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d2dfd05357295810 | Max Planck
Max Planck
Max planck.jpg
Max Karl Ernst Ludwig Planck
April 23, 1858
Kiel, Germany
Died October 4, 1947
Göttingen, Germany
Residence Flag of Germany.svg Germany
Nationality Flag of Germany.svg German
Field Physicist
Institutions University of Kiel
Humboldt-Universität zu Berlin
Georg-August-Universität Göttingen
Alma mater Ludwig-Maximilians-Universität München
Academic advisor Philipp von Jolly
Notable students Gustav Ludwig Hertz Nobel.svg
Erich Kretschmann
Walther Meißner
Walter Schottky
Max von Laue Nobel.svg
Max Abraham
Moritz Schlick
Walther Bothe Nobel.svg
Known for Planck's constant, quantum theory
Notable prizes Nobel.svg Nobel Prize in Physics (1918)
He was the father of Erwin Planck.
Max Karl Ernst Ludwig Planck (April 23, 1858 – October 4, 1947) was a German physicist who is widely regarded as one of the most significant scientists in history. He developed a simple but revolutionary concept that was to become the foundation of a new way of looking at the world, called quantum theory.
In 1900, to solve a vexing problem concerning the radiation emitted by a glowing body, he introduced the radical view that energy is transmitted not in the form of an unbroken (infinitely subdivisible) continuum, but in discrete, particle-like units. He called each such unit a quantum (the plural form being quanta). This concept was not immediately accepted by physicists, but it ultimately changed the very foundations of physics. Planck himself did not quite believe in the reality of this concept—he considered it a mathematical construct. In 1905, Albert Einstein used that concept to explain the photoelectric effect, and in 1913, Niels Bohr used the same idea to explain the structures of atoms. From then on, Planck's idea became central to all of physics. He received the Nobel Prize in 1918, and both Einstein and Bohr received the prize a few years later.
Planck was also a deeply religious man who believed that religion and science were mutually compatible, both leading to a larger, universal truth. By basing his convictions on seeking the higher truth, not on doctrine, he was able to stay open-minded when it came to formulating scientific concepts and being tolerant toward alternative belief systems.
Life and work
Early childhood
Planck came from a traditional, intellectual family. His paternal great-grandfather and grandfather were both theology professors in Göttingen, his father was a law professor in Kiel and Munich, and his paternal uncle was a judge.
Planck was born in Kiel to Johann Julius Wilhelm Planck and his second wife, Emma Patzig. He was the sixth child in the family, including two siblings from his father's first marriage. Among his earliest memories was the marching of Prussian and Austrian troops into Kiel during the Danish-Prussian War in 1864. In 1867, the family moved to Munich, and Planck enrolled in the Maximilians gymnasium. There he came under the tutelage of Hermann Müller, a mathematician who took an interest in the youth and taught him astronomy and mechanics as well as mathematics. It was from Müller that Planck first learned the principle of conservation of energy. Planck graduated early, at age 16. This is how Planck first came in contact with the field of physics.
Planck was extremely gifted when it came to music: He took singing lessons and played the piano, organ, and cello, and composed songs and operas. However, instead of music, he chose to study physics.
Munich physics professor Philipp von Jolly advised him against going into physics, saying, "in this field, almost everything is already discovered, and all that remains is to fill a few holes." Planck replied that he did not wish to discover new things, only to understand the known fundamentals of the field. In 1874, he began his studies at the University of Munich. Under Jolly's supervision, Planck performed the only experiments of his scientific career: Studying the diffusion of hydrogen through heated platinum. He soon transferred to theoretical physics.
In 1877, he went to Berlin for a year of study with the famous physicists Hermann von Helmholtz and Gustav Kirchhoff, and the mathematician Karl Weierstrass. He wrote that Helmholtz was never quite prepared (with his lectures), spoke slowly, miscalculated endlessly, and bored his listeners, while Kirchhoff spoke in carefully prepared lectures, which were, however, dry and monotonous. Nonetheless, he soon became close friends with Helmholtz. While there, he mostly undertook a program of self-study of Rudolf Clausius's writings, which led him to choose heat theory as his field.
In October 1878, Planck passed his qualifying exams and in February 1879, defended his dissertation, Über den zweiten Hauptsatz der mechanischen Wärmetheorie (On the second fundamental theorem of the mechanical theory of heat). He briefly taught mathematics and physics at his former school in Munich. In June 1880, he presented his habilitation thesis, Gleichgewichtszustände isotroper Körper in verschiedenen Temperaturen (Equilibrium states of isotropic bodies at different temperatures).
Academic career
With the completion of his habilitation thesis, Planck became an unpaid private lecturer in Munich, waiting until he was offered an academic position. Although he was initially ignored by the academic community, he furthered his work on the field of heat theory and discovered one after the other the same thermodynamical formalism as Josiah Willard Gibbs without realizing it. Clausius's ideas on entropy occupied a central role in his work.
In April 1885, the University of Kiel appointed Planck an associate professor of theoretical physics. Further work on entropy and its treatment, especially as applied in physical chemistry, followed. He proposed a thermodynamic basis for Arrhenius's theory of electrolytic dissociation.
Within four years, he was named the successor to Kirchhoff's position at the University of Berlin—presumably thanks to Helmholtz's intercession—and by 1892 became a full professor. In 1907, Planck was offered Boltzmann's position in Vienna, but turned it down to stay in Berlin. During 1909, he was the Ernest Kempton Adams Lecturer in Theoretical Physics at Columbia University in New York City. He retired from Berlin on January 10, 1926, and was succeeded by Erwin Schrödinger.
In March 1887, Planck married Marie Merck (1861-1909), sister of a school fellow, and moved with her into a sublet apartment in Kiel. They had four children: Karl (1888-1916), the twins Emma (1889-1919) and Grete (1889-1917), and Erwin (1893-1945).
After the appointment to Berlin, the Planck family lived in a villa in Berlin-Grunewald, Wangenheimstraße 21. Several other professors of Berlin University lived nearby, among them the famous theologian Adolf von Harnack, who became a close friend of Planck. Soon the Planck home became a social and cultural center. Numerous well-known scientists—such as Albert Einstein, Otto Hahn, and Lise Meitner—were frequent visitors. The tradition of jointly playing music had already been established in the home of Helmholtz.
After several happy years, the Planck family was struck by a series of disasters: In July 1909, Marie Planck died, possibly from tuberculosis. In March 1911, Planck married his second wife, Marga von Hoesslin (1882-1948); in December his third son, Herrmann, was born.
During the First World War, Planck's son Erwin was taken prisoner by the French in 1914, and his son Karl was killed in action at Verdun in 1916. His daughter Grete died in 1917 while giving birth to her first child; her sister lost her life two years later under the same circumstances, after marrying Grete's widower. Both granddaughters survived and were named after their mothers. Planck endured all these losses with stoic submission to fate.
During World War II, Planck's house in Berlin was completely destroyed by bombs in 1944, and his youngest son, Erwin, was implicated in the attempt made on Hitler's life on July 20, 1944. Consequently, Erwin died a horrible death at the hands of the Gestapo in 1945.
Professor at Berlin University
In Berlin, Planck joined the local Physical Society. He later wrote about this time: "In those days I was essentially the only theoretical physicist there, whence things were not so easy for me, because I started mentioning entropy, but this was not quite fashionable, since it was regarded as a mathematical spook." Thanks to his initiative, the various local Physical Societies of Germany merged in 1898 to form the German Physical Society (Deutsche Physikalische Gesellschaft, DPG), and Planck was its president from 1905 to 1909.
Planck started a six semester course of lectures on theoretical physics. Lise Meitner described the lectures as "dry, somewhat impersonal." An English participant, James R. Partington, wrote, "using no notes, never making mistakes, never faltering; the best lecturer I ever heard." He continues: "There were always many standing around the room. As the lecture-room was well heated and rather close, some of the listeners would from time to time drop to the floor, but this did not disturb the lecture."
Planck did not establish an actual "school," the number of his graduate students was only about 20 altogether. Among his students were the following individuals. The year in which each individual achieved the highest degree is indicated after the person's name (outside the parentheses); the individual's year of birth and year of death are given within parentheses.
Max Abraham 1897 (1875-1922)
Moritz Schlick 1904 (1882-1936)
Walther Meißner 1906 (1882-1974)
Max von Laue 1906 (1879-1960)
Fritz Reiche 1907 (1883-1960)
Walter Schottky 1912 (1886-1976)
Walther Bothe 1914 (1891-1957)
Black-body radiation
In 1894, Planck had been commissioned by electricity companies to discover how to generate the greatest luminosity from light bulbs with the minimum energy. To approach that question, he turned his attention to the problem of black-body radiation. In physics, a black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. Black bodies below around 700 K (430 °C) produce very little radiation at visible wavelengths and appear black (hence the name). Above this temperature, however, they produce radiation at visible wavelengths, starting at red and going through orange, yellow, and white before ending up at blue, as the temperature is raised. The light emitted by a black body is called black-body radiation (or cavity radiation). The amount and wavelength (color) of electromagnetic radiation emitted by a black body is directly related to its temperature. The problem, stated by Kirchhoff in 1859, was: How does the intensity of the electromagnetic radiation emitted by a black body depend on the frequency of the radiation (correlated with the color of the light) and the temperature of the body?
This question had been explored experimentally, but the Rayleigh-Jeans law, derived from classical physics, failed to explain the observed behavior at high frequencies, where it predicted a divergence of the energy density toward infinity (the "ultraviolet catastrophe"). Wilhelm Wien proposed Wien's law, which correctly predicted the behavior at high frequencies but failed at low frequencies. By interpolating between the laws of Wien and Rayleigh-Jeans, Planck formulated the now-famous Planck's law of black-body radiation, which described the experimentally observed black-body spectrum very well. It was first proposed in a meeting of the DPG on October 19, 1900, and published in 1901.
By December 14, 1900, Planck was already able to present a theoretical derivation of the law, but this required him to use ideas from statistical mechanics, as introduced by Boltzmann. So far, he had held a strong aversion to any statistical interpretation of the second law of thermodynamics, which he regarded as having an axiomatic nature. Compelled to use statistics, he noted: "… an act of despair … I was ready to sacrifice any of my previous convictions about physics …"
The central assumption behind his derivation was the supposition that electromagnetic energy could be emitted only in quantized form. In other words, the energy could only be a multiple of an elementary unit. Mathematically, this was expressed as:
E = h \nu
where h is a constant that came to be called Planck's constant (or Planck's action quantum), first introduced in 1899, and \nu is the frequency of the radiation. Planck's work on quantum theory, as it came to be known, was published in the journal Annalen der Physik. His work is summarized in two books Thermodynamik (Thermodynamics) (1897) and Theorie der Wärmestrahlung (theory of heat radiation) (1906).
At first, Planck considered that quantization was only "a purely formal assumption … actually I did not think much about it…" This assumption, incompatible with classical physics, is now regarded as the birth of quantum physics and the greatest intellectual accomplishment of Planck's career. (However, in a theoretical paper published in 1877, Ludwig Boltzmann had already been discussing the possibility that the energy states of a physical system could be discrete.) In recognition of this accomplishment, Planck was awarded the Nobel prize for physics in 1918.
The discovery of Planck's constant enabled him to define a new universal set of physical units—such as Planck length and Planck mass—all based on fundamental physical constants.
Subsequently, Planck tried to integrate the concept of energy quanta with classical physics, but to no avail. "My unavailing attempts to somehow reintegrate the action quantum into classical theory extended over several years and caused me much trouble." Even several years later, other physicists—including Lord Rayleigh, James Jeans, and Hendrik Lorentz—set Planck's constant to zero, in an attempt to align with classical physics, but Planck knew well that this constant had a precise, nonzero value. "I am unable to understand Jeans' stubbornness—he is an example of a theoretician as should never be existing, the same as Hegel was for philosophy. So much the worse for the facts, if they are wrong."
Max Born wrote about Planck: "He was by nature and by the tradition of his family conservative, averse to revolutionary novelties and skeptical towards speculations. But his belief in the imperative power of logical thinking based on facts was so strong that he did not hesitate to express a claim contradicting to all tradition, because he had convinced himself that no other resort was possible."
Einstein and the theory of relativity
In 1905, the three epochal papers of the hitherto completely unknown Albert Einstein were published in the journal Annalen der Physik. Planck was among the few who immediately recognized the significance of the special theory of relativity. Thanks to his influence, this theory was soon widely accepted in Germany. Planck also contributed considerably to extend the special theory of relativity.
To explain the photoelectric effect (discovered by Philipp Lenard in 1902), Einstein proposed that light consists of quanta, which he called photons. Planck, however, initially rejected this theory, as he was unwilling to completely discard Maxwell's theory of electrodynamics. Planck wrote, "The theory of light would be thrown back not by decades, but by centuries, into the age when Christian Huygens dared to fight against the mighty emission theory of Isaac Newton …"
In 1910, Einstein pointed out the anomalous behavior of specific heat at low temperatures as another example of a phenomenon that defies explanation by classical physics. To resolve the increasing number of contradictions, Planck and Walther Nernst organized the First Solvay Conference in Brussels in 1911. At this meeting, Einstein was finally able to convince Planck.
Meanwhile, Planck had been appointed dean of Berlin University. Thereby, it was possible for him to call Einstein to Berlin and establish a new professorship for him in 1914. Soon the two scientists became close friends and met frequently to play music together.
World War I and the Weimar Republic
At the onset of the First World War Planck was not immune to the general excitement of the public: "… besides of much horrible also much unexpectedly great and beautiful: The swift solution of the most difficult issues of domestic policy through arrangement of all parties… the higher esteem for all that is brave and truthful…"
He refrained from the extremes of nationalism. For instance, he voted successfully for a scientific paper from Italy to receive a prize from the Prussian Academy of Sciences in 1915, (Planck was one of its four permanent presidents), although at that time Italy was about to join the Allies. Nevertheless, the infamous "Manifesto of the 93 intellectuals," a polemic pamphlet of war propaganda, was also signed by Planck. Einstein, on the other hand, retained a strictly pacifist attitude, which almost led to his imprisonment, from which he was saved only by his Swiss citizenship. But already in 1915, Planck revoked parts of the Manifesto, (after several meetings with Dutch physicist Lorentz), and in 1916, he signed a declaration against the German policy of annexation.
In the turbulent post-war years, Planck, by now the highest authority of German physics, issued the slogan "persevere and continue working" to his colleagues. In October 1920, he and Fritz Haber established the Notgemeinschaft der Deutschen Wissenschaft (Emergency Organization of German Science), which aimed at providing support for the destitute scientific research. They obtained a considerable portion of their funds from abroad. In this time, Planck held leading positions also at Berlin University, the Prussian Academy of Sciences, the German Physical Society, and the Kaiser Wilhelm Gesellschaft (KWG, which in 1948 became the Max Planck Gesellschaft). Under such circumstances, he himself could hardly conduct any more research.
He became a member of the Deutsche Volks-Partei (German People's Party), the party of peace Nobel prize laureate Gustav Stresemann, which aspired to liberal aims for domestic policy and rather revisionist aims for international politics. He disagreed with the introduction of universal suffrage and expressed later the view that the Nazi dictatorship was the result of "the ascent of the rule of the crowds."
Quantum mechanics
At the end of the 1920s, Bohr, Werner Heisenberg, and Wolfgang Pauli had worked out the Copenhagen interpretation of quantum mechanics. It was, however, rejected by Planck, as well as Schrödinger and Laue. Even Einstein had rejected Bohr's interpretation. Planck called Heisenberg's matrix mechanics "disgusting," but he gave the Schrödinger equation a warmer reception. He expected that wave mechanics would soon render quantum theory—his own brainchild—unnecessary.
Nonetheless, scientific progress ignored Planck's concerns. He experienced the truth of his own earlier concept, after his struggle with the older views. He wrote, "A new scientific truth does not establish itself by its enemies being convinced and expressing their change of opinion, but rather by its enemies gradually dying out and the younger generation being taught the truth from the beginning."
Nazi dictatorship and World War II
When the Nazis seized power in 1933, Planck was 74. He witnessed many Jewish friends and colleagues expelled from their positions and humiliated, and hundreds of scientists emigrated from Germany. Again he tried the "persevere and continue working" slogan and asked scientists who were considering emigration to stay in Germany. He hoped the crisis would abate soon and the political situation would improve again. There was also a deeper argument against emigration: Emigrating non-Jewish scientists would need to look for academic positions abroad, but these positions better served Jewish scientists, who had no chance of continuing to work in Germany.
