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.

Who Invented the Quantum? Einstein vs. Planck

Albert Einstein defies condensation—it is impossible to condense his approach, his insight, his motivation—into a single word like “genius”.  He was complex, multifaceted, contradictory, revolutionary as well as conservative.  Some of his work was so simple that it is hard to understand why no-one else did it first, even when they were right in the middle of it.  Lorentz and Poincaré spring to mind—they had been circling the ideas of spacetime for decades—but never stepped back to see what the simplest explanation could be.  Einstein did, and his special relativity was simple and beautiful, and the math is just high-school algebra.  On the other hand, parts of his work—like gravitation—are so embroiled in mathematics and the religion of general covariance that it remains opaque to physics neophytes 100 years later and is usually reserved for graduate study. 

            Yet there is a third thread in Einstein’s work that relies on pure intuition—neither simple nor complicated—but almost impossible to grasp how he made his leap.  This is the case when he proposed the real existence of the photon—the quantum particle of light.  For ten years after this proposal, it was considered by almost everyone to be his greatest blunder. It even came up when Planck was nominating Einstein for membership in the German Academy of Science. Planck said

That he may sometimes have missed the target of his speculations, as for example, in his hypothesis of light quanta, cannot really be held against him.

In this single statement, we have the father of the quantum being criticized by the father of the quantum discontinuity.

Max Planck’s Discontinuity

In histories of the development of quantum theory, the German physicist Max Planck (1858—1947) is characterized as an unlikely revolutionary.  He was an establishment man, in the stolid German tradition, who was already embedded in his career, in his forties, holding a coveted faculty position at the University of Berlin.  In his research, he was responding to a theoretical challenge issued by Kirchhoff many years ago in 1860 to find the function of temperature and wavelength that described and explained the observed spectrum of radiating bodies.  Planck was not looking for a revolution.  In fact, he was looking for the opposite.  One of his motivations in studying the thermodynamics of electromagnetic radiation was to rebut the statistical theories of Boltzmann.  Planck had never been convinced by the atomistic and discrete approach Boltzmann had used to explain entropy and the second law of thermodynamics.  With the continuum of light radiation he thought he had the perfect system that would show how entropy behaved in a continuous manner, without the need for discrete quantities. 

Therefore, Planck’s original intentions were to use blackbody radiation to argue against Boltzmann—to set back the clock.  For this reason, not only was Planck an unlikely revolutionary, he was a counter-revolutionary.  But Planck was a revolutionary because that is what he did, whatever his original intentions were, and he accepted his role as a revolutionary when he had the courage to stand in front of his scientific peers and propose a quantum hypothesis that lay at the heart of physics.

            Blackbody radiation, at the end of the nineteenth century, was a topic of keen interest and had been measured with high precision.  This was in part because it was such a “clean” system, having fundamental thermodynamic properties independent of any of the material properties of the black body, unlike the so-called ideal gases, which always showed some dependence on the molecular properties of the gas. The high-precision measurements of blackbody radiation were made possible by new developments in spectrometers at the end of the century, as well as infrared detectors that allowed very precise and repeatable measurements to be made of the spectrum across broad ranges of wavelengths. 

In 1893 the German physicist Wilhelm Wien (1864—1928) had used adiabatic expansion arguments to derive what became known as Wien’s Displacement Law that showed a simple linear relationship between the temperature of the blackbody and the peak wavelength.  Later, in 1896, he showed that the high-frequency behavior could be described by an exponential function of temperature and wavelength that required no other properties of the blackbody.  This was approaching the solution of Kirchhoff’s challenge of 1860 seeking a universal function.  However, at lower frequencies Wien’s approximation failed to match the measured spectrum.  In mid-year 1900, Planck was able to define a single functional expression that described the experimentally observed spectrum.  Planck had succeeded in describing black-body radiation, but he had not satisfied Kirchhoff’s second condition—to explain it. 

            Therefore, to describe the blackbody spectrum, Planck modeled the emitting body as a set of ideal oscillators.  As an expert in the Second Law, Planck derived the functional form for the radiation spectrum, from which he found the entropy of the oscillators that produced the spectrum.  However, once he had the form for the entropy, he needed to explain why it took that specific form.  In this sense, he was working backwards from a known solution rather than forwards from first principles.  Planck was at an impasse.  He struggled but failed to find any continuum theory that could work. 

