The Solvay Debates: Einstein versus Bohr

Einstein is the alpha of the quantum. Einstein is also the omega. Although he was the one who established the quantum of energy and matter (see my Blog Einstein vs Planck), Einstein pitted himself in a running debate against Niels Bohr’s emerging interpretation of quantum physics that had, in Einstein’s opinion, severe deficiencies. Between sessions during a series of conferences known as the Solvay Congresses over a period of eight years from 1927 to 1935, Einstein constructed a challenges of increasing sophistication to confront Bohr and his quasi-voodoo attitudes about wave-function collapse. To meet the challenge, Bohr sharpened his arguments and bested Einstein, who ultimately withdrew from the field of battle. Einstein, as quantum physics’ harshest critic, played a pivotal role, almost against his will, establishing the Copenhagen interpretation of quantum physics that rules to this day, and also inventing the principle of entanglement which lies at the core of almost all quantum information technology today.

Debate Timeline

  • Fifth Solvay Congress: 1927 October Brussels: Debate Round 1
    • Einstein and ensembles
  • Sixth Solvay Congress: 1930 Debate Round 2
    • Photon in a box
  • Seventh Solvay Congress: 1933
    • Einstein absent (visiting the US when Hitler takes power…decides not to return to Germany.)
  • Physical Review 1935: Debate Round 3
    • EPR paper and Bohr’s response
    • Schrödinger’s Cat
  • Notable Nobel Prizes
    • 1918 Planck
    • 1921 Einstein
    • 1922 Bohr
    • 1932 Heisenberg
    • 1933 Dirac and Schrödinger

The Solvay Conferences

The Solvay congresses were unparalleled scientific meetings of their day.  They were attended by invitation only, and invitations were offered only to the top physicists concerned with the selected topic of each meeting.  The Solvay congresses were held about every three years always in Belgium, supported by the Belgian chemical industrialist Ernest Solvay.  The first meeting, held in 1911, was on the topic of radiation and quanta. 

Fig. 1 First Solvay Congress (1911). Einstein (standing second from right) was one of the youngest attendees.

The fifth meeting, held in 1927, was on electrons and photons and focused on the recent rapid advances in quantum theory.  The old quantum guard was invited—Planck, Bohr and Einstein.  The new quantum guard was invited as well—Heisenberg, de Broglie, Schrödinger, Born, Pauli, and Dirac.  Heisenberg and Bohr joined forces to present a united front meant to solidify what later became known as the Copenhagen interpretation of quantum physics.  The basic principles of the interpretation include the wavefunction of Schrödinger, the probabilistic interpretation of Born, the uncertainty principle of Heisenberg, the complementarity principle of Bohr and the collapse of the wavefunction during measurement.  The chief conclusion that Heisenberg and Bohr sought to impress on the assembled attendees was that the theory of quantum processes was complete, meaning that unknown or uncertain  characteristics of measurements could not be attributed to lack of knowledge or understanding, but were fundamental and permanently inaccessible.

Fig. 2 Fifth Solvay Congress (1927). Einstein front and center. Bohr on the far right middle row.

Einstein was not convinced with that argument, and he rose to his feet to object after Bohr’s informal presentation of his complementarity principle.  Einstein insisted that uncertainties in measurement were not fundamental, but were caused by incomplete information, that , if known, would accurately account for the measurement results.  Bohr was not prepared for Einstein’s critique and brushed it off, but what ensued in the dining hall and the hallways of the Hotel Metropole in Brussels over the next several days has become one of the most famous scientific debates of the modern era, known as the Bohr-Einstein debate on the meaning of quantum theory.  The debate gently raged night and day through the fifth congress, and was renewed three years later at the 1930 congress.  It finished, in a final flurry of published papers in 1935 that launched some of the central concepts of quantum theory, including the idea of quantum entanglement and, of course, Schrödinger’s cat.