Hahn asked Planck to gather well-known German professors, to issue a public proclamation against the treatment of Jewish professors. Planck, however, replied, "If you are able to gather today 30 such gentlemen, then tomorrow 150 others will come and speak against it, because they are eager to take over the positions of the others." Although, in a slightly different translation, Hahn remembers Planck saying: "If you bring together 30 such men today, then tomorrow 150 will come to denounce them because they want to take their places." Under Planck's leadership, the KWG avoided open conflict with the Nazi regime. One exception was Fritz Haber. Planck tried to discuss the issue with Adolf Hitler but was unsuccessful. In the following year, 1934, Haber died in exile.
One year later, Planck, having been the president of the KWG since 1930, organized in a somewhat provocative style an official commemorative meeting for Haber. He also succeeded in secretly enabling a number of Jewish scientists to continue working in institutes of the KWG for several years. In 1936, his term as president of the KWG ended, and the Nazi government put pressure on him to refrain from running for another term.
As the political climate in Germany gradually became more hostile, Johannes Stark, prominent exponent of Deutsche Physik ("German Physics," also called "Aryan Physics") attacked Planck, Arnold Sommerfeld, and Heisenberg for continuing to teach the theories of Einstein, calling them "white Jews." The "Hauptamt Wissenschaft" (Nazi government office for science) started an investigation of Planck's ancestry, but all they could find out was that he was "1/16 Jewish."
In 1938, Planck celebrated his 80th birthday. The DPG held an official celebration, during which the Max Planck medal (founded as the highest medal by the DPG in 1928) was awarded to French physicist Louis de Broglie. At the end of 1938, the Prussian Academy lost its remaining independence and was taken over by Nazis (Gleichschaltung). Planck protested by resigning his presidency. He continued to travel frequently, giving numerous public talks, such as his famous talk on "Religion and Science." Five years later, he was still sufficiently fit to climb 3,000-meter peaks in the Alps.
During the Second World War, the increasing number of Allied bombing campaigns against Berlin forced Planck and his wife to leave the city temporarily and live in the countryside. In 1942, he wrote: "In me an ardent desire has grown to persevere this crisis and live long enough to be able to witness the turning point, the beginning of a new rise." In February 1944, his home in Berlin was completely destroyed by an air raid, annihilating all his scientific records and correspondence. Finally, he was in a dangerous situation in his rural retreat during the rapid advance of Allied armies from both sides. After the end of the war, Planck, his second wife, and their son Herrmann moved to Göttingen, where he died on October 4, 1947.
Max Planck commemorated on the German 2 Mark Coin
Religious views
Max Planck was a devoted Christian from early life to death. As a scientist, however, he was very tolerant toward other religions and alternate views, and was discontent with the church organization's demands for unquestioning belief. He noted that "natural laws … are the same for men of all races and nations."
Planck regarded the search for universal truth as the loftiest goal of all scientific activity. Perhaps foreseeing the central role it now plays in current thinking, Planck made great note of the fact that the quantum of action retained its significance in relativity because of the relativistic invariance of the Principle of Least Action.
Max Planck's view of God can be regarded as pantheistic, with an almighty, all-knowing, benevolent but unintelligible God who permeates everything, manifest by symbols, including physical laws. His view may have been motivated by an opposition—like that of Einstein and Schrödinger—to the positivist, statistical, subjective universe of scientists such as Bohr, Heisenberg, and others. Planck was interested in truth and the Universe beyond observation, and he objected to atheism as an obsession with symbols.[1]
Planck was the very first scientist to contradict the physics established by Newton. This is why all physics before Planck is called "classical physics," while all physics after him is called "quantum physics." In the classical world, energy is continuous; in the quantum world, it is discrete. On this simple insight of Planck's was constructed all of the new physics of the twentieth century.
Unlike religion with its great leaps, science proceeds by baby steps. The small step taken by Planck was the first of the many needed to reach the current "internal wave and external particle" view of modern physics a century later.
Honors and medals
• "Pour le Mérite" for Science and Arts 1915 (in 1930 he became chancellor of this order)
• Nobel Prize in Physics 1918 (awarded 1919)
• Lorentz Medal 1927
• Adlerschild des Deutschen Reiches (1928)
• Max Planck medal (1929, together with Einstein)
• Planck received honorary doctorates from the universities of Frankfurt, Munich (TH), Rostock, Berlin(TH), Graz, Athens, Cambridge, London, and Glasgow
• The asteroid 1069 was given the name "Stella Planckia" (1938)
Planck units
• Planck time
• Planck length
• Planck temperature
• Planck current
• Planck power
• Planck density
• Planck mass
1., The Religious Affiliation of Physicist Max Planck. Retrieved July 16, 2007.
Selected publications by Planck
• Gamow, George. 1966. Thirty Years That Shook Physics: The Story of Quantum Theory. Garden City, NY: Doubleday.
• Heilbron, J. L. 2000. The Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science. Cambridge, MA: Harvard University Press. ISBN 0-674-00439-6
• Rosenthal-Schneider, Ilse. 1980. Reality and Scientific Truth: Discussions with Einstein, von Laue, and Planck. Wayne State University. ISBN 0-8143-1650-6
External links
All links retrieved May 6, 2015.
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435bb07a721c19fd | Quantum Logic in Algebraic Approach (Fundamental Theories of
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Schrödinger's insight,[ citation needed ] late in 1925, was to express the phase of a plane wave as a complex phase factor using these relations: As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. For a plane wave in three space dimensions, the wave is represented in a similar way, A(x, t) = A0 sin(k · x − ωt), (5.2) where x is now the position vector and k is the wave vector. If the matter wave to the left of the discontinuity is ψ1 = sin(k1x x+k1y y−ω1 t) and to the right is ψ2 = sin(k2x x + k2y y − ω2 t), then the wavefronts of the waves will match across the discontinuity for all time only if ω1 = ω2 ≡ ω and k1y = k2y ≡ ky.
Pages: 243
Publisher: Springer; 1998 edition (January 31, 1998)
ISBN: 0792349032
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Tsunami and Nonlinear Waves
Elementary Wave Optics
The fact that we are talking about light beams is only for convenience Quantum Electrodynamics Quantum Electrodynamics. This theory seems to apply to every process occurring between any kinds of matter or energy in the universe. (The actual details have undergone some revision since the Copenhagen Interpretation was formulated, but the essential ideas presented here remain the same.) But of course, the theory also raises a lot of disturbing questions , e.g. The Method of Moments in read online http://warholprints.com/library/the-method-of-moments-in-electromagnetics-second-edition. Such is the case for photons, for example, but also whole atoms may be bosons. Bosons are social beasts that like to be on the same wavelength � or, as physicists put it, they like to be in the same quantum state Electromagnetics (McGraw-Hill read here dh79.com. And then you go about to prove that actually this complex solution implies the existence of two real solutions. So this complex solution implies existence of two real solutions , cited: Adaptive Filters http://elwcoaching.com/library/adaptive-filters. Effective current: DC current that would produce the same heating effects. Effective voltage: DC potential difference that would produce the same heating effects. Efficiency: ratio of output work to input work. Effort force: force extended on a machine. Elastic collision: interaction between two objects in which the total energy is the same before and after the interaction. Elasticity: ability of object to original shape after deforming forces are removed Elements of Engineering download online download online. So it’s safe to apply Maxwell’s equations to the full complex functions transverse, they are also perpendicular to each other. Now suppose we want a monochromatic, plane wave that travels in some arbitrary direction given by onto the direction of propagation, since for a plane wave, the wave function depends only on the distance we’ve moved along this direction download. In each case a is a constant. dxa = axa−1 (1.27) dx d exp(ax) = a exp(ax) (1.28) dx d 1 log(ax) = (1.29) dx x d sin(ax) = a cos(ax) (1.30) dx d cos(ax) = −a sin(ax) (1.31) dx The product and chain rules are used to compute the derivatives of complex functions. For instance, d sin(x) d cos(x) d (sin(x) cos(x)) = cos(x) + sin(x) = cos2 (x) − sin2 (x) dx dx dx and 1 d sin(x) cos(x) d log(sin(x)) = =. dx sin(x) dx sin(x) We now ask the following question: How fast do wave packets move online? University of Hawaii Lots of midterms and finals from Physics 151, a class that covers mechanics and thermodynamics. University of Rochestor Tests with answers from Physics 141, a mechanics course that also covers special relativity. CSU Fresno Multiple choice sample exams with answers and quizzes from Physics 2A , cited: Vibrations and Waves in Physics Vibrations and Waves in Physics. Even for the high temperatures in the center of a star, fusion requires the quantum tunneling of a neutron or proton to overcome the repulsive electrostatic forces of an atomic nuclei. Notice that both fission and fusion release energy by converting some of the nuclear mass into gamma-rays, this is the famous formulation by Einstein that E=mc2 Theory of Many-Particle Systems (AIP Translation Series) download here. After a measurement is made, the wave function is permanently changed in such a way that any successive measurement will certainly return the same value. This is called the collapse of the wave function , source: Lightwave Technology: Telecommunication Systems warholprints.com. The shape of a sine wave is given by its amplitude, phase, wavelength and frequency , e.g. Supersymmetry and String read pdf read pdf. |
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Nanoelectronic Modeling Lecture 09: Open 1D Systems - Reflection at and Transmission over 1 Step
By Gerhard Klimeck1, Dragica Vasileska2, Samarth Agarwal3
Published on
One of the most elemental quantum mechanical transport problems is the solution of the time independent Schrödinger equation in a one-dimensional system where one of the two half spaces has a higher potential energy than the other. The analytical solution is readily obtained using a scattering matrix approach where wavefunction amplitude and slope are matched at the interface between the two half-spaces. Of particular interest are the wave/particle injection from the lower potential energy half-space. In a classical system a particle will suffer from complete reflection at the half-space border if its kinetic energy is not larger than the potential energy difference at the barrier. The classical particle will be completely transmitted if its kinetic energy exceeds the potential barrier difference. A quantum mechanical particle or wave however exhibits a few interesting features: 1) it can penetrate into the potential barrier when ints kinetic energy is lower than the potential step energy, and 2) transmission over the barrier is not complete and energy dependent. Incomplete transmission implies a reflection probability for the wave even though its kinteci energy exceeds the potential barrier difference. This simple example shos the extended nature of wavefunctions and the non-local effects of local potential variations in its simplest form.
Cite this work
Researchers should cite this work as follows:
• Gerhard Klimeck; Dragica Vasileska; Samarth Agarwal (2010), "Nanoelectronic Modeling Lecture 09: Open 1D Systems - Reflection at and Transmission over 1 Step,"
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Università di Pisa, Pisa, Italy
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7bc4b3e985c0b65b | Monday, March 20, 2006
A universe of Qualia
In my previous posting I applied Tegmark's idea that every mathematical model is a universe, to humans. This leads to the conclusion that we can think of our minds as universes in their own right. If we think of the universe we live in, we usually think of the objects we see around us, their properties and how they behave.
In case of our mind considered as a universe, the laws of physics are contained in an exact description of the way our neurons in our brain interact with each other. This description is, of course, enormously complicated. Alternatively, we could think of the neurons in our brain as simulating ''emergent laws of physics'' that describe the qualia we experience.
Just like one can do organic chemistry without solving the Schrödinger equation for complex organic molecules, we can talk about how we feel, what we see etc. without referring to what exactly our neurons are doing in our brains. We can thus think of the qualia as ''events'' in our personal universe. These are described by ''effective laws of physics'', analogously to the imprecise laws of, say, organic chemistry or biology.
Since we experience the qualia and not the fundamental processes that give rise to the qualia (this follows from the Simulation Argument: If the brain were simulated on some computer, it would have the same consciousness), we should consider the qualia as fundamental objects of our personal universe. The universe on the level of the qualia is where the mind really resides. It is here that the notions of pain, anger, happiness, colors etc. exist.
Blogger QUASAR9 said...
Hi Count seeing the universe with the eyes is relative, our eyes can and do deceive us. The same with thoughts we can like Quixote be fighting windmills.
But physical 'reality' ie a brick wall, no matter whether we have 20/20 vision, whether we are partially sighted, whether we are totally blind, or whether our minds are troubled or otherwise distracted, if we walk into a brick wall we shall know we have walked into one.
You'll be surprised how many people walk into lampposts, even among those with 20/20 vision, no not because they weren't looking in that direction (in front of them) but because they didn't see it (didn't even see it coming).
Not because of the 'blind' spot, but because their focus, or thoughts were on something other than wahat was in front of them.
Incidentally, have you ever pulled up at a roundabout, there is a car in front, you look (left) in EU (right) in uk, no traffic on roundabout, so you start to move forward, only to slam the brakes on when you realise the vehicle in front has not moved. You (brain) just assumed that because you could see it was ok to go the chap in front would see it too, and respond at the same speed as you.
Of course some people travel thru life seldom encountering a red light, or gettin caught in traffic, whilst others go from one red light to the next.
And some people drive accelerating braking accelerating braking in urban traffic, whilst taxi drivers have developed the skill of going with the 'flow' and often arriving at the destination in less time, with less stress and less wear on their selves and vehicles.
But I digress, what I meant is that if there is something solid there, there are no X-men that can walk through it, the wall is there whether you can see or whether you are blind. Ram raiders used to and do get over the problem of walls or reinforced glass by using 4x4's with bull bars. lol! Laters ... Q
Wed Jun 14, 06:23:00 AM PDT
Blogger Faust said...
Hi Count,
I have just opened up my own blog; you may find it interesting.
Name: 'Space - Time - Matter'
p.s.: Are you a physics (grad) student? I want to get primarily physics and math students to my site.
Fri Oct 06, 07:03:00 AM PDT
Blogger Count Iblis said...
Quasar9, I agree with your analyses. An interesting question is why is there a physical world and why can't be like the X-men you mention? I'll elaborate on that in a next posting.
Mon Oct 09, 06:56:00 PM PDT
Blogger Count Iblis said...
Hi Faust, I'll visit your Blog. I've a Ph.D. in physics. On this blog I only explore metaphysical ideas that are not (yet) publishable :)
Mon Oct 09, 06:57:00 PM PDT
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Following up on the previous MO question "Are there any important mathematical concepts without discrete analogue?", I'd like to ask the opposite: what are examples of notions in math that were not originally discrete, but have good discrete analogues? While a few examples arose in the answers to that earlier MO question, this wasn't what that question was asking, so I'm sure there are many more examples not mentioned there or at least not really explained there. What reminded me of this older MO question was seeing an MO question "Why is the Laplacian ubiquitous?", since that is an instance of an important notion which has a discrete analgoue.
In an answer, it would be interesting to hear about the relationship between the continuous and discrete versions of the notion, if possible, and references could also be helpful. Thanks!
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I don't actually know if this is true, but I would guess that the Fourier transform was discovered before the discrete Fourier transform. – Qiaochu Yuan Aug 19 '12 at 16:50
That's a great example -- I actually wasn't so concerned about chronology, rather was interested in understanding better the interesting relationships between the discrete and continuous versions of things and thought it might be nice if there were a list of examples. You could certainly write an answer about the Fourier transform. – Patricia Hersh Aug 19 '12 at 17:01
Patricia, since you are asking for examples: please make this community wiki. There is no "right" answer here. – Vidit Nanda Aug 19 '12 at 17:53
@Vel: I thought some people could be more motivated to go to the trouble to write a good answer if I didn't make this CW -- there have been other questions like this that aren't CW, especially when a good answer could involve substantial mathematics. So I wanted to see if I could hold off on that, at least for awhile. – Patricia Hersh Aug 19 '12 at 18:20
Some people do not answer big list questions until they are CW. I have flagged the mods because only they can make existing answers CW. – Benjamin Steinberg Aug 20 '12 at 0:08
12 Answers 12
Negative curvature of Riemannian manifolds, originally a differentiable theory, has been discretized in several phases. The first phase might have been Dehn's algorithm for the word problem in a surface group; I am guessing that at the time this might have seemed more an "application" of hyperbolic geometry than a discretization of it. But then comes the next big phase, the development of small cancellation theory, in which Dehn's algorithm (and related tools) were applied to many abstractly defined groups. The culminating phase was the development (by Gromov among others) of the theory of hyperbolic groups.
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I'll give one answer to get things started: discrete Morse theory.
A discrete Morse function assigns a real number to each face in a simplicial complex or more generally to each cell in a regular CW complex. (With care, one can also work with non-regular CW complexes.) While in Morse theory there are critical points, each having an index, the discrete Morse theoretic analogue is a critical cell, with the dimension of a critical cell playing the role of index of a critical point. The Morse inequalities still hold, and one can still calculate Euler characteristic as alternating sum of Morse numbers (i.e. alternating sum of the number of critical cells of each dimension). The original regular CW complex will be (simple) homotopy equivalent to a CW complex having fewer cells (unless all cells are critical), namely a CW complex whose cells are indexed by the critical cells.