Then Planck turned to Boltzmann’s statistical theory of entropy, the same theory that he had previously avoided and had hoped to discredit.  He described this as “an act of despair … I was ready to sacrifice any of my previous convictions about physics.”  In Boltzmann’s expression for entropy, it was necessary to “count” possible configurations of states.  But counting can only be done if the states are discrete.  Therefore, he lumped the energies of the oscillators into discrete ranges, or bins, that he called “quanta”.  The size of the bins was proportional to the frequency of the oscillator, and the proportionality constant had the units of Maupertuis’ quantity of action, so Planck called it the “quantum of action”. Finally, based on this quantum hypothesis, Planck derived the functional form of black-body radiation.

            Planck presented his findings at a meeting of the German Physical Society in Berlin on November 15, 1900, introducing the word quantum (plural quanta) into physics from the Latin word that means quantity [1].  It was a casual meeting, and while the attendees knew they were seeing an intriguing new physical theory, there was no sense of a revolution.  But Planck himself was aware that he had created something fundamentally new.  The radiation law of cavities depended on only two physical properties—the temperature and the wavelength—and on two constants—Boltzmann’s constant kB and a new constant that later became known as Planck’s constant h = ΔE/f = 6.6×10-34 J-sec.  By combining these two constants with other fundamental constants, such as the speed of light, Planck was able to establish accurate values for long-sought constants of nature, like Avogadro’s number and the charge of the electron.

            Although Planck’s quantum hypothesis in 1900 explained the blackbody radiation spectrum, his specific hypothesis was that it was the interaction of the atoms and the light field that was somehow quantized.  He certainly was not thinking in terms of individual quanta of the light field.

Figure. Einstein and Planck at a dinner held by Max von Laue in Berlin on Nov. 11, 1931.

Einstein’s Quantum

When Einstein analyzed the properties of the blackbody radiation in 1905, using his deep insight into statistical mechanics, he was led to the inescapable conclusion that light itself must be quantized in amounts E = hf, where h is Planck’s constant and f is the frequency of the light field.  Although this equation is exactly the same as Planck’s from 1900, the meaning was completely different.  For Planck, this was the discreteness of the interaction of light with matter.  For Einstein, this was the quantum of light energy—whole and indivisible—just as if the light quantum were a particle with particle properties.  For this reason, we can answer the question posed in the title of this Blog—Einstein takes the honor of being the inventor of the quantum.

            Einstein’s clarity of vision is a marvel to behold even to this day.  His special talent was to take simple principles, ones that are almost trivial and beyond reproach, and to derive something profound.  In Special Relativity, he simply assumed the constancy of the speed of light and derived Lorentz’s transformations that had originally been based on obtuse electromagnetic arguments about the electron.  In General Relativity, he assumed that free fall represented an inertial frame, and he concluded that gravity must bend light.  In quantum theory, he assumed that the low-density limit of Planck’s theory had to be consistent with light in thermal equilibrium in thermal equilibrium with the black body container, and he concluded that light itself must be quantized into packets of indivisible energy quanta [2].  One immediate consequence of this conclusion was his simple explanation of the photoelectric effect for which the energy of an electron ejected from a metal by ultraviolet irradiation is a linear function of the frequency of the radiation.  Einstein published his theory of the quanta of light [3] as one of his four famous 1905 articles in Annalen der Physik in his Annus Mirabilis

Figure. In the photoelectric effect a photon is absorbed by an electron state in a metal promoting the electron to a free electron that moves with a maximum kinetic energy given by the difference between the photon energy and the work function W of the metal. The energy of the photon is absorbed as a whole quantum, proving that light is composed of quantized corpuscles that are today called photons.

            Einstein’s theory of light quanta was controversial and was slow to be accepted.  It is ironic that in 1914 when Einstein was being considered for a position at the University in Berlin, Planck himself, as he championed Einstein’s case to the faculty, implored his colleagues to accept Einstein despite his ill-conceived theory of light quanta [4].  This comment by Planck goes far to show how Planck, father of the quantum revolution, did not fully grasp, even by 1914, the fundamental nature and consequences of his original quantum hypothesis.  That same year, the American physicist Robert Millikan (1868—1953) performed a precise experimental measurement of the photoelectric effect, with the ostensible intention of proving Einstein wrong, but he accomplished just the opposite—providing clean experimental evidence confirming Einstein’s theory of the photoelectric effect. 

