WOLFRAM NOTEBOOK

This is part of live presentation series called Mathematical Games in which we explore a variety of games and puzzles using Wolfram Language. In this episode, we explore the math and physics behind the Manhattan Project.
The movie Oppenheimer features a wide range of mathematicians and physicists. For this talk, I thought I’d touch on a dozen of these that have had a widespread impact. If you have used a computer, the internet, a microwave or a refrigerator, then your life has been impacted by one of these people.

Demonstrations related to Feynman

More on Feynman

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Pioneered the field of quantum computing; also known for his work in particle physics, quantum mechanics, and electrodynamicsCalled the "Great Explainer," he became one of the world's best-known scientists through his popular physics books and lecturesParticipated in the creation of the atomic bomb, developing with Hans Bethe a formula for calculating the yield of a fission bombIntroduced the concept of nanotechnology in a 1959 lectureMember of the panel that investigated the Space Shuttle Challenger disasterHeld the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology
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Chadwick Rutherford Thomson or ...
Neutron Proton Electron

Thomson taught Rutherford, who taught Chadwick.
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Physicist who discovered the neutronMain British scientist to be involved in the Manhattan Project to develop the atomic bombMade possible the fission of uranium 235 and the laboratory creation of elements heavier than uranium

The Einstein Letter

Eugene Wigner

Helped to write the letter
This Demonstration shows the Wigner quasiprobability distribution for 101 energy states of the quantum harmonic oscillator.
The Wigner 3j symbols give amplitudes for pairs of quantum spins to be in different quantum states. Change the magnitudes of the quantum spins and see the distributions of amplitudes for different states. For large spins the distributions become smooth and have semiclassical features.

Leó Szilárd

Wikipedia: He conceived the nuclear chain reaction in 1933, patented the idea in 1936, and in late 1939 wrote the letter for Albert Einstein’s signature that resulted in the Manhattan Project that built the atomic bomb. Together with Enrico Fermi, he applied for a nuclear reactor patent in 1944. In addition to the nuclear reactor, Szilard coined and submitted the earliest known patent applications and the first publications for the concepts of electron microscope (1928), the linear accelerator (1928), and the cyclotron (1929) in Germany, proving him as the originator of the idea of these devices. Between 1926 and 1930, he worked with Einstein on the development of the Einstein refrigerator.
In 1867 the Scottish physicist James Clerk Maxwell formulated this thought experiment that seems to violate the second law of thermodynamics. The apparent violation was later explained by Leó Szilárd and Léon Brillouin.

Edward Teller

21 Oct 1939 -- Eugene Wigner, Leó Szilárd and Edward Teller report to Roosevelt that uranium “would provide a possible source of bombs with a destructiveness vastly greater than anything now known.”
Nuclear weapons using plutonium-239 (an alternative to uranium-235) are triggered by an implosion mechanism, in which a spherical assembly of conventional high explosives compresses a subcritical core of a plutonium compound to supercritical density. A number of famous scientists working at Los Alamos, including Richard Tolman, John von Neumann, and Edward Teller contributed to this design. The first such device was successfully exploded near Alamogordo, NM in 1945, in a location now known as the Trinity Site. This Demonstration gives a highly simplified account of the implosion mechanism. (Some details are still classified.) It is important that the implosion geometry must be spherically symmetrical to high accuracy, otherwise ineffective preignition can occur. One design is based on a spherical array of 32 explosive charges.

Edward Teller

Exact solutions of the nonrelativistic wave equations contain all the necessary information for the quantum system and have important applications in particle physics. This Demonstration discusses a solution of the Schrödinger equation in three-dimensional configuration space with the trigonometric Pöschl–Teller potential in the Bohm approach.

