December 3, 2025 by David D. Nolte

100 Years of Quantum Physics: Rise of the Matrix (1925)

History of Dynamics
Hamiltonian Mechanics, Heisenberg, History of Physics, Matrix mechanics, Max Born, Niels Bohr, Paul Dirac, philosophy, physics, Quantum Mechanics, quantum-physics, science, Wolfgang Pauli

Niels Bohr’s atom, by late 1925, was a series of kludges cobbled together into a Rube Goldberg type of construction. On the one hand, there was the Bohr-Sommerfeld quantization conditions that let Bohr’s originally circular orbits morph into ellipses like planets in the solar system. On the other hand, there was Pauli’s exclusion principle that partially explained the building up of electron orbits in many-electron atoms, but to many it seemed like an ad hoc rule.

The time was ripe for a new perspective. Enter the wunderkind, Werner Heisenberg.

Heisenberg’s Trajectory

Werner Heisenberg (1901 – 1976) was the golden boy—smart, dashing, ambitious. He excelled at everything he did and was a natural leader among his group of young friends. He entered the University of Munich in 1920 at the age of 19 to begin working towards his doctorate degree in mathematics, but he quickly became entranced with an advanced seminar course given by Arnold Sommerfeld (1868 – 1951) on quantum mechanics. His studies under Sommerfeld advanced quickly, and he was proficient enough to be “lent out” to the group of Max Born and David Hilbert at the University of Göttingen for the 1922-1923 semester when Sommerfeld was on sabbatical at the University of Wisconsin, Madison, in the United States. Born was impressed with the young student and promised him a post-doc position upon his graduation with a doctoral degree in theoretical physics the next year (when Heisenberg would be just 22 years old).

Unfortunately, his brilliantly ascending career ran headlong into “Willy” Wien who had won the Nobel Prize in 1911 for his displacement law of black body radiation. Wien was a hard-baked experimentalist who had little patience with the speculative flights of theoretical physics. Heisenberg, in contrast, had little patience with the mundane details of experimental science. The two were heading for an impasse.

The collision came during the oral examination for Heisenberg’s doctoral degree. Wien, determined to put Heisenberg in his place, opened with a difficult question about experimental methods. Heisenberg could not answer, so Wien asked a slightly less difficult but still detailed question that Heisenberg also could not answer. The examination went on like this until finally Wien asked Heisenberg to derive the resolving power of a simple microscope. Heisenberg was so flustered by this time that he could not do even that. Wien, in disgust, turned to Sommerfeld and pronounced a failing grade for Heisenberg. After Heisenberg stepped out of the room, the professors wrangled over the single committee grade that would need to be recorded. Sommerfeld’s top grade for Heisenberg’s mathematical performance and Wien’s bottom grade for his experimental performance led to the compromise grade of a “C” for the exam—the minimum grade sufficient to pass.

Heisenberg was mortified. Accustomed always to excelling and being lauded for his talents, Heisenberg left town that night, taking the late train to Göttingen where a surprised Born found him outside his office early the next morning—fully two months ahead of schedule. Heisenberg told him everything and asked if Born would still have him. After learning more about Wien’s “ambush”, Born assured Heisenberg that he still had a place for him.

Heisenberg was so successful at Göttingen, that when Born planned to spend a year sabbatical at MIT in the United States for the 1924-1925 semester, Heisenberg was “lent out” to Niels Bohr in Copenhagen. While there, Heisenberg, Bohr, Pauli and Kramers had intense discussions about the impending crisis in quantum theory. Bohr was fully aware of the precarious patches that made up the quantum theory of the many-electron atom, and the four physicists attempted to patch it yet again with a theoretical effort led by Kramers to try to reconcile optical transitions in the atomic spectra. But no one was satisfied, and the theory had serious internal inconsistencies, not the least of which was a need to sacrifice the sacrosanct principle of conservation of energy.

Through it all, Heisenberg was thrilled by his deep involvement in the most fundamental questions of physics of the day and was even more thrilled by his interactions with the great minds he found in Copenhagen. When he returned to Göttingen on April 27, 1925, the arguments and inconsistencies were ringing in his head, infecting the group at Göttingen with the challenging physics, especially Max Born and Pascual Jordan.

Little headway could be made, until Heisenberg had a serious attack of hay fever that sent him for respite on June 7 to the remote island of Helgoland in the North Sea far off of the coast from Bremerhaven. The trip cleared Heisenberg’s head—literally and figuratively—as he had time to come to grips with the core difficulties of quantum theory.

Trajectory’s End

The Mythology of Physics recounts the tale of when Heisenberg had his epiphany, watching from the beach as the sun rose over the sea. The repeated retelling has solidified the moment into revealed “truth”, but the origins are probably more prosaic. Strip a problem bare of all its superficial coverings and what remains must be the minimal set of what can be known. Yet to do so requires courage, for much of the superficial coverings are established dogma, embedded so deeply in the thought of the day that angry reactions must be expected.

Map of Heligoland, Germany — Fig. 1 Heligoland Germany. *(From Google Maps and Wikipedia)*

At some moment, Heisenberg realized that the superficial covering of atomic theory was the slavish devotion to the electron trajectory—to the Bohr-Sommerfeld electron orbits. Ever since Kepler, the mental image of masses in orbit around their force center had dominated physical theory. Quantum theory likewise was obsessed with images of trajectories—it persists to this day in the universal logo of atomic energy. Heisenberg now rejected this image as unknowable and hence not relevant for a successful theory. But if electron orbits were out, what was the minimal set of what can be known to be true? Heisenberg decided that it was simply the wavelengths and intensities of light absorbed and emitted by atoms. But what then? How do you create a theory constructed on transition energies and intensities alone? The epiphany was the answer—construct a dynamics by which the quantum system proceeds step-by-step, transition-by-transition, while retaining the sacrosanct energy conservation that had been discarded by Kramer’s unsuccessful theory.

The result, after returning to Göttingen, is Heisenberg’s paper, submitted July 29, 1925 to Zeitschrift für Physik titled Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen (Over the quantum theoretical meaning of kinematic and mechanical relationships).

Heisenberg 1925 Over quantum theoretical meaning of kinematic and mechanical relationships — Fig. 2 The heading of Heisenberg’s 1925 Zeitschrift für Physik article. The abstract reads: “This work seeks to find fundamental principles for a quantum theoretical mechanics that is based exclusively on relationships among principal observable magnitudes.” [1]

Heisenberg begins with the fundamental energy relationship between the frequency of light and the energy difference in a transition

Heisenberg 1925 Zeitschrift fur Physik paper on transition energies — *Fig. 3 Dynamics emerges from the transitions among different energy states in the atom [1].*

His goal is to remove the electron orbit from the theory, yet positions cannot be removed entirely, so he takes the step of transforming position into a superposition of amplitudes with frequencies related to the optical transitions.

Heisenberg replacing the electron orbits with their Fourier coefficients based on transition frequencies. — *Fig. 4 Replace the electron orbits with their Fourier coefficients based on transition frequencies [1].*

Armed with the amplitude coefficients and the transition frequencies, he constructs sums of transitions that proceed step by step between initial and final states.

Heisenberg connecting initial and final states with series of steps that conserve energy. — *Fig. 5 Consider all the possible paths between initial and final states that obey energy conservation [1].*

After introducing the electric field, Heisenberg calculates the polarizability of the atom, the induced moment, using Kramer’s dispersion formula combined with his new superposition.

Heisenberg transition amplitude based on sums over possible energy steps — *Fig. 6 Transition amplitude between initial and final states based on a series of energy-conserving transition steps [1].*

Heisenberg applied his new theoretical approach to one-dimensional quantum systems, using as an explicit example the anharmonic oscillator, and it worked! Heisenberg had invented a new theoretical approach to quantum physics that relied only on transition frequencies and amplitudes—only what could be measured without any need to speculate on what types of motions electrons might be executing. Heisenberg published his new theory on his own, as sole author befitting his individual breakthrough. Yet it was done under the guidance of his supervisor Max Born, who recognized something within Heisenberg’s mathematics.

The Matrix

Heisenberg’s derivations involved numerous summations as amplitudes multiplied amplitudes in complicated sequences. The mathematical steps themselves were straightforward—just products and sums—but the numbers of permutations were daunting, and their sequential order mattered, requiring skill and care not to miss terms or to get minus signs wrong.

Yet Born recognized within Heisenberg’s mathematics the operations of matrix multiplication. The different permutations with sums of alternating signs were exactly what one obtained by taking determinants of matrices, and it was well known that the order of matrix multiplication mattered, where a*b ≠ b*a. With his assistant Pacual Jordan, the two reworked Heisenberg’s paper in the mathematical language of matrices, submitting their “mirror” paper to Zeitschrift on Sept. 27, 1925. Their title was prophetic: Towards Quantum Mechanics. This was the first time that the phrase “quantum mechanics” was used to encompass all of the widely varying aspects of quantum systems.

Born and Jordan 1925 Zur Quantenmechanik — *Fig. 7 The header for Born and Jordan’s reworking of Heisenberg’s paper into matrix mathematics [2].*

In the abstract, they state:

The approaches recently put forward by Heisenberg (initially for systems with one degree of freedom) are developed into a systematic theory of quantum mechanics. The mathematical tool is matrix calculus. After this is briefly outlined, the mechanical equations of motion are derived from a variational principle, and the proof is carried out that, on the basis of Heisenberg’s quantum condition, the energy theorem and Bohr’s frequency condition follow from the mechanical equations. Using the example of the anharmonic oscillator, the question of the uniqueness of the solution and the significance of the phases in the partial oscillations are discussed. The conclusion describes an attempt to incorporate the laws of the electromagnetic field into the new theory.

Born and Jordan begin by creating a matrix form for the Hamiltonian subject to Hamilton’s dynamical equations

Born Hamiltonian and Hamilton's equations — *Fig. 8 Defining the Hamiltonian with matrix operators [2].*

Armed with matrix quantities for position and momentum, Born and Jordan construct the commutator of p with q to arrive at one of the most fundamental quantum relationships: the non-zero difference in the permuted products related to Planck’s constant. This commutation relationship would become the foundation for many quantum theories to come.

Born 1925 matrix commutator — *Fig. 9 Page 871 of Born and Jordan’s 1925 Zeitschrift article that introduces the commutation relationship between p and q [2].*

As Heisenberg had done in his paper, Born and Jordan introduce the electric field of light to derive the dispersion of an atomic gas.

Max Born quantum transition amplitude from matrix elements — *Fig. 10 Expression for the dispersion of light in an atomic gas [2].*

The Born and Jordan paper appeared in the November issue of Zeitschrift für Physik, although a pre-print was picked up in England by Paul Dirac, who was working towards his doctoral degree under the mentorship of Ralph Fowler (1889 – 1944) at Cambridge. Dirac was deeply knowledgable in classical mechanics, and he recognized as soon as he saw it that the new quantum commutator was intimately connected to a quantity in classical mechanics known as a Poisson bracket. The Poisson bracket is part of Hamiltonian mechanics that defines how two variables, known as conjugate variables, are connected. For instance, the Poisson bracket of x with px is non-zero, meaning that these are conjugate variables, while the Poisson bracket of x with py is zero, meaning that these variables are fully independent. Conjugate variables are not “dependent” in an algebraic sense, but are linked through the structure of Hamilton’s equations—they are the “p’s and q’s” of phase space.

Dirac 1926 Proceedings of the Royal Society comparison of Poisson bracket to quantum commutator. — *Fig. 11 The Poisson bracket in Dirac’s paper submitted on Nov. 7, 1925 [3].*

Dirac submitted a paper on Nov. 7, 1925 to the Proceedings of the Royal Society of London where he showed that the Heisenberg commutator (a quantum quantity) directly proportional to the Poisson bracket (the classical quantity) with a proportionality factor that depended on Planck’s constant.

Dirac quantum commutation relationship — *Fig. 12 Dirac relating the quantum commutator to the classical Poisson (Jacobi) bracket [3].*

The Drei-Männer Quantum Mechanics Paper: Born, Heisenberg, and Jordan

Meanwhile, back in Göttingen, the three quantum physicists Born, Heisenberg and Jordan now combined forces to write a third foundational paper that established the full range of the new matrix mechanics. Heisenberg’s first paper had been the insight. Born and Jordan’s following paper had re-expressed Heistenberg’s formulas into matrix algebra. But both papers had used simple one-dimensional problems as test examples. Working together, they extended the new quantum mechanics to systems with many degrees of freedom.