The Stimulated Emission of Light

About a year after Millikan proved that the quantum of energy associated with light absorption was absorbed as a whole quantum of energy that was not divisible, Einstein took a step further in his theory of the light quantum. In 1916 he published a paper in the proceedings of the German Physical Society that explored how light would be in a state of thermodynamic equilibrium when interacting with atoms that had discrete energy levels. Once again he used simple arguments, this time using the principle of detailed balance, to derive a new and unanticipated property of light—stimulated emission!

Figure. The stimulated emission of light. An excited state is stimulated to emit an identical photon when the electron transitions to its ground state.

The stimulated emission of light occurs when an electron is in an excited state of a quantum system, like an atom, and an incident photon stimulates the emission of a second photon that has the same energy and phase as the first photon. If there are many atoms in the excited state, then this process leads to a chain reaction as 1 photon produces 2, and 2 produce 4, and 4 produce 8, etc. This exponential gain in photons with the same energy and phase is the origin of laser radiation. At the time that Einstein proposed this mechanism, lasers were half a century in the future, but he was led to this conclusion by extremely simple arguments about transition rates.

Figure. Section of Einstein’s 1916 paper that describes the absorption and emission of light by atoms with discrete energy levels [5].

Detailed balance is a principle that states that in thermal equilibrium all fluxes are balanced. In the case of atoms with ground states and excited states, this principle requires that as many transitions occur from the ground state to the excited state as from the excited state to the ground state. The crucial new element that Einstein introduced was to distinguish spontaneous emission from stimulated emission. Just as the probability to absorb a photon must be proportional to the photon density, there must be an equivalent process that de-excites the atom that also must be proportional the photon density. In addition, an electron must be able to spontaneously emit a photon with a rate that is independent of photon density. This leads to distinct coefficients in the transition rate equations that are today called the “Einstein A and B coefficients”. The B coefficients relate to the photon density, while the A coefficient relates to spontaneous emission.

Figure. Section of Einstein’s 1917 paper that derives the equilibrium properties of light interacting with matter. The “B”-coefficient for transition from state m to state n describes stimulated emission. [6]

Using the principle of detailed balance together with his A and B coefficients as well as Boltzmann factors describing the number of excited states relative to ground state atoms in equilibrium at a given temperature, Einstein was able to derive an early form of what is today called the Bose-Einstein occupancy function for photons.

Derivation of the Einstein A and B Coefficients

Detailed balance requires the rate from m to n to be the same as the rate from n to m

where the first term is the spontaneous emission rate from the excited state m to the ground state n, the second term is the stimulated emission rate, and the third term (on the right) is the absorption rate from n to m. The numbers in each state are Nm and Nn, and the density of photons is ρ. The relative numbers in the excited state relative to the ground state is given by the Boltzmann factor

By assuming that the stimulated transition coefficient from n to m is the same as m to n, and inserting the Boltzmann factor yields

The Planck density of photons for ΔE = hf is

which yields the final relation between the spontaneous emission coefficient and the stimulated emission coefficient

The total emission rate is

where the p-bar is the average photon number in the cavity. One of the striking aspects of this derivation is that no assumptions are made about the physical mechanisms that determine the coefficient B. Only arguments of detailed balance are required to arrive at these results.

Einstein’s Quantum Legacy

Einstein was awarded the Nobel Prize in 1921 for the photoelectric effect, not for the photon nor for any of Einstein’s other theoretical accomplishments.  Even in 1921, the quantum nature of light remained controversial.  It was only in 1923, after the American physicist Arthur Compton (1892—1962) showed that energy and momentum were conserved in the scattering of photons from electrons, that the quantum nature of light began to be accepted.  The very next year, in 1924, the quantum of light was named the “photon” by the American American chemical physicist Gilbert Lewis (1875—1946). 

            A blog article like this, that attributes the invention of the quantum to Einstein rather than Planck, must say something about the irony of this attribution.  If Einstein is the father of the quantum, he ultimately was led to disinherit his own brain child.  His final and strongest argument against the quantum properties inherent in the Copenhagen Interpretation was his famous EPR paper which, against his expectations, launched the concept of entanglement that underlies the coming generation of quantum computers.