Einstein’s strategy, to refute Bohr, was to construct careful thought experiments that envisioned perfect experiments, without errors, that measured properties of ideal quantum systems.  His aim was to paint Bohr into a corner from which he could not escape, caught by what Einstein assumed was the inconsistency of complementarity.  Einstein’s “thought experiments” used electrons passing through slits, diffracting as required by Schrödinger’s theory, but being detected by classical measurements.  Einstein would present a thought experiment to Bohr, who would then retreat to consider the way around Einstein’s arguments, returning the next hour or the next day with his answer, only to be confronted by yet another clever device of Einstein’s clever imagination that would force Bohr to retreat again.  The spirit of this back and forth encounter between Bohr and Einstein is caught dramatically in the words of Paul Ehrenfest who witnessed the debate first hand, partially mediating between Bohr and Einstein, both of whom he respected deeply.

“Brussels-Solvay was fine!… BOHR towering over everybody.  At first not understood at all … , then  step by step defeating everybody.  Naturally, once again the awful Bohr incantation terminology.  Impossible for anyone else to summarise … (Every night at 1 a.m., Bohr came into my room just to say ONE SINGLE WORD to me, until three a.m.)  It was delightful for me to be present during the conversation between Bohr and Einstein.  Like a game of chess, Einstein all the time with new examples.  In a certain sense a sort of Perpetuum Mobile of the second kind to break the UNCERTAINTY RELATION.  Bohr from out of philosophical smoke clouds constantly searching for the tools to crush one example after the other.  Einstein like a jack-in-the-box; jumping out fresh every morning.  Oh, that was priceless.  But I am almost without reservation pro Bohr and contra Einstein.  His attitude to Bohr is now exacly like the attitude of the defenders of absolute simultaneity towards him …” [1]

The most difficult example that Einstein constructed during the fifth Solvary Congress involved an electron double-slit apparatus that could measure, in principle, the momentum imparted to the slit by the passing electron, as shown in Fig.3.  The electron gun is a point source that emits the electrons in a range of angles that illuminates the two slits.  The slits are small relative to a de Broglie wavelength, so the electron wavefunctions diffract according to Schrödinger’s wave mechanics to illuminate the detection plate.  Because of the interference of the electron waves from the two slits, electrons are detected clustered in intense fringes separated by dark fringes. 

So far, everyone was in agreement with these suggested results.  The key next step is the assumption that the electron gun emits only a single electron at a time, so that only one electron is present in the system at any given time.  Furthermore, the screen with the double slit is suspended on a spring, and the position of the screen is measured with complete accuracy by a displacement meter.  When the single electron passes through the entire system, it imparts a momentum kick to the screen, which is measured by the meter.  It is also detected at a specific location on the detection plate.  Knowing the position of the electron detection, and the momentum kick to the screen, provides information about which slit the electron passed through, and gives simultaneous position and momentum values to the electron that have no uncertainty, apparently rebutting the uncertainty principle.             

Fig. 3 Einstein’s single-electron thought experiment in which the recoil of the screen holding the slits can be measured to tell which way the electron went. Bohr showed that the more “which way” information is obtained, the more washed-out the interference pattern becomes.

This challenge by Einstein was the culmination of successively more sophisticated examples that he had to pose to combat Bohr, and Bohr was not going to let it pass unanswered.  With ingenious insight, Bohr recognized that the key element in the apparatus was the fact that the screen with the slits must have finite mass if the momentum kick by the electron were to produce a measurable displacement.  But if the screen has finite mass, and hence a finite momentum kick from the electron, then there must be an uncertainty in the position of the slits.  This uncertainty immediately translates into a washout of the interference fringes.  In fact the more information that is obtained about which slit the electron passed through, the more the interference is washed out.  It was a perfect example of Bohr’s own complementarity principle.  The more the apparatus measures particle properties, the less it measures wave properties, and vice versa, in a perfect balance between waves and particles. 

Einstein grudgingly admitted defeat at the end of the first round, but he was not defeated.  Three years later he came back armed with more clever thought experiments, ready for the second round in the debate.

The Sixth Solvay Conference: 1930

At the Solvay Congress of 1930, Einstein was ready with even more difficult challenges.  His ultimate idea was to construct a box containing photons, just like the original black bodies that launched Planck’s quantum hypothesis thirty years before.  The box is attached to a weighing scale so that the weight of the box plus the photons inside can be measured with arbitrarily accuracy. A shutter over a hole in the box is opened for a time T, and a photon is emitted.  Because the photon has energy, it has an equivalent weight (Einstein’s own famous E = mc2), and the mass of the box changes by an amount equal to the photon energy divided by the speed of light squared: m = E/c2.  If the scale has arbitrary accuracy, then the energy of the photon has no uncertainty.  In addition, because the shutter was open for only a time T, the time of emission similarly has no uncertainty.  Therefore, the product of the energy uncertainty and the time uncertainty is much smaller than Planck’s constant, apparently violating Heisenberg’s precious uncertainty principle.

