two-particle entanglement was observed through the statistical properties of many measurements, the new 3-particle entanglement could show violations on single measurements, further strengthening the arguments for quantum non-locality. This new state is called the GHZ state (after Greenberger, Horne and Zeilinger) [12].

As the Zeilinger group in Vienna was working towards experimental demonstrations of the GHZ state, Charles Bennett of IBM proposed the possibility for quantum teleportation, using entanglement as a core quantum information resource [13]. Zeilinger realized that his experimental set-up could perform an experimental demonstration of the effect, and in a rapid re-tooling of the experimental apparatus [14], the Zeilinger group was the first to demonstrate quantum teleportation that satisfied the conditions of the Bennett teleportation proposal [15]. An Italian-UK collaboration also made an early demonstration of a related form of teleportation in a paper that was submitted first, but published after Zeilinger’s, due to delays in review [16]. But teleportation was just one of a widening array of quantum applications for entanglement that was pursued by the Zeilinger group over the succeeding 30 years [17], including entanglement swapping, quantum repeaters, and entanglement-based quantum cryptography. Perhaps most striking, he has worked on projects at astronomical observatories that entangle photons coming from cosmic sources.

By David D. Nolte Nov. 26, 2022

Video Lectures

Physics Colloquium on the Backstory of the 2023 Nobel Prize in Physics

Timeline

1935 – Einstein EPR

1935 – Bohr EPR

1935 – Schrödinger: Entanglement and Cat

1950 – Madam Wu positron decay

1952 – David Bohm and Non-local hidden variables

1957 – Bohm and Ahronov version of EPR

1963 – Bell’s inequalities

1967 – Clauser reads Bell’s paper

1967 – Commins experiment with Calcium

1969 – CHSH inequality: measurable with detection inefficiencies

1972 – Clauser and Freedman experiment

1975 – Aspect reads Bell’s paper

1976 – Zeilinger reads Bell’s paper

1981 – Aspect two-photon generation source

1982 – Aspect time variable analyzers

1988 – Parametric down-conversion of EPR pairs (Shih and Alley, Ou and Mandel)

1989 – GHZ state proposed

1993 – Bennett quantum teleportation proposal

1995 – High-intensity down-conversion source of EPR pairs (Kwiat and Zeilinger)

1997 – Zeilinger quantum teleportation experiment

1999 – Observation of the GHZ state

Bibliography

[1] See the full details in: David D. Nolte, Interference: A History of Interferometry and the Scientists Who Tamed Light (Oxford University Press, July 2023)

[2] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Physical Review 47, 0777-0780 (1935).

[3] N. Bohr, Can quantum-mechanical description of physical reality be considered complete? Physical Review 48, 696-702 (1935).

[4] E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik. Die Naturwissenschaften 23, 807-12; 823-28; 844-49 (1935).

[5] D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .1. Physical Review 85, 166-179 (1952); D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables .2. Physical Review 85, 180-193 (1952).

[6] J. Bell, On the Einstein-Podolsky-Rosen paradox. Physics 1, 195 (1964).

[7] 1. J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Proposed experiment to test local hidden-variable theories. Physical Review Letters 23, 880-& (1969).

[8] S. J. Freedman, J. F. Clauser, Experimental test of local hidden-variable theories. Physical Review Letters 28, 938-& (1972).

[9] A. Aspect, P. Grangier, G. Roger, EXPERIMENTAL TESTS OF REALISTIC LOCAL THEORIES VIA BELLS THEOREM. Physical Review Letters 47, 460-463 (1981).

[10] Alain Aspect, Bell’s Theorem: The Naïve Veiw of an Experimentalit. (2004), hal- 00001079

[11] A. Aspect, J. Dalibard, G. Roger, EXPERIMENTAL TEST OF BELL INEQUALITIES USING TIME-VARYING ANALYZERS. Physical Review Letters 49, 1804-1807 (1982).

[12] D. M. Greenberger, M. A. Horne, A. Zeilinger, in 1988 Fall Workshop on Bells Theorem, Quantum Theory and Conceptions of the Universe. (George Mason Univ, Fairfax, Va, 1988), vol. 37, pp. 69-72.