Einstein’s Quantum Timeline

1900 – Planck’s quantum discontinuity for the calculation of the entropy of blackbody radiation.

1905 – Einstein’s “Miracle Year”. Proposes the light quantum.

1911 – First Solvay Conference on the theory of radiation and quanta.

1913 – Bohr’s quantum theory of hydrogen.

1914 – Einstein becomes a member of the German Academy of Science.

1915 – Millikan measurement of the photoelectric effect.

1916 – Einstein proposes stimulated emission.

1921 – Einstein receives Nobel Prize for photoelectric effect and the light quantum. Third Solvay Conference on atoms and electrons.

1927 – Heisenberg’s uncertainty relation. Fifth Solvay International Conference on Electrons and Photons in Brussels. “First” Bohr-Einstein debate on indeterminancy in quantum theory.

1930 – Sixth Solvay Conference on magnetism. “Second” Bohr-Einstein debate.

1935 – Einstein-Podolsky-Rosen (EPR) paper on the completeness of quantum mechanics.


Selected Einstein Quantum Papers

Einstein, A. (1905). “Generation and conversion of light with regard to a heuristic point of view.” Annalen Der Physik 17(6): 132-148.

Einstein, A. (1907). “Die Plancksche Theorie der Strahlung und die Theorie der spezifischen W ̈arme.” Annalen der Physik 22: 180–190.

Einstein, A. (1909). “On the current state of radiation problems.” Physikalische Zeitschrift 10: 185-193.

Einstein, A. and O. Stern (1913). “An argument for the acceptance of molecular agitation at absolute zero.” Annalen Der Physik 40(3): 551-560.

Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.

Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.

Einstein, A., B. Podolsky and N. Rosen (1935). “Can quantum-mechanical description of physical reality be considered complete?” Physical Review 47(10): 0777-0780.


Notes

[1] M. Planck, “Elementary quanta of matter and electricity,” Annalen Der Physik, vol. 4, pp. 564-566, Mar 1901.

[2] Klein, M. J. (1964). Einstein’s First Paper on Quanta. The natural philosopher. D. A. Greenberg and D. E. Gershenson. New York, Blaidsdell. 3.

[3] A. Einstein, “Generation and conversion of light with regard to a heuristic point of view,” Annalen Der Physik, vol. 17, pp. 132-148, Jun 1905.

[4] Chap. 2 in “Mind at Light Speed”, by David Nolte (Free Press, 2001)

[5] Einstein, A. (1916). “Strahlungs-Emission un -Absorption nach der Quantentheorie.” Verh. Deutsch. Phys. Ges. 18: 318.

[6] Einstein, A. (1917). “Quantum theory of radiation.” Physikalische Zeitschrift 18: 121-128.

Science 1916: A Hundred-year Time Capsule

In one of my previous blog posts, as I was searching for Schwarzschild’s original papers on Einstein’s field equations and quantum theory, I obtained a copy of the January 1916 – June 1916 volume of the Proceedings of the Royal Prussian Academy of Sciences through interlibrary loan.  The extremely thick volume arrived at Purdue about a week after I ordered it online.  It arrived from Oberlin College in Ohio that had received it as a gift in 1928 from the library of Professor Friedrich Loofs of the University of Halle in Germany.  Loofs had been the Haskell Lecturer at Oberlin for the 1911-1912 semesters. 

As I browsed through the volume looking for Schwarzschild’s papers, I was amused to find a cornucopia of turn-of-the-century science topics recorded in its pages.  There were papers on the overbite and lips of marsupials.  There were papers on forgotten languages.  There were papers on ancient Greek texts.  On the origins of religion.  On the philosophy of abstraction.  Histories of Indian dramas.  Reflections on cancer.  But what I found most amazing was a snapshot of the field of physics and mathematics in 1916, with historic papers by historic scientists who changed how we view the world. Here is a snapshot in time and in space, a period of only six months from a single journal, containing papers from authors that reads like a who’s who of physics.