This analogue with Morse theory was established by Robin Forman in his paper "Morse theory for cell complexes", Adv. Math., 134 (1998), no. 1, 90-145. Another nice reference is his paper "A user's guide to discrete Morse theory". The idea has proven quite useful in the study of various simplicial complexes e.g. in combinatorics, and the idea appeared independently in work of Ken Brown under the name "collapsing scheme".
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You might add as a reference the paper of Bestvina and Brady, Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), no. 3, 445–470. – Lee Mosher Aug 19 '12 at 16:02
Thanks! My description of the analogy was for Forman's notion, but it's a good idea to add this reference. – Patricia Hersh Aug 19 '12 at 17:46
We have to be careful with the simple homotopy claim. Given $f:X \to \mathbb{R}$ and setting $X^a = \lbrace \sigma \in X~|~f(\sigma) < a\rbrace $ there is a simple homotopy equivalence between $X^a$ and $X^b$ provided there are no critical values in $(a,b)$. On the other hand, when we cross a critical value, then we only have homotopy equivalence coming from the attaching map of the boundary of the critical cell: this need not be a simple homotopy equivalence. – Vidit Nanda Aug 19 '12 at 17:50
@Vel: I think one can also handle the critical cells by using some anticollapses, but I don't know a reference for this. Idea: once one removes critical cell $C$, one can see what elementary collapses to do to get down to $X^a$ and how they would carry the boundary of $C$ to have new attaching map $f_{C_a}$. Therefore, we first do an anticollapse by adding in cell $C'$ with attaching map $f_{C_a}$ along with a cell $D$ of dimension one higher that has $C'$ as a free face and also attaches to the cells $\sigma $ with $a\le f(\sigma ) \le b$. Now collapse away $C,D$ and the noncritical cells. – Patricia Hersh Aug 19 '12 at 18:49
Vel, my recollection is that Robin Forman made this statement various times at conferences that a discrete Morse function implies a simple homotopy equivalence, so probably that's how it became folklore. I think it's true, and hopefully the argument I gave above explains why. – Patricia Hersh Aug 20 '12 at 12:27
A simplicial set is a discrete analogue (and in many ways a generalizaion) of a topological space, giving rise to discrete notions of fibration, homotopy groups, etc etc.
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Trees (in particular, homogeneous) are discrete analogues of Cartan-Hadamard manifolds (in particular, of simply connected manifolds of constant negative curvature). Although dealing with trees is much easier technically, they were considered much later: function theory, harmonic analysis, automorphism groups, random walks vs Brownian motion, representation theory etc. One has to admit that mostly (not always, though) it was done by direct translation (sometimes almost verbatim) from continuous into discrete language.
Another example is provided by the discrete potential theory (sometimes interpreted as the theory of resistive electrical networks). Here, once again, in spite of being much more elementary it was developed significantly later than the continuous theory. I would say that in the latter case the discrete theory is more independent than in the case of geometry on trees.
Yet another example (where the discrete part is much more original) is buildings vs Riemannian symmetric spaces.
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One of my favorite examples of this is the "q-calculus", which is like a multiplicative version of the classical subject of calculus of finite differences. One can, using suitably defined "q" versions of the derivative, integral, and so on, recover analogues of most of the usual theorems in calculus. But what's more interesting is that this all ties in with noncommutative geometry and the field with one element (see John Baez's This Weeks Finds in Mathematical Physics).
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That last sentence needs some justification – Yemon Choi Aug 19 '12 at 21:25
You're right. I added a reference. – Aleksandar Bahat Aug 19 '12 at 22:02
Finite graphs are a rich source of discrete analogues (I will be partially repeating the OP and some other answers here):
• The Laplacian on a finite graph is a discrete analogue of the Laplacian on a Riemannian manifold. In particular, it is possible to formulate the heat equation, the wave equation, and the Schrödinger equation on a finite graph. There are actually two Laplacians, a vertex Laplacian and an edge Laplacian, which give a discrete analogue of Hodge theory.
• The Ihara zeta function of a finite graph is a discrete analogue of the Selberg zeta function of a Riemannian manifold. A regular graph satisfies an analogue of the Riemann hypothesis if and only if it is a Ramanujan graph. There is also an analogue of the Selberg trace formula in this setting; Terras has written extensively about this kind of thing.
• The Picard group (or critical group, or sandpile group) of a finite graph is a discrete analogue of the Picard group of an algebraic curve. More generally a lot of the theory of algebraic curves can be transported to this setting, e.g. the Riemann-Roch theorem.
(Finite graphs are also a rich source of other kinds of analogues; for example the Ihara zeta function is also analogous to the Dedekind zeta function of a number field, with coverings of graphs analogous to extensions of number fields and the Picard group analogous to the class group. There is even an analogue of the analytic class number formula in this setting although I have forgotten the reference.)
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Thanks! Great answer! – Patricia Hersh Aug 24 '12 at 13:47
I would consider symbolic dynamics as a discrete version of usual dynamical systems. This may depend on whether you view infinite words on finite alphabets as discrete.
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Discrete difference equations generalize differential equations. In a similar spirit, divided difference operators generalize partial differentiation operators. Though such operators go back to Newton, there has a been a resurgence of interest in them since the work of Lascoux and Schutzenberger on Schubert polynomials. While partial differentiation operators satisfy commutativity relations $\partial_x \partial_y = \partial_y \partial_x$, the divided difference operators satisfy the nilHecke relations. This gives the discrete operators a certain richness that is not present in the continuous operators.
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As one reference on this topic, I really like the paper: Sergey Fomin and Richard Stanley, "Schubert polynomials and the nilCoxeter algebra", Adv. Math. 103 (1994), 196--207. – Patricia Hersh Aug 20 '12 at 14:16
A more or less elementary example: Sperner's lemma is a discrete/combinatorial analog to the Brouwer fixed point theorem. Furthermore, its one-dimensional case is a discrete analog to the intermediate value theorem.
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Continuous-time random walks on graphs are in some sense a discrete analogue of diffusions on a Riemannian manifold (of course, the reverse can be argued, but I think that diffusions play a more central role in modern probability theory). Of course, the most important diffusion is Brownian motion, i.e., the Markov process associated with the Laplace-Beltrami operator. From my perspective, the natural analogue of Brownian motion is the operator $\mathcal{L}_V$ given by (we use unweighted graphs for simplicity)
\begin{equation*} (\mathcal{L}_Vf)(x) := \sum\_{y\sim x}(f(y)-f(x)). \end{equation*}
A more 'common' choice might be the rate-1 continuous time random walk with generator $\mathcal{L}_C$ given by
\begin{equation*} (\mathcal{L}_Cf)(x) := \frac{1}{\deg(x)}\sum\_{y\sim x}(f(y)-f(x)). \end{equation*}
However, this choice of generator has several 'bad' properties if you want to view it as an analogue of Brownian motion -- for example, the generator is always bounded on $L^2(\deg)$, it cannot have discrete spectrum, and the associated random walk cannot explode; in contrast, the operator $\mathcal{L}_V$ may be unbounded, and discrete spectrum and explosiveness are possible.
Once you have this discrete (space) analogue of Brownian motion on a Riemannian manifold, a natural question is to ask what the discrete analogue of the Riemannian metric should be for this process. It is not too hard to find examples that show that the graph metric is a bad analogue, since the Riemannian metric governs heat flow (in some sense) on a Riemannian manifold (see e.g. here), but Gaussian heat kernel estimates do not hold for the random walk associated with $\mathcal{L}_V$ if you take the manifold heat kernel estimates and replace the distance function with the graph metric. A reasonable analogue has been formulated recently, see e.g. here and here.
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If you have a discrete data structure (say a tree), and you want to make a small change to it (i.e. insert a node at some location), it turns out that the original datatype can be described as a function, and the "small change" datatype is the derivative of the original datatype's function, that you can calculate with the usual rules for derivatives. The original article is here:
and it's been extended in various ways since then. Amazing stuff.
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ACUTE—Assembly for Computational Electronics
by Dragica Vasileska, Gerhard Klimeck, Xufeng Wang, Stephen M. Goodnick, Margaret Shepard Morris, Michael Anderson, Philathia Rufaro Bolton, Cristina Leal Gonzalez, Craig Titus, Jamie E Hickner
This nanoHUB “topic page” provides an easy access to selected nanoHUB educational material on computational electronics that is openly accessible.
We invite users to participate in this open source, interactive educational initiative:
• content by uploading it to the nanoHUB. (See “Contribute Content”) on the nanoHUB mainpage.
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Thank you for using the nanoHUB, and be sure to your nanoHUB success stories with us. We like to hear from you, and our sponsors need to know that the nanoHUB is having impact.
The purpose of the ACUTE tool-based curriculum is to introduce interested scientists from academia and industry to the advanced methods of simulation needed for the proper modeling of state-of-the-art nanoscale devices. The multiple scale transport in doped semiconductors is summarized in the figure below, in terms of the transport regimes, relative importance of the scattering mechanisms, and possible applications.
intro1.png intro2.png
ACUTE begins with a discussion of the energy band structure that enters as an input to any device simulator. The next section offers a discussion of simulators that involve the drift-diffusion model, and then simulations that involve hydrodynamic and energy-balance transport, and conclude the semi-classical transport modeling with application of particle-based device simulation methods.
After the study and utilization of the semiclassical simulation tools and their applications, the next step includes quantum corrections into the classical simulators. The final set of tools is dedicated to the far-from equilibrium transport, where the concept of pure and mixed states and the distribution function is introduced. Several tools that utilize different methods will be used for that purpose, such as tools that use the recursive Green’s-function method and its variant, the Usuki method, as well as the Contact Block Reduction tool, as the most efficient and complete way of solving the quantum-transport problem because this method allows users to simultaneously calculate source-drain current and gate leakage (which is not the case, for example, with the Usuki and the recursive Green’s function techniques that are in fact quasi one-dimensional in nature for transport through a device). A table that shows the advantages and the limitation of various semi-classical and quantum-transport simulation tools is presented below.
More details on the actual tool design and information on commercial tool usage can be found on the web pages:
Computational Electronics
Computational Electronics HW Set
Energy Bands and Effective Masses
Piecewise Constant Potential Barrier Tool in ACUTE– Open Systems
pcpbt.png The Piecewise Constant Potential Barrier Tool in ACUTE allows users to calculate the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2n-segment piecewise constant potential energy profile. For the case of a multi-well structure, it also calculates the quasi-bound states. Thus the Piecewise Constant Potential Tool can be used as a simple demonstration tool for the formation of energy bands.
Other uses include: 1) in the case of stationary perturbation theory, as an exercise to test the validity of the first-order and the second-order correction to the ground state energy of the system due to small perturbations of the confining potential, and 2) as a test of the validity of the Wentzel–Kramers–Brillouin (WKB) approximation for triangular potential barriers.
Periodic Potential Lab in ACUTE
The Periodic Potential Lab in ACUTE solves the time-independent Schrödinger Equation in a one-dimensional spatial potential variation. Rectangular, triangular, parabolic (harmonic), and Coulomb potential confinements can be considered. The user can determine energetic and spatial details of the potential profiles, compute the allowed and forbidden bands, plot the bands in a compact as well as an expanded zone, and compare the results against a simple effective-mass parabolic band. Transmission is also calculated. This lab also allows students to become familiar with the reduced-zone and expanded-zone representation of the dispersion relation (i. e. the E-k relation for carriers).
Band Structure Lab in ACUTE
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is “forbidden” or “allowed” to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties. The Band Structure Lab in ACUTE enables the study of bulk dispersion relationships of silicon (Si), gallium arsenide (GaAs), and indium arsenide (InAs). Plotting the full dispersion relation of different materials, students first get familiar with a band structure of direct bandgap (gallium arsenide and indium arsenide) and indirect band-gap semiconductors (silicon). In the case of multiple conduction band valleys, the user has first to determine the Miller indices of one of the equivalent valleys and then from that information it immediately follows, e. g., how many equivalent conduction bands one has in silicon and germanium (Ge).
In advanced applications, the users can apply tensile and compressive strain and observe the variation in the band structure, bandgaps, and effective masses. Advanced users can also study band-structure effects in ultra-scaled (thin body) quantum wells, and nanowires of different cross sections. Band Structure Lab uses the sp3s*d5 tight binding method to compute dispersion (E-k) for bulk, planar, and nanowire semiconductors.
Drift-Diffusion and Energy Balance Simulations
PADRE Tool in ACUTE—Modeling of silicon-based devices
PADRE Tool in ACUTE is a two-dimensional/three-dimensional simulator for electronic devices, such as MOSFET transistors.
PADRE Tool in ACUTE is a 2D/3D simulator for electronic devices, such as MOSFETs. With PADRE, users can simulate physical structures of arbitrary geometry—including heterostructures—with arbitrary doping profiles, which can be obtained using analytical functions or directly from such multidimentional process simulators as Prophet For each electrical bias, PADRE Tool in ACUTE solves coupled sets of partial differential equations (PDEs). The variety of PDE systems supported in PADRE form a hierarchy of accuracy: 1) electrostatic (Poisson equations), 2) drift-diffusion, including carrier continuity equations, 3) energy balance, including carrier temperature, and 4) electrothermal, including lattice heating.
Listed below are tools, exercises, and sets of problems that utilize the PADRE Tool in ACUTE:
Supplemental documentation:
A set of course notes on computational electronics with detailed explanations on band structure, pseudopotentials, numerical issues, and drift diffusion is also available.
• Introduction to DD Modeling with PADRE
SILVACO Simulator—Modeling of Silicon-Based and III-V Devices
In preparation.
Particle-Based Simulators
Bulk Monte Carlo Lab in ACUTE
scattering.png mc.png
The Bulk Monte Carlo Lab in ACUTE calculates the bulk values of the electron drift velocity, electron average energy, and electron mobility for electric fields applied in arbitrary crystallographic direction in both column 4 (silicon and germanium) and III-V (gallium arsenide, silicon carbide and gallium nitride) materials. All relevant scattering mechanisms for the materials being considered have been included in the model.
Bulk Monte Carlo Code Described
An A/V presentation is also available:
Ensemble Monte Carlo Method Described
Consistent Parameter Set for an Ensemble Monte Carlo Simulation of 4H-SiC
Quamc2D Lab in ACUTE
quamc2d1.png quamc2d2.png
Quamc2D Lab in ACUTE
QuaMC 2D (pronounced “quam-see”) is a quasi three-dimensional quantum-corrected semi-classical Monte-Carlo transport simulator for conventional and non-conventional MOSFET devices.
Thermal Particle-Based Device Simulator
In preparation.
Exercises and Other Resources:
Inclusion of Quantum Corrections in Semiclassical Simulation Tools
Schred in ACUTE
schred.png Schred in ACUTE calculates the envelope wavefunctions and the corresponding bound-state energies in a typical MOS (Metal-Oxide-Semiconductor) or SOS (Semiconductor-Oxide- Semiconductor) structure and a typical SOI structure by solving self-consistently the one-dimensional Poisson and Schrödinger equations.
To better understand the operation of Schred in ACUTE and the physics of MOS capacitors please refer to:
• Quantum Size Effects and the Need for Schred
1D Heterostructure Tool in ACUTE
1dhet1.png 1dhet2.png
The 1D Heterostructure Tool in ACUTE simulates the confined states in one-dimentional heterostructures by self-consistently calculating their charge based on a quantum-mechanical description of the one-dimensional device. Increased interest in high electron mobility transistors (HEMTs) is due to the eventual limitations reached by scaling conventional transistors. The 1D Heterostructure Tool in ACUTE is a very valuable tool for the design of HEMTs because the user can determine such components as the position and the magnitude of the delta-doped layer, the thickness of the barrier, and the spacer layer, for which the user can maximize the amount of free carriers in the channel, which, in turn, leads to a larger drive current.
The most commonly used semiconductor devices for applications in the GHz range now are gallium arsenide based MESFETs, HEMTs and HBTs. Although MESFETs are the cheapest devices because they can be realized with bulk material, i.e. without epitaxially grown layers, HEMTs and HBTs are promising devices for the near future. The advantage of HEMTs and HBTs compared to MESFETs is a higher power density (by a factor of two to three), which leads to a significantly smaller chip size.
HEMTs are field-effect transistors wherein the flow of the current between two ohmic contacts, known as the source and the drain, is controlled by a third contact, the gate. Such gates are usually Schottky contacts. In contrast to ion-implanted MESFETs, HEMTs are based on epitaxial layers with different band gaps.
Quantum Transport
Recursive Green’s Function Method for Modeling RTD’s
in preparation.
nanoMOS in ACUTE
nanoMOS in ACUTE is a two-dimensional simulator for thin body (less than 5 nm), fully depleted, double-gated n-MOSFETs. Five transport models is available (drift-diffusion, classical ballistic, energy transport, quantum ballistic, and quantum diffusive). The transport models treat quantum effects in the confinement direction exactly, and the names indicate the technique used to account for carrier transport along the channel. Each of these transport models is solved self-consistently with Poisson’s equation. Several internal quantities such as subband profiles, subband areal electron densities, potential profiles, and current-voltage (I/V) information can be obtained from the source code.