Edward Teller

Ernest Lawrence

25 Feb 1941 -- Plutonium discovered by Glenn Seaborg.
6 Dec 1941 -- Ernest Lawrence brought in to investigate electromagnetic separation methods.
Made with 14700 tons of silver!
The cyclotron was invented in 1932 by Ernest O. Lawrence and M. S. Livingston at Berkeley. Particles or ions are injected into the center of two hollow D-shaped objects called “dees”. A magnetic field is applied to them that is perpendicular to the plane in which they move and they accelerate across a gap between the dees by a potential difference. The orbit radius increases and eventually the particles gain energy and are ejected to hit a target. It is one of the earliest types of accelerators in use today.
Centrifugal separation of isotopes came years later.

Oppenheimer’s Car

Robert Oppenheimer and Ernest Lawrence
Sy Blinder: On a personal note, I spent the summer of 1956 in T-Division at Los Alamos where Oppenheimer once reigned. For a month I lived in the house of my supervisor Rolf Landshoff, while he was away. He was in possession of a beat up 1937 Plymouth that had once belonged to Oppenheimer. And I drove that car to work for a month. The one Oppy drove to San Francisco to see Jean Tatlock.

J. Robert Oppenheimer

J. Robert Oppenheimer

Born–Oppenheimer approximation

Hans Bethe

Conventional screws have right-handed threads. To tighten a screw, turn the head or bolt clockwise, moving its top to the right. To loosen, turn counterclockwise, moving the top to the left. The famous physicist Hans Bethe was reputed to remember the convention by relating it to the right-hand rule for vector products:

Hans Bethe

Applied double groups to Physics.
The Molien equation [1, 2] determines symmetry correlation tables [3]. The tables presented here describe correlations between representations of the unitary group and various finite subgroups: double trigonal , double octahedral , and double icosahedral . In the history of science, double groups were introduced by Klein [4] and subsequently applied to physics by Bethe [5]. Both German works are available in English.
[5] H. A. Bethe, “Termaufspaltung in Kristallen,” Annalen der Physik, 395(2), 1929 pp. 133–208. doi:10.1002/andp.19293950202.

Hans Bethe

Bethe ansatz of the quantum integrable models
A recent breakthrough is the discovery of the connection with the rigged configurations (RC for short) that were originally discovered by Kerov–Kirillov–Reshetikhin in 1986 during their study of the Bethe ansatz of the quantum integrable models (Kuniba–Okado–Sakamoto–Takagi–Yamada 2006 for BBS and Kuniba–Takagi–Takenouchi 2006 for PBBS). RC are depicted by Young diagrams associated with integers. Roughly speaking, each row represents a soliton of the length equal to the row and integers determine the positions of such solitons. Similar machinery is established for a wider class of algebras and yields general solutions of BBS and PBBS. Note that in this Demonstration, the periodic boundary condition leads us to consider the quotient space of the set of RC.

Hans Bethe

Bethe lattices

Isidor Isaac Rabi

Won the Nobel Prize in Physics in 1944 for his discovery of nuclear magnetic resonance, which is used in magnetic resonance imaging. He was also one of the first scientists in the United States to work on the cavity magnetron, which is used in microwave radar and microwave ovens.
Rabi frequency
The state of polarization of light can be represented on the so-called Poincaré sphere, which provides a graphic representation of the Stokes parameters [1]. The circular-right polarization R corresponds to the north pole of the sphere and the circular-left polarization L to the south pole. The other cardinal points define the diagonal D, antidiagonal A, horizontal H, and vertical V polarizations. The Poincaré sphere can be used to track the state of polarization in a variety of contexts, such as spatial variations [2].
This Demonstration lets you tune these parameters as well as the system parameters (Rabi frequency , polariton lifetime ) and track the state of polarization in time. Without decay (), the polarization describes a green circle, whereas with decay (), the polarization drifts around the sphere to converge to an asymptotic value, pointed to by the red arrow; the blue line shows the polarization trajectory between these two points, which can be tracked in time with the purple arrow.

Werner Heisenberg

W. Heisenberg’s real matrix group provides a noncommutative translation group of an affine three-space. The Nil-geometry, which is one of the eight Thurston three-geometries, can be derived from this group. E. Molnár proved that the homogeneous three-spaces have a unified interpretation in the projective three-sphere ). Here, the tori of the Nil-space are visualized.