In 1916 there were three major centers of science in the world with leading science publications: London with the Philosophical Magazine and Proceedings of the Royal Society; Paris with the Comptes Rendus of the Académie des Sciences; and Berlin with the Proceedings of the Royal Prussian Academy of Sciences and Annalen der Physik. In Russia, there were the scientific Journals of St. Petersburg, but the Bolshevik Revolution was brewing that would overwhelm that country for decades.  And in 1916 the academic life of the United States was barely worth noticing except for a few points of light at Yale and Johns Hopkins. 

Berlin in 1916 was embroiled in war, but science proceeded relatively unmolested.  The six-month volume of the Proceedings of the Royal Prussian Academy of Sciences contains a number of gems.  Schwarzschild was one of the most prolific contributors, publishing three papers in just this half-year volume, plus his obituary written by Einstein.  But joining Schwarzschild in this volume were Einstein, Planck, Born, Warburg, Frobenious, and Rubens among others—a pantheon of German scientists mostly cut off from the rest of the world at that time, but single-mindedly following their individual threads woven deep into the fabric of the physical world.

Karl Schwarzschild (1873 – 1916)

Schwarzschild had the unenviable yet effective motivation of his impending death to spur him to complete several projects that he must have known would make his name immortal.  In this six-month volume he published his three most important papers.  The first (pg. 189) was on the exact solution to Einstein’s field equations to general relativity.  The solution was for the restricted case of a point mass, yet the derivation yielded the Schwarzschild radius that later became known as the event horizon of a non-roatating black hole.  The second paper (pg. 424) expanded the general relativity solutions to a spherically symmetric incompressible liquid mass. 

Schwarzschild’s solution to Einstein’s field equations for a point mass.

          

Schwarzschild’s extension of the field equation solutions to a finite incompressible fluid.

The subject, content and success of these two papers was wholly unexpected from this observational astronomer stationed on the Russian Front during WWI calculating trajectories for German bombardments.  He would not have been considered a theoretical physicist but for the importance of his results and the sophistication of his methods.  Within only a year after Einstein published his general theory, based as it was on the complicated tensor calculus of Levi-Civita, Christoffel and Ricci-Curbastro that had taken him years to master, Schwarzschild found a solution that evaded even Einstein.

Schwarzschild’s third and final paper (pg. 548) was on an entirely different topic, still not in his official field of astronomy, that positioned all future theoretical work in quantum physics to be phrased in the language of Hamiltonian dynamics and phase space.  He proved that action-angle coordinates were the only acceptable canonical coordinates to be used when quantizing dynamical systems.  This paper answered a central question that had been nagging Bohr and Einstein and Ehrenfest for years—how to quantize dynamical coordinates.  Despite the simple way that Bohr’s quantized hydrogen atom is taught in modern physics, there was an ambiguity in the quantization conditions even for this simple single-electron atom.  The ambiguity arose from the numerous possible canonical coordinate transformations that were admissible, yet which led to different forms of quantized motion. 

Schwarzschild’s proposal of action-angle variables for quantization of dynamical systems.

 Schwarzschild’s doctoral thesis had been a theoretical topic in astrophysics that applied the celestial mechanics theories of Henri Poincaré to binary star systems.  Within Poincaré’s theory were integral invariants that were conserved quantities of the motion.  When a dynamical system had as many constraints as degrees of freedom, then every coordinate had an integral invariant.  In this unexpected last paper from Schwarzschild, he showed how canonical transformation to action-angle coordinates produced a unique representation in terms of action variables (whose dimensions are the same as Planck’s constant).  These action coordinates, with their associated cyclical angle variables, are the only unambiguous representations that can be quantized.  The important points of this paper were amplified a few months later in a publication by Schwarzschild’s friend Paul Epstein (1871 – 1939), solidifying this approach to quantum mechanics.  Paul Ehrenfest (1880 – 1933) continued this work later in 1916 by defining adiabatic invariants whose quantum numbers remain unchanged under slowly varying conditions, and the program started by Schwarzschild was definitively completed by Paul Dirac (1902 – 1984) at the dawn of quantum mechanics in Göttingen in 1925.

Albert Einstein (1879 – 1955)

In 1916 Einstein was mopping up after publishing his definitive field equations of general relativity the year before.  His interests were still cast wide, not restricted only to this latest project.  In the 1916 Jan. to June volume of the Prussian Academy Einstein published two papers.  Each is remarkably short relative to the other papers in the volume, yet the importance of the papers may stand in inverse proportion to their length.