Bohr was stopped in his tracks with this challenge.  Although he sensed immediately that Einstein had missed something (because Bohr had complete confidence in the uncertainty principle), he could not put his finger immediately on what it was.  That evening he wandered from one attendee to another, very unhappy, trying to persuade them and saying that Einstein could not be right because it would be the end of physics.  At the end of the evening, Bohr was no closer to a solution, and Einstein was looking smug.  However, by the next morning Bohr reappeared tired but in high spirits, and he delivered a master stroke.  Where Einstein had used special relaitivity against Bohr, Bohr now used Einstein’s own general relativity against him. 

The key insight was that the weight of the box must be measured, and the process of measurement was just as important as the quantum process being measured—this was one of the cornerstones of the Copenhagen interpretation.  So Bohr envisioned a measuring apparatus composed of a spring and a scale with the box suspended in gravity from the spring.  As the photon leaves the box, the weight of the box changes, and so does the deflection of the spring, changing the height of the box.  This change in height, in a gravitational potential, causes the timing of the shutter to change according to the law of gravitational time dilation in general relativity.  By calculating the the general relativistic uncertainty in the time, coupled with the special relativistic uncertainty in the weight of the box, produced a product that was at least as big as Planck’s constant—Heisenberg’s uncertainty principle was saved!

Fig. 4 Einstein’s thought experiment that uses special relativity to refute quantum mechanics. Bohr then invoked Einstein’s own general relativity to refute him.

Entanglement and Schrödinger’s Cat

Einstein ceded the point to Bohr but was not convinced. He still believed that quantum mechanics was not a “complete” theory of quantum physics and he continued to search for the perfect thought experiment that Bohr could not escape. Even today when we have become so familiar with quantum phenomena, the Copenhagen interpretation of quantum mechanics has weird consequences that seem to defy common sense, so it is understandable that Einstein had his reservations.

After the sixth Solvay congress Einstein and Schrödinger exchanged many letters complaining to each other about Bohr’s increasing strangle-hold on the interpretation of quantum mechanics. Egging each other on, they both constructed their own final assault on Bohr. The irony is that the concepts they devised to throw down quantum mechanics have today become cornerstones of the theory. For Einstein, his final salvo was “Entanglement”. For Schrödinger, his final salvo was his “cat”. Today, Entanglement and Schrödinger’s Cat have become enshrined on the alter of quantum interpretation even though their original function was to thwart that interpretation.

The final round of the debate was carried out, not at a Solvay congress, but in the Physical review journal by Einstein [2] and Bohr [3], and in the Naturwissenshaften by Schrödinger [4].

In 1969, Heisenberg looked back on these years and said,

To those of us who participated in the development of atomic theory, the five years following the Solvay Conference in Brussels in 1927 looked so wonderful that we often spoke of them as the golden age of atomic physics. The great obstacles that had occupied all our efforts in the preceding years had been cleared out of the way, the gate to an entirely new field, the quantum mechanics of the atomic shells stood wide open, and fresh fruits seemed ready for the picking. [5]

References

[1] A. Whitaker, Einstein, Bohr, and the quantum dilemma : from quantum theory to quantum information, 2nd ed. Cambridge University Press, 2006. (pg. 210)

[2] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 47, no. 10, pp. 0777-0780, May (1935)

[3] N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 48, no. 8, pp. 696-702, Oct (1935)

[4] E. Schrodinger, “The current situation in quantum mechanics,” Naturwissenschaften, vol. 23, pp. 807-812, (1935)

[5] W Heisenberg, Physics and beyond : Encounters and conversations (Harper, New York, 1971)

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.

Niels Bohr (Wikipedia)

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.

Arnold Sommerfeld (Wikipedia)

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 .  

Paul Ehrenfest (Wikipedia)

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.

Karl Schwarzschild (Wikipedia)

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.

Quantization conditions.

 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.