[13] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Physical Review Letters 70, 1895-1899 (1993).

[14] J. Gea-Banacloche, Optical realizations of quantum teleportation, in Progress in Optics, Vol 46, E. Wolf, Ed. (2004), vol. 46, pp. 311-353.

[15] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Experimental quantum teleportation. Nature 390, 575-579 (1997).

[16] D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-podolsky-Rosen Channels. Phys. Rev. Lett. 80, 1121-1125 (1998).

[17] A. Zeilinger, Light for the quantum. Entangled photons and their applications: a very personal perspective. Physica Scripta 92, 1-33 (2017).

February 18, 2020March 19, 2023 by David D. Nolte

Bohr’s Orbits

The first time I ran across the Bohr-Sommerfeld quantization conditions I admit that I laughed! I was a TA for the Modern Physics course as a graduate student at Berkeley in 1982 and I read about Bohr-Sommerfeld in our Tipler textbook. I was familiar with Bohr orbits, which are already the wrong way of thinking about quantized systems. So the Bohr-Sommerfeld conditions, especially for so-called “elliptical” orbits, seemed like nonsense.

But it’s funny how a a little distance gives you perspective. Forty years later I know a little more physics than I did then, and I have gained a deep respect for an obscure property of dynamical systems known as “adiabatic invariants”. It turns out that adiabatic invariants lie at the core of quantum systems, and in the case of hydrogen adiabatic invariants can be visualized as … elliptical orbits!

Quantum Physics in Copenhagen

Niels Bohr (1885 – 1962) was born in Copenhagen, Denmark, the middle child of a physiology professor at the University in Copenhagen. Bohr grew up with his siblings as a faculty child, which meant an unconventional upbringing full of ideas, books and deep discussions. Bohr was a late bloomer in secondary school but began to show talent in Math and Physics in his last two years. When he entered the University in Copenhagen in 1903 to major in physics, the university had only one physics professor, Christian Christiansen, and had no physics laboratories. So Bohr tinkered in his father’s physiology laboratory, performing a detailed experimental study of the hydrodynamics of water jets, writing and submitting a paper that was to be his only experimental work. Bohr went on to receive a Master’s degree in 1909 and his PhD in 1911, writing his thesis on the theory of electrons in metals. Although the thesis did not break much new ground, it uncovered striking disparities between observed properties and theoretical predictions based on the classical theory of the electron. For his postdoc studies he applied for and was accepted to a position working with the discoverer of the electron, Sir J. J. Thompson, in Cambridge. Perhaps fortunately for the future history of physics, he did not get along well with Thompson, and he shifted his postdoc position in early 1912 to work with Ernest Rutherford at the much less prestigious University of Manchester.

Ernest Rutherford had just completed a series of detailed experiments on the scattering of alpha particles on gold film and had demonstrated that the mass of the atom was concentrated in a very small volume that Rutherford called the nucleus, which also carried the positive charge compensating the negative electron charges. The discovery of the nucleus created a radical new model of the atom in which electrons executed planetary-like orbits around the nucleus. Bohr immediately went to work on a theory for the new model of the atom. He worked closely with Rutherford and the other members of Rutherford’s laboratory, involved in daily discussions on the nature of atomic structure. The open intellectual atmosphere of Rutherford’s group and the ready flow of ideas in group discussions became the model for Bohr, who would some years later set up his own research center that would attract the top young physicists of the time. Already by mid 1912, Bohr was beginning to see a path forward, hinting in letters to his younger brother Harald (who would become a famous mathematician) that he had uncovered a new approach that might explain some of the observed properties of simple atoms.