Quantum Mechanics. Born, Heisenberg and Jordan three-man paper heading. — *Fig. 13 Header for the completed new theory on quantum mechanics by Born, Heisenberg and Jordan [4].*

With this paper, the matrix properties of dynamical variables are defined and used in their full form.

Quantum Mechanics. Born, Heisenberg and Jordan. View of a matrix. — *Fig. 14 An explicit form for a dynamical matrix in the “three-man” paper [4].*

With the theory out in the open, Pauli in Hamburg and Dirac at Cambridge used the new quantum mechanics to derive the transition energies of hydrogen, while Lucy Mensing and J. Robert Oppenheimer in Göttingen extended it to the spectra of more complicated molecules.

Open Issues

Heisenberg’s matrix mechanics might have exclusively taken hold of the quantum theory community and we would all be using matrices today to perform all our calculations. But within one month of the success of matrix mechanics, an alternative quantum theory would be proposed by Erwin Schrödinger based on waves, a theory that came to be called wave mechanics. There was a minor battle fought over matrix mechanics versus wave mechanics, but in the end, Bohr compromised with his complementarity principle, allowing each to stand as equivalent viewpoints of quantum phenomena (but more about Schrödinger and his waves in my next Blog).

References

[1] Heisenberg, W. (1925). “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen”. Zeitschrift für Physik, 33(1), 879–893.

[2] Born, M., & Jordan, P. (1925). “Zur Quantenmechanik”. Zeitschrift für Physik, 34(1), 858–888.

[3] Dirac, P. A. M. (1925). The fundamental equations of quantum mechanics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 109(752), 642–653

[4] Born, M., W. Heisenberg and P. Jordan (1926). “Quantum mechanics II.” Zeitschrift Fur Physik, 35, (8/9): 557–615.

[5] Dirac, P. A. M. (1926). “Quantum mechanics and a preliminary investigation of the hydrogen atom.” Proceedings of the Royal Society of London Series A, 110(755): 561–79.

[6] Pauli, W. (1926). “The hydrogen spectrum from the view point of the new quantal mechanics.” Zeitschrift Fur Physik, 36(5): 336–63.

[7] Mensing, L. (1926). “Die Rotations-Schwingungsbanden nach der Quantenmechanik”. Zeitschrift für Physik, 36(11), 814–823.

[8] Born, M., & Oppenheimer, J. R. (1927). “Zur Quantentheorie der Molekeln”. Annalen der Physik, 389(20), 457–484.

December 11, 2024September 5, 2025 by David D. Nolte

100 Years of Quantum Physics: Pauli’s Exclusion Principle (1924)

One hundred years ago this month, in December 1924, Wolfgang Pauli submitted a paper to Zeitschrift für Physik that provided the final piece of the puzzle that connected Bohr’s model of the atom to the structure of the periodic table. In the process, he introduced a new quantum number into physics that governs how matter as extreme as neutron stars, or as perfect as superfluid helium, organizes itself.

He was led to this crucial insight, not by his superior understanding of quantum physics, which he was grappling with as much as Bohr and Born and Sommerfeld were at that time, but through his superior understanding of relativistic physics that convinced him that the magnetism of atoms in magnetic fields could not be explained through the orbital motion of electrons alone.

Encyclopedia Article on Relativity

Bored with the topics he was being taught in high school in Vienna, Pauli was already reading Einstein on relativity and Emil Jordan on functional analysis before he arrived at the university in Munich to begin studying with Arnold Sommerfeld. Pauli was still merely a student when Felix Klein approached Sommerfeld to write an article on relativity theory for his Encyclopedia of Mathematical Sciences. Sommerfeld by that time was thoroughly impressed with Pauli’s command of the subject and suggested that he write the article.

PauliRelativity Download

Pauli’s encyclopedia article on relativity expanded to 250 pages and was published in Klein’s fifth volume in 1921 when Pauli was only 21 years old—just 5 years after Einstein had published his definitive work himself! Pauli’s article is still considered today one of the clearest explanations of both special and general relativity.

Pauli’s approach established the methodical use of metric space concepts that is still used today when teaching introductory courses on the topic. This contrasts with articles written only a few years earlier that seem archaic by comparison—even Einstein’s paper itself. As I recently read through his article, I was struck by how similar it is to what I teach from my textbook on modern dynamics to my class at Purdue University for junior physics majors.

Anomalous Zeeman Effect

In 1922, Pauli completed his thesis on the properties of water molecules and began studying a phenomenon known as the anomalous Zeeman effect. The Zeeman effect is the splitting of optical transitions in atoms under magnetic fields. The electron orbital motion couples with the magnetic field through a semi-classical interaction between the magnetic moment of the orbital and the applied magnetic field, producing a contribution to the energy of the electron that is observed when it absorbs or emits light.

The Bohr model of the atom had already concluded that the angular momentum of electron orbitals was quantized into integer units. Furthermore, the Stern-Gerlach experiment of 1922 had shown that the projection of these angular momentum states onto the direction of the magnetic field was also quantized. This was known at the time as “space quantization”. Therefore, in the Zeeman effect, the quantized angular momentum created quantized energy interactions with the magnetic field, producing the splittings in the optical transitions.

File:Breit-rabi-Zeeman-en.svg — Fig. 2 The magnetic Zeeman splitting of Rb-87 from the weak field to the strong-field (Pachen-Back) effect

So far so good. But then comes the problem with the anomalous Zeeman effect.

In the Bohr model, all angular momenta have integer values. But in the anomalous Zeeman effect, the splittings could only be explained with half integers. For instance, if total angular momentum were equal to one-half, then in a magnetic field it would produce a “doublet” with +1/2 and -1/2 space quantization. An integer like L = 1 would produce a triplet with +1, 0, and -1 space quantization. Although doublets of the anomalous Zeeman effect were often observed, half-integers were unheard of (so far) in the quantum numbers of early quantum physics.

But half integers were not the only problem with “2”s in the atoms and elements. There was also the problem of the periodic table. It, too, seemed to be constructed out of “2”s, multiplying a sequence of the difference of squares.

The Difference of Squares

The difference of squares has a long history in physics stretching all the way back to Galileo Galilei who performed experiments around 1605 on the physics of falling bodies. He noted that the distance traveled in successive time intervals varied as the difference 1² – 0² = 1, then 2²-1² = 3, then 3²-2² = 5, then 4²-3² = 7 and so on. In other words, the distances traveled in each successive time interval varied as the odd integers. Galileo, ever the astute student of physics, recognized that the distance traveled by an accelerating body in a time t varied as the square of time t². Today, after Newton, we know that this is simply the dependence of distance for an accelerating body on the square of time s = (1/2)gt².

By early 1924 there was another law of the difference of squares. But this time the physics was buried deep inside the new science of the elements, put on graphic display through the periodic table.

The periodic table is constructed on the difference of squares. First there is 2 for hydrogen and helium. Then another 2 for lithium and beryllium, followed by 6 for B, C, N, O, F and Ne to make a total of 8. After that there is another 8 plus 10 for the sequence of Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn to make a total of 18. The sequence of 2-8-18 is 2•1² = 2, 2•2² = 8, 2•3² = 18 for the sequence 2n².

Why the periodic table should be constructed out of the number 2 times the square of the principal quantum number n was a complete mystery. Sommerfeld went so far as to call the number sequence of the periodic table a “cabalistic” rule.

The Bohr Model for Many Electrons

It is easy to picture how confusing this all was to Bohr and Born and others at the time. From Bohr’s theory of the hydrogen atom, it was clear that there were different energy levels associated with the principal quantum number n, and that this was related directly to angular momentum through the motion of the electrons in the Bohr orbitals.

But as the periodic table is built up from H to He and then to Li and Be and B, adding in successive additional electrons, one of the simplest questions was why the electrons did not all reside on the lowest energy level? But even if that question could not be answered, there was the question of why after He the elements Li and Be behaved differently than B, N, O and F, leading to the noble gas Ne. From normal Zeeman spectroscopy as well as x-ray transitions, it was clear that the noble gases behaved as the core of succeeding elements, like He for Li and Be and Ne for Na and Mg.

To grapple with all of this, Bohr had devised a “building up” rule for how electrons were “filling” the different energy levels as each new electron of the next element was considered. The noble-gas core played a key role in this model, and the core was also assumed to be contributing to both the normal Zeeman effect as well as the anomalous Zeeman effect with its mysterious half-integer angular momenta.

But frankly, this core model was a mess, with ad hoc rules on how the additional electrons were filling the energy levels and how they were contributing to the total angular momentum.

This was the state of the problem when Pauli, with his exceptional understanding of special relativity, began to dig deep into the problem. Since the Zeeman splittings were caused by the orbital motion of the electrons, the strongly bound electrons in high-Z atoms would be moving at speeds near the speed of light. Pauli therefore calculated what the systematic effects would be on the Zeeman splittings as the Z of the atoms got larger and the relativistic effects got stronger.

He calculated this effect to high precision, and then waited for Landé to make the measurements. When Landé finally got back to him, it was to say that there was absolutely no relativistic corrections for Thallium (Z = 90). The splitting remained simply fixed by the Bohr magneton value with no relativistic effects.

Pauli had no choice but to reject the existing core model of angular momentum and to ascribe the Zeeman effects to the outer valence electron. But this was just the beginning.

Pauli’s Breakthrough

https://onionesquereality.wordpress.com/wp-content/uploads/2012/07/wolfgang-pauli.jpg — Fig. 5 Wolfgang Pauli [Image]

By November of 1924 Pauli had concluded, in a letter to Landé

“In a puzzling, non-mechanical way, the valence electron manages to run about in two states with the same k but with different angular momenta.”

And in December of 1924 he submitted his work on the relativistic effects (or lack thereof) to Zeitschrift für Physik,

“From this viewpoint the doublet structure of the alkali spectra as well as the failure of Larmor’s theorem arise through a specific, classically non-describable sort of Zweideutigkeit (two-foldness) of the quantum-theoretical properties of the valence electron. (Pauli, 1925a, pg. 385)

Around this time, he read a paper by Edmund Stoner in the Philosophical Magazine of London published in October of 1924. Stoner’s insight was a connection between the number of states observed in a magnetic field and the number of states filled in the successive positions of elements in the periodic table. Stoner’s insight led naturally to the 2-8-18 sequence for the table, although he was still thinking in terms of the quantum numbers of the core model of the atoms.

This is when Pauli put 2 plus 2 together: He realized that the states of the atom could be indexed by a set of 4 quantum numbers: n-the principal quantum number, k1-the angular momentum, m1-the space quantization number, and a new fourth quantum number m2 that he introduced but that had, as yet, no mechanistic explanation. With these four quantum numbers enumerated, he then made the major step:

It should be forbidden that more than one electron, having the same equivalent quantum numbers, can be in the same state. When an electron takes on a set of values for the four quantum numbers, then that state is occupied.

This is the Exclusion Principle: No two electrons can have the same set of quantum numbers. Or equivalently, no electron state can be occupied by two electrons.

Fig. 6 Level filling for Krypton using the Pauli Exclusion Principle

Today, we know that Pauli’s Zweideutigkeit is electron spin, a concept first put forward in 1925 by the American physicist Ralph Kronig and later that year by George Uhlenbeck and Samuel Goudsmit.

The “Famous” Pauli Exclusion Paper Download

And Pauli’s Exclusion Principle is a consequence of the antisymmetry of electron wavefunctions first described by Paul Dirac in 1926 after the introduction of wavefunctions into quantum theory by Erwin Schrödinger earlier that year.

Timeline:

1845 – Faraday effect (rotation of light polarization in a magnetic field)

1896 – Zeeman effect (splitting of optical transition in a magnetic field)

1897 – Anomalous Zeeman effect (half-integer splittings)

1902 – Lorentz and Zeeman awarded Nobel prize (for electron theory)

1921 – Paschen-Back effect (strong-field Zeeman effect)

1922 – Stern-Gerlach (space quantization)

1924 – de Broglie matter waves

1924 – Bose statistics of photons

1924 – Stoner (conservation of number of states)

1924 – Pauli Exclusion Principle

References:

E. C. Stoner (Philosophical Magazine, 48 [1924], 719) Issue 286 October 1924

M. Jammer, The conceptual development of quantum mechanics (Los Angeles, Calif.: Tomash Publishers, Woodbury, N.Y. : American Institute of Physics, 1989).

M. Massimi, Pauli’s exclusion principle: The origin and validation of a scientific principle (Cambridge University Press, 2005).

