Tag / History of Quantum Physics

Intersystem crossing (ISC) for singlet-to-triplet transition and generation of singlet oxygen
by David D. Nolte

A Story of Singlets and Triplets: Relativistic Biology

The tortoise and the hare.  The snail and the falcon.  The sloth and the cheetah.

Comparing the slowest animals to the fastest, the peregrine falcon wins in the air at 200 mph diving on a dove.  The cheetah wins on the ground at 75 mph running down dinner.  That’s fast!

Einstein’s theory of relativity says that fast things behave differently than slow things.  So how fast do things need to move to see these effects?

The measure is the ratio of the speed of the object relative to the speed of light at 670 million miles per hour.  This ratio is called beta

The cheetah has β = 1×10-7, and the peregrine falcon has β = 4×10-7—which are less than one part in a million. 

And what about that snail?  At a speed of 0.003 mph it has β = 4×10-12 for a just few parts per trillion.

Yet relativistic physics is present in biology, despite these slow speeds, and they can help keep us alive.  How?

The Boon and Bane of Oxygen

All animal life on Earth needs oxygen to survive.  Oxygen is the energy storehouse that fuels mitochondria—the tiny batteries inside our cells that generate the energetic molecules that make us run.

Of all the elements, oxygen is second only to flourine as the most chemically reactive.  But fluorine is too one-dimensional to help much with life.  It needs only one extra electron to complete its outer-shell octet, leaving nothing in reserve to attach to much else.  Oxygen, on the other hand, needs two electrons to complete its outer shell, making it an easy bridge between two other atoms, usually carbons or hydrogens, and there you have it—organic chemistry is off and running.

Yet the same chemical reactivity that makes oxygen central to life also makes it corrosive to life.  When oxygen is not doing its part keeping things alive, it is tearing things apart.

The problem is reactive oxygen species, or ROS.  These are activated forms of oxygen that damage biological molecules, including DNA, putting wear and tear on (aging) the cellular machinery and introducing mutations into the genetic codes.  And one of the ROS is an active form of simple diatomic oxygen O2 — the very air we breathe.

The Saga of the Singlet and the Triplet

Diatomic oxygen is deceptively simple: two oxygen atoms, each two electrons short of a full shell, coming together to share two of each other’s electrons in a double bond, satisfying the closed shell octet for the outer valence electrons for this element.

Fig. 1 The Lewis diagram of the oxygen double bond.  Each oxygen shares two electrons with the other, filling the valence shell.

Bonds are based on energy levels, and the individual energy levels of the two separate oxygen atoms combine and shift as they form molecular orbitals (MO).  These orbitals are occupied by the 6 + 6 = 12 electrons of the diatomic molecule.  The first 10 electrons fill up the lower MOs until the last two go into the next highest state.  Here, for interesting reasons associated with how electrons interact with each other in confined spaces, the last two electrons both have the same orientation of their spins.  The total electron spin of the final full ground-state configuration of 12 electrons has S = 1. 

The unreactive triplet ground state molecular orbital diagram of diatomic oxygen (dioxygen). The two last electron spins are unpaired.
Fig. 2 Molecular orbital diagram for diatomic oxygen.  The 2s and 2p levels of the separated oxygen hybridize into new orbitals that are filled with 6 + 6 = 12 electrons.  The last two electron states have the same spin with a total spin S = 1.  This is the unreactive triplet ground state.

When a many-electron system has a spin S = 1, quantum mechanics prescribes that it has 2S+1 spin projections, so that the ground state of diatomic oxygen has three possible spin projections—known as a triplet.  But this is just for the lowest energy configuration of O2.

It is also possible to put both electrons into the final levels with opposite spins, giving S = 0.  This is known as the singlet, and there are two different ways these spins can be anti-aligned, creating two excited states.  The first excited state is about 1 eV in energy above the ground state, and the second excited state is about 0.7 eV higher than the first. 

The spin excited states of diatomic oxygen showing the pairing of spins to produce reactive singlet oxygen.
Fig. 3 Ground and excited states of O2.  The two singlet excited states have paired electron spins with a total spin of S = 0.  The ground state has S = 1.  The transitions between ground and excited states are “spin forbidden”.

Now we come the heart of our story.  It hinges on how reactive the different forms of O2 are, and how easily the singlet and the triplet can transform into each other.

Don’t Mix Your Spins

What happens if you mix hydrogen and oxygen in a 2:1 ratio in a plastic bag?

Nothing!  Unless you touch a match to it, and then it “pops” and creates water H2O.

Despite the thermodynamic instability of oxygen and hydrogen, the mixture is kinetically stable because it takes an additional input of energy to make the reaction go.  This is because oxygen at room temperature is almost exclusively in its ground state, the triplet state.  The triplet oxygen accepts electrons from other molecules, or atoms like hydrogen, but the local environment of organic molecules are almost exclusively in S = 0 states because of their stability. To accept two electrons from organic molecules requires spins to flip, which has an energy associated with the process, creating an energy barrier. This is known as the spin conservation rule of organic chemistry. Therefore, the reaction with the triplet is unlikely to go, unless you give it a hand with extra energy or a catalyst to get over the barrier.