By the end of 1912 his postdoc travel stipend was over, and he returned to Copenhagen, where he completed his work on the hydrogen atom. One of the key discrepancies in the classical theory of the electron in atoms was the requirement, by Maxwell’s Laws, for orbiting electrons to continually radiate because of their angular acceleration. Furthermore, from energy conservation, if they radiated continuously, the electron orbits must also eventually decay into the nuclear core with ever-decreasing orbital periods and hence ever higher emitted light frequencies. Experimentally, on the other hand, it was known that light emitted from atoms had only distinct quantized frequencies. To circumvent the problem of classical radiation, Bohr simply assumed what was observed, formulating the idea of stationary quantum states. Light emission (or absorption) could take place only when the energy of an electron changed discontinuously as it jumped from one stationary state to another, and there was a lowest stationary state below which the electron could never fall. He then took a critical and important step, combining this new idea of stationary states with Planck’s constant h. He was able to show that the emission spectrum of hydrogen, and hence the energies of the stationary states, could be derived if the angular momentum of the electron in a Hydrogen atom was quantized by integer amounts of Planck’s constant h.

Bohr published his quantum theory of the hydrogen atom in 1913, which immediately focused the attention of a growing group of physicists (including Einstein, Rutherford, Hilbert, Born, and Sommerfeld) on the new possibilities opened up by Bohr’s quantum theory [1]. Emboldened by his growing reputation, Bohr petitioned the university in Copenhagen to create a new faculty position in theoretical physics, and to appoint him to it. The University was not unreceptive, but university bureaucracies make decisions slowly, so Bohr returned to Rutherford’s group in Manchester while he awaited Copenhagen’s decision. He waited over two years, but he enjoyed his time in the stimulating environment of Rutherford’s group in Manchester, growing steadily into the role as master of the new quantum theory. In June of 1916, Bohr returned to Copenhagen and a year later was elected to the Royal Danish Academy of Sciences.

Although Bohr’s theory had succeeded in describing some of the properties of the electron in atoms, two central features of his theory continued to cause difficulty. The first was the limitation of the theory to single electrons in circular orbits, and the second was the cause of the discontinuous jumps. In response to this challenge, Arnold Sommerfeld provided a deeper mechanical perspective on the origins of the discrete energy levels of the atom.

Quantum Physics in Munich

Arnold Johannes Wilhem Sommerfeld (1868—1951) was born in Königsberg, Prussia, and spent all the years of his education there to his doctorate that he received in 1891. In Königsberg he was acquainted with Minkowski, Wien and Hilbert, and he was the doctoral student of Lindemann. He also was associated with a social group at the University that spent too much time drinking and dueling, a distraction that lead to his receiving a deep sabre cut on his forehead that became one of his distinguishing features along with his finely waxed moustache. In outward appearance, he looked the part of a Prussian hussar, but he finally escaped this life of dissipation and landed in Göttingen where he became Felix Klein’s assistant in 1894. He taught at local secondary schools, rising in reputation, until he secured a faculty position of theoretical physics at the University in Münich in 1906. One of his first students was Peter Debye who received his doctorate under Sommerfeld in 1908. Later famous students would include Peter Ewald (doctorate in 1912), Wolfgang Pauli (doctorate in 1921), Werner Heisenberg (doctorate in 1923), and Hans Bethe (doctorate in 1928). These students had the rare treat, during their time studying under Sommerfeld, of spending weekends in the winter skiing and staying at a ski hut that he owned only two hours by train outside of Münich. At the end of the day skiing, discussion would turn invariably to theoretical physics and the leading problems of the day. It was in his early days at Münich that Sommerfeld played a key role aiding the general acceptance of Minkowski’s theory of four-dimensional space-time by publishing a review article in Annalen der Physik that translated Minkowski’s ideas into language that was more familiar to physicists.

Around 1911, Sommerfeld shifted his research interest to the new quantum theory, and his interest only intensified after the publication of Bohr’s model of hydrogen in 1913. In 1915 Sommerfeld significantly extended the Bohr model by building on an idea put forward by Planck. While further justifying the black body spectrum, Planck turned to descriptions of the trajectory of a quantized one-dimensional harmonic oscillator in phase space. Planck had noted that the phase-space areas enclosed by the quantized trajectories were integral multiples of his constant. Sommerfeld expanded on this idea, showing that it was not the area enclosed by the trajectories that was fundamental, but the integral of the momentum over the spatial coordinate [2]. This integral is none other than the original action integral of Maupertuis and Euler, used so famously in their Principle of Least Action almost 200 years earlier. Where Planck, in his original paper of 1901, had recognized the units of his constant to be those of action, and hence called it the quantum of action, Sommerfeld made the explicit connection to the dynamical trajectories of the oscillators. He then showed that the same action principle applied to Bohr’s circular orbits for the electron on the hydrogen atom, and that the orbits need not even be circular, but could be elliptical Keplerian orbits.