NanoMOS 3.0 includes an improved treatment of carrier scattering. Some important information about nanoMOS in ACUTE can be found on the following links:
in preparation.
Atomistic Modeling
modeling_agenda5.gif qdot.png
Users of NEMO3D in ACUTE can analyze quantum dots, alloyed quantum dots, long-range strain effects on quantum dots, the effects of wetting layers, piezo-electric effects in quantum dots, quantum-dot nuclear-spin interaction, quantum-dot phonon spectra, coupled quantum-dot systems, miscut silicon quantum wells with silicon-germanium alloy buffers, core-shell nanowires, alloyed nanowires, phosphorous impurities in silicon (P:Si qubits), and buck alloys.
Boundary conditions to treat the effects of surface states have been developed. Direct and exchange interactions and interactions with electromagnetic fields can be computed in a post-processing approach based on the NEMO 3D single particle states.
* Quantum Dot Lab
* Coulomb Blockade Simulation
Collection of tools that comprise ACUTE
Piece-Wise Constant Potential Barriers Tool
Periodic Potential Lab
Band Structure Lab
Bulk Monte Carlo Lab
1D Heterostructure Tool
Quantum Dot Lab
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I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on photons rather than wave physics.
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Photons are wavelike, in that they are quanta of wave functions. The two descriptions are inexorably linked. – rajb245 Oct 2 '13 at 22:19
@rajb245 What if we have first photon, and then his "replica" is generated due to stimulated emission in gain medium. Will we have 2 photons flying in exactly same direction? – BarsMonster Oct 2 '13 at 22:38
The point is that even a single photon traveling through an aperture (hole at the end of the laser cavity) is scattered (deflected) by it. The wave function exists over the entire aperture and diffracts through it. You can't decouple wave and particle views. – rajb245 Oct 3 '13 at 4:01
@Ruslan and gregsan have attributed it to different effects, but I haven't seen any convincing evidence offered as to which is right. – Ben Crowell Oct 3 '13 at 15:41
6 Answers 6
Due to Heisenberg uncertainty principle $\Delta x\Delta p\gtrsim \frac{\hbar}2$, one can't really make a quantum have zero momentum in any direction. So you can't say that photons go in the same direction - this is just a simplified description of laser operation. In reality, the thinner the beam, the higher the divergence.
Compare e.g. a DPSS laser (e.g. green laser pointer) with a diode laser (e.g. a red laser pointer).
• In a DPSS laser the active material will have diameter of order of hundreds of micrometer, and the exiting beam will start from even smaller diameter for various reasons. The divergence is quite small: if you remove the collimating lens, your light image from a green laser pointer will be several centimeters after the light goes several meters. Divergence angle would be $\sim \lambda/d=532\text{nm}/100\operatorname{\mu m}\sim0.3^\circ$.
• If you try doing the same with a red laser pointer, you'll see that its light diverges quite a lot: after going several centimeters in direction of propagation, it'll already give image of several centimeters. The reason for this is that active zone of diode laser has diameter of order of several micrometers. This makes output beam quite thin, making $\Delta x$ small and thus $\Delta p$ high, and this is what leads to high divergence. Divergence angle would be $\sim \lambda/d=640\text{nm}/1\operatorname{\mu m}\sim 40^\circ$. Actual angle would depend on which transverse direction you select, because active zone is $\sim 10\times$ longer in one direction than in another.
In general, the thicker your starting laser beam, the more collimated it is, so if you manage to make a (visible wavelength) laser with beam starting at 1cm thickness, you'll have almost perfectly collimated laser beam.
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It would be nice to see an order of magnitude estimate showing that this effect really is of the right size to explain what is observed. A separate issue, and really just a matter of taste, is that I don't like the unnecessary invocation of the Heisenberg uncertainty principle to explain a fact about classical optics. – Ben Crowell Oct 3 '13 at 15:42
@BenCrowell I added uncertainty principle to fit OP's requirement of working in terms of photons rather than waves. Otherwise, of course, it's better to talk about diffraction. As for estimates, I'll add them a bit later. – Ruslan Oct 3 '13 at 17:08
@BenCrowell I actually think that steering clear of the HUP is more than taste - see the second half of my answer. Ruslan's answer is valid because the mathematics of diffraction and of the HUP are the same, as I talk about. also give an estimate of divergence grounded on a diffraction calculation, and it is exactly what is used to engineer Gaussian or other beams. Namely, you get about a $10^{−5}\mathrm{radian}$ cone angle for a high quality (diffraction limited) 1mm square laser chip at 500nm wavelength. – WetSavannaAnimal aka Rod Vance Oct 5 '13 at 2:26
Electromagnetic waves are diffracted, so a plane wave can only exist at a single location along the axis of propagation (in a uniform homogeneous medium) . In a semiconductor laser, the end mirrors might be planar crystal faces; but they aren't always; for example they aren't in VCSELs; where Bragg mirrors are often used.
Smaller source diameters lead to larger diffraction angles, which depend on the ratio of source diameter and wavelength, so semiconductor lasers can have very large beam angles.
A cavity resonator with parallel end mirrors is unstable, so is a poor choice for a laser. In practice, there is a physical "gain medium" that the waves are propagating in in the resonator, and inhomogeneities in that medium will render the effective cavity not parallel; particularly in semiconductor lasers, where impurity doping, will render the refractive index non uniform.
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Interesting +1: the lasers I have worked with are wont to be small, high quality crystals outputting low power and therefore owing to the small size and quality, they tend to be diffraction limited. So I've not had experience with nonparallel situations - could you briefly say something about how the spherical end mirror achieves its stability - or a reference? – WetSavannaAnimal aka Rod Vance Oct 8 '13 at 8:50
PS: I asked the same question of Ruslan too. – WetSavannaAnimal aka Rod Vance Oct 8 '13 at 8:58
Rod, There is a CRC handbook on laser physics, that has a very good section on Resonators regarding stability. I have that book, but will have to dig for it, so it may take some time; but I WON'T forget that you asked. A popular stable resonator for small lasers (HeNe) is the "Confocal " resonator consisting of a plane output mirror, which is the beam waist, and a spherical back mirror, with its center of curvature on the plane mirror axis. There always IS a normal to the plane, that is a radius of the sphere, so it is robust re alignment. Mr comment is terse, so I'll get back to you. – user26165 Oct 10 '13 at 0:16
Awesome, George, I've been meaning to look this up for a long time. Actually your plane and sphere explanation is very clear to me - I have in the past spent quite a bit of time building interferometers just because I became a bit obsessed - maybe neurotically so - with the idea of "seeing complex quantities" through interferometry and a sphere's alignment properties - something for which rotations can be effected by equivalent translations and contrawise - is something you quickly get a grasp of trying to align things. I never knew the plane was the beam waist -that was the key idea I lacked. – WetSavannaAnimal aka Rod Vance Oct 10 '13 at 0:39
Rod, If you think about it, for the TEM00 mode, in order to have a stable wave propagating back and forth inside the cavity, the resonator end mirrors MUST be the shape of the local wave front. Ergo, the wave front must be planar at the plane mirror, and that of course only occurs at the beam waist, in the middle of the Raleigh Range. In general, you can make a cavity with two spherical mirrors, convex or concave, relative to the spacing. Logically, you can replace the plane mirror, with the mirror image; so two concaves. at twice the spacing. Only some forms are stable. Ejection coming. – user26165 Oct 10 '13 at 22:40
Talking about photons doesn't mean giving up the concept of a spatial mode. If you look at a laser beam, which is diverging, and attenuate it to the level of single photons it still has the same spatial properties. Attenuation doesn't change the way light (or photons) are propagating. The assumption that all photons propagate in the same direction is wrong.
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To add to Ruslan's answer:
1. Whether you speak of photons or classical fields, the explanation is precisely the same. Maxwell's equations are the exact, single quantized description of photon propagation; I bang on about this topic ad nauseam here (How can we interpret polarization and frequency when we are dealing with one single photon?) and here (Electromagnetic radiation and quanta), so if you want more info, please see these answers;
2. So now we get to the mechanism that sets a lower bound to a beam's divergence, to wit diffraction, and the minimum divergence is described by exactly the same mathematics (more on this further on) as the Heisenberg Uncertainty Principle, but I believe it is misleading to think of these two phenomenons as the same thing, even though their mathematics is the same.
So let's set our minds to diffraction: first a quick summary of what I mean by this word. Consider a field on a plane, say $z = 0$ and split it up using Fourier decomposition of the field variation over the plane $z=0$ into constituent plane waves, which are "modes" of Maxwell's equations insofar that their propagation description is simply that the fields become phase delayed by a simple scale factor $\exp(i\,\mathbf{k}\cdot\mathbf{\Delta r})$ under the action of a translation $\mathbf{\Delta r}$. Each constituent plane wave has a different direction defined by the wavevector $\left(k_x, k_y, k_z\right)$ with $k^2 = k_x^2 + k_y^2 + k_z^2$ (i.e. the Fourier space equivalent of Helmholtz's equation), that is, all the wavevectors have the same magnitude but different directions. So, when we ask what the field looks like at a different value of $z$, we build the field up from our plane wave constituents at this point (use an inverse Fourier transform). However, now, because the wavevectors are all in different directions, the plane waves have all undergone different phase delays in reaching the new value of $z$ (even though their phase advances by $k$ radians per unit length in the direction of the respective wave vector). Therefore, the field's configuration gets scrambled by all these different phase delays. I sketch this idea in a drawing below:
Plane waves with the same phase speed but in different directions undergo different phase delays in running from $z=0$ to $z=L$
Now to study diffraction in some detail. Think of a one-dimensional problem, so we have a uniformly lit slit of some finite width $w$ modeling the laser output; in this simplified system that there are only 2D wave vectors. The screen with the slit is in the $z = 0$ plane and the one orthogonal direction is the $x$ axis. All the Cartesian components of the fields fulfil the same (Helmholtz) equation, so we can discuss the principles by just looking at one scalar field $\psi$ (say, the electric field's $x$-component). Each plane wave has the form $\psi(k_x) = \exp\left(i \,(k_x\, x + k_z\, z)\right)$ The Fourier transform of the field output from the slit is then (I'll leave out factors of $2\pi$ in the unitary FT because scale factors don't affect the following):
$$\frac{\sin\left(\frac{w\, k_x}{2}\right)}{k_x} \quad\quad\quad(1)$$
where $w$ is the slit width, and unless the slit is very wide, the Fourier transform has a wide spread of frequencies. This means that for $z = 0^+$ ("immediately downstream" of the the slit's output) the field is the superposition
$$\int\limits_{-\infty}^\infty \frac{\sin\left(\frac{w\, k_x}{2}\right)}{k_x} \exp\left(i\, (k_x\, x + k_z\, z)\right) \mathrm{d} k_x\quad\quad\quad(2)$$
When we plug $z = 0$ in, the integral is simply the inverse FT of (1) and we get our original slit field. But now put some nonzero value of $z$ in: because $k_x^2 + k_z^2 = k^2$, we have $k_z = \sqrt{k^2 - k_x^2}$ (assuming the field is running in the $+z$ direction), we get
$$\int\limits_{-\infty}^\infty \frac{\sin\left(\frac{w\, k_x}{2}\right)}{k_x} \exp\left(i\, (k_x\, x + \sqrt{k^2 - k_x^2}\, z)\right)\, \mathrm{d} k_x\quad\quad\quad(3)$$
You can see the "scrambling", $k_x$-dependent phase factor $\exp(i\, \sqrt{k^2 - k_x^2}\, z) = \exp\left(i\, k\, \cos\theta_x\,\right)$ (where $\theta_x$ is the angle that the plane wave with wavevector $(k_x, k_z)$ makes with the $z$-axis) will yield the complicated scrambling you see as "diffraction". Various approximations, notably Fraunhofer and Fresnel, are applied to this integral. The angle a Fourier component with $x$ component of wavenumber $k_x$ makes with the $z$-axis is $\theta = \arcsin (k_z/k)\approx k_s/k$. So we see that the Fourier transform of the transverse field dependence defines the divergence. In the above, we see a reciprocal relationship between a rough measure $2\pi/w$ of the maximum skew angle of the constituent plane waves and the "confinement" $w$ of the light field to the slit. The beam divergence and the beamwidth are indeed related by a Heisenberg-like inequality, and if we measure beam divergence and confinement by RMS values, we can indeed show the following from the basic properties of Fourier transforms. If $f(x)\in \mathbf{L}^2(\mathbb{R})$ and $F(k_x)$ is its Fourier transform, then the product of the root mean square spreads of both functions is bounded as follows. Without loss of generality, assume that $f(x)$ is real and $\int_{-\infty}^\infty x\,f(x)\,\mathrm{d}\,x = \int_{-\infty}^\infty k_x\,F(k_x)\,\mathrm{d}\,k_x= 0$, then:
$$\sqrt{\frac{\int_{-\infty}^\infty x^2\,|f(x)|^2\,\mathrm{d}\,x}{\int_{-\infty}^\infty |f(x)|^2\,\mathrm{d}\,x}}\;\sqrt{\frac{\int_{-\infty}^\infty k_x^2\,|F(k_x)|^2\,\mathrm{d}\,k_x}{\int_{-\infty}^\infty |F(k_x)|^2\,\mathrm{d}\,k_x}} \leq \frac{1}{2}\quad\quad\quad\quad(4)$$
and moreover the inequality is saturated by Gaussian $f(x)$ $f(x) \propto \exp\left(-\frac{x^2}{2\,\sigma^2}\right)\,e^{-i\,k_0\,x}$ for some real constants $\sigma$ and $k_0>0$, i.e. such functions (their Fourier transforms are also Gaussian) achieve equality in the above bound.
So we have, since $\theta \approx k_x / k$:
$$\Delta k_x \Delta x = \geq \frac{1}{2} \;\;\Rightarrow \;\;\frac{2\pi}{\lambda}\, \Delta\theta \,w \approx \frac{1}{2}\quad\quad\quad\quad(5)$$
Plugging in a $w = 1\mathrm{mm}$ beamwidth for $\lambda =500\mathrm{nm}$ wavelength light, we get a beam divergence of $\Delta \theta \approx 10^{-5} \mathrm{radian}$. This is the typical beam divergence for a high quality 1mm laser chip. There is some arbitrariness in what measures we use for beam divergence (since Gaussian beams have theoretically infinite breadth): often it is the vertex angle of the cone containing $1 - e^{-2}$ of the beam's power. But I have equally seen the Gaussian RMS $\sigma$ or twice this value (one can speak of cone vertex angles or halfangles) used as the beamwidth; these are the $1 - e^{-2}$ beamwidth divided by $2\sqrt{2}$ and $\sqrt{2}$, respectively. You have to be a little bit careful how the beam divergence is defined.
Applying the Heisenberg Uncertainty Principle to Light
Let's finish with the Heisenberg uncertainty principle. The second part of my answer Time duration for pulse of single electron viewed as a wave shows how we can derive the following from the canonical commutation relationship $X\,P - P\,X = i\,\hbar\,I$ between conjugate quantum observables alone:
We can always find co-ordinates for our quantum state Hilbert space such that $X$ is a simple multiplication operator and $P$ is the simple derivative operator $-i\hbar \mathrm{d}_x$
and as such position and momentum co-ordinates are mapped into one-another by the Fourier transform (because the eigenfunctions of $\mathrm{d}_x$ are of the form $e^{i\,k_x\,x}$). Therefore, exactly the same techniques and ideas apply as above, which is why the Heisenberg uncertainty principle seems so like the ideas in my answer. But it is most assuredly not the same thing. The HUP can't be applied to light for position-momentum because there are problems defining a position observable for the photon. This has to do with the fact that if $(\vec{E}, \vec{B})$ is a solution of Maxwell's equations, then things like $(x_j \vec{E}, x_j \vec{B})$ (where $x_j$ are the Cartesian co-ordinates) generally aren't (the Gauss laws showing divergencelessness in freespace are violated). Of course the HUP always applies to noncommuting (conjugate) observables and there are many pairs of those in QED. Contrast this with the scalar quantum electron state in the scalar massive particle nonrelativistic Schrödinger equation where the scalar eigenstates as $\mathbf{L}^2$ complete, so that if $\psi(x)$ is a quantum state in position co-ordinates, then $x \psi(x)$ is also in the Hilbert space of states. One can of course define an intensity field which yields a probability distribution to (destructively) photodetect a photon, but this is different from asking where (position observable) an electron electron is in an orbital. Electrons can be detected nondestructively - it is very hard to do this for photons. Also, position observables are readily defined only for scalar quantum states in nonrelativistic first quantized descriptions: there is of course no nonrelativitistic first quantised description of the photon. The bispinor valued electron state is also weird and the question of where the electron is cannot be addressed by a simple position observable either. Now you can still define the momentum with the usual observable, because the eigenfunctions of $-i\,\hbar\,\partial_j$ are plane waves, i.e. well defined momentum states. But when you talk of localization of photons - probability distributions of where to detect them - you are talking about diffraction. This has exactly the same mathematics as the HUP, as I have shown in my answer above. Having said this, Margaret Hawton is one of a few researchers who have taken a step back and looked at ways wherein we can meaningfully talk about photon positions, i.e. what we can salvage from the wreckage of the above problems: she derives a "position" observable with commuting components essentially by concocting something which has canonical commutation relationships with the momentum observable by definition and goes on to build a second quantized theory with these ideas. One finds that one gets what would wontedly be defined as a position observable PLUS some interesting and weird terms related to the photon's topological (Berry) phase. In other words, she explicitly shows how the wonted "no-go" theorems that forbid a photon position observable manifest themselves as extremely interesting terms that have to be added to the "wonted" and defective position observable. See her personal website for her papers.