However, this “spin blockade” that makes triplet (S = 1) oxygen unreactive (kinetically stable) is lifted in the case of singlet (S = 0) oxygen.  The singlet still accepts two electrons from other molecules, but these can be taken up easily by accepting electrons from the S = 0 organic molecules around it, conserving spin.  This makes singlet oxygen extremely reactive organically and hence is why it is an ROS. Singlet oxygen reacts with organic molecules, damaging them.

Despite the deleterious effects of singlet oxygen, it is produced as a side effect of lipids (fats) immersed in an oxygen environment. In a sense, fat slowly “burns”, creating reactive organic species that further react with lipids (primarily polyunsaturated fatty acids (PUFAs)), generating more ROS and creating a chain reaction. The chain reaction is terminated by the Russell mechanism that generates singlet oxygen as a byproduct.

Singlet oxygen, even though it is an excited state, cannot convert directly back to the benign triplet ground state for the same spin conservation reasons as organic chemistry. The singlet (S = 0) cannot emit a photon to get to its ground state triplet (S = 1) because photons cannot change the spin of the electrons in a transition. So once the singlet oxygen is formed, it can stick around and react chemically, eating up an organic molecule.

Therefore, the oxygen environment we are immersed in, so necessary for our survival, is slowly killing us with oxidative stress. In response, mechanisms to de-excite singlet oxygen evolved, using organic molecules like carotenoids that are excellent physical quenchers of the singlet state. How does this happen? Enter Einstein’s theory of special relativity.

Relativistic Origins of Spin-Orbit Coupling

When I was a sophomore at Cornell University in the late 1970’s, I took the introductory class in electricity and magnetism. The text was an oddball thing from the Berkeley Physics Series authored by the Nobel Laureate, Edward Purcell, from Harvard.

Cover and front page of Purcell's E&M volume of the Berkeley Series.
Fig. 4 The cover and front page of my 1978 copy of Edward Purcell’s Berkeley Physics volume on Electricity and Magnetism.

(The Berkeley Series was a set of 5 volumes to accompany a 5-semester introduction to physics. It was the brainchild of Charles Kittel from Berkeley and Philip Morrison from Cornell who, in 1961, were responding to the Sputnik crisis and the need to improve the teaching of physics in the West.)

Purcell’s book had the quirky misfortune to use Gaussian units based on centimeters and grams (cgs) instead of meters and kilograms (MKS). Physicists tend to like cgs units, especially in the teaching of electromagnetism, because it places electric fields and magnetic fields on equal footing. Unfortunately, it creates a weird set of units like “statvolts” and “statcoulombs” that are a nightmare to calculate with.

Nonetheless, Purcell’s book was revolutionary as a second-semester intro physics book, in that it used Special Relativity to explain the transformations between electric and magnetic fields. For instance, starting with the static electric field of a stationary parallel plate capacitor in one frame, Purcell showed how an observer moving relatively to the capacitor in a moving frame detects the electric field as expected, but also detects a slight magnetic field. As the speed of the observer increases, the strength of the magnetic field increases, becoming comparable to the electric field in strength as the relative frame speed approaches the speed of light.

In this way, there is one frame in which there is no magnetic field at all, and another frame in which there is a strong magnetic field. The only difference is the point of view—yet the consequences are profound, especially when the quantum spin of an electron is involved.

Conversion of E fields to B fields through relativistically moving frames
Fig. 5 Purcell’s use of a parallel plate capacitor in his demonstration of the origin of magnetic fields from electric fields in relatively moving frames.

The spin of the electron is like a tiny magnet, and if the spin is in an external magnetic field, it feels a torque as well as an interaction energy. The torque makes it precess, and the interaction energy shifts its energy levels depending on the orientation of the spin to the field.

When an electron is in a quantum orbital around a stationary nucleus, attracted by the electric field of the nucleus, it would seem that there is no magnetic field for the spin to interact with, which indeed is true if the electron has no angular momentum. But if the electron orbital does have angular momentum with a non-zero expectation value for its velocity, then this moving electron does experience a magnetic field—the electron is moving relative to the electric field of the nucleus, and special relativity dictates that it experiences a magnetic field. The resulting magnetic field interacts with the magnetic moment of the electron and shifts its energy a tiny amount.

Spin-orbit coupling and spin-forbidden transitions. Spin-orbit mixes singlet and triplet spin states.
Fig. 6 Transitions to the singlet excited state are allowed but to the triplet state are spin-forbidden. The spin-orbit coupling mixes the spin states, giving a little triplet character to the singlet and vice versa

This is called the spin-orbit interaction and leads to the fine structure of the electron energy levels in atoms and molecules. More importantly for our story of Singlets and Triplets, the slight shift in energy also mixes the spin states. Quantum mechanical superposition of states mixes in a little S = 0 into the triplet excited state and a little S = 1 into the singlet ground state, and the transition is no longer strictly spin forbidden. The spin-orbit effect is relatively small in oxygen, contributing little to the quenching of singlet oxygen. But spin-orbit in other molecules, especially dye-like organic molecules, can be strong, creating a protective antioxidant mechanism to remove singlet oxygen.