The quantum condition for this otherwise classical trajectory was the requirement for the action integral over the motion to be equal to integer units of the quantum of action. Furthermore, Sommerfeld showed that there must be as many action integrals as degrees of freedom for the dynamical system. In the case of Keplerian orbits, there are radial coordinates as well as angular coordinates, and each action integral was quantized for the discrete electron orbits. Although Sommerfeld’s action integrals extended Bohr’s theory of quantized electron orbits, the new quantum conditions also created a problem because there were now many possible elliptical orbits that all had the same energy. How was one to find the “correct” orbit for a given orbital energy?

Quantum Physics in Leiden

In 1906, the Austrian Physicist Paul Ehrenfest (1880 – 1933), freshly out of his PhD under the supervision of Boltzmann, arrived at Göttingen only weeks before Boltzmann took his own life. Felix Klein at Göttingen had been relying on Boltzmann to provide a comprehensive review of statistical mechanics for the Mathematical Encyclopedia, so he now entrusted this project to the young Ehrenfest. It was a monumental task, which was to take him and his physicist wife Tatyana nearly five years to complete. Part of the delay was the desire by Ehrenfest to close some open problems that remained in Boltzmann’s work. One of these was a mechanical theorem of Boltzmann’s that identified properties of statistical mechanical systems that remained unaltered through a very slow change in system parameters. These properties would later be called adiabatic invariants by Einstein. Ehrenfest recognized that Wien’s displacement law, which had been a guiding light for Planck and his theory of black body radiation, had originally been derived by Wien using classical principles related to slow changes in the volume of a cavity. Ehrenfest was struck by the fact that such slow changes would not induce changes in the quantum numbers of the quantized states, and hence that the quantum numbers must be adiabatic invariants of the black body system. This not only explained why Wien’s displacement law continued to hold under quantum as well as classical considerations, but it also explained why Planck’s quantization of the energy of his simple oscillators was the only possible choice. For a classical harmonic oscillator, the ratio of the energy of oscillation to the frequency of oscillation is an adiabatic invariant, which is immediately recognized as Planck’s quantum condition .

Ehrenfest published his observations in 1913 [3], the same year that Bohr published his theory of the hydrogen atom, so Ehrenfest immediately applied the theory of adiabatic invariants to Bohr’s model and discovered that the quantum condition for the quantized energy levels was again the adiabatic invariants of the electron orbits, and not merely a consequence of integer multiples of angular momentum, which had seemed somewhat ad hoc. Later, when Sommerfeld published his quantized elliptical orbits in 1916, the multiplicity of quantum conditions and orbits had caused concern, but Ehrenfest came to the rescue with his theory of adiabatic invariants, showing that each of Sommerfeld’s quantum conditions were precisely the adabatic invariants of the classical electron dynamics [4]. The remaining question was which coordinates were the correct ones, because different choices led to different answers. This was quickly solved by Johannes Burgers (one of Ehrenfest’s students) who showed that action integrals were adiabatic invariants, and then by Karl Schwarzschild and Paul Epstein who showed that action-angle coordinates were the only allowed choice of coordinates, because they enabled the separation of the Hamilton-Jacobi equations and hence provided the correct quantization conditions for the electron orbits. Schwarzshild’s paper was published the same day that he died on the Eastern Front. The work by Schwarzschild and Epstein was the first to show the power of the Hamiltonian formulation of dynamics for quantum systems, which foreshadowed the future importance of Hamiltonians for quantum theory.