Engineering Endnote
As well as diffraction, there are also very definite engineering reasons why small divergence is deliberately introduced into beams, so that the cavity is easier to realize with stable modes, as noted in George E. Smith's answer. As a result, some lasers have divergences rather above my figures above (they are far from saturating the Heisenber-like inequality), but, by the same token, there are many lasers around that come very near to saturating this inequality. Needless to say, these latter are not the "entry level" ones used in laser pointers.
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the parallel mirrors cannot be perfectly parallel. they only need to be aligned enough so that photons can bounce between them long enough for lasing to occur. in practice this is not easy, but using intuitive geometry, a shorter and wider (radius) optical cavity allows more tolerance for misaligned mirrors (photons can reflect off axis for multiple passes without missing either mirror) with the draw back of producing a larger beam waist laser.
in contrast a narrow and long cavity requires stricter alignment as a small deviation angle in photon travel within the cavity will quickly cause it to escape the medium after a few passes.
the use of concave mirrors aids the situation greatly. but as long as there is a nonzero beam radius, there will be divergence. for flat mirrors the perception of perfect collimation in the cavity is an illusion owing to the fact that the cavity is simply too short to observe any divergence.
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Well in fact one can think of semiconductor laser's mirrors as perfectly parallel - to the same accuracy as they are flat - because their orientation is limited to crystal planes. Yet divergence is much higher in such lasers. So non-parallelity of mirrors isn't real reason for divergence. – Ruslan Oct 3 '13 at 9:17
Ruslan is partly right, but it IS a reason in SOME cases which are not wonted to me (so +1); could you please say something about how the spherical end mirror achieves its stability - or a reference? I asked George E Smith the same thing. – WetSavannaAnimal aka Rod Vance Oct 8 '13 at 8:52
I'm posting this as a second answer, since comments are limited in length. For a laser cavity (cylindrical geometry) of mirror separation (d), the end mirrors are generally spherical having radii (R1) and (R2). We have to be sign conscious here. Concave mirrors have positive radii (for these purposes) while convex mirrors have negative radii. This is NOT the normal protocol in ordinary ray optics.
We can define two variables: g1 = 1-d/R1 , and g2 = 1-d/R2 . With two concave mirrors, d, R1, and R2 are ALL positive.
It can be shown that the resonator is stable if, and only if 0 < g1.g2 <1
Hence both g1 and g2 must be of the same sign, either positive or negative.
A stability diagram of g2 plotted against g1 (y & x) shows that all stable resonators are in either the first or third quadrants; with the confocal resonator, R1 = R2 = d at the origin (g1 = g2 = 0).
The plano plano Fabry Perot resonator, is the point 1,1 on the diagram, and the concentric resonator, R1 = R2 = d/2 is the point -1,-1.
All second and fourth quadrant resonators are unstable, and g1.g2 = 1 plots as first and third quadrant rectangular hyperbolas, beyond which other unstable resonators can be found.
The origin confocal resonator is considered the most efficient in most situations, as having the least losses, and the smallest mirror diameters. The beam waist is in the center of the cavity, and the end mirrors are identical geometrically, but they normally would have different reflectance coatings, to let some energy out.
The half confocal cavity has g1 = 1 and g2 = 1/2 typically, giving the plane output mirror.
An extensive expose, appeared in "Applied Optics", 5, 1550, October 1966, and simultaneously in Proc IEEE, 54, 1312 , October 1966, and has been widely cited since.
Some cautions. In lasers, the cavity is always filled (not necessarily completely) with some "Gain medium", solid, liquid or gas, so one must consider the actual refractive index of the gain medium, in doing Maxwell wave equation calculations, and use the right in cavity wavelength, which will surely change, when the beam exits the laser.
Sometimes the active laser medium, will have Brewster angle end mirrors, which render the laser plane polarized, and then the actual laser resonator mirrors are external, so operate in "air".
The mathematics of Gaussian beam laser modes, is very interesting stuff, and quite fun to work with (was for me anyway).
Very high power lasers will generally stay away from the region containing the beam waist, to keep the EM fields down on the end mirrors to prevent damage.
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Fractional calculus
and of the integration operator J[Note 1]
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator D to a function f, that is, repeatedly composing D with itself, as in.
For example, one may ask for a meaningful interpretation of:
as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that when applied twice to any function will have the same effect as differentiation. More generally, one can look at the question of defining a linear functional
for every real-number a in such a way that, when a takes an integer value n ∈ ℤ, it coincides with the usual n-fold differentiation D if n > 0, and with the −nth power of J when n < 0.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator D is that the sets of operator powers { Da |a ∈ ℝ } defined in this way are continuous semigroups with parameter a, of which the original discrete semigroup of { Dn | n ∈ ℤ } for integer n is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations,[1] are a generalization of differential equations through the application of fractional calculus.
Historical notesEdit
In applied mathematics and mathematical analysis, fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in 1695.[2] As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832.[3] The autodidact Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890.[4]
Nature of the fractional derivativeEdit
The ath derivative of a function f (x) at a point x is a local property only when a is an integer; this is not the case for non-integer power derivatives. In other words, it is not correct to say that the fractional derivative at x of a function f (x) depends only on values of f very near x, in the way that integer-power derivatives certainly do. Therefore, it is expected that the theory involves some sort of boundary conditions, involving information on the function further out.[5]
Repeating this process gives
and this can be extended arbitrarily.
The Cauchy formula for repeated integration, namely
leads in a straightforward way to a generalization for real n.
This is in fact a well-defined operator.
It is straightforward to show that the J operator satisfies
Fractional derivative of a basic power functionEdit
The animation shows the derivative operator oscillating between the antiderivative (α = −1: y = 1/2x2) and the derivative (α = +1: y = 1) of the simple power function y = x continuously.
The first derivative is as usual
Repeating this gives the more general result that
For k = 1 and a = 1/2, we obtain the half-derivative of the function x as
To demonstrate that this is, in fact, the "half derivative" (where H2f (x) = Df (x)), we repeat the process to get:
(because Γ(3/2) = 1/2π and Γ(1) = 1) which is indeed the expected result of
This extension of the above differential operator need not be constrained only to real powers. For example, the (1 + i)th derivative of the (1 − i)th derivative yields the 2nd derivative. Also setting negative values for a yields integrals.
Laplace transformEdit
We can also come at the question via the Laplace transform. Knowing that
and so on, we assert
For example,
as expected. Indeed, given the convolution rule
which is what Cauchy gave us above.
Fractional integralsEdit
Riemann–Liouville fractional integralEdit
The classical form of fractional calculus is given by the Riemann–Liouville integral, which is essentially what has been described above. The theory for periodic functions (therefore including the 'boundary condition' of repeating after a period) is the Weyl integral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to 0). The Riemann-Liouville integral exists in two forms, upper and lower. Considering the interval [a,b], the integrals are defined as
Where the former is valid for t > a and the latter is valid for t < b.[8]
Hadamard fractional integralEdit
The Hadamard fractional integral is introduced by Jacques Hadamard[9] and is given by the following formula,
Atangana–Baleanu fractional integralEdit
Recently, using the generalized Mittag-Leffler function, Atangana and Baleanu suggested a new formulation of the fractional derivative with a nonlocal and nonsingular kernel. The integral is defined as:
where AB(α) is a normalization function such that AB(0) = AB(1) = 1.[10]
Fractional derivativesEdit
Unlike classical Newtonian derivatives, a fractional derivative is defined via a fractional integral.
Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative.
Riemann–Liouville fractional derivativeEdit
The corresponding derivative is calculated using Lagrange's rule for differential operators. Computing nth order derivative over the integral of order (nα), the α order derivative is obtained. It is important to remark that n is the smallest integer greater than α ( that is, n = ⌈α). Similar to the definitions for the Riemann-Liouville integral, the derivative has upper and lower variants.[11]
Caputo fractional derivativeEdit
Another option for computing fractional derivatives is the Caputo fractional derivative. It was introduced by Michele Caputo in his 1967 paper.[12] In contrast to the Riemann-Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows.
There is the Caputo fractional derivative defined as:
which has the advantage that is zero when f (t) is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as
where φ(ν) is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.
Atangana–Baleanu derivativeEdit
Like the integral, there is also a fractional derivative using the general Mittag-Leffler function as a kernel.[10] The authors introduced two versions, the Atangana–Baleanu in Caputo sense (ABC) derivative, which is the convolution of a local derivative of a given function with the generalized Mittag-Leffler function, and the Atangana–Baleanu in Riemann–Liouville sense (ABR) derivative, which is the derivative of a convolution of a given function that is not differentiable with the generalized Mittag-Leffler function.[13] The Atangana-Baleanu fractional derivative in Caputo sense is defined as:
Riesz derivativeEdit
where F denotes the Fourier transform.[14][15]
Other typesEdit
Classical fractional derivatives include:
• Grünwald–Letnikov derivative
• Sonin–Letnikov derivative
• Liouville derivative
• Caputo derivative
• Hadamard derivative
• Marchaud derivative
• Riesz derivative
• Riesz–Miller derivative
• Miller–Ross derivative
• Weyl derivative
• Erdélyi–Kober derivative
New fractional derivatives include:
• Machado derivative[citation needed] (This derivative does not exist anywhere in the literature)
• Chen–Machado derivative
• Coimbra derivative
• Katugampola derivative
• Caputo–Katugampola derivative
• Hilfer derivative
• Hilfer–Katugampola derivative
• Davidson derivative
• Chen derivative
• Caputo Fabrizio derivative
• Atangana–Baleanu derivative
• Pichaghchi derivative
Erdélyi–Kober operatorEdit
The Erdélyi–Kober operator is an integral operator introduced by Arthur Erdélyi (1940).[16] and Hermann Kober (1940)[17] and is given by
Katugampola operatorsEdit
A recent generalization introduced by Udita Katugampola is the following, which generalizes the Riemann–Liouville fractional integral and the Hadamard fractional integral. The integral is now known as the Katugampola fractional integral and is given by,[2][18]
Functional calculusEdit
Fractional conservation of massEdit
As described by Wheatcraft and Meerschaert (2008),[19] a fractional conservation of mass equation is needed to model fluid flow when the control volume is not large enough compared to the scale of heterogeneity and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:
Groundwater flow problemEdit
In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of derivative with fractional order.[20][21] In these works, The classical Darcy law is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.
Fractional advection dispersion equationEdit
This equation[clarification needed] has been shown useful for modeling contaminant flow in heterogenous porous media.[22][23][24]
Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a variational order derivative. The modified equation was numerically solved via the Crank–Nicolson method. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives[25]
Time-space fractional diffusion equation modelsEdit
Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.[26][27] The time derivative term is corresponding to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as
A simple extension of fractional derivative is the variable-order fractional derivative, α and β are changed into α(x, t) and β(x, t). Its applications in anomalous diffusion modeling can be found in reference.[25][28][29]
Structural damping modelsEdit
Fractional derivatives are used to model viscoelastic damping in certain types of materials like polymers.[30]
PID controllersEdit
Generalizing PID controllers to use fractional orders can increase their degree of freedom. The new equation relating the control variable u(t) in terms of a measured error value e(t) can be written as
where α and β are positive fractional orders and Kp, Ki, and Kd, all non-negative, denote the coefficients for the proportional, integral, and derivative terms, respectively (sometimes denoted P, I, and D).[31]
Acoustical wave equations for complex mediaEdit
The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:
See also Holm & Näsholm (2011)[32] and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in Näsholm & Holm (2011b)[33] and in the survey paper,[34] as well as the acoustic attenuation article. See Holm & Nasholm (2013)[35] for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.[36]
Fractional Schrödinger equation in quantum theoryEdit
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form:[37][38]
Further, Δ = 2/r2 is the Laplace operator, and Dα is a scale constant with physical dimension [Dα] = J1 − α·mα·sα = kg1 − α·m2 − α·sα − 2, (at α = 2, D2 = 1/2m for a particle of mass m), and the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by
Variable-order fractional Schrödinger equationEdit
As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:[39]
See alsoEdit
1. ^ The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, such as identities.
1. ^ Daniel Zwillinger (12 May 2014). Handbook of Differential Equations. Elsevier Science. ISBN 978-1-4832-2096-3.
2. ^ a b c d Katugampola, Udita N. (15 October 2014). "A New Approach To Generalized Fractional Derivatives" (PDF). Bulletin of Mathematical Analysis and Applications. 6 (4): 1–15. arXiv:1106.0965. Bibcode:2011arXiv1106.0965K.
4. ^ For a historical review of the subject up to the beginning of the 20th century, see: Bertram Ross (1977). "The development of fractional calculus 1695-1900". Historia Mathematica. 4: 75–89. doi:10.1016/0315-0860(77)90039-8.
5. ^ "Fractional Calculus". Retrieved 2018-01-03.
6. ^ Kilbas, Srivastava & Trujillo 2006, p. 75 (Property 2.4)
8. ^ Hermann, Richard (2014). Fractional Calculus: An Introduction for Physicists (2nd ed.). New Jersey: World Scientific Publishing. p. 46. doi:10.1142/8934. ISBN 978-981-4551-07-6.
9. ^ Hadamard, J. (1892). "Essai sur l'étude des fonctions données par leur développement de Taylor" (PDF). Journal de Mathématiques Pures et Appliquées. 4 (8): 101–186.
10. ^ a b Atangana, Abdon; Baleanu, Dumitru (2016). "New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model". arXiv:1602.03408 [math.GM].
11. ^ Hermann, Richard, ed. (2014). Fractional Calculus. Fractional Calculus: An Introduction for Physicists (2nd ed.). New Jersey: World Scientific Publishing Co. p. 54. doi:10.1142/8934. ISBN 978-981-4551-07-6.
12. ^ Caputo, Michele (1967). "Linear model of dissipation whose Q is almost frequency independent. II". Geophysical Journal International. 13 (5): 529–539. Bibcode:1967GeoJ...13..529C. doi:10.1111/j.1365-246x.1967.tb02303.x..
13. ^ Atangana, Abdon; Koca, Ilknur (2016). "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order". Chaos, Solitons & Fractals. 89: 447–454. Bibcode:2016CSF....89..447A. doi:10.1016/j.chaos.2016.02.012.
14. ^ Chen, YangQuan; Li, Changpin; Ding, Hengfei (22 May 2014). "High-Order Algorithms for Riesz Derivative and Their Applications". Abstract and Applied Analysis. 2014: 1–17. doi:10.1155/2014/653797.
15. ^ Bayın, Selçuk Ş. (5 December 2016). "Definition of the Riesz derivative and its application to space fractional quantum mechanics". Journal of Mathematical Physics. 57 (12): 123501. arXiv:1612.03046. Bibcode:2016JMP....57l3501B. doi:10.1063/1.4968819.
16. ^ Erdélyi, Arthur (1950–51). "On some functional transformations". Rendiconti del Seminario Matematico Dell'Università e del Politecnico di Torino. 10: 217–234. MR 0047818.
17. ^ Kober, Hermann (1940). "On fractional integrals and derivatives". The Quarterly Journal of Mathematics. os-11 (1): 193–211. Bibcode:1940QJMat..11..193K. doi:10.1093/qmath/os-11.1.193.
18. ^ Katugampola, Udita N. (2011). "New approach to a generalized fractional integral". Applied Mathematics and Computation. 218 (3): 860–865. arXiv:1010.0742. CiteSeerX doi:10.1016/j.amc.2011.03.062.