The first paper (pg. 184) is placed right before Schwarzschild’s first paper on February 3.  The subject of the paper is the expression of Maxwell’s equations in four-dimensional space time.  It is notable and ironic that Einstein mentions Hermann Minkowski (1864 – 1909) in the first sentence of the paper.  When Minkowski proposed his bold structure of spacetime in 1908, Einstein had been one of his harshest critics, writing letters to the editor about the absurdity of thinking of space and time as a single interchangeable coordinate system.  This is ironic, because Einstein today is perhaps best known for the special relativity properties of spacetime, yet he was slow to adopt the spacetime viewpoint. Einstein only came around to spacetime when he realized around 1910 that a general approach to relativity required the mathematical structure of tensor manifolds, and Minkowski had provided just such a manifold—the pseudo-Riemannian manifold of space time.  Einstein subsequently adopted spacetime with a passion and became its greatest champion, calling out Minkowski where possible to give him his due, although he had already died tragically of a burst appendix in 1909.

Relativistic energy density of electromagnetic fields.

The importance of Einstein’s paper hinges on his derivation of the electromagnetic field energy density using electromagnetic four vectors.  The energy density is part of the source term for his general relativity field equations.  Any form of energy density can warp spacetime, including electromagnetic field energy.  Furthermore, the Einstein field equations of general relativity are nonlinear as gravitational fields modify space and space modifies electromagnetic fields, producing a coupling between gravity and electromagnetism.  This coupling is implicit in the case of the bending of light by gravity, but Einstein’s paper from 1916 makes the connection explicit. 

Einstein’s second paper (pg. 688) is even shorter and hence one of the most daring publications of his career.  Because the field equations of general relativity are nonlinear, they are not easy to solve exactly, and Einstein was exploring approximate solutions under conditions of slow speeds and weak fields.  In this “non-relativistic” limit the metric tensor separates into a Minkowski metric as a background on which a small metric perturbation remains.  This small perturbation has the properties of a wave equation for a disturbance of the gravitational field that propagates at the speed of light.  Hence, in the June 22 issue of the Prussian Academy in 1916, Einstein predicts the existence and the properties of gravitational waves.  Exactly one hundred years later in 2016, the LIGO collaboration announced the detection of gravitational waves generated by the merger of two black holes.

Einstein’s weak-field low-velocity approximation solutions of his field equations.
Einstein’s prediction of gravitational waves.

Max Planck (1858 – 1947)

Max Planck was active as the secretary of the Prussian Academy in 1916 yet was still fully active in his research.  Although he had launched the quantum revolution with his quantum hypothesis of 1900, he was not a major proponent of quantum theory even as late as 1916.  His primary interests lay in thermodynamics and the origins of entropy, following the theoretical approaches of Ludwig Boltzmann (1844 – 1906).  In 1916 he was interested in how to best partition phase space as a way to count states and calculate entropy from first principles.  His paper in the 1916 volume (pg. 653) calculated the entropy for single-atom solids.

Counting microstates by Planck.

Max Born (1882 – 1970)

Max Born was to be one of the leading champions of the quantum mechanical revolution based at the University of Göttingen in the 1920’s. But in 1916 he was on leave from the University of Berlin working on ranging for artillery.  Yet he still pursued his academic interests, like Schwarzschild.  On pg. 614 in the Proceedings of the Prussian Academy, Born published a paper on anisotropic liquids, such as liquid crystals and the effect of electric fields on them.  It is astonishing to think that so many of the flat-panel displays we have today, whether on our watches or smart phones, are technological descendants of work by Born at the beginning of his career.

Born on liquid crystals.

Ferdinand Frobenius (1849 – 1917)

Like Schwarzschild, Frobenius was at the end of his career in 1916 and would pass away one year later, but unlike Schwarzschild, his career had been a long one, receiving his doctorate under Weierstrass and exploring elliptic functions, differential equations, number theory and group theory.  One of the papers that established him in group theory appears in the May 4th issue on page 542 where he explores the series expansion of a group.

Frobenious on groups.