Pauli, W. Über den Einfluß der Geschwindigkeitsabhängigkeit der Elektronenmasse auf den Zeemaneffekt. Z. Physik 31, 373–385 (1925). https://doi.org/10.1007/BF02980592

Pauli, W. (1925). “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren”. Zeitschrift für Physik. 31 (1): 765–783

Read more in Books by David Nolte at Oxford University Press

September 25, 2024September 5, 2025 by David D. Nolte

Edward Purcell: From Radiation to Resonance

As the days of winter darkened in 1945, several young physicists huddled in the basement of Harvard’s Research Laboratory of Physics, nursing a high field magnet to keep it from overheating and dumping its field. They were working with bootstrapped equipment—begged, borrowed or “stolen” from various labs across the Harvard campus. The physicist leading the experiment, Edward Mills Purcell, didn’t even work at Harvard—he was still on the payroll of the Radiation Laboratory at MIT, winding down from its war effort on radar research for the military in WWII, so the Harvard experiment was being done on nights and weekends.

Just before Christmas, 1945, as college students were fleeing campus for the first holiday in years without war, the signal generator, borrowed from a psychology lab, launched an electromagnetic pulse into simple paraffin—and disappeared! It had been absorbed by the nuclear spins of the copious number of hydrogen nuclei (protons) in the wax.

The experiment was simple, unfunded, bootstrapped—and it launched a new field of physics that ultimately led to magnetic resonance imaging (MRI) that is now the workhorse of 3D medical imaging.

This is the story, in Purcell’s own words, of how he came to the discovery of nuclear magnetic resonance in solids, for which he was awarded the Nobel Prize in Physics in 1952.

Early Days

Edward Mills Purcell (1912 – 1997) was born in a small town in Illinois, the son of a telephone businessman, and some of his earliest memories were of rummaging around in piles of telephone equipment—wires and transformers and capacitors. He especially like the generators:

“You could always get plenty of the bell-ringing generators that were in the old telephones, which consisted of a series of horseshoe magnets making the stator field and an armature that was wound with what must have been a mile of number 39 wire or something like that… These made good shocking machines if nothing else.”

His science education in the small town was modest, mostly chemistry, but he had a physics teacher, a rare woman at that time, who was open to searching minds. When she told the students that you couldn’t pull yourself up using a single pulley, Purcell disagreed and got together with a friend:

“So we went into the barn after school and rigged this thing up with a seat and hooked the spring scales to the upgoing rope and then pulled on the downcoming rope.”

The experiment worked, of course, with the scale reading half the weight of the boy. When they rushed back to tell the physics teacher, she accepted their results immediately—demonstration trumped mere thought, and Purcell had just done his first physics experiment.

However, physics was not a profession in the early 1920’s.

“In the ’20s the idea of chemistry as a science was extremely well publicized and popular, so the young scientist of shall we say 1928 — you’d think of him as a chemist holding up his test tube and sighting through it or something…there was no idea of what it would mean to be a physicist.

The name Steinmetz was more familiar and exciting than the name Einstein, because Steinmetz was the famous electrical engineer at General Electric and was this hunchback with a cigar who was said to know the four-place logarithm table by heart.”

Purdue University and Prof. Lark-Horowitz

Purcell entered Purdue University in the Fall of 1929. The University had only 4500 students who paid $50 a year to attend. He chose a major in electrical engineering, because

“Being a physicist…I don’t remember considering that at that time as something you could be…you couldn’t major in physics. You see, Purdue had electrical, civil, mechanical and chemical engineering. It had something called the School of Science, and you could graduate, having majored in science.”

But he was drawn to physics. The Physics Department at Purdue was going through a Renaissance under the leadership of its new department head Prof. Lark-Horovitz

“His [Lark-Horovitz] coming to Purdue was really quite important for American physics in many ways… It was he who subsequently over the years brought many important and productive European physicists to this country; they came to Purdue, passed through. And he began teaching; he began having graduate students and teaching really modern physics as of 1930, in his classes.”

Purcell attended Purdue during the early years of the depression when some students didn’t have enough money to find a home:

“People were also living down there in the cellar, sleeping on cots in the research rooms, because it was the Depression and some of the graduate students had nowhere else to live. I’d come in in the morning and find them shaving.”

Lark-Horovitz was a demanding department chair, but he was bringing the department out of the dark ages and into the modern research world.

“Lark-Horovitz ran the physics department on the European style: a pyramid with the professor at the top and everybody down below taking orders and doing what the professor thought ought to be done. This made working for him rather difficult. I was insulated by one layer from that because it was people like Yearian, for whom I was working, who had to deal with the Lark. “

Hubert Yearian had built a 20-kilovolt electron diffraction camera, a Debye-Scherrer transmission camera, just a few years after Davisson and Germer had performed the Nobel-prize winning experiment at Bell Labs that proved the wavelike nature of electrons. Purcell helped Yearian build his own diffraction system, and recalled:

“When I turned on the light in the dark room, I had Debye-Scherrer rings on it from electron diffraction — and that was only five years after electron diffraction had been discovered. So it really was right in the forefront. And as just an undergraduate, to be able to do that at that time was fantastic.”

Purcell graduated from Purdue in 1933 and from contacts through Lark-Horovitz he was able to spend a year in the physics department at Karlsruhe in Germany. He returned to the US in 1934 to enter graduate scool in physics at Harvard, working under Kenneth Bainbridge. His thesis topic was a bit of a bust, a dusty old problem in classical electrostatics that was a topic far older than the electron diffraction he worked on at Purdue. But it was enough to get him his degree in 1938, and he stayed on at Harvard as a faculty instructor until the war broke out.

Radiation Laboratory, MIT

In the Fall at the end of 1940 the Radiation Lab at MIT was launched and began vacuuming up all the unattached physicists in the United States, and Purcell was one of them. The radiation lab also vacuumed up some of the top physicists in the country, like Isidor Rabi from Columbia, to supervise the growing army of scientists that were committed to the war effort—even before the US was in the war.

“Our mission was to make a radar for a British night fighter using 10-centimeter magnetron that had been discovered at Birmingham.”

This research turned Purcell and his cohort into experts in radio-frequency electronics and measurement. He worked closely with Rabi (Nobel Prize 1944) and Norman Ramsey (Nobel Prize 1989) and Jerrold Zacharias, who were in the midst of measuring resonances in molecular beams. The names at the Rad Lab was like reading a Who’s Who of physics at that time:

“And then there was the theoretical group, which was also under Rabi. Most of their theory was concerned with electromagnetic fields and signal to noise, things of that sort. George Uhlenbeck was in charge of it for quite a long time, and Bethe was in it for a while; Schwinger was in it; Frank Carlson; David Saxon, now president of the University of California; Goudsmit also.”

Nuclear Magnetic Resonance

The research by Rabi had established the physics of resonances in molecular beams, but there were serious doubts that such phenomena could exist in solids. This became one of the Holy Grails of physics, with only a few physicists across the country with the skill and understanding to make a try to observe it in the solid state.

Many of the physicists at the Rad Lab were wondering what they should do next, after the war was over.

“Came the end of the war and we were all thinking about what shall we do when we go back and start doing physics. In the course of knocking around with these people, I had learned enough about what they had done in molecular beams to begin thinking about what can we do in the way of resonance with what we’ve learned. And it was out of that kind of talk that I was struck with the idea for what turned into nuclear magnetic resonance.”

“Well, that’s how NMR started, with that idea which, as I say, I can trace back to all those indirect influences of talking with Rabi, Ramsey and Zacharias, thinking about what we should do next.

“We actually did the first NMR experiment here [Harvard], not at MIT. But I wasn’t officially back. In fact, I went around MIT trying to borrow a magnet from somebody, a big magnet, get access to a big magnet so we could try it there and I didn’t have any luck. So I came back and talked to Curry Street, and he invited us to use his big old cosmic ray magnet which was out in the shed. So I didn’t ask anybody else’s permission. I came back and got the shop to make us some new pole pieces, and we borrowed some stuff here and there. We borrowed our signal generator from the Psycho Acoustic Lab that Smitty Stevens had. I don’t know that it ever got back to him. And some of the apparatus was made in the Radiation Lab shops. Bob Pound got the cavity made down there. They didn’t have much to do — things were kind of closing up — and so we bootlegged a cavity down there. And we did the experiment right here on nights and week-ends.

This was in December, 1945.

“Our first experiment was done on paraffin, which I bought up the street at the First National store between here and our house. For paraffin we thought we might have to deal with a relaxation time as long as several hours, and we were prepared to detect it with a signal which was sufficiently weak so that we would not upset the spin temperature while applying the r-f field. And, in fact, in the final time when the experiment was successful, I had been over here all night … nursing the magnet generator along so as to keep the field on for many hours, that being in our view a possible prerequisite for seeing the resonances. Now, it turned out later that in paraffin the relaxation time is actually 10^-4 seconds. So I had the magnet on exactly 10⁸ times longer than necessary!

The experiment was completed just before Christmas, 1945.

E. M. Purcell, H. C. Torrey, and R. V. Pound, “RESONANCE ABSORPTION BY NUCLEAR MAGNETIC MOMENTS IN A SOLID,” Physical Review 69, 37-38 (1946).

“But the thing that we did not understand, and it gradually dawned on us later, was really the basic message in the paper that was part of Bloembergen’s thesis … came to be known as BPP (Bloembergen, Purcell and Pound). [This] was the important, dominant role of molecular motion in nuclear spin relaxation, and also its role in line narrowing. So that after that was cleared up, then one understood the physics of spin relaxation and understood why we were getting lines that were really very narrow.”

Diagram of the microwave cavity filled with paraffin.

This was the discovery of nuclear magnetic resonance (NMR) for which Purcell shared the 1952 Nobel Prize in physics with Felix Bloch.

David D. Nolte is the Edward M. Purcell Distinguished Professor of Physics and Astronomy, Purdue University. Sept. 25, 2024

References and Notes

• The quotes from EM Purcell are from the “Living Histories” interview in 1977 by the AIP.

• K. Lark-Horovitz, J. D. Howe, and E. M. Purcell, “A new method of making extremely thin films,” Review of Scientific Instruments 6, 401-403 (1935).

• E. M. Purcell, H. C. Torrey, and R. V. Pound, “RESONANCE ABSORPTION BY NUCLEAR MAGNETIC MOMENTS IN A SOLID,” Physical Review 69, 37-38 (1946).

• National Academy of Sciences Biographies: Edward Mills Purcell

July 24, 2024September 5, 2025 by David D. Nolte

The Vital Virial of Rudolph Clausius: From Stat Mech to Quantum Mech

I often joke with my students in class that the reason I went into physics is because I have a bad memory. In biology you need to memorize a thousand things, but in physics you only need to memorize 10 things … and you derive everything else!

Of course, the first question they ask me is “What are those 10 things?”.

That’s a hard question to answer, and every physics professor probably has a different set of 10 things. Obviously, energy conservation would be first on the list, followed by other conservation laws for various types of momentum. Inverse-square laws probably come next. But then what? What do you need to memorize to be most useful when you are working out physics problems on the back of an envelope, when your phone is dead, and you have no access to your laptop or books?

One of my favorites is the Virial Theorem because it rears its head over and over again, whether you are working on problems in statistical mechanics, orbital mechanics or quantum mechanics.

The Virial Theorem

The Virial Theorem makes a simple statement about the balance between kinetic energy and potential energy (in a conservative mechanical system). It summarizes in a single form many different-looking special cases we learn about in physics. For instance, everyone learns early in their first mechanics course that the average kinetic energy <T> of a mass on a spring is equal to the average potential energy <V>. But this seems different than the problem of a circular orbit in gravitation or electrostatics where the average kinetic energy is equal to half the average potential energy, but with the opposite sign.

Yet there is a unity to these two—it is the Virial Theorem:

for cases where the potential energy V has power law dependence V ≈ rⁿ. The harmonic oscillator has n = 2, leading to the well-known equality between average kinetic and potential energy as

The inverse square force law has a potential that varies with n = -1, leading to the flip in sign. For instance, for a circular orbit in gravitation, it looks like

and in electrostatics it looks like

where a is the radius of the orbit.

Yet orbital mechanics is hardly the only place where the Virial Theorem pops up. It began its life with statistical mechanics.

Rudolph Clausius and his Virial Theorem

The pantheon of physics is a somewhat exclusive club. It lets in the likes of Galileo, Lagrange, Maxwell, Boltzmann, Einstein, Feynman and Hawking, but it excludes many worthy candidates, like Gilbert, Stevin, Maupertuis, du Chatelet, Arago, Clausius, Heaviside and Meitner all of whom had an outsized influence on the history of physics, but who often do not get their due. Of this later group, Rudolph Clausius stands above the others because he was an inventor of whole new worlds and whole new terminologies that permeate physics today.