Paul Dirac

Made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger “for the discovery of new productive forms of atomic theory”.[11] He also made significant contributions to the reconciliation of general relativity with quantum mechanics.
This Demonstration shows a simulation of the Dirac belt trick, which is a physical analogue of the homotopy classes created following the standard procedure for defining the universal covering group of the Lie group . “homotopy” sets whether the ribbon encodes the homotopy class of the identity loop (homotopy = 1) or of the noncontractible path (homotopy = -1) between the identity (time=0) and the rotation group member at any given “time” in SO(3) . Use “belts” to define whether one or two belts are used; two belts shows that a 4 π twist in the middle of an infinite Dirac belt can be undone without untwisting the central “doll” (representing the spinorial object). The “showdoll” checkbox puts an orientation-marking matrioshka doll at the ribbon’s end.

Enrico Fermi

Maxwell–Boltzmann statistics apply where quantum-mechanical effects do not play a role and the particles of the gas can be considered “distinguishable”. Both Fermi–Dirac and Bose–Einstein statistics become Maxwell–Boltzmann statistics at high temperatures and low chemical potentials.

Albert Einstein

Move a point source lens of a given mass over different types of grids to see the effects of gravitational lensing.

Albert Einstein

The 2D array on the left is a simplified model of a solid, where each cell represents an oscillator whose energy is some multiple of ℏω. In the initial state, the energy of each cell is exactly ℏω. At each successive step, one quantum of energy is transferred between two randomly chosen cells. After about 20,000 steps, the distribution of energies closely approximates the Boltzmann distribution and thermal equilibrium is obtained.

Richard Feynman

In his eccentric collection of autobiographical stories (see reference), Richard Feynman recounts: “I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling. I had nothing to do, so I start figuring out the motion of the rotating plate. I discovered that when the angle is very slight, the medallion rotates twice as fast as the wobble rate—two to one. It came out of a complicated equation! I went on to work out equations for wobbles. Then I thought about how the electron orbits start to move in relativity. Then there’s the Dirac equation in electrodynamics. And then quantum electrodynamics. And before I knew it the whole business that I got the Nobel prize for came from that piddling around with the wobbling plate.” A replica of the Cornell plate is now part of an exhibit marking the centennial of the Nobel Prize.

Richard Feynman

Newton showed this construction in Book 1, Section 4, Lemma 15, of Principia. On March 13, 1964, Feynman resurrected the construction and used it in a lecture, “The Motion of Planets Around the Sun”. The lecture is detailed in a book with audio CD, Feynman’s Lost Lecture, by David and Judith Goodstein. In the lecture, Feynman used the diagram and differential geometry to prove the planetary laws of motion.

Richard Feynman

Stanisław Ulam

He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion.
A cube of the next generation is formed if:
(1) It is contiguous to one and only one cube of the current generation.
(2) It touches no other previously occupied cube except if the cube should be its “grandparent”.
(3) Of the generation satisfying the previous condition, all those that would touch each other are eliminated. There is an exception for those cubes that have the same parent; these are allowed to touch.
Ulam’s prime spiral arranges the positive integers in a spiral, marking primes with dark pixels.
This is a simplified version of chess, played on a 6×6 (rather than 8×8) board without bishops. Pawns can move only one space forward, and there is consequently no en passant capture. In this modified version, castling is allowed, along with pawn promotion to a previously lost piece. There is no automation in this Demonstration.
You might want also to mention the mathematician Stanislaw Ulam, who first designed the implosion mechanism for triggering the plutonium bomb. I also worked for him for a while that summer. As a side interest, he was trying to program the MANIAC I computer to play chess. This is when I invented the 6x6 reduced version known as Los Alamos Chess to make the programming more feasible. See https://demonstrations.wolfram.com/LosAlamosChess/

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