19. ^ Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008). "Fractional conservation of mass" (PDF). Advances in Water Resources. 31 (10): 1377–1381. Bibcode:2008AdWR...31.1377W. doi:10.1016/j.advwatres.2008.07.004. ISSN 0309-1708.
22. ^ Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "Application of a fractional advection-dispersion equation". Water Resources Res. 36 (6): 1403–1412. Bibcode:2000WRR....36.1403B. CiteSeerX doi:10.1029/2000wr900031.
23. ^ Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "The fractional-order governing equation of Lévy motion". Water Resources Res. 36 (6): 1413–1423. Bibcode:2000WRR....36.1413B. doi:10.1029/2000wr900032.
24. ^ Wheatcraft, Stephen W.; Meerschaert, Mark M.; Schumer, Rina; Benson, David A. (2001-01-01). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests". Transport in Porous Media. 42 (1–2): 211–240. CiteSeerX doi:10.1023/A:1006733002131. ISSN 1573-1634.
26. ^ Metzler, R.; Klafter, J. (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Phys. Rep. 339 (1): 1–77. Bibcode:2000PhR...339....1M. doi:10.1016/s0370-1573(00)00070-3.
27. ^ Mainardi, F.; Luchko, Y.; Pagnini, G. (2001). "The fundamental solution of the space-time fractional diffusion equation". Fractional Calculus and Applied Analysis. 4 (2): 153–192. arXiv:cond-mat/0702419. Bibcode:2007cond.mat..2419M.
28. ^ Atangana, Abdon; Baleanu, Dumitru (2007). "Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk". In Rangarajan, G.; Ding, M. (eds.). Processes with Long-Range Correlations. Processes with Long-Range Correlations. Lecture Notes in Physics. 621. p. 148. arXiv:0709.3990. Bibcode:2003LNP...621..148G.
29. ^ Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017). "Scaling laws of passive-scalar diffusion in the interstellar medium". Monthly Notices of the Royal Astronomical Society. 467 (2): 2421–2429. arXiv:1610.06590. Bibcode:2017MNRAS.467.2421C. doi:10.1093/mnras/stx261.
30. ^ Mainardi, Francesco (May 2010). Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press. doi:10.1142/p614. ISBN 9781848163294.
31. ^ Tenreiro Machado, J. A.; Silva, Manuel F.; Barbosa, Ramiro S.; Jesus, Isabel S.; Reis, Cecília M.; Marcos, Maria G.; Galhano, Alexandra F. (2010). "Some Applications of Fractional Calculus in Engineering". Mathematical Problems in Engineering. 2010: 1–34. doi:10.1155/2010/639801.
32. ^ Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media". Journal of the Acoustical Society of America. 130 (4): 2195–2201. Bibcode:2011ASAJ..130.2195H. doi:10.1121/1.3631626. PMID 21973374.
33. ^ Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations". Journal of the Acoustical Society of America. 130 (5): 3038–3045. Bibcode:2011ASAJ..130.3038N. doi:10.1121/1.3641457. PMID 22087931.
34. ^ Näsholm, S. P.; Holm, S. (2012). "On a Fractional Zener Elastic Wave Equation". Fract. Calc. Appl. Anal. arXiv:1212.4024. Bibcode:2012arXiv1212.4024N. doi:10.2478/s13540-013-0003-1.
35. ^ Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography". Ultrasound in Medicine & Biology. 40 (4): 695–703. arXiv:1306.6507. Bibcode:2013arXiv1306.6507H. CiteSeerX doi:10.1016/j.ultrasmedbio.2013.09.033. PMID 24433745.
36. ^ Holm, S. (2019). Waves with Power-Law Attenuation. Springer and Acoustical Society of America Press.
37. ^ Laskin, N. (2002). "Fractional Schrodinger equation". Phys. Rev. E. 66 (5): 056108. arXiv:quant-ph/0206098. Bibcode:2002PhRvE..66e6108L. CiteSeerX doi:10.1103/PhysRevE.66.056108. PMID 12513557.
38. ^ Laskin, Nick (2018). Fractional Quantum Mechanics. CiteSeerX doi:10.1142/10541. ISBN 978-981-322-379-0.
39. ^ Bhrawy, A.H.; Zaky, M.A. (2017). "An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations". Applied Numerical Mathematics. 111: 197–218. doi:10.1016/j.apnum.2016.09.009.
• Kilbas, Anatolii Aleksandrovich; Srivastava, Hari Mohan; Trujillo, Juan J. (2006). Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier. ISBN 978-0-444-51832-3.
Further readingEdit
Articles regarding the history of fractional calculusEdit
• Ross, B. (1975). A brief history and exposition of the fundamental theory of fractional calculus. Fractional Calculus and Its Applications. Lecture Notes in Mathematics. Lecture Notes in Mathematics. 457. pp. 1–36. doi:10.1007/BFb0067096. ISBN 978-3-540-07161-7.
• Tenreiro Machado, J.; Kiryakova, V.; Mainardi, F. (2011). "Recent history of fractional calculus". Communications in Nonlinear Science and Numerical Simulation. 16 (3): 1140–1153. Bibcode:2011CNSNS..16.1140M. doi:10.1016/j.cnsns.2010.05.027. hdl:10400.22/4149.
• Tenreiro Machado, J.A.; Galhano, A.M.; Trujillo, J.J. (2013). "Science metrics on fractional calculus development since 1966". Fractional Calculus and Applied Analysis. 16 (2): 479–500. doi:10.2478/s13540-013-0030-y.
External linksEdit |
448ad58bea5c924f | The high-frequency Floquet theory describing the interaction of a two-electron atom with a linearly polarized laser field is applied to the case when the characteristic parameter a0=E0w-2 a.u. is large, corresponding to the dichotomy regime of the one-electron problem. We first revisit this case and extend the large- a0 energy-level formula obtained earlier to higher order in a0-1. We then prove the existence of a dichotomy regime also for the two-electron atom, characterized by the two electrons being situated in disjoint electronic clouds separated by an average distance of 2a0. We obtain the first four terms in the expansion of the related energy-level formula in fractional powers of a0-1. The coefficients entering this expansion have been expressed in terms of the eigenvalues of a nonseparable Schrödinger equation containing the end-point potential and of integrals over its eigenfunctions. The equation was solved using the finite element method. An infinite sequence of levels emerges. In the case of H- this implies the existence of a large number of light-induced excited states, some of them corresponding to two-electron excitations not subject to autodetachment. Finally, we prove that in the dichotomy regime a two-electron atom undergoes stabilization and that the ionization rates are essentially twice those for a one-electron atom with the same nuclear charge. |
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Eigenvalues and eigenvectors
(Redirected from Eigenvalues)
In linear algebra, an eigenvector (/ˈɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it.
There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. For this reason, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations.[1][2]
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.[3] Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. However, in a one-dimensional vector space, the concept of rotation is meaningless.
Formal definitionEdit
If T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation
where λ is a scalar in the field F, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector v.
If the vector space V is finite-dimensional, then the linear transformation T can be represented as a square matrix A, and the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left-hand side and a scaling of the column vector on the right-hand side in the equation
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen for "proper", "characteristic".[4] Originally utilized to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.
In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.
In this shear mapping the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is 1.
The Mona Lisa example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right and points in the bottom half are moved to the left proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one because the mapping does not change their length, either.
Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like , in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as
Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices that are also referred to as eigenvectors. If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation above for a linear transformation can be rewritten as the matrix multiplication
where the eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix, for example by diagonalizing it.
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them:
• The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.[5][6]
• The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T associated with that eigenvalue.[7]
• If a set of eigenvectors of T forms a basis of the domain of T, then this basis is called an eigenbasis.
In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes.[8] Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.[9] In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.[10] Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.[11]
Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur.[12] Sturm developed Fourier's ideas further and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues.[13] This was extended by Hermite in 1855 to what are now called Hermitian matrices.[14] Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,[13] and Clebsch found the corresponding result for skew-symmetric matrices.[14] Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.[13]
In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.[15] Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.[16]
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.[17] He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904,[18] though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.[19]
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis[20] and Vera Kublanovskaya[21] in 1961.[22][23]
Eigenvalues and eigenvectors of matricesEdit
Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.[24][25] Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices,[26][2] which is especially common in numerical and computational applications.[27]
Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.
Consider n-dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors
These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar λ such that
In this case λ = −1/20.
Now consider the linear transformation of n-dimensional vectors defined by an n by n matrix A,
where, for each row,
If it occurs that v and w are scalar multiples, that is if
then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector. Equation (1) is the eigenvalue equation for the matrix A.
Equation (1) can be stated equivalently as
(2 )
where I is the n by n identity matrix and 0 is the zero vector.
Eigenvalues and the characteristic polynomialEdit
Equation (2) has a nonzero solution v if and only if the determinant of the matrix (AλI) is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation
(3 )
Using Leibniz' rule for the determinant, the left-hand side of Equation (3) is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1)nλn. This polynomial is called the characteristic polynomial of A. Equation (3) is called the characteristic equation or the secular equation of A.
The fundamental theorem of algebra implies that the characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, can be factored into the product of n linear terms,
(4 )
where each λi may be real but in general is a complex number. The numbers λ1, λ2, ... λn, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of A.
As a brief example, which is described in more detail in the examples section later, consider the matrix
Taking the determinant of (AλI), the characteristic polynomial of A is
Setting the characteristic polynomial equal to zero, it has roots at λ = 1 and λ = 3, which are the two eigenvalues of A. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation Av = λv. In this example, the eigenvectors are any nonzero scalar multiples of
If the entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be irrational numbers even if all the entries of A are rational numbers or even if they are all integers. However, if the entries of A are all algebraic numbers, which include the rationals, the eigenvalues are complex algebraic numbers.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
Algebraic multiplicityEdit
Let λi be an eigenvalue of an n by n matrix A. The algebraic multiplicity μA(λi) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λλi)k divides evenly that polynomial.[7][28][29]
Suppose a matrix A has dimension n and dn distinct eigenvalues. Whereas Equation (4) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
If d = n then the right-hand side is the product of n linear terms and this is the same as Equation (4). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as
If μA(λi) = 1, then λi is said to be a simple eigenvalue.[29] If μA(λi) equals the geometric multiplicity of λi, γA(λi), defined in the next section, then λi is said to be a semisimple eigenvalue.
Eigenspaces, geometric multiplicity, and the eigenbasis for matricesEdit
Given a particular eigenvalue λ of the n by n matrix A, define the set E to be all vectors v that satisfy Equation (2),
On one hand, this set is precisely the kernel or nullspace of the matrix (AλI). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (AλI). E is called the eigenspace or characteristic space of A associated with λ.[30][7] In general λ is a complex number and the eigenvectors are complex n by 1 matrices. A property of the nullspace is that it is a linear subspace, so E is a linear subspace of ℂn.
Because the eigenspace E is a linear subspace, it is closed under addition. That is, if two vectors u and v belong to the set E, written (u,v) ∈ E, then (u + v) ∈ E or equivalently A(u + v) = λ(u + v). This can be checked using the distributive property of matrix multiplication. Similarly, because E is a linear subspace, it is closed under scalar multiplication. That is, if vE and α is a complex number, (αv) ∈ E or equivalently Av) = λ(αv). This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as u + v and αv are not zero, they are also eigenvectors of A associated with λ.
The dimension of the eigenspace E associated with λ, or equivalently the maximum number of linearly independent eigenvectors associated with λ, is referred to as the eigenvalue's geometric multiplicity γA(λ). Because E is also the nullspace of (AλI), the geometric multiplicity of λ is the dimension of the nullspace of (AλI), also called the nullity of (AλI), which relates to the dimension and rank of (AλI) as
Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n.
To prove the inequality , consider how the definition of geometric multiplicity implies the existence of orthonormal eigenvectors , such that . We can therefore find a (unitary) matrix whose first columns are these eigenvectors, and whose remaining columns can be any orthonormal set of vectors orthogonal to these eigenvectors of . Then has full rank and is therefore invertible, and with a matrix whose top left block is the diagonal matrix . This implies that . In other words, is similar to , which implies that . But from the definition of we know that contains a factor , which means that the algebraic multiplicity of must satisfy .
Suppose has distinct eigenvalues , where the geometric multiplicity of is . The total geometric multiplicity of ,
is the dimension of the sum of all the eigenspaces of 's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of . If , then
• The direct sum of the eigenspaces of all of 's eigenvalues is the entire vector space .
• A basis of can be formed from linearly independent eigenvectors of ; such a basis is called an eigenbasis
• Any vector in can be written as a linear combination of eigenvectors of .
Additional properties of eigenvaluesEdit
Let be an arbitrary matrix of complex numbers with eigenvalues . Each eigenvalue appears times in this list, where is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
• The trace of , defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
• The determinant of is the product of all its eigenvalues,
• The eigenvalues of the th power of ; i.e., the eigenvalues of , for any positive integer , are .
• The matrix is invertible if and only if every eigenvalue is nonzero.
• If is invertible, then the eigenvalues of are and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity.
• If is equal to its conjugate transpose , or equivalently if is Hermitian, then every eigenvalue is real. The same is true of any symmetric real matrix.
• If is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
• If is unitary, every eigenvalue has absolute value .
• if is a matrix and are its eigenvalues, then the eigenvalues of matrix (where is the identity matrix) are . Moreover, if , the eigenvalues of are . More generally, for a polynomial the eigenvalues of matrix are .
Left and right eigenvectorsEdit
Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a column vector that right multiplies the matrix in the defining equation, Equation (1),
The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix . In this formulation, the defining equation is
where is a scalar and is a matrix. Any row vector satisfying this equation is called a left eigenvector of and is its associated eigenvalue. Taking the transpose of this equation,
Comparing this equation to Equation (1), it follows immediately that a left eigenvector of is the same as the transpose of a right eigenvector of , with the same eigenvalue. Furthermore, since the characteristic polynomial of is the same as the characteristic polynomial of , the eigenvalues of the left eigenvectors of are the same as the eigenvalues of the right eigenvectors of .
Diagonalization and the eigendecompositionEdit
Suppose the eigenvectors of A form a basis, or equivalently A has n linearly independent eigenvectors v1, v2, ..., vn with associated eigenvalues λ1, λ2, ..., λn. The eigenvalues need not be distinct. Define a square matrix Q whose columns are the n linearly independent eigenvectors of A,
Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue,
With this in mind, define a diagonal matrix Λ where each diagonal element Λii is the eigenvalue associated with the ith column of Q. Then
Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q−1,
or by instead left multiplying both sides by Q−1,
A can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix A is said to be similar to the diagonal matrix Λ or diagonalizable. The matrix Q is the change of basis matrix of the similarity transformation. Essentially, the matrices A and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.
Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P−1AP is some diagonal matrix D. Left multiplying both by P, AP = PD. Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix A has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.
Variational characterizationEdit
In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of is the maximum value of the quadratic form . A value of that realizes that maximum, is an eigenvector.
Matrix examplesEdit
Two-dimensional matrix exampleEdit
The transformation matrix A = preserves the direction of purple vectors parallel to vλ=1 = [1 −1]T and blue vectors parallel to vλ=3 = [1 1]T. The red vectors are not parallel to either eigenvector, so, their directions are changed by the transformation. The lengths of the purple vectors are unchanged after the transformation (due to their eigenvalue of 1), while blue vectors are three times the length of the original (due to their eigenvalue of 3). See also: An extended version, showing all four quadrants.
Consider the matrix
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors v of this transformation satisfy Equation (1), and the values of λ for which the determinant of the matrix (A − λI) equals zero are the eigenvalues.
Taking the determinant to find characteristic polynomial of A,
For λ = 1, Equation (2) becomes,
Any nonzero vector with v1 = −v2 solves this equation. Therefore,
is an eigenvector of A corresponding to λ = 1, as is any scalar multiple of this vector.
For λ = 3, Equation (2) becomes
Any nonzero vector with v1 = v2 solves this equation. Therefore,
is an eigenvector of A corresponding to λ = 3, as is any scalar multiple of this vector.
Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues λ = 1 and λ = 3, respectively.
Three-dimensional matrix exampleEdit
Consider the matrix
The characteristic polynomial of A is
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of A. These eigenvalues correspond to the eigenvectors and , or any nonzero multiple thereof.
Three-dimensional matrix example with complex eigenvaluesEdit
Consider the cyclic permutation matrix
This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − λ3, whose roots are
where is an imaginary unit with
For the real eigenvalue λ1 = 1, any vector with three equal nonzero entries is an eigenvector. For example,
For the complex conjugate pair of imaginary eigenvalues,
Therefore, the other two eigenvectors of A are complex and are and with eigenvalues λ2 and λ3, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
Diagonal matrix exampleEdit
Matrices with entries only along the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
The characteristic polynomial of A is
which has the roots λ1 = 1, λ2 = 2, and λ3 = 3. These roots are the diagonal elements as well as the eigenvalues of A.
Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Triangular matrix exampleEdit
A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
Consider the lower triangular matrix,
The characteristic polynomial of A is
These eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Matrix with repeated eigenvalues exampleEdit
As in the previous example, the lower triangular matrix
has a characteristic polynomial that is the product of its diagonal elements,
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of each distinct eigenvalue is μA = 4 = n, the order of the characteristic polynomial and the dimension of A.
On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector . The total geometric multiplicity γA is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
Eigenvalues and eigenfunctions of differential operatorsEdit
The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator on the space C of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation
The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions.
Derivative operator exampleEdit
Consider the derivative operator with eigenvalue equation
This differential equation can be solved by multiplying both sides by dt/f(t) and integrating. Its solution, the exponential function
is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for λ = 0 the eigenfunction f(t) is a constant.
The main eigenfunction article gives other examples.
General definitionEdit
The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let V be any vector space over some field K of scalars, and let T be a linear transformation mapping V into V,
We say that a nonzero vector vV is an eigenvector of T if and only if there exists a scalar λK such that
( 5 )
This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v.[36][37]
Eigenspaces, geometric multiplicity, and the eigenbasisEdit
Given an eigenvalue λ, consider the set
which is the union of the zero vector with the set of all eigenvectors associated with λ. E is called the eigenspace or characteristic space of T associated with λ.
By definition of a linear transformation,
for (x,y) ∈ V and α ∈ K. Therefore, if u and v are eigenvectors of T associated with eigenvalue λ, namely u,vE, then
So, both u + v and αv are either zero or eigenvectors of T associated with λ, namely u + v, αvE, and E is closed under addition and scalar multiplication. The eigenspace E associated with λ is therefore a linear subspace of V.[38] If that subspace has dimension 1, it is sometimes called an eigenline.[39]
The geometric multiplicity γT(λ) of an eigenvalue λ is the dimension of the eigenspace associated with λ, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue.[7][29] By the definition of eigenvalues and eigenvectors, γT(λ) ≥ 1 because every eigenvalue has at least one eigenvector.
The eigenspaces of T always form a direct sum. As a consequence, eigenvectors of different eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the vector space on which T operates, and there cannot be more than n distinct eigenvalues.[40]
Any subspace spanned by eigenvectors of T is an invariant subspace of T, and the restriction of T to such a subspace is diagonalizable. Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is diagonalizable.
Zero vector as an eigenvectorEdit
While the definition of an eigenvector used in this article excludes the zero vector, it is possible to define eigenvalues and eigenvectors such that the zero vector is an eigenvector.[41]
Consider again the eigenvalue equation, Equation (5). Define an eigenvalue to be any scalar λK such that there exists a nonzero vector vV satisfying Equation (5). It is important that this version of the definition of an eigenvalue specify that the vector be nonzero, otherwise by this definition the zero vector would allow any scalar in K to be an eigenvalue. Define an eigenvector v associated with the eigenvalue λ to be any vector that, given λ, satisfies Equation (5). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation (5), so the zero vector is included among the eigenvectors by this alternate definition.
Spectral theoryEdit
If λ is an eigenvalue of T, then the operator (TλI) is not one-to-one, and therefore its inverse (TλI)−1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (TλI) may not have an inverse even if λ is not an eigenvalue.
For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator T as the set of all scalars λ for which the operator (TλI) has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
Associative algebras and representation theoryEdit
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
The representation-theoretical concept of weight is an analog of eigenvalues, while weight vectors and weight spaces are the analogs of eigenvectors and eigenspaces, respectively.
Dynamic equationsEdit
The simplest difference equations have the form
The solution of this equation for x in terms of t is found by using its characteristic equation
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations giving a k-dimensional system of the first order in the stacked variable vector in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots for use in the solution equation
A similar procedure is used for solving a differential equation of the form
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
Classical methodEdit
The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as floating-point.
The eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial. This is easy for matrices, but the difficulty increases rapidly with the size of the matrix.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.[42] However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial).[42] Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the determinant, which for an is a sum of different products.[note 1]
Explicit algebraic formulas for the roots of a polynomial exist only if the degree is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree is the characteristic polynomial of some companion matrix of order .) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods. Even the exact formula for the roots of a degree 3 polynomial is numerically impractical.
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
we can find its eigenvectors by solving the equation , that is
This matrix equation is equivalent to two linear equations
that is
Both equations reduce to the single linear equation . Therefore, any vector of the form , for any nonzero real number , is an eigenvector of with eigenvalue .
The matrix above has another eigenvalue . A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of , that is, any vector of the form , for any nonzero real number .
Simple iterative methodsEdit
The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalising the vector to keep its elements of reasonable size); surprisingly this makes the vector converge towards an eigenvector. A variation is to instead multiply the vector by ; this causes it to converge to an eigenvector of the eigenvalue closest to .
If is (a good approximation of) an eigenvector of , then the corresponding eigenvalue can be computed as
where denotes the conjugate transpose of .
Modern methodsEdit
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961.[42] Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm.[citation needed] For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.[42]
Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.
Eigenvalues of geometric transformationsEdit
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.
Scaling Unequal scaling Rotation Horizontal shear Hyperbolic rotation
Algebraic mult.,
Geometric mult.,
Eigenvectors All nonzero vectors
The characteristic equation for a rotation is a quadratic equation with discriminant , which is a negative number whenever θ is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, ; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.
Schrödinger equationEdit
An example of an eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
where , the Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which and can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
The bra–ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by . In this notation, the Schrödinger equation is:
where is an eigenstate of and represents the eigenvalue. is an observable self adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above is understood to be the vector obtained by application of the transformation to .
Molecular orbitalsEdit
In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a non-orthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations.
Geology and glaciologyEdit
In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram,[43][44] or as a Stereonet on a Wulff Net.[45]
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered by their eigenvalues ;[46] then is the primary orientation/dip of clast, is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of , , and are dictated by the nature of the sediment's fabric. If , the fabric is said to be isotropic. If , the fabric is said to be planar. If , the fabric is said to be linear.[47]
Principal component analysisEdit
PCA of the multivariate Gaussian distribution centered at with a standard deviation of 3 in roughly the direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal components analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthonormal eigen-basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
Principal component analysis is used to study large data sets, such as those encountered in bioinformatics, data mining, chemical research, psychology, and in marketing. PCA is also popular in psychology, especially within the field of psychometrics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of practical significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.
Vibration analysisEdit
Mode shape of a tuning fork at eigenfrequency 440.09 Hz
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
that is, acceleration is proportional to position (i.e., we expect to be sinusoidal in time).
In dimensions, becomes a mass matrix and a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
where is the eigenvalue and is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of alone. Furthermore, damped vibration, governed by
leads to a so-called quadratic eigenvalue problem,
This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.
Eigenfaces as examples of eigenvectors
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.[48] The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal component analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.
Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation.
Tensor of moment of inertiaEdit
In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
Stress tensorEdit
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix , or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or (sometimes called the normalized Laplacian), where is a diagonal matrix with equal to the degree of vertex , and in , the th diagonal entry is . The th principal eigenvector of a graph is defined as either the eigenvector corresponding to the th largest or th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
Basic reproduction numberEdit
The basic reproduction number ( ) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, , from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time has passed. is then the largest eigenvalue of the next generation matrix.[49][50]
See alsoEdit
1. ^ By doing Gaussian elimination over formal power series truncated to terms it is possible to get away with operations, but that does not take combinatorial explosion into account.
1. ^ Herstein 1964, pp. 228, 229.
2. ^ a b Nering 1970, p. 38.
3. ^ Burden & Faires 1993, p. 401.
4. ^ Betteridge 1965.
5. ^ Press 2007, p. 536.
6. ^ Weisstein, Eric W. "Eigenvector". Retrieved 2019-08-04.
7. ^ a b c d Nering 1970, p. 107.
8. ^ Note:
• In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751 ; published: 1760) "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile" (On the movement of any solid body while it rotates around a moving axis), Histoire de l'Académie royale des sciences et des belles lettres de Berlin, pp. 176–227. On p. 212, Euler proves that any body contains a principal axis of rotation: "Théorem. 44. De quelque figure que soit le corps, on y peut toujours assigner un tel axe, qui passe par son centre de gravité, autour duquel le corps peut tourner librement & d'un mouvement uniforme." (Theorem. 44. Whatever be the shape of the body, one can always assign to it such an axis, which passes through its center of gravity, around which it can rotate freely and with a uniform motion.)
• In 1755, Johann Andreas Segner proved that any body has three principal axes of rotation: Johann Andreas Segner, Specimen theoriae turbinum [Essay on the theory of tops (i.e., rotating bodies)] ( Halle ("Halae"), (Germany) : Gebauer, 1755). p. xxviiii [29], Segner derives a third-degree equation in t, which proves that a body has three principal axes of rotation. He then states (on the same page): "Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores." (However, it is not inconsistent [that there] be three such positions of the plane HM, because in cubic equations, [there] can be three roots, and three values of the tangent t.)
• The relevant passage of Segner's work was discussed briefly by Arthur Cayley. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics," Report of the Thirty-second meeting of the British Association for the Advancement of Science; held at Cambridge in October 1862, 32 : 184–252 ; see especially 225–226.
9. ^ Hawkins 1975, §2.
10. ^ Hawkins 1975, §3.
11. ^ Kline 1972, pp. 807–808 Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), Comptes rendus, 8 : 827–830, 845–865, 889–907, 931–937. From p. 827: "On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai l'équation caractéristique, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer." (One knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.)
12. ^ Kline 1972, p. 673.
13. ^ a b c See Hawkins 1975, §3
14. ^ a b See Kline 1972, pp. 807–808
15. ^ Kline 1972, pp. 715–716.
16. ^ Kline 1972, pp. 706–707.
17. ^ Kline 1972, p. 1063.
18. ^ See:
• David Hilbert (1904) "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)" (Fundamentals of a general theory of linear integral equations. (First report)), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49–91. From p. 51: "Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich Eigenfunktionen nenne, liefern: … (In particular, in this first report I arrive at formulas that provide the [series] development of an arbitrary function in terms of some distinctive functions, which I call eigenfunctions: … ) Later on the same page: "Dieser Erfolg ist wesentlich durch den Umstand bedingt, daß ich nicht, wie es bisher geschah, in erster Linie auf den Beweis für die Existenz der Eigenwerte ausgehe, … " (This success is mainly attributable to the fact that I do not, as it has happened until now, first of all aim at a proof of the existence of eigenvalues, … )
• For the origin and evolution of the terms eigenvalue, characteristic value, etc., see: Earliest Known Uses of Some of the Words of Mathematics (E)
19. ^ Aldrich 2006.
20. ^ Francis, J. G. F. (1961), "The QR Transformation, I (part 1)", The Computer Journal, 4 (3): 265–271, doi:10.1093/comjnl/4.3.265 and Francis, J. G. F. (1962), "The QR Transformation, II (part 2)", The Computer Journal, 4 (4): 332–345, doi:10.1093/comjnl/4.4.332
21. ^ Kublanovskaya, Vera N. (1961), "On some algorithms for the solution of the complete eigenvalue problem", USSR Computational Mathematics and Mathematical Physics, 3: 637–657. Also published in: "О некоторых алгорифмах для решения полной проблемы собственных значений" [On certain algorithms for the solution of the complete eigenvalue problem], Журнал вычислительной математики и математической физики (Journal of Computational Mathematics and Mathematical Physics), 1 (4): 555–570, 1961
22. ^ Golub & van Loan 1996, §7.3.
23. ^ Meyer 2000, §7.3.
24. ^ Cornell University Department of Mathematics (2016) Lower-Level Courses for Freshmen and Sophomores. Accessed on 2016-03-27.
25. ^ University of Michigan Mathematics (2016) Math Course Catalogue Archived 2015-11-01 at the Wayback Machine. Accessed on 2016-03-27.
26. ^ Herstein 1964, pp. 228,229.
27. ^ Press 2007, pp. 38.
28. ^ Fraleigh 1976, p. 358.
29. ^ a b c Golub & Van Loan 1996, p. 316.
30. ^ Anton 1987, pp. 305,307.
31. ^ a b Beauregard & Fraleigh 1973, p. 307.
32. ^ Herstein 1964, p. 272.
33. ^ Nering 1970, pp. 115–116.
34. ^ Herstein 1964, p. 290.
35. ^ Nering 1970, p. 116.
36. ^ Korn & Korn 2000, Section 14.3.5a.
37. ^ et al. 1989, p. 217.
38. ^ Nering 1970, p. 107; Shilov 1977, p. 109 Lemma for the eigenspace
39. ^ Lipschutz, Seymour; Lipson, Marc (2002-08-12). Schaum's Easy Outline of Linear Algebra. McGraw Hill Professional. p. 111. ISBN 9780071398800.
40. ^ For a proof of this lemma, see Roman 2008, Theorem 8.2 on p. 186; Shilov 1977, p. 109; Hefferon 2001, p. 364; Beezer 2006, Theorem EDELI on p. 469; and Lemma for linear independence of eigenvectors
41. ^ Axler, Sheldon, "Ch. 5", Linear Algebra Done Right (2nd ed.), p. 77
42. ^ a b c d Trefethen, Lloyd N.; Bau, David (1997), Numerical Linear Algebra, SIAM
43. ^ Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method", Earth Surface Processes and Landforms, 25 (13): 1473–1477, Bibcode:2000ESPL...25.1473G, doi:10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
44. ^ Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis", Journal of Geology, 66 (2): 114–150, Bibcode:1958JG.....66..114S, doi:10.1086/626490
45. ^ Knox-Robinson, C.; Gardoll, Stephen J. (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system", Computers & Geosciences, 24 (3): 243, Bibcode:1998CG.....24..243K, doi:10.1016/S0098-3004(97)00122-2
46. ^ Stereo32 software
47. ^ Benn, D.; Evans, D. (2004), A Practical Guide to the study of Glacial Sediments, London: Arnold, pp. 103–107
48. ^ Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004), Estimation of 3D motion and structure of human faces (PDF), National Technical University of Athens
49. ^ Diekmann O, Heesterbeek JA, Metz JA (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations", Journal of Mathematical Biology, 28 (4): 365–382, doi:10.1007/BF00178324, PMID 2117040
50. ^ Odo Diekmann; J. A. P. Heesterbeek (2000), Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
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481cda2c7c300d09 | D Bandrauk, A Fillion-Gourdeau, F Lorin, E Electronic wavefunction delocalization <p><strong>Figure 1.</strong> Electronic wavefunction delocalization.</p> <p><strong>Abstract</strong></p> <p>Gauge invariance was discovered in the development of classical electromagnetism and was required when the latter was formulated in terms of the scalar and vector potentials. It is now considered to be a fundamental principle of nature, stating that different forms of these potentials yield the same physical description: they describe the same electromagnetic field as long as they are related to each other by gauge transformations. Gauge invariance can also be included into the quantum description of matter interacting with an electromagnetic field by assuming that the wavefunction transforms under a given local unitary transformation. The result of this procedure is a quantum theory describing the coupling of electrons, nuclei and photons. Therefore, it is a very important concept: it is used in almost every field of physics and it has been generalized to describe electroweak and strong interactions in the standard model of particles. A review of quantum mechanical gauge invariance and general unitary transformations is presented for atoms and molecules in interaction with intense short laser pulses, spanning the perturbative to highly nonlinear non-perturbative interaction regimes. Various unitary transformations for a single spinless particle time-dependent Schrödinger equation (TDSE) are shown to correspond to different time-dependent Hamiltonians and wavefunctions. Accuracy of approximation methods involved in solutions of TDSEs such as perturbation theory and popular numerical methods depend on gauge or representation choices which can be more convenient due to faster convergence criteria. We focus on three main representations: length and velocity gauges, in addition to the acceleration form which is not a gauge, to describe perturbative and non-perturbative radiative interactions. Numerical schemes for solving TDSEs in different representations are also discussed. A final brief discussion of these issues for the relativistic time-dependent Dirac equation for future super-intense laser field problems is presented.</p> transformation;representation;electromagnetic field;potential;wavefunction;perturbative;tdse;equation;laser;interaction;method;quantum;Abstract Gauge invariance 2013-07-09 |
2a5f2d54e92cadab | Schrödinger's wave equation
Also found in: Dictionary, Thesaurus.
Schrödinger's wave equation
A linear, homogeneous partial differential equation that determines the evolution with time of a quantum-mechanical wave function.