Within the German Confederation dominated by Prussia in the mid 1800’s, Clausius was among the first wave of the “modern” physicists who emerged from new or reorganized German universities that integrated mathematics with practical topics. Carl Neumann at Königsberg, Carl Gauss and Max Weber at Göttingen, and Hermann von Helmholtz at Berlin were transforming physics from a science focused on pure mechanics and astronomy to one focused on materials and their associated phenomena, applying mathematics to these practical problems.

Clausius was educated at Berlin under Heinrich Gustav Magnus beginning in 1840, and he completed his doctorate at the University of Halle in 1847. His doctoral thesis on light scattering in the atmosphere represented an early attempt at treating statistical fluctuations. Though his initial approach was naïve, it helped orient Clausius to physics problems of statistical ensembles and especially to gases. The sophistication of his physics matured rapidly and already in 1850 he published his famous paper Über die bewegende Kraft der Wärme, und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen (About the moving power of heat and the laws that can be derived from it for the theory of heat itself).

This was the fundamental paper that overturned the archaic theory of caloric, which had assumed that heat was a form of conserved quantity. Clausius proved that this was not true, and he introduced what are today called the first and second laws of thermodynamics. This early paper was one in which he was still striving to simplify thermodynamics, and his second law was mostly a qualitative statement that heat flows from higher temperatures to lower. He refined the second law four years later in 1854 with Über eine veranderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie (On a modified form of the second law of the mechanical theory of heat). He gave his concept the name Entropy in 1865 from the Greek word τροπη (transformation or change) with a prefix similar to Energy.

Clausius was one of the first to consider the kinetic theory of heat where heat was understood as the average kinetic energy of the atoms or molecules that comprised the gas. He published his seminal work on the topic in 1857 expanding on earlier work by Augustus Krönig. Maxwell, in turn, expanded on Clausius in 1860 by introducing probability distributions. By 1870, Clausius was fully immersed in the kinetic theory as he was searching for mechanical proofs of the second law of thermodynamics. Along the way, he discovered a quantity based on action-reaction pairs of forces that was related to the kinetic energy.

At that time, kinetic energy was often called vis viva, meaning “living force”. The singular of force (vis) had a plural (virias), so Clausius—always happy to coin new words—called the action-reaction pairs of forces the virial, and hence he proved the Virial Theorem.

The argument is relatively simple. Consider the action of a single molecule of the gas subject to a force F that is applied reciprocally from another molecule. Also, for simplicity consider only a single direction in the gas. The change of the action over time is given by the derivative

The average over all action-reaction pairs is

but by the reciprocal nature of action-reaction pairs, the left-hand side balances exactly to zero, giving

This expression is expanded to include the other directions and to all N bodies to yield the Virial Theorem

where the sum is over all molecules in the gas, and Clausius called the term on the right the Virial.

An important special case is when the force law derives from a power law

Then the Virial Theorem becomes (again in just one dimension)

This is often the most useful form of the theorem. For a spring force, it leads to <T> = <V>. For gravitational or electrostatic orbits it is <T> = -1/2<V>.

The Virial in Astrophysics

Clausius originally developed the Virial Theorem for the kinetic theory of gases, but it has applications that go far beyond. It is already useful for simple orbital systems like masses interacting through central forces, and these can be scaled up to N-body systems like star clusters or galaxies.

Star clusters are groups of hundreds or thousands of stars that are gravitationally bound. Such a cluster may begin in a highly non-equilibrium configuration, but the mutual interactions among the stars causes a relaxation to an equilibrium configuration of positions and velocities. This process is known as Virialization. The time scale for virializaiton depends on the number of stars and on the initial configuration, such as whether there is a net angular momentum in the cluster.

A gravitational simulation of 700 stars is shown in Fig. 2. The stars are distributed uniformly with zero velocities. The cluster collapses under gravitational attraction, rebounds and approaches a steady state. The Virial Theorem applies at long times. The simulation assumed all motion was in the plane, and a regularization term was added to the gravitational potential to keep forces bounded.

Simulation of the virial theorem for a star cluster with kinetic and potential energy graphs — Fig. 2 A numerical example of the Virial Theorem for a star cluster of 700 stars beginning in a uniform initial state, collapsing under gravitational attraction, rebounding and then approaching a steady state. The kinetic energy and the potential energy of the system satisfy the Virial Theorem at long times.

The Virial in Quantum Physics

Quantum theory holds strong analogs to classical mechanics. For instance, the quantum commutation relations have strong similarities to Poisson Brackets. Similarly, the Virial in classical physics has a direct quantum analog.

Begin with the commutator between the Hamiltonian H and the action composed as the product of the position operator and the momentum operator X_nP_n

Expand the two commutators on the right to give

Now recognize that the commutator with the Hamiltonian is Ehrenfest’s Theorem on the time dependence of the operators

which equals zero when the system become stationary or steady state. All that remains is to take the expectation value of the equation (which can include many-body interactions as well)

which is the quantum form of the Virital Theorem which is identical to the classical form when the expectation value is replaced by the ensemble average.

For the hydrogen atom this is

for principal quantum number n and Bohr radius a_B. The quantum energy levels of the hydrogen atom are

By David D. Nolte, July 24, 2024

References

“Ueber die bewegende Kraft der Warme and die Gesetze welche sich daraus für die Warmelehre selbst ableiten lassen,” in Annalen der Physik, 79 (1850), 368–397, 500–524.

Über eine veranderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie, Annalen der Physik, 93 (1854), 481–506.

Ueber die Art der Bewegung, welche wir Warmenennen, Annalen der Physik, 100 (1857), 497–507.

Clausius, RJE (1870). “On a Mechanical Theorem Applicable to Heat”. Philosophical Magazine. Series 4. 40 (265): 122–127.

Matlab Code

function [y0,KE,Upoten,TotE] = Nbody(N,L)   %500, 100, 0

A = -1;        % Grav factor
eps = 1;        % 0.1
K = 0.00001;    %0.000025

format compact

mov_flag = 1;
if mov_flag == 1
    moviename = 'DrawNMovie';
    aviobj = VideoWriter(moviename,'MPEG-4');
    aviobj.FrameRate = 10;
    open(aviobj);
end

hh = colormap(jet);
%hh = colormap(gray);
rie = randintexc(255,255);       % Use this for random colors
%rie = 1:64;                     % Use this for sequential colors
for loop = 1:255
    h(loop,:) = hh(rie(loop),:);
end
figure(1)
fh = gcf;
clf;
set(gcf,'Color','White')
axis off

thet = 2*pi*rand(1,N);
rho = L*sqrt(rand(1,N));
X0 = rho.*cos(thet);
Y0 = rho.*sin(thet);

Vx0 = 0*Y0/L;   %1.5 for 500   2.0 for 700
Vy0 = -0*X0/L;
% X0 = L*2*(rand(1,N)-0.5);
% Y0 = L*2*(rand(1,N)-0.5);
% Vx0 = 0.5*sign(Y0);
% Vy0 = -0.5*sign(X0);
% Vx0 = zeros(1,N);
% Vy0 = zeros(1,N);

for nloop = 1:N
    y0(nloop) = X0(nloop);
    y0(nloop+N) = Y0(nloop);
    y0(nloop+2*N) = Vx0(nloop);
    y0(nloop+3*N) = Vy0(nloop);
end

T = 300;  %500
xp = zeros(1,N); yp = zeros(1,N);

for tloop = 1:T
    tloop
    
    delt = 0.005;
    tspan = [0 loop*delt];
    opts = odeset('RelTol',1e-2,'AbsTol',1e-5);
    [t,y] = ode45(@f5,tspan,y0,opts);
    
    %%%%%%%%% Plot Final Positions
    
    [szt,szy] = size(y);
    
    % Set nodes
    ind = 0; xpold = xp; ypold = yp;
    for nloop = 1:N
        ind = ind+1;
        xp(ind) = y(szt,ind+N);
        yp(ind) = y(szt,ind);
    end
    delxp = xp - xpold;
    delyp = yp - ypold;
    maxdelx = max(abs(delxp));
    maxdely = max(abs(delyp));
    maxdel = max(maxdelx,maxdely);
    
    rngx = max(xp) - min(xp);
    rngy = max(yp) - min(yp);
    maxrng = max(abs(rngx),abs(rngy));
    
    difepmx = maxdel/maxrng;
    
    crad = 2.5;
    subplot(1,2,1)
    gca;
    cla;
    
    % Draw nodes
    for nloop = 1:N
        rn = rand*63+1;
        colorval = ceil(64*nloop/N);
        
        rectangle('Position',[xp(nloop)-crad,yp(nloop)-crad,2*crad,2*crad],...
            'Curvature',[1,1],...
            'LineWidth',0.1,'LineStyle','-','FaceColor',h(colorval,:))
        
    end
    
    [syy,sxy] = size(y);
    y0(:) = y(syy,:);
    
    rnv = (2.0 + 2*tloop/T)*L;    % 2.0   1.5
    
    axis equal
    axis([-rnv rnv -rnv rnv])
    box on
    drawnow
    pause(0.01)
    
    KE = sum(y0(2*N+1:4*N).^2);
    
    Upot = 0;
    for nloop = 1:N
        for mloop = nloop+1:N
            dx = y0(nloop)-y0(mloop);
            dy = y0(nloop+N) - y0(mloop+N);
            dist = sqrt(dx^2+dy^2+eps^2);
            Upot = Upot + A/dist;
        end
    end
    
    Upoten = Upot;
    
    TotE = Upoten + KE;
    
    if tloop == 1
        TotE0 = TotE;
    end

    Upotent(tloop) = Upoten;
    KEn(tloop) = KE;
    TotEn(tloop) = TotE;
    
    xx = 1:tloop;
    subplot(1,2,2)
    plot(xx,KEn,xx,Upotent,xx,TotEn,'LineWidth',3)
    legend('KE','Upoten','TotE')
    axis([0 T -26000 22000])     % 3000 -6000 for 500   6000 -8000 for 700
    
    
    fh = figure(1);
    
    if mov_flag == 1
        frame = getframe(fh);
        writeVideo(aviobj,frame);
    end
    
end

if mov_flag == 1
    close(aviobj);
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    function yd = f5(t,y)
        
        for n1loop = 1:N
            
            posx = y(n1loop);
            posy = y(n1loop+N);
            momx = y(n1loop+2*N);
            momy = y(n1loop+3*N);
            
            tempcx = 0; tempcy = 0;
            
            for n2loop = 1:N
                if n2loop ~= n1loop
                    cposx = y(n2loop);
                    cposy = y(n2loop+N);
                    cmomx = y(n2loop+2*N);
                    cmomy = y(n2loop+3*N);
                    
                    dis = sqrt((cposy-posy)^2 + (cposx-posx)^2 + eps^2);
                    CFx = 0.5*A*(posx-cposx)/dis^3 - 5e-5*momx/dis^4;
                    CFy = 0.5*A*(posy-cposy)/dis^3 - 5e-5*momy/dis^4;
                    
                    tempcx = tempcx + CFx;
                    tempcy = tempcy + CFy;
                    
                end
            end
                        
            ypp(n1loop) = momx;
            ypp(n1loop+N) = momy;
            ypp(n1loop+2*N) = tempcx - K*posx;
            ypp(n1loop+3*N) = tempcy - K*posy;
        end
        
        yd=ypp'; 
     
    end     % end f5

end     % end Nbody

Read more in Books by David D. Nolte at Oxford University Press

June 26, 2024September 5, 2025 by David D. Nolte

100 Years of Quantum Physics: The Statistics of Satyendra Nath Bose (1924)

One hundred years ago, in July of 1924, a brilliant Indian physicist changed the way that scientists count. Satyendra Nath Bose (1894 – 1974) mailed a letter to Albert Einstein enclosed with a manuscript containing a new derivation of Planck’s law of blackbody radiation. Bose had used a radical approach that went beyond the classical statistics of Maxwell and Boltzmann by counting the different ways that photons can fill a volume of space. His key insight was the indistinguishability of photons as quantum particles.

Today, the indistinguishability of quantum particles is the foundational element of quantum statistics that governs how fundamental particles combine to make up all the matter of the universe. At the time, neither Bose nor Einstein realized just how radical his approach was, until Einstein, using Bose’s idea, derived the behavior of material particles under conditions similar black-body radiation, predicting a new state of condensed matter [1]. It would take scientists 70 years to finally demonstrate “Bose-Einstein” condensation in a laboratory in Boulder, Colorado in 1995.