Heinrich Rubens (1865 – 1922)

Max Planck owed his quantum breakthrough in part to the exquisitely accurate experimental measurements made by Heinrich Rubens on black body radiation.  It was only by the precise shape of what came to be called the Planck spectrum that Planck could say with such confidence that his theory of quantized radiation interactions fit Rubens spectrum so perfectly.  In 1916 Rubens was at the University of Berlin, having taken the position vacated by Paul Drude in 1906.  He was a specialist in infrared spectroscopy, and on page 167 of the Proceedings he describes the spectrum of steam and its consequences for the quantum theory.

Rubens and the infrared spectrum of steam.

Emil Warburg (1946 – 1931)

Emil Warburg’s fame is primarily as the father of Otto Warburg who won the 1931 Nobel prize in physiology.  On page 314 Warburg reports on photochemical processes in BrH gases.     In an obscure and very indirect way, I am an academic descendant of Emil Warburg.  One of his students was Robert Pohl who was a famous early researcher in solid state physics, sometimes called the “father of solid state physics”.  Pohl was at the physics department in Göttingen in the 1920’s along with Born and Franck during the golden age of quantum mechanics.  Robert Pohl’s son, Robert Otto Pohl, was my professor when I was a sophomore at Cornell University in 1978 for the course on introductory electromagnetism using a textbook by the Nobel laureate Edward Purcell, a quirky volume of the Berkeley Series of physics textbooks.  This makes Emil Warburg my professor’s father’s professor.

Warburg on photochemistry.

Papers in the 1916 Vol. 1 of the Prussian Academy of Sciences

Schulze,  Alt– und Neuindisches

Orth,  Zur Frage nach den Beziehungen des Alkoholismus zur Tuberkulose

Schulze,  Die Erhabunen auf der Lippin- und Wangenschleimhaut der Säugetiere

von Wilamwitz-Moellendorff, Die Samie des Menandros

Engler,  Bericht über das >>Pflanzenreich<<

von Harnack,  Bericht über die Ausgabe der griechischen Kirchenväter der dri ersten Jahrhunderte

Meinecke,  Germanischer und romanischer Geist im Wandel der deutschen Geschichtsauffassung

Rubens und Hettner,  Das langwellige Wasserdampfspektrum und seine Deutung durch die Quantentheorie

Einstein,  Eine neue formale Deutung der Maxwellschen Feldgleichungen der Electrodynamic

Schwarschild,  Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie

Helmreich,  Handschriftliche Verbesserungen zu dem Hippokratesglossar des Galen

Prager,  Über die Periode des veränderlichen Sterns RR Lyrae

Holl,  Die Zeitfolge des ersten origenistischen Streits

Lüders,  Zu den Upanisads. I. Die Samvargavidya

Warburg,  Über den Energieumsatz bei photochemischen Vorgängen in Gasen. VI.

Hellman,  Über die ägyptischen Witterungsangaben im Kalender von Claudius Ptolemaeus

Meyer-Lübke,  Die Diphthonge im Provenzaslischen

Diels,  Über die Schrift Antipocras des Nikolaus von Polen

Müller und Sieg,  Maitrisimit und >>Tocharisch<<

Meyer,  Ein altirischer Heilsegen

Schwarzschild,  Über das Gravitationasfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie

Brauer,  Die Verbreitung der Hyracoiden

Correns,  Untersuchungen über Geschlechtsbestimmung bei Distelarten

Brahn,  Weitere Untersuchungen über Fermente in der Lever von Krebskranken

Erdmann,  Methodologische Konsequenzen aus der Theorie der Abstraktion

Bang,  Studien zur vergleichenden Grammatik der Türksprachen. I.

Frobenius,  Über die  Kompositionsreihe einer Gruppe

Schwarzschild,  Zur Quantenhypothese

Fischer und Bergmann,  Über neue Galloylderivate des Traubenzuckers und ihren Vergleich mit der Chebulinsäure

Schuchhardt,  Der starke Wall und die breite, zuweilen erhöhte Berme bei frügeschichtlichen Burgen in Norddeutschland

Born,  Über anisotrope Flüssigkeiten

Planck,  Über die absolute Entropie einatomiger Körper

Haberlandt,  Blattepidermis und Lichtperzeption

Einstein,  Näherungsweise Integration der Feldgleichungen der Gravitation

Lüders,  Die Saubhikas.  Ein Beitrag zur Gecschichte des indischen Dramas