Relativistic Biology: Antioxidants

Do you eat blueberries? Or carrots? Or other brightly colored foods? If so, you are enlisting relativistic biology to help keep you healthy. These fruits and vegetables are colored because they contain organic molecules that absorb certain wavelengths of light but not others, giving them their strong colors. The same organic complexes that absorb light are also good at interacting with singlet oxygen, taking the singlet spin onto themselves while converting the oxygen back to its benign triplet.

With the reactive oxygen singlet gone, the organic singlet still has stored energy that must be dissipated. This happens through intersystem crossing allowed by the spin-orbit coupling (relativistic biology), converting the antioxidant to its triplet state that can de-excite through normal vibrational processes, harmlessly releasing the energy as heat that diffuses away.

Antioxidant molecules like the carotenoids, that color food, accept the singlet spin state from the oxygen and de-excite through intersystem crossing and thermal dissipation. Other antioxidants, like vitamin E (tocopherols) and amines, provide high electron density directly to the singlet oxygen, enhancing the otherwise weak spin-orbit effect of isolated oxygen, allowing the oxygen to convert back to it triplet ground state. For both of these types of antioxidants, the molecule is unaltered chemically by this process, and they keep sweeping up reactive singlet oxygen, removing them from the local biological environment and protecting your health.

But let’s flip the problem of the singlet oxygen, and rather than get rid of it, ask what it takes to selectively harness singlet oxygen production to act as a targeted therapy to kill cancer cells.

Relativistic Biology: Photodynamic Cancer Therapy

The physics of light propagation through living tissue is a fascinating topic. Light is easily transported through tissue through multiple scattering. This is why your whole hand glows red when you cover a flashlight at night. The photons bounce around but are not easily absorbed. This surprising translucence of tissue can be used to help treat cancer.

Photodynamic cancer therapy uses photosensitizer molecules, typically organic dyes, that absorb light, transforming the molecule from a singlet ground state to a singlet excited state (spin-allowed transition). Although singlet-to-triplet conversion is spin-forbidden, the spin-orbit coupling slightly mixes the spin states which allows a transformation known as intersystem crossing (ISC). The excited singlet crosses over (usually through a vibrational state associated with thermal energy) to an excited triplet state of the photosensitizer molecule. Oxygen triplet molecules, that are always prevalent in tissue, collide with the triplet photosensitizer—and they swap their energies in a spin-allowed transfer. The photosensitizer triplet returns to its singlet ground state, while the triplet oxygen converts to highly reactive singlet oxygen. The swap doesn’t change total spin, so it is allowed and fast, generating large amounts of reactive oxygen singlets.

Intersystem crossing (ISC) for singlet-to-triplet transition and generation of singlet oxygen
Fig. 7 Intersystem crossing diagram. A photosensitizer molecule in a singlet ground state absorbs a photon that creates a singlet excited state of the molecule. The spin-orbit mixing of spin states allows intersystem crossing (ISC) to generate a triplet excited state of the photosensitizer. This triplet can then exchange its energy with oxygen to create highly-reactive singlet oxygen.

In photodynamic therapy, the photosensitizer is taken up by the patient’s body, but the doctor only shines light around the tumor area, letting the light multiply-scatter through the tissue, exciting the photosensitizer and locally producing singlet oxygens that kill the local cancer cells. Other parts of the body remain in the dark and have none of the ill-effects that are so common for the conventional systemic cytotoxic anti-cancer drugs. This local treatment of the tumor using localized light can be much more benign to overall patient health while still delivering effective treatment to the tumor.

Photodynamic therapy has been approved by the FDA for some early stage cancers, and continuing research is expanding its applications. Because light transport through tissue is limited to about a centimeter, most of these applications are for “shallow” tumors that can be accessed by light through the skin or through internal surfaces (like the esophagus or colon).

Postscript: Relativistic Spin

I can’t leave this story of relativistic biology without going into a final historical detail from the early days of relativistic quantum theory. When Uhlenbeck and Goudsmit first proposed in 1925 that the electron has spin, there were two immediate objections. First, Lorentz showed that in a semiclassical model of the spinning electron, the surface of the electron would be moving at 300 times the speed of light. The solution to this problem came three years later in 1928 with the relativistic quantum theory of Paul Dirac, which took an entirely different view of quantum versus classical physics. In quantum theory, there is no “radius of the electron” as there was in semiclassical theory.

The second objection was more serious. The original predictions from electron spin and the spin-orbit interaction in Bohr’s theory of the atom predicted a fine-structure splitting that was a factor of 2 times larger than observed experimentally. In 1926 Llewellyn Thomas showed that a relativistically precessing spin was in a non-inertial frame, requiring a continuous set of transformations from one instantaneously inertial frame to another (a mathematical phenomenon related to parallel transport known as holonomy). These continuously shifting transformations introduced a factor of 1/2 into the spin precession, exactly matching the experimental values. This effect is known as Thomas precession. Interestingly, the fully relativistic theory of Dirac in 1928 automatically incorporated this precession within the theory, so once again Dirac’s equation superseded the old quantum theory.

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 50th 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).

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[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).

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