Early Days of the Photon

As outlined in a previous blog (see Who Invented the Quantum? Einstein versus Planck), Max Planck was a reluctant revolutionary. He was led, almost against his will, in 1900 to postulate a quantized interaction between electromagnetic radiation and the atoms in the walls of a black-body enclosure. He could not break free from the hold of classical physics, assuming classical properties for the radiation and assigning the quantum only to the “interaction” with matter. It was Einstein, five years later in 1905, who took the bold step of assigning quantum properties to the radiation field itself, inventing the idea of the “photon” (named years later by the American chemist Gilbert Lewis) as the first quantum particle.

Despite the vast potential opened by Einstein’s theory of the photon, quantum physics languished for nearly 20 years from 1905 to 1924 as semiclassical approaches dominated the thinking of Niels Bohr in Copenhagen, and Max Born in Göttingen, and Arnold Sommerfeld in Munich, as they grappled with wave-particle duality.

The existence of the photon, first doubted by almost everyone, was confirmed in 1915 by Robert Millikan’s careful measurement of the photoelectric effect. But even then, skepticism remained until Arthur Compton demonstrated experimentally in 1923 that the scattering of photons by electrons could only be explained if photons carried discrete energy and momentum in precisely the way that Einstein’s theory required.

Despite the success of Einstein’s photon by 1923, derivations of the Planck law still used a purely wave-based approach to count the number of electromagnetic standing waves that a cavity could support. Bose would change that by deriving the Planck law using purely quantum methods.

The Quantum Derivation by Bose

Satyendra Nath Bose was born in 1894 in Calcutta, the old British capital city of India, now Kolkata. He excelled at his studies, especially in mathematics, and received a lecturer post at the University of Calcutta from 1916 to 1921, when he moved into a professorship position at the new University of Dhaka.

One day, as he was preparing a class lecture on the derivation of Planck’s law,

he became dissatisfied with the usual way it was presented in textbooks, based on standing waves in the cavity, and he flipped the problem.

Rather than deriving the number of standing wave modes in real space, he considered counting the number of ways a photon would fill up phase space.

Phase space is the natural dynamical space of Hamiltonian systems [2], such as collections of quantum particles like photons, in which the axes of the space are defined by the positions and momenta of the particles. The differential volume of phase space dV_PS occupied by a single photon is given by

Using Einstein’s formula for the relationship between momentum and frequency

where h is Planck’s constant, yields

No quantum particle can have its position and momentum defined arbitrarily precisely because of Heisenberg’s uncertainty principle, requiring phase space volumes to be resolvable only to within a minimum reducible volume element given by h³.

Therefore, the number of states in phase space occupied by the single photon are obtained by dividing dV_PS by h³ to yield

which is half of the prefactor in the Planck law. Several comments are now necessary.

First, when Bose did this derivation, there was no Heisenberg Uncertainty relationship—that would come years later in 1927. Bose was guided, instead, by the work of Bohr and Sommerfeld and Ehrenfest who emphasized the role played by the action principle in quantum systems. Phase space dimensions are counted in units of action, and the quantized unit of action is given by Planck’s constant h, hence quantized volumes of action in phase space are given by h³. By taking this step, Bose was anticipating Heisenberg by nearly three years.

Second, Bose knew that his phase space volume was half of the prefactor in Planck’s law. But since he was counting states, he reasoned that this meant that each photon had two internal degrees of freedom. A possibility he considered to account for this was that the photon might have a spin that could be aligned, or anti-aligned, with the momentum of the photon [3, 4]. How he thought of spin is hard to fathom, because the spin of the electron, proposed by Uhlenbeck and Goudsmit, was still two years away.

But Bose was not finished. The derivation, so far, is just how much phase space volume is accessible to a single photon. The next step is to count the different ways that many photons can fill up phase space. For this he used (bringing in the factor of 2 for spin)

where p_n is the probability that a volume of phase space contains n photons, plus he used the usual conditions on energy and number

The probability for all the different permutations for how photons can occupy phase space is then given by

A third comment is now necessary: By assuming this probability, Bose was discounting situations where the photons could be distinguished from one another. This indistinguishability of quantum particles is absolutely fundamental to our understanding today of quantum statistics, but Bose was using it implicitly for the first time here.

The final distribution of photons at a given temperature T is found by maximizing the entropy of the system

subject to the conditions of photon energy and number. Bose found the occupancy probabilities to be

with a coefficient B to be found next by using this in the expression for the geometric series

yielding

Also, from the total number of photons

And, from the total energy

Bose obtained, finally

which is Planck’s law.

This derivation uses nothing by the counting of quanta in phase space. There are no standing waves. It is a purely quantum calculation—the first of its kind.

Enter Einstein

As usual with revolutionary approaches, Bose’s initial manuscript submitted to the British Philosophical Magazine was rejected. But he was convinced that he had attained something significant, so he wrote his letter to Einstein containing his manuscript, asking that if Einstein found merit in the derivation, then perhaps he could have it translated into German and submitted to the Zeitschrift für Physik. (That Bose would approach Einstein with this request seems bold, but they had communicated some years before when Bose had translated Einstein’s theory of General Relativity into English.)

Indeed, Einstein recognized immediately what Bose had accomplished, and he translated the manuscript himself into German and submitted it to the Zeitschrift on July 2, 1924 [5].

During his translation, Einstein did not feel that Bose’s conjecture about photon spin was defensible, so he changed the wording to attribute the factor of 2 in the derivation to the two polarizations of light (a semiclassical concept), so Einstein actually backtracked a little from what Bose originally intended as a fully quantum derivation. The existence of photon spin was confirmed by C. V. Raman in 1931 [6].

In late 1924, Einstein applied Bose’s concepts to an ideal gas of material atoms and predicted that at low temperatures the gas would condense into a new state of matter known today as a Bose-Einstein condensate [1]. Matter differs from photons because the conservation of atom number introduces a finite chemical potential to the problem of matter condensation that is not present in the Planck law.

Fig. 1 Experimental evidence for the Bose-Einstein condensate in an atomic vapor [7].

Paul Dirac, in 1945, enshrined the name of Bose by coining the phrase “Boson” to refer to a particle of integer spin, just as he coined “Fermion” after Enrico Fermi to refer to a particle of half-integer spin. All quantum statistics were encased by these two types of quantum particle until 1982, when Frank Wilczek coined the term “Anyon” to describe the quantum statistics of particles confined to two dimensions whose behaviors vary between those of a boson and of a fermion.

By David D. Nolte, June 26, 2024

References

[1] A. Einstein. “Quantentheorie des einatomigen idealen Gases”. Sitzungsberichte der Preussischen Akademie der Wissenschaften. 1: 3. (1925)

[2] D. D. Nolte, “The tangled tale of phase space,” Physics Today 63, 33-38 (2010).

[3] Partha Ghose, “The Story of Bose, Photon Spin and Indistinguishability” arXiv:2308.01909 [physics.hist-ph]

[4] Barry R. Masters, “Satyendra Nath Bose and Bose-Einstein Statistics“, Optics and Photonics News, April, pp. 41-47 (2013)

[5] S. N. Bose, “Plancks Gesetz und Lichtquantenhypothese”, Zeitschrift für Physik , 26 (1): 178–181 (1924)

[6] C. V. Raman and S. Bhagavantam, Ind. J. Phys. vol. 6, p. 353, (1931).

[7] Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A. (14 July 1995). “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor”. Science. 269 (5221): 198–201.

Read more in Books by David Nolte at Oxford University Press

April 24, 2024September 5, 2025 by David D. Nolte

The Ubiquitous George Uhlenbeck

There are sometimes individuals who seem always to find themselves at the focal points of their times. The physicist George Uhlenbeck was one of these individuals, showing up at all the right times in all the right places at the dawn of modern physics in the 1920’s and 1930’s. He studied under Ehrenfest and Bohr and Born, and he was friends with Fermi and Oppenheimer and Oskar Klein. He taught physics at the universities at Leiden, Michigan, Utrecht, Columbia, MIT and Rockefeller. He was a wide-ranging theoretical physicist who worked on Brownian motion, early string theory, quantum tunneling, and the master equation. Yet he is most famous for the very first thing he did as a graduate student—the discovery of the quantum spin of the electron.

Electron Spin

G. E. Uhlenbeck, and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature 117, 264-265 (1926).

George Uhlenbeck (1900 – 1988) was born in the Dutch East Indies, the son of a family with a long history in the Dutch military [1]. After the father retired to The Hague, George was expected to follow the family tradition into the military, but he stumbled onto a copy of H. Lorentz’ introductory physics textbook and was hooked. Unfortunately, to attend university in the Netherlands at that time required knowledge of Greek and Latin, which he lacked, so he entered the Institute of Technology in Delft to study chemical engineering. He found the courses dreary.

Fortunately, he was only a few months into his first semester when the language requirement was dropped, and he immediately transferred to the University of Leiden to study physics. He tried to read Boltzmann, but found him opaque, but then read the famous encyclopedia article by the husband and wife team of Paul and Tatiana Ehrenfest on statistical mechanics (see my Physics Today article [2]), which became his lifelong focus.

After graduating, he continued into graduate school, taking classes from Ehrenfest, but lacking funds, he supported himself by teaching classes at a girls high school, until he heard of a job tutoring the son of the Dutch ambassador to Italy. He was off to Rome for three years, where he met Enrico Fermi and took classes from Tullio Bevi-Cevita and Vito Volterra.

However, he nearly lost his way. Surrounded by the rich cultural treasures of Rome, he became deeply interested in art and was seriously considering giving up physics and pursuing a degree in art history. When Ehrenfest got wind of this change in heart, he recalled Uhlenbeck in 1925 to the Netherlands and shrewdly paired him up with another graduate student, Samuel Goudsmit, to work on a new idea proposed by Wolfgang Pauli a few months earlier on the exclusion principle.

Pauli had explained the filling of the energy levels of atoms by introducing a new quantum number that had two values. Once an energy level was filled by two electrons, each carrying one of the two quantum numbers, this energy level “excluded” any further filling by other electrons.

To Uhlenbeck, these two quantum numbers seemed as if they must arise from some internal degree of freedom, and in a flash of insight he imagined that it might be caused if the electron were spinning. Since spin was a form of angular momentum, the spin degree of freedom would combine with orbital angular momentum to produce a composite angular momentum for the quantum levels of atoms.

The idea of electron spin was not immediately embraced by the broader community, and Bohr and Heisenberg and Pauli had their reservations. Fortunately, they all were traveling together to attend the 50^th anniversary of Lorentz’ doctoral examination and were met at the train station in Leiden by Ehrenfest and Einstein. As usual, Einstein had grasped the essence of the new physics and explained how the moving electron feels an induced magnetic field which would act on the magnetic moment of the electron to produce spin-orbit coupling. With that, Bohr was convinced.

Uhlenbeck and Goudsmit wrote up their theory in a short article in Nature, followed by a short note by Bohr. A few months later, L. H. Thomas, while visiting Bohr in Copenhagen, explained the factor of two that appears in (what later came to be called) Thomas precession of the electron, cementing the theory of electron spin in the new quantum mechanics.

5-Dimensional Quantum Mechanics

P. Ehrenfest, and G. E. Uhlenbeck, “Graphical illustration of De Broglie’s phase waves in the five-dimensional world of O Klein,” Zeitschrift Fur Physik 39, 495-498 (1926).

Around this time, the Swedish physicist Oskar Klein visited Leiden after returning from three years at the University of Michigan where he had taken advantage of the isolation to develop a quantum theory of 5-dimensional spacetime. This was one of the first steps towards a grand unification of the forces of nature since there was initial hope that gravity and electromagnetism might both be expressed in terms of the five-dimensional space.

An unusual feature of Klein’s 5-dimensional relativity theory was the compactness of the fifth dimension, in which it was “rolled up” into a kind of high-dimensional string with a tiny radius. If the 4-dimensional theory of spacetime was sometimes hard to visualize, here was an even tougher problem.

Uhlenbeck and Ehrenfest met often with Klein during his stay in Leiden, discussing the geometry and consequences of the 5-dimensional theory. Ehrenfest was always trying to get at the essence of physical phenomena in the simplest terms. His famous refrain was “Was ist der Witz?” (What is the point?) [1]. These discussions led to a simple paper in Zeitschrift für Physik published later that year in 1926 by Ehrenfest and Uhlenbeck with the compelling title “Graphical Illustration of De Broglie’s Phase Waves in the Five-Dimensional World of O Klein”. The paper provided the first visualization of the 5-dimensional spacetime with the compact dimension. The string-like character of the spacetime was one of the first forays into modern day “string theory” whose dimensions have now expanded to 11 from 5.