Bohr-Sommerfeld

Emboldened by Ehrenfest’s adiabatic principle, which demonstrated a close connection between classical dynamics and quantization conditions, Bohr formalized a technique that he had used implicitly in his 1913 model of hydrogen, and now elevated it to the status of a fundamental principle of quantum theory. He called it the Correspondence Principle, and published the details in 1920. The Correspondence Principle states that as the quantum number of an electron orbit increases to large values, the quantum behavior converges to classical behavior. Specifically, if an electron in a state of high quantum number emits a photon while jumping to a neighboring orbit, then the wavelength of the emitted photon approaches the classical radiation wavelength of the electron subject to Maxwell’s equations.

Bohr’s Correspondence Principle cemented the bridge between classical physics and quantum physics. One of the biggest former questions about the physics of electron orbits in atoms was why they did not radiate continuously because of the angular acceleration they experienced in their orbits. Bohr had now reconnected to Maxwell’s equations and classical physics in the limit. Like the theory of adiabatic invariants, the Correspondence Principle became a new tool for distinguishing among different quantum theories. It could be used as a filter to distinguish “correct” quantum models, that transitioned smoothly from quantum to classical behavior, from those that did not. Bohr’s Correspondence Principle was to be a powerful tool in the hands of Werner Heisenberg as he reinvented quantum theory only a few years later.

By the end of 1920, all the elements of the quantum theory of electron orbits were apparently falling into place. Bohr’s originally ad hoc quantization condition was now on firm footing. The quantization conditions were related to action integrals that were, in turn, adiabatic invariants of the classical dynamics. This meant that slight variations in the parameters of the dynamics systems would not induce quantum transitions among the various quantum states. This conclusion would have felt right to the early quantum practitioners. Bohr’s quantum model of electron orbits was fundamentally a means of explaining quantum transitions between stationary states. Now it appeared that the condition for the stationary states of the electron orbits was an insensitivity, or invariance, to variations in the dynamical properties. This was analogous to the principle of stationary action where the action along a dynamical trajectory is invariant to slight variations in the trajectory. Therefore, the theory of quantum orbits now rested on firm foundations that seemed as solid as the foundations of classical mechanics.

From the perspective of modern quantum theory, the concept of elliptical Keplerian orbits for the electron is grossly inaccurate. Most physicists shudder when they see the symbol for atomic energy—the classic but mistaken icon of electron orbits around a nucleus. Nonetheless, Bohr and Ehrenfest and Sommerfeld had hit on a deep thread that runs through all of physics—the concept of action—the same concept that Leibniz introduced, that Maupertuis minimized and that Euler canonized. This concept of action is at work in the macroscopic domain of classical dynamics as well as the microscopic world of quantum phenomena. Planck was acutely aware of this connection with action, which is why he so readily recognized his elementary constant as the quantum of action.

However, the old quantum theory was running out of steam. For instance, the action integrals and adiabatic invariants only worked for single electron orbits, leaving the vast bulk of many-electron atomic matter beyond the reach of quantum theory and prediction. The literal electron orbits were a crutch or bias that prevented physicists from moving past them and seeing new possibilities for quantum theory. Orbits were an anachronism, exerting a damping force on progress. This limitation became painfully clear when Bohr and his assistants at Copenhagen–Kramers and Slater–attempted to use their electron orbits to explain the refractive index of gases. The theory was cumbersome and exhausted. It was time for a new quantum revolution by a new generation of quantum wizards–Heisenberg, Born, Schrödinger, Pauli, Jordan and Dirac.

References

[1] N. Bohr, “On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus,” Philosophical Magazine, vol. 26, pp. 476–502, 1913.

[2] A. Sommerfeld, “The quantum theory of spectral lines,” Annalen Der Physik, vol. 51, pp. 1-94, Sep 1916.

[3] P. Ehrenfest, “Een mechanische theorema van Boltzmann en zijne betrekking tot de quanta theorie (A mechanical theorem of Boltzmann and its relation to the theory of energy quanta),” Verslag van de Gewoge Vergaderingen der Wis-en Natuurkungige Afdeeling, vol. 22, pp. 586-593, 1913.

[4] P. Ehrenfest, “Adiabatic invariables and quantum theory,” Annalen Der Physik, vol. 51, pp. 327-352, Oct 1916.

Galileo Unbound

Companion blog to the book "Galileo Unbound" published by Oxford University Press (David D. Nolte)

Tag / History of Quantum Mechanics