During his visit, Klein also told Uhlenbeck about the relativistic Schrödinger equation that he was working on, which would later become the Klein-Gordon equation. This was a near miss, because what the Klein-Gordon equation was missing was electron spin—which Uhlenbeck himself had introduced into quantum theory—but it would take a few more years before Dirac showed how to incorporate spin into the theory.

Brownian Motion

G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review 36, 0823-0841 (1930).

After spending time with Bohr in Copenhagen while finishing his PhD, Uhlenbeck visited Max Born at Göttingen where he met J. Robert Oppenheimer who was also visiting Born at that time. When Uhlenbeck traveled to the United States in late summer of 1927 to take a position at the University of Michigan, he was met at the dock in New York by Oppenheimer.

Uhlenbeck was a professor of physics at Michigan for eight years from 1927 to 1935, and he instituted a series of Summer Schools [3] in theoretical physics that attracted international participants and introduced a new generation of American physicists to the rigors of theory that they previously had to go to Europe to find.

In this way, Uhlenbeck was part of a great shift that occurred in the teaching of graduate-level physics of the 1930’s that brought European expertise to the United States. Just a decade earlier, Oppenheimer had to go to Göttingen to find the kind of education that he needed for graduate studies in physics. Oppenheimer brought the new methods back with him to Berkeley, where he established a strong theory department to match the strong experimental activities of E. O. Lawrence. Now, European physicists too were coming to America, an exodus accelerated by the increasing anti-Semitism in Europe under the rise of fascism.

During this time, one of Uhlenbeck’s collaborators was L. S. Ornstein, the director of the Physical Laboratory at the University of Utrecht and a founding member of the Dutch Physical Society. Uhlenbeck and Ornstein were both interested in the physics of Brownian motion, but wished to establish the phenomenon on a more sound physical basis. Einstein’s famous paper of 1905 on Brownian motion had made several Einstein-style simplifications that stripped the complicated theory to its bare essentials, but had lost some of the details in the process, such as the role of inertia at the microscale.

Uhlenbeck and Ornstein published a paper in 1930 that developed the stochastic theory of Brownian motion, including the effects of particle inertia. The stochastic differential equation (SDE) for velocity is

where γ is viscosity, Γ is a fluctuation coefficient, and dw is a “Wiener process”. The Wiener differential dw has unusual properties such that

Uhlenbeck and Ornstein solived this SDE to yield an average velocity

which decays to zero at long times, and a variance

that asymptotes to a finite value at long times. The fluctuation coefficient is thus given by

for a process with characteristic speed v₀. An estimate for the fluctuation coefficient can be obtained by considering the force F on an object of size a

For instance, for intracellular transport [4], the fluctuation coefficient has a rough value of Γ = 2 Hz μm²/sec².

Quantum Tunneling

D. M. Dennison and G. E. Uhlenbeck, “The two-minima problem and the ammonia molecule,” Physical Review 41, 313-321 (1932).

By the early 1930’s, quantum tunnelling of the electron through classically forbidden regions of potential energy was well established, but electrons did not have a monopoly on quantum effects. Entire atoms—electrons plus nucleus—also have quantum wave functions and can experience regions of classically forbidden potential.

Uhlenbeck, with David Dennison, a fellow physicist at Ann Arbor, Michigan, developed the first quantum theory of molecular tunneling for the molecular configuration of ammonia NH₃ that can tunnel between the two equivalent configurations. Their use of the WKB approximation in the paper set the standard for subsequent WKB approaches that would play an important role in the calculation of nuclear decay rates.

Master Equation

A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I. The furry model and the fluctuation problem,” Physica 7, 344-360 (1940)

In 1935, Uhlenbeck left Michigan to take up the physics chair recently vacated by Kramers at Utrecht. However, watching the rising Nazism in Europe, he decided to return to the United States, beginning as a visiting professor at Columbia University in New York in 1940. During his visit, he worked with W. E. Lamb and A. Nordsieck on the problem of cosmic ray showers.

Their publication on the topic included a rate equation that is encountered in a wide range of physical phenomena. They called it the “Master Equation” for ease of reference in later parts of the paper, but this phrase stuck, and the “Master Equation” is now a standard tool used by physicists when considering the balances among multiples transitions.

Uhlenbeck never returned to Europe, moving among Michigan, MIT, Princeton and finally settling at Rockefeller University in New York from where he retired in 1971.

By David D. Nolte, April 24, 2024

Selected Works by George Uhlenbeck:

G. E. Uhlenbeck, and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature 117, 264-265 (1926).

P. Ehrenfest, and G. E. Uhlenbeck, “On the connection of different methods of solution of the wave equation in multi dimensional spaces,” Proceedings of the Koninklijke Akademie Van Wetenschappen Te Amsterdam 29, 1280-1285 (1926).

P. Ehrenfest, and G. E. Uhlenbeck, “Graphical illustration of De Broglie’s phase waves in the five-dimensional world of O Klein,” Zeitschrift Fur Physik 39, 495-498 (1926).

G. E. Uhlenbeck, and L. S. Ornstein, “On the theory of the Brownian motion,” Physical Review 36, 0823-0841 (1930).

D. M. Dennison, and G. E. Uhlenbeck, “The two-minima problem and the ammonia molecule,” Physical Review 41, 313-321 (1932).

E. Fermi, and G. E. Uhlenbeck, “On the recombination of electrons and positrons,” Physical Review 44, 0510-0511 (1933).

A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I The furry model and the fluctuation problem,” Physica 7, 344-360 (1940).

M. C. Wang, and G. E. Uhlenbeck, “On the Theory of the Brownian Motion-II,” Reviews of Modern Physics 17, 323-342 (1945).

G. E. Uhlenbeck, “50 Years of Spin – Personal Reminiscences,” Physics Today 29, 43-48 (1976).

Notes:

[1] George Eugene Uhlenbeck: A Biographical Memoire by George Ford (National Academy of Sciences, 2009). https://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/uhlenbeck-george.pdf

[2] D. D. Nolte, “The tangled tale of phase space,” Physics Today 63, 33-38 (2010).

NoltePT10 Download

[3] One of these was the famous 1948 Summer School session where Freeman Dyson met Julian Schwinger after spending days on a cross-country road trip with Richard Feynman. Schwinger and Feynman had developed two different approaches to quantum electrodynamics (QED), which Dyson subsequently reconciled when he took up his position later that year at Princeton’s Institute for Advanced Study, helping to launch the wave of QED that spread out over the theoretical physics community.

[4] D. D. Nolte, “Coherent light scattering from cellular dynamics in living tissues,” Reports on Progress in Physics 87 (2024).

February 14, 2024September 5, 2025 by David D. Nolte

100 Years of Quantum Physics: de Broglie’s Wave (1924)

One hundred years ago this month, in Feb. 1924, a hereditary member of the French nobility, Louis Victor Pierre Raymond, the 7th Duc de Broglie, published a landmark paper in the Philosophical Magazine of London [1] that revolutionized the nascent quantum theory of the day.

Prior to de Broglie’s theory of quantum matter waves, quantum physics had been mired in ad hoc phenomenological prescriptions like Bohr’s theory of the hydrogen atom and Sommerfeld’s theory of adiabatic invariants. After de Broglie, Erwin Schrödinger would turn the concept of matter waves into the theory of wave mechanics that we still practice today.

*Fig. 1 The 1924 paper by de Broglie in the Philosophical Magazine.*

The story of how de Broglie came to his seminal idea had an odd twist, based on an initial misconception that helped him get the right answer ahead of everyone else, for which he was rewarded with the Nobel Prize in Physics.

de Broglie’s Early Days

When Louis de Broglie was a student, his older brother Maurice (the 6^th Duc de Broglie) was already a practicing physicist making important discoveries in x-ray physics. Although Louis initially studied history in preparation for a career in law, and he graduated from the Sorbonne with a degree in history, his brother’s profession drew him like a magnet. He also read Poincaré at this critical juncture in his career, and he was hooked. He enrolled in the Faculty of Sciences for his advanced degree, but World War I side-tracked him into the signal corps, where he was assigned to the wireless station on top of the Eiffel Tower. He may have participated in the famous interception of a coded German transmission in 1918 that helped turn the tide of the war.

Beginning in 1919, Louis began assisting his brother in the well-equiped private laboratory that Maurice had outfitted in the de Broglie ancestral home. At that time Maurice was performing x-ray spectroscopy of the inner quantum states of atoms, and he was struck by the duality of x-ray properties that made them behave like particles under some conditions and like waves in others.

*Fig. 2 Maurice de Broglie in his private laboratory (Figure credit).*

*Fig. 3 Louis de Broglie (Figure credit)*

Through his close work with his brother, Louis also came to subscribe to the wave-particle duality of x-rays and chose the topic for his PhD thesis—and hence the twist that launched de Broglie backwards towards his epic theory.

de Broglie’s Massive Photons

Today, we say that photons have energy and momentum although they are massless. The momentum is a simple consequence of Einstein’s special relativity

And if m = 0, then

and momentum requires energy but not necessarily mass.

But de Broglie started out backwards. He was so convinced of the particle-like nature of the x-ray photons, that he first considered what would happen if the photons actually did have mass. He constructed a massive photon and compared its proper frequency with a Lorentz-boosted frequency observed in a laboratory. The frequency he set for the photon was like an internal clock, set by its rest-mass energy and by Bohr’s quantization condition

He then boosted it into the lab frame by time dilation

But the energy would be transformed according to

with a corresponding frequency

which is in direct contradiction with Bohr’s quantization condition. What is the resolution of this seeming paradox?

de Broglie’s Matter Wave

de Broglie realized that his “massive photon” must satisfy a condition relating the observed lab frequency to the transformed frequency, such that

This only made sense if his “massive photon” could be represented as a wave with a frequency

that propagated with a phase velocity given by c/β. (Note that β < 1 so that the phase velocity is greater than the speed of light, which is allowed as long as it does not transmit any energy.)

To a modern reader, this all sounds alien, but only because this work in early 1924 represented his first pass at his theory. As he worked on this thesis through 1924, finally defending it in November of that year, he refined his arguments, recognizing that when he combined his frequency with his phase velocity,

it yielded the wavelength for a matter wave to be

where p was the relativistic mechanical momentum of a massive particle.

Using this wavelength, he explained Bohr’s quantization condition as a simple standing wave of the matter wave. In the light of this derivation, de Broglie wrote

We are then inclined to admit that any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave.
pg. 450, Philosophical Magazine of London (1924)

Here was the strongest statement yet of the wave-particle duality of quantum particles. de Broglie went even further and connected the ideas of waves and rays through the Hamilton-Jacobi formalism, an approach that Dirac would extend several years later, establishing the formal connection between Hamiltonian physics and wave mechanics. Furthermore, de Broglie conceived of a “pilot wave” interpretation that removed some of Einstein’s discomfort with the random character of quantum measurement that ultimately led Einstein to battle Bohr in their famous debates, culminating in the iconic EPR paper that has become a cornerstone for modern quantum information science. After the wave-like nature of particles was confirmed in the Davisson-Germer experiments, de Broglie received the Nobel Prize in Physics in 1929.

*Fig. 4 A standing matter wave is a stationary state of constructive interference. This wavefunction is in the L = 5 quantum manifold of the hydrogen atom.*

Louis de Broglie was clearly ahead of his times. His success was partly due to his isolation from the dogma of the day. He was able to think without the constraints of preconceived ideas. But as soon as he became a regular participant in the theoretical discussions of his day, and bowed under the pressure from Copenhagen, his creativity essentially ceased. The subsequent development of quantum mechanics would be dominated by Heisenberg, Born, Pauli, Bohr and Schrödinger, beginning at the 1927 Solvay Congress held in Brussels.

By David D. Nolte, Feb. 14, 2024

[1] L. de Broglie, “A tentative theory of light quanta,” Philosophical Magazine 47, 446-458 (1924).

December 17, 2022September 5, 2025 by David D. Nolte

Paul Dirac’s Delta Function

Physical reality is nothing but a bunch of spikes and pulses—or glitches. Take any smooth phenomenon, no matter how benign it might seem, and decompose it into an infinitely dense array of infinitesimally transient, infinitely high glitches. Then the sum of all glitches, weighted appropriately, becomes the phenomenon. This might be called the “glitch” function—but it is better known as Green’s function in honor of the ex-millwright George Green who taught himself mathematics at night to became one of England’s leading mathematicians of the age.

The δ function is thus merely a convenient notation … we perform operations on the abstract symbols, such as differentiation and integration …
PAM Dirac (1930)

The mathematics behind the “glitch” has a long history that began in the golden era of French analysis with the mathematicians Cauchy and Fourier, was employed by the electrical engineer Heaviside, and ultimately fell into the fertile hands of the quantum physicist, Paul Dirac, after whom it is named.

Augustin-Louis Cauchy (1815)

The French mathematician and physicist Augustin-Louis Cauchy (1789 – 1857) has lent his name to a wide array of theorems, proofs and laws that are still in use today. In mathematics, he was one of the first to establish “modern” functional analysis and especially complex analysis. In physics he established a rigorous foundation for elasticity theory (including the elastic properties of the so-called luminiferous ether).

In the early days of the 1800’s Cauchy was exploring how integrals could be used to define properties of functions. In modern terminology we would say that he was defining kernel integrals, where a function is integrated over a kernel to yield some property of the function.

In 1815 Cauchy read before the Academy of Paris a paper with the long title “Theory of wave propagation on a surface of a fluid of indefinite weight”. The paper was not published until more than ten years later in 1827 by which time it had expanded to 300 pages and contained numerous footnotes. The thirteenth such footnote was titled “On definite integrals and the principal values of indefinite integrals” and it contained one of the first examples of what would later become known as a generalized distribution. The integral is a function F(μ) integrated over a kernel

Cauchy lets the scale parameter α be “an infinitely small number”. The kernel is thus essentially zero for any values of μ “not too close to α”. Today, we would call the kernel given by

in the limit that α vanishes, “the delta function”.

Cauchy’s approach to the delta function is today one of the most commonly used descriptions of what a delta function is. It is not enough to simply say that a delta function is an infinitely narrow, infinitely high function whose integral is equal to unity. It helps to illustrate the behavior of the Cauchy function as α gets progressively smaller, as shown in Fig. 1.

Fig. 1 Cauchy function for decreasing scale factor α approaches a delta function in the limit.

In the limit as α approaches zero, the function grows progressively higher and progressively narrower, but the integral over the function remains unity.

Joseph Fourier (1822)

The delayed publication of Cauchy’s memoire kept it out of common knowledge, so it can be excused if Joseph Fourier (1768 – 1830) may not have known of it by the time he published his monumental work on heat in 1822. Perhaps this is why Fourier’s approach to the delta function was also different than Cauchy’s.

Fourier noted that an integral over a sinusoidal function, as the argument of the sinusoidal function went to infinity, became independent of the limits of integration. He showed

when ε << 1/p as p went to infinity. In modern notation, this would be the delta function defined through the “sinc” function

and Fourier noted that integrating this form over another function f(x) yielded the value of the function f(α) evaluated at α, rediscovering the results of Cauchy, but using a sinc(x) function in Fig. 2 instead of the Cauchy function of Fig. 1.

Fig. 2 Sinc function for increasing scale factor p approaches a delta function in the limit.

George Green’s Function (1829)

A history of the delta function cannot be complete without mention of George Green, one of the most remarkable British mathematicians of the 1800’s. He was a miller’s son who had only one year of education and spent most of his early life tending to his father’s mill. In his spare time, and to cut the tedium of his work, he read the most up-to-date work of the French mathematicians, reading the papers of Cauchy and Poisson and Fourier, whose work far surpassed the British work at that time. Unbelievably, he mastered the material and developed new material of his own, that he eventually self published. This is the mathematical work that introduced the potential function and introduced fundamental solutions to unit sources—what today would be called point charges or delta functions. These fundamental solutions are equivalent to the modern Green’s function, although they were developed rigorously much later by Courant and Hilbert and by Kirchhoff.

George Green’s flour mill in Sneinton, England.

The modern idea of a Green’s function is simply the system response to a unit impulse—like throwing a pebble into a pond to launch expanding ripples or striking a bell. To obtain the solutions for a general impulse, one integrates over the fundamental solutions weighted by the strength of the impulse. If the system response to a delta function impulse at x = a, that is, a delta function δ(x-a), is G(x-a), then the response of the system to a distributed force f(x) is given by

where G(x-a) is called the Green’s function.

Fig. Principle of Green’s function. The Green’s function is the system response to a delta-function impulse. The net system response is the integral over all the individual system responses summed over each of the impulses.

Oliver Heaviside (1893)

Oliver Heaviside (1850 – 1925) tended to follow his own path, independently of whatever the mathematicians were doing. Heaviside took particularly pragmatic approaches based on physical phenomena and how they might behave in an experiment. This is the context in which he introduced once again the delta function, unaware of the work of Cauchy or Fourier.

Heaviside was an engineer at heart who practiced his art by doing. He was not concerned with rigor, only with what works. This part of his personality may have been forged by his apprenticeship in telegraph technology helped by his uncle Charles Wheatstone (of the Wheatstone bridge). While still a young man, Heaviside tried to tackle Maxwell on his new treatise on electricity and magnetism, but he realized his mathematics were lacking, so he began a project of self education that took several years. The product of those years was his development of an idiosyncratic approach to electronics that may be best described as operator algebra. His algebra contained mis-behaved functions, such as the step function that was later named after him. It could also handle the derivative of the step function, which is yet another way of defining the delta function, though certainly not to the satisfaction of any rigorous mathematician—but it worked. The operator theory could even handle the derivative of the delta function.

The Heaviside function (step function) and its derivative the delta function.

Perhaps the most important influence by Heaviside was his connection of the delta function to Fourier integrals. He was one of the first to show that

which states that the Fourier transform of a delta function is a complex sinusoid, and the Fourier transform of a sinusoid is a delta function. Heaviside wrote several influential textbooks on his methods, and by the 1920’s these methods, including the Heaviside function and its derivative, had become standard parts of the engineer’s mathematical toolbox.

Given the work by Cauchy, Fourier, Green and Heaviside, what was left for Paul Dirac to do?

Paul Dirac (1930)

Paul Dirac (1902 – 1984) was given the moniker “The Strangest Man” by Niels Bohr during his visit to Copenhagen shortly after he had received his PhD. In part, this was because of Dirac’s internal intensity that could make him seem disconnected from those around him. When he was working on a problem in his head, it was not unusual for him to start walking, and by the time he he became aware of his surroundings again, he would have walked the length of the city of Copenhagen. And his solutions to problems were ingenious, breaking bold new ground where others, some of whom were geniuses themselves, were fumbling in the dark.

Among his many influential works—works that changed how physicists thought of and wrote about quantum systems—was his 1930 textbook on quantum mechanics. This was more than just a textbook, because it invented new methods by unifying the wave mechanics of Schrödinger with the matrix mechanics of Born and Heisenberg.

In particular, there had been a disconnect between bound electron states in a potential and free electron states scattering off of the potential. In the one case the states have a discrete spectrum, i.e. quantized, while in the other case the states have a continuous spectrum. There were standard quantum tools for decomposing discrete states by a projection onto eigenstates in Hilbert space, but an entirely different set of tools for handling the scattering states.

Yet Dirac saw a commonality between the two approaches. Specifically, eigenstate decomposition on the one hand used discrete sums of states, while scattering solutions on the other hand used integration over a continuum of states. In the first format, orthogonality was denoted by a Kronecker delta notation, but there was no equivalent in the continuum case—until Dirac introduced the delta function as a kernel in the integrand. In this way, the form of the equations with sums over states multiplied by Kronecker deltas took on the same form as integrals over states multiplied by the delta function.

Page 64 of Dirac’s 1930 edition of Quantum Mechanics.

In addition to introducing the delta function into the quantum formulas, Dirac also explored many of the properties and rules of the delta function. He was aware that the delta function was not a “proper” function, but by beginning with a simple integral property as a starting axiom, he could derive virtually all of the extended properties of the delta function, including properties of its derivatives.

Mathematicians, of course, were appalled and were quick to point out the insufficiency of the mathematical foundation for Dirac’s delta function, until the French mathematician Laurent Schwartz (1915 – 2002) developed the general theory of distributions in the 1940’s, which finally put the delta function in good standing.

Dirac’s introduction, development and use of the delta function was the first systematic definition of its properties. The earlier work by Cauchy, Fourier, Green and Heaviside had all touched upon the behavior of such “spiked” functions, but they had used it in passing. After Dirac, physicists embraced it as a powerful new tool in their toolbox, despite the lag in its formal acceptance by mathematicians, until the work of Schwartz redeemed it.

By David D. Nolte Feb. 17, 2022

Bibliography

V. Balakrishnan, “All about the Dirac Delta function(?)”, Resonance, Aug., pg. 48 (2003)

M. G. Katz. “Who Invented Dirac’s Delta Function?”, Semantic Scholar (2010).

J. Lützen, The prehistory of the theory of distributions. Studies in the history of mathematics and physical sciences ; 7 (Springer-Verlag, New York, 1982).

November 26, 2022August 16, 2025 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).

November 6, 2022August 16, 2025 by David D. Nolte

A Short History of Quantum Tunneling

Quantum physics is often called “weird” because it does things that are not allowed in classical physics and hence is viewed as non-intuitive or strange. Perhaps the two “weirdest” aspects of quantum physics are quantum entanglement and quantum tunneling. Entanglement allows a particle state to extend across wide expanses of space, while tunneling allows a particle to have negative kinetic energy. Neither of these effects has a classical analog.

Quantum entanglement arose out of the Bohr-Einstein debates at the Solvay Conferences in the 1920’s and 30’s, and it was the subject of a recent Nobel Prize in Physics (2022). The quantum tunneling story is just as old, but it was recognized much earlier by the Nobel Prize in 1972 when it was awarded to Brian Josephson, Ivar Giaever and Leo Esaki—each of whom was a graduate student when they discovered their respective effects and two of whom got their big idea while attending a lecture class.

Always go to class, you never know what you might miss, and the payoff is sometimes BIG
Ivar Giaever

Of the two effects, tunneling is the more common and the more useful in modern electronic devices (although entanglement is coming up fast with the advent of quantum information science). Here is a short history of quantum tunneling, told through a series of publications that advanced theory and experiments.

Double-Well Potential: Friedrich Hund (1927)

The first analysis of quantum tunneling was performed by Friedrich Hund (1896 – 1997), a German physicist who studied early in his career with Born in Göttingen and Bohr in Copenhagen. He published a series of papers in 1927 in Zeitschrift für Physik [1] that solved the newly-proposed Schrödinger equation for the case of the double well potential. He was particularly interested in the formation of symmetric and anti-symmetric states of the double well that contributed to the binding energy of atoms in molecules. He derived the first tunneling-frequency expression for a quantum superposition of the symmetric and anti-symmetric states

where f is the coherent oscillation frequency, V is the height of the potential and hν is the quantum energy of the isolated states when the atoms are far apart. The exponential dependence on the potential height V made the tunnel effect extremely sensitive to the details of the tunnel barrier.

Electron Emission: Lothar Nordheim and Ralph Fowler (1927 – 1928)

The first to consider quantum tunneling from a bound state to a continuum state was Lothar Nordheim (1899 – 1985), a German physicist who studied under David Hilbert and Max Born at Göttingen and worked with John von Neumann and Eugene Wigner and later with Hans Bethe. In 1927 he solved the problem of a particle in a well that is separated from continuum states by a thin finite barrier [2]. Using the new Schrödinger theory, he found transmission coefficients that were finite valued, caused by quantum tunneling of the particle through the barrier. Nordheim’s use of square potential wells and barriers are now, literally, textbook examples that every student of quantum mechanics solves. (For a quantum simulation of wavefunction tunneling through a square barrier see the companion Quantum Tunneling YouTube video.) Nordheim later escaped the growing nationalism and anti-semitism in Germany in the mid 1930’s to become a visiting professor of physics at Purdue University in the United States, moving to a permanent position at Duke University.

Fig. 2 Nordheim square tunnel barrier and Fowler-Nordheim triangular tunnel barrier for electron tunneling from bound states into the continuum.

One of the giants of mathematical physics in the UK from the 1920s through the 1930’s was Ralph Fowler (1889 – 1944). Three of his doctoral students went on to win Nobel Prizes (Chandrasekhar, Dirac and Mott) and others came close (Bhabha, Hartree, Lennard-Jones). In 1928 Fowler worked with Nordheim on a more realistic version of Nordheim’s surface electron tunneling that could explain thermionic emission of electrons from metals under strong electric fields. The electric field modified Nordheim’s square potential barrier into a triangular barrier (which they treated using WKB theory) to obtain the tunneling rate [3]. This type of tunnel effect is now known as Fowler-Nordheim tunneling.

Nuclear Alpha Decay: George Gamow (1928)

George Gamov (1904 – 1968) is one of the icons of mid-twentieth-century physics. He was a substantial physicist who also had a solid sense of humor that allowed him to achieve a level of cultural popularity shared by a few of the larger-than-life physicists of his time, like Richard Feynman and Stephen Hawking. His popular books included One Two Three … Infinity as well as a favorite series of books under the rubric of Mr. Tompkins (Mr. Tompkins in Wonderland and Mr. Tompkins Explores the Atom, among others). He also wrote a history of the early years of quantum theory (Thirty Years that Shook Physics).

In 1928 Gamow was in Göttingen (the Mecca of early quantum theory) with Max Born when he realized that the radioactive decay of Uranium by alpha decay might be explained by quantum tunneling. It was known that nucleons were bound together by some unknown force in what would be an effective binding potential, but that charged alpha particles would also feel a strong electrostatic repulsive potential from a nucleus. Gamow combined these two potentials to create a potential landscape that was qualitatively similar to Nordheim’s original system of 1927, but with a potential barrier that was neither square nor triangular (like the Fowler-Nordheim situation).

Gamow was able to make an accurate approximation that allowed him to express the decay rate in terms of an exponential term

where Z_α is the atomic charge of the alpha particle, Z is the nuclear charge of the Uranium decay product and v is the speed of the alpha particle detected in external measurements [4].

The very next day after Gamow submitted his paper, Ronald Gurney and Edward Condon of Princeton University submitted a paper [5] that solved the same problem using virtually the same approach … except missing Gamow’s surprisingly concise analytic expression for the decay rate.

Molecular Tunneling: George Uhlenbeck (1932)

Because tunneling rates depend inversely on the mass of the particle tunneling through the barrier, electrons are more likely to tunnel through potential barriers than atoms. However, hydrogen is a particularly small atom and is therefore the most amenable to experiencing tunneling.

The first example of atom tunneling is associated with hydrogen in the ammonia molecule NH₃. The molecule has a pyramidal structure with the Nitrogen hovering above the plane defined by the three hydrogens. However, an equivalent configuration has the Nitrogen hanging below the hydrogen plane. The energies of these two configurations are the same, but the Nitrogen must tunnel from one side of the hydrogen plane to the other through a barrier. The presence of light-weight hydrogen that can “move out of the way” for the nitrogen makes this barrier very small (infrared energies). When the ammonia is excited into its first vibrational excited state, the molecular wavefunction tunnels through the barrier, splitting the excited level by an energy associated with a wavelength of 1.2 cm which is in the microwave. This tunnel splitting was the first microwave transition observed in spectroscopy and is used in ammonia masers.

Fig. 4 Nitrogen inversion in the ammonia molecule is achieved by excitation to a vibrational excited state followed by tunneling through the barrier, proposed by George Uhlenbeck in 1932.

One of the earliest papers [6] written on the tunneling of nitrogen in ammonia was published by George Uhlenbeck in 1932. George Uhlenbeck (1900 – 1988) was a Dutch-American theoretical physicist. He played a critical role, with Samuel Goudsmit, in establishing the spin of the electron in 1925. Both Uhlenbeck and Goudsmit were close associates of Paul Ehrenfest at Leiden in the Netherlands. Uhlenbeck is also famous for the Ornstein-Uhlenbeck process which is a generalization of Einstein’s theory of Brownian motion that can treat active transport such as intracellular transport in living cells.

Solid-State Electron Tunneling: Leo Esaki (1957)

Although the tunneling of electrons in molecular bonds and in the field emission from metals had been established early in the century, direct use of electron tunneling in solid state devices had remained elusive until Leo Esaki (1925 – ) observed electron tunneling in heavily doped Germanium and Silicon semiconductors. Esaki joined an early precursor of Sony electronics in 1956 and was supported to obtain a PhD from the University of Tokyo. In 1957 he was working with heavily-doped p-n junction diodes and discovered a phenomenon known as negative differential resistance where the current through an electronic device actually decreases as the voltage increases.

Because the junction thickness was only about 100 atoms, or about 10 nanometers, he suspected and then proved that the electronic current was tunneling quantum mechanically through the junction. The negative differential resistance was caused by a decrease in available states to the tunneling current as the voltage increased.

Fig. 5 Esaki tunnel diode with heavily doped p- and n-type semiconductors. At small voltages, electrons and holes tunnel through the semiconductor bandgap across a junction that is only about 10 nm wide. Ht higher voltage, the electrons and hole have no accessible states to tunnel into, producing negative differential resistance where the current decreases with increasing voltage.

Esaki tunnel diodes were the fastest semiconductor devices of the time, and the negative differential resistance of the diode in an external circuit produced high-frequency oscillations. They were used in high-frequency communication systems. They were also radiation hard and hence ideal for the early communications satellites. Esaki was awarded the 1973 Nobel Prize in Physics jointly with Ivar Giaever and Brian Josephson.

Superconducting Tunneling: Ivar Giaever (1960)

Ivar Giaever (1929 – ) is a Norwegian-American physicist who had just joined the GE research lab in Schenectady New York in 1958 when he read about Esaki’s tunneling experiments. He was enrolled at that time as a graduate student in physics at Rensselaer Polytechnic Institute (RPI) where he was taking a course in solid state physics and learning about superconductivity. Superconductivity is carried by pairs of electrons known as Cooper pairs that spontaneously bind together with a binding energy that produced an “energy gap” in the electron energies of the metal, but no one had ever found a way to directly measure it. The Esaki experiment made him immediately think of the equivalent experiment in which Cooper pairs might tunnel between two superconductors (through a thin oxide layer) and yield a measurement of the energy gap. The idea actually came to him during the class lecture.

The experiments used a junction between aluminum and lead (Al—Al₂O₃—Pb). At first, the temperature of the system was adjusted so that Al remained a normal metal and Pb was superconducting, and Giaever observed a tunnel current with a threshold related to the gap in Pb. Then the temperature was lowered so that both Al and Pb were superconducting, and a peak in the tunnel current appeared at the voltage associated with the difference in the energy gaps (predicted by Harrison and Bardeen).

Fig. 6 Diagram from Giaever “The Discovery of Superconducting Tunneling” at https://conferences.illinois.edu/bcs50/pdf/giaever.pdf

The Josephson Effect: Brian Josephson (1962)

In Giaever’s experiments, the external circuits had been designed to pick up “ordinary” tunnel currents in which individual electrons tunneled through the oxide rather than the Cooper pairs themselves. However, in 1962, Brian Josephson (1940 – ), a physics graduate student at Cambridge, was sitting in a lecture (just like Giaever) on solid state physics given by Phil Anderson (who was on sabbatical there from Bell Labs). During lecture he had the idea to calculate whether it was possible for the Cooper pairs themselves to tunnel through the oxide barrier. Building on theoretical work by Leo Falicov who was at the University of Chicago and later at Berkeley (years later I was lucky to have Leo as my PhD thesis advisor at Berkeley), Josephson found a surprising result that even when the voltage was zero, there would be a supercurrent that tunneled through the junction (now known as the DC Josephson Effect). Furthermore, once a voltage was applied, the supercurrent would oscillate (now known as the AC Josephson Effect). These were strange and non-intuitive results, so he showed Anderson his calculations to see what he thought. By this time Anderson had already been extremely impressed by Josephson (who would often come to the board after one of Anderson’s lectures to show where he had made a mistake). Anderson checked over the theory and agreed with Josephson’s conclusions. Bolstered by this reception, Josephson submitted the theoretical prediction for publication [9].

As soon as Anderson returned to Bell Labs after his sabbatical, he connected with John Rowell who was making tunnel junction experiments, and they revised the external circuit configuration to be most sensitive to the tunneling supercurrent, which they observed in short time and submitted a paper for publication. Since then, the Josephson Effect has become a standard element of ultra-sensitive magnetometers, measurement standards for charge and voltage, far-infrared detectors, and have been used to construct rudimentary qubits and quantum computers.

By David D. Nolte: Nov. 6, 2022

YouTube Video

YouTube Video of Quantum Tunneling Systems

References:

[1] F. Hund, Z. Phys. 40, 742 (1927). F. Hund, Z. Phys. 43, 805 (1927).

[2] L. Nordheim, Z. Phys. 46, 833 (1928).

[3] R. H. Fowler, L. Nordheim, Proc. R. Soc. London, Ser. A 119, 173 (1928).

[4] G. Gamow, Z. Phys. 51, 204 (1928).

[5] R. W. Gurney, E. U. Condon, Nature 122, 439 (1928). R. W. Gurney, E. U. Condon, Phys. Rev. 33, 127 (1929).

[6] Dennison, D. M. and G. E. Uhlenbeck. “The two-minima problem and the ammonia molecule.” Physical Review 41(3): 313-321. (1932)

[7] L. Esaki, New Phenomenon in Narrow Germanium Para-Normal-Junctions, Phys. Rev., 109, 603-604 (1958); L. Esaki, (1974). Long journey into tunneling, disintegration, Proc. of the Nature 123, IEEE, 62, 825.

[8] I. Giaever, Energy Gap in Superconductors Measured by Electron Tunneling, Phys. Rev. Letters, 5, 147-148 (1960); I. Giaever, Electron tunneling and superconductivity, Science, 183, 1253 (1974)

[9] B. D. Josephson, Phys. Lett. 1, 251 (1962); B.D. Josephson, The discovery of tunneling supercurrent, Science, 184, 527 (1974).

[10] P. W. Anderson, J. M. Rowell, Phys. Rev. Lett. 10, 230 (1963); Philip W. Anderson, How Josephson discovered his effect, Physics Today 23, 11, 23 (1970)

[11] Eugen Merzbacher, The Early History of Quantum Tunneling, Physics Today 55, 8, 44 (2002)

[12] Razavy, Mohsen. Quantum Theory Of Tunneling, World Scientific Publishing Company, 2003.

Interference (New from Oxford University Press, 2023)

Read the stories of the scientists and engineers who tamed light and used it to probe the universe.

Available from Amazon.

Available from Oxford U Press

Available from Barnes & Nobles

Heisenberg’s Trajectory

Trajectory’s End

The Matrix

The Drei-Männer Quantum Mechanics Paper: Born, Heisenberg, and Jordan

Open Issues

Further Reading

References

Encyclopedia Article on Relativity

Anomalous Zeeman Effect

The Difference of Squares

The Bohr Model for Many Electrons

Pauli’s Breakthrough

Timeline:

References:

Early Days

Purdue University and Prof. Lark-Horowitz

Radiation Laboratory, MIT

Nuclear Magnetic Resonance

References and Notes

The Virial Theorem

Rudolph Clausius and his Virial Theorem

The Virial in Astrophysics

The Virial in Quantum Physics

References

Matlab Code

Early Days of the Photon

The Quantum Derivation by Bose

Enter Einstein

References

Electron Spin

5-Dimensional Quantum Mechanics

Brownian Motion

Quantum Tunneling

Master Equation

Selected Works by George Uhlenbeck:

Notes:

de Broglie’s Early Days

de Broglie’s Massive Photons

de Broglie’s Matter Wave

Augustin-Louis Cauchy (1815)

Joseph Fourier (1822)

George Green’s Function (1829)

Oliver Heaviside (1893)

Paul Dirac (1930)

Bibliography

The EPR Papers of 1935

Bohm and Ahronov Tackle EPR

John Bell’s Inequalities

John Clauser’s Tenacious Pursuit

Alain Aspect’s Non-locality

Anton Zeilinger: Master of Entanglement

Read more about the history of quantum entanglement in Interference (New From Oxford University Press, 2023)

Video Lectures

Timeline

Bibliography

Double-Well Potential: Friedrich Hund (1927)

Electron Emission: Lothar Nordheim and Ralph Fowler (1927 – 1928)

Nuclear Alpha Decay: George Gamow (1928)

Molecular Tunneling: George Uhlenbeck (1932)

Solid-State Electron Tunneling: Leo Esaki (1957)

Superconducting Tunneling: Ivar Giaever (1960)

The Josephson Effect: Brian Josephson (1962)

YouTube Video

References:

Interference (New from Oxford University Press, 2023)