Quantum mechanics was developed in the 1920s along two different lines, by W. Heisenberg and by E. Schrödinger. Schrödinger's approach can be traced to the notion of wave-particle duality that flowed from A. Einstein's association of particlelike energy bundles (photons, as they were later called) with electromagnetic radiation, which, classically, is a wavelike phenomenon. For radiation of definite frequency f, each bundle carries energy hf. The proportionality factor, h = 6.626 × 10-34 joule-second, is a fundamental constant of nature, introduced by M. Planck in his empirical fit to the spectrum of blackbody radiation. This notion of wave-particle duality was extended in 1923 by L. de Broglie, who postulated the existence of wavelike phenomena associated with material particles such as electrons. See Photon, Wave mechanics
There are certain purely mathematical similarities between classical particle dynamics and the so-called geometric optics approximation to propagation of electromagnetic signals in material media. For the case of a single (nonrelativistic) particle moving in a potential V( r ), this analogy leads to the association with the system of a wave function, &PSgr;( r ), which obeys
Eq. (1). Here m is the mass of the particle, E its energy, ℏ = h/(2&pgr;), and ∇2 is the laplacian operator. See Geometrical optics
It is possible to ask what more general equation a time- as well as space-dependent wave function, &PSgr;( r , t), might obey. What suggests itself is Eq. (2),
which is now called the Schrödinger equation.
The wave function can be generalized to a system of more than one particle, say N of them. A separate wave function is not assigned to each particle. Instead, there is a single wave function, &PSgr;( r1, r2, …, rN , t), which depends at once on all the position coordinates as well as time. This space of position variables is the so-called configuration space. The generalized Schrödinger equation is Eq. (3),
where the potential V may now depend on all the position variables. Three striking features of this equation are to be noted:
1. The complex number i (the square root of minus one) appears in the equation. Thus &PSgr; is in general complex.
2. The time derivative is of first order. Thus, if the wave function is known as a function of the position variables at any one instant, it is fully determined for all later times.
3. The Schrödinger equation is linear and homogeneous in &PSgr;, which means that if &PSgr; is a solution so is c&PSgr;, where c is an arbitrary complex constant. More generally, if &PSgr;1 and &PSgr;2 are solutions, so too is the linear combination c1&PSgr;1 + c2&PSgr;2, where c1 and c2 are arbitrary complex constants. This is the superposition principle of quantum mechanics. See Superposition principle
The Schrödinger equation suggests an interpretation in terms of probabilities. Provided that the wave function is square integrable over configuration space, it follows from Eq. (3) that the norm, 〈&PSgr;&PSgr;〉, is independent of time, where the norm is defined by Eq. (4). (4) It is possible to normalize &PSgr; (multiply it by a suitable constant) to arrange that this norm is equal to unity. With that done, the Schrödinger equation itself suggests that expression (5)
is the joint probability distribution at time t for finding particle 1 in the volume element d3x1, particle 2 in d3x2, and so forth.
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0f66c4042f7d03da | About states, observables and the wave functional interpretation in QFT with gauge fields | PhysicsOverflow
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+ 4 like - 0 dislike
First of all, I'm a mathematician, so forgive me for my possible trivial mistakes and poor knowledge of physics.
In a QFT, we just start with a field (scalar, vectorial, sponsorial, gauge etc), so I would like to know what are the observables and the states in this context.
In QFT, the general approach would be by using the Fock space (for the free field case, since I don't really know if this would be true for the interacting one) and getting down, by using the particles associated to the operators $a$ and $a^{\dagger}$, to QM particles (I don't really know if this is true, because the number of particles is not constant and depends on the observer) or by using the wave functional interpretation (a functional on the space of field configurations satisfying Schrödinger equation), though I've heard that this functional is not Lorentz covariant (by the way, any proof?). However, according to this article (http://core.ac.uk/download/pdf/11921990.pdf) the wave functional interpretation is equivalent to the Fock space, so, in any case, this interpretation is not physically reasonable.
In AQFT, in contrast, the operators are already given (so we already have the observables). Furthermore, if the Lorentzian manifold is globally hyperbolic, a Cauchy hyper surface would be a possible interpretation for a state.
In other aspect, are the quantized fields of a given QFT really observables in the sense that they measure?
Now, adding gauge fields, everything will be grupoid valued and observables would be defined on quotients by the gauge group. In this context, I haven't really seen anything written about states and I have no idea on how the Fock space would be. The naive approach would be to consider the wave functional interpretation with domain in a grupoid.
Furthermore, if we restrict ourselves to TQFT, CFT or other specific class of field theories, would all this problem be solved?
Thanks in advance.
This post imported from StackExchange Physics at 2015-05-01 12:37 (UTC), posted by SE-user user40276
asked Apr 17, 2015 in Theoretical Physics by user40276 (140 points) [ revision history ]
edited May 1, 2015 by Dilaton
An historical remark: the pdf reference you cite seems to be quite out of date concerning the references...the interpretations of QFT provided there and the related discussions/problems are known since the end of the fifties of the last century ;-)
2 Answers
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The algebraic approach gives the better idea of what the states and observables of a quantum theory are, and this holds in infinite dimensional systems as well.
In the modern mathematical terminology, observables of quantum mechanics are the elements of a topological $*$-algebra, and states are objects of its topological dual that are positive and have norm one. The most usual case is to take the $*$-algebra to be a $C^*$ or $W^*$ (von Neumann) algebra; however with such choice unbounded operators are not, strictly speaking, observables (but they can be "affiliated" to the algebra if their spectral projections are in the algebra). The advantage of this abstract approach is that, by the GNS construction, one can immediately associate an Hilbert space to the given $*$-algebra (and a particular state), where the elements of the algebra act as linear operators, and the given state as the average w.r.t. a specific Hilbert space vector.
In usual physical terms, only self-adjoint operators are considered to be observables, for an observable should have real spectrum (and could be associated to a strongly continuous group of unitary operators). The quantum field is, usually, considered to be an observable in a QFT (it is self-adjoint but unbounded, so often it would be affiliated to the $W^*$ algebra generated by its family of exponentials, the Weyl operators); and it is perfectly possible, theoretically, to measure its average value on states (to do it really in experiments, that is all another problem).
Quantum field theories are almost always represented in Fock spaces. However, since the Heisenberg group associated with an infinite dimensional symplectic space is not locally compact, the Stone-von Neumann theorem does not hold and there are infinitely many irreducible inequivalent representations of the Weyl relations, the Fock space being only one of them. To complicate things more, the Haag's theorem states that, roughly speaking, the free and interacting Fock representations are unitarily inequivalent (but that is a problem mostly for scattering theory, not at a fundamental level).
The "wave functional interpretation" (never heard this terminology) is just the functorial nature of the second quantization procedure that can associate to each Hilbert space the corresponding Fock space. This is due to Segal and you may also consult Nelson. The idea is that to each Hilbert space $\mathscr{H}$ one can associate a Gaussian probability space $(\Omega,\mu)$ such that the Fock space $\Gamma(\mathscr{H})$ is unitarily equivalent to $L^2(\Omega,\mu)$, and the map between $\mathscr{H}$ and $\Gamma(\mathscr{H})$($L^2(\Omega,\mu)$) is a functor in the category of Hilbert spaces with self-adjoint and unitary maps as morphisms. The $L^2(\Omega,\mu)$ point of view becomes very natural if one is interested to study QFTs by means of the stochastic integral approach (Feynman-Kac formulas) in euclidean time.
answered Apr 17, 2015 by yuggib (360 points) [ no revision ]
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Thanks for your answer. I've never heard about interacting Fock space, is there any reference? About the wave functional, I don't really know how can I get an Hamiltonian to construct a Schrödinger equation to this functional. Furthermore, in the case of gauge fields, do you know how observables and states would be defined? Actually, I've never seen Wightman axioms for the case of a gauge fields (any reference?), so I don't really know what's a QFT with gauge fields.
The interacting Fock space cannot be rigorously constructed in most interesting QFTs; however you may take a look to the second book of Bratteli-Robinson to get an idea (applied on a different context) of the Haag's theorem and the inequivalent vacuum/ground-state representations associated to different QFTs. Also the book by Derezinski and Gerard gives some detail (in the end) on quantization of interacting theories. Finally, you may also try to take a direct look at the original works by Haag himself.
Concerning the wave functional, the Hamiltonian in that case would be, roughly speaking, the same as in the Fock representation but with the field replaced by the multiplication by the gaussian functional, and the momentum replaced by the derivative w.r.t. to the aforementioned functional. In general the Hamiltonian has to be a self-adjoint operator on the $L^2(\Omega,\mu)$ space. Anyways I am not completely familiar with this type of description, so take these informations with benefit of the doubt ;-)
Finally, gauge theories are not different, in principle, to other field theories. I am not an expert on this context either, but I suggest you to take again a look at the second volume of the Bratteli-Robinson where gauge fields are studied in the language of AQFT, even if the application they have in mind are mostly in statistical mechanics (anyways this should be not so different from what you look for).
Sorry, but what do you mean by the Fock representation. There is no sympletic space at the beginning of the construction, so ,given a QFT, how can you associate a Fock representation?
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Sorry, but, again, I can't see what do you mean by the Hamiltonian in the Fock representation.
Let us continue this discussion in chat.
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From the rigorous point of view, the observable vacuum sector of a relativistic quantum field theory (QFT) on flat Minkowski space is defined by the Wightman axioms. (There are also variations of these in terms of nets of local algebras, but the Wightman axioms are considered most basic; they are also the criterion to be met for a solution of the Clay Millenium problem to construct a QFT for Yang-Mills. There you can also see how the vacuum sector of a gauge theory fits in conceptually. The unsolved conceptual problems that you allude to concern the charged sectors only.)
Given the Wightman axioms, the observables (in the sense of potentially measurable operators) are the smeared fields obtained by integrating the distribution-operator valued fields with an arbitrary Schwartz test function, their products, the linear combinations of these, and their weak limits, as far as they exist.
The state vectors are the products $\psi=A|0\rangle$ where $A$ is an observable and $|0\rangle$ is the vacuum state. (Of course, many different $A$ produce the same $\psi$; e.g., for a free QFT, one can change $A$ by adding any operator of the form $Ba(f)$ where $a(f)$ is a smeared annihilation operator, without changing the state.)
The dynamics is dependent on the choice of a time direction along positive multiples of a timelike vector $v$, and is given by $\psi(t):=A(t)|0\rangle$, where $A(t)$ is obtained from $A$ by replacing all arguments $x$ of field operators in the expression defining $A$ by $x-tv$. The latter operation is an algebra automorphism believed to be always inner, i.e., induced by conjugation with a strongly continuous 1-parameter group generated by a $v$-dependent Hamiltonian $H$ with $H|0\rangle=0$. Assuming this, the Schroedinger equation holds.
To get a more concrete view of the Hilbert space and the dynamics one must either consider exactly solvable QFTs (of which nontrivial examples currently are known only in spacetime dimensions $<4$, and indeed, in 2-dimensional conformal field theory one can give a much more specific picture.), or sacrifice rigor and consider renormalized perturbation theory. In 4 dimensions, the latter builds the Hilbert space as a formal deformation of a Fock space and the fields as formal power series in $\hbar$ or a renormalized coupling constant, although to get physical results one hopes that these formal power series can be evaluated numerically by appropriate trickery. In case of QED this works exceedingly well, but less so in other QFTs.
Alternatively, one discretizes the QFT on a finite lattice, and reduces the problem in this way to one of ordinary quantum mechanics, hoping that for a fine enough and large enough lattice, the results close to the continuum results.
One can also use the functional Schroedinger representation, though this is not mathematically well-defined. Note that contrary to the false claim unlike the functional field equation discussed in the article cited by the OP (which is a philosophical, not a physics paper), the functional Schroedinger equation is in general not equivalent to the Fock representation. In particular, unlike the Fock representation, the functional Schroedinger equation is able to explain many nonperturbative features of interacting QFT. See the discussion of Jackiw's work.
For nonrelativistic QFTs, the situation is somewhat simpler, as particle number is conserved. In the vacuum representation, the Hilbert space is a proper Fock space, and splits into a direct sum of $N$-particle spaces to which standard quantum mechanics applies. However other representations such as those relevant for equilibrium thermodynamics, some of the problems from the relativistic case recur, since the appropriate Hilbert space is no longer a Fock space.
In curved space, no good system of axioms is known, and one generally uses a Fock space perturbation approach with all its limitations.
answered May 1, 2015 by Arnold Neumaier (14,019 points) [ revision history ]
edited May 2, 2015 by Arnold Neumaier
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The perturbation theory is an approach based on the Fock representation, but it is not Fock representation by itself (which is just one of the infinitely many possible unitarily inequivalent irreducible representations of the CCR). So if you say that the perturbative approach has its limitations I agree with you, but if you say that the limitations are proper of the Fock representation in its generality I don't agree.
@yuggib: Of the infinitely many inequivalent representations only one is a Fock representations, according to standard terminology.
Yes there is only a representation called like that, fixed what CCR (CAR) you are representing: i.e. fixed the complex Hilbert space (one-particle space) on which the Weyl's relations are written. Nevertheless, as I said, it is not tied to perturbation theory, is just a representation of the CCR (CAR).
@yuggib: If you still think the functional Schroedinger representation is not more powerful than Fock space and CCR, please tell me how you find instantons or theta angles in a Fock representation. The point is that many nontrivial field theories (and maybe all) need a Hilbert space that is strictly larger than the limiting Fock space that is obtained when the renormalized coupling constant tends to zero. Perturbation theory cannot see the missing part.
Concerning 1), my intuition (but it may be also wrong) is that if there are other non-fock representations of the free dynamics they should satisfy the Wightman axioms as well, for these axioms seem "representation-independent", at least to me.
Concerning 3), I think that even if you cannot say for sure that the interacting representations are non-Fock, this may indeed be the case for many (or maybe most) interesting theories. Looking a little bit around, I found for example that the representation associated to the renormalized $\phi^4_3$ Hamiltonian given by Glimm is non-Fock (link to where I found the assertion: https://projecteuclid.org/euclid.cmp/1103857837, in page 2); however Glimm's hamiltonian has still the volume cutoff, so this seems unrelated to Haag's theorem. I think that the possibilities are many, and it is difficult to know a priori in which type of representation one may end up after renormalization.
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@yuggib: I couldn't find an appropriate reference; so I retract my claims about 1) and 3). They reflected my intuition rather than definite knowledge, and after the present discussion I am no longer convinced that my intuition was correct.
I now found a weak reference; Arthur Jaffe, shorthly after minute 04:00 in http://media.scgp.stonybrook.edu/video/video.php?f=20120117_1_qtp.mp4 says that Fock space is not appropriate for interacting fields.
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76e65d3386f93b91 | Foundations of Physics
, Volume 43, Issue 1, pp 46–53 | Cite as
On the Foundations of Superstring Theory
• Gerard ’t HooftEmail author
String theory Black holes Determinism QCD Local conformal symmetry Hidden variables
The question “what are superstring theory’s foundations?” is not just a philosophical one. Just because it is still not understood how to formulate concisely any chain of logical arguments that could reveal what its basic assumptions are, how the theory is constructed part-by-part, what its strengths and limitations are, how many string theory scenarios one can imagine, and how, at least in principle, accurate calculations can be performed to decide unambiguously how initial configurations evolve into the future, it is of tantamount importance to carry out as many critical investigations as is possible, to analyze this situation and to reach an agreement that is no longer disputed by a vast majority of the experts.1
Yet we see disappointing reluctance in the practitioners of superstring theory to do so. They appear to prefer to discover more and more new “stringy miracles”, such as new miraculous matches of black hole microstates, or new cosmological scenarios. If any logical jumps appear to be too large to comprehend, we call these “conjectures”, find tests to corroborate the conjectures, and continue our way. These are easier ways to score successes but only deepen and widen the logical depths that block any true understanding.
The situation in standard quantum mechanics, and its extension that incorporates special relativity, now known as quantum field theory, is very different and much more mature. We know what quantum mechanics tells us and what not; we know what quantum field theory can do for us and what it cannot, and why this is so. We do not know this for superstring theory, while any support for the naive expectation that this theory will “solve everything” is rapidly fading.
This short note is not intended to be a critique of superstring theory. The theory has not led to genuine explanations of well-known features of the Standard Model, such as the number of quark and lepton generations, let alone the values of constants such as the finestructure constant or the electron mass, and no definitely testable predictions could be arrived at, but by itself there is nothing wrong with this; such explanations and predictions are still way out of reach for respectable theories of physics. Superstring theory has come closer to potential explanations and predictions than any of its competitors such as loop quantum gravity. Superstring theory does provide for a natural looking framework for the gravitational force acting on fermions, scalar fields and gauge fields, and it does, cautiously, predict an important role for supersymmetry in extensions of the Standard Model that may well be in reach of experiments.
Even if our desire for better foundations of the theory may appear to be a “philosophical” one, it is intended to be much more than that. Our present lack of a deeper, genuine, understanding has not really been an obstacle against progress in the past; physicists are guessing their way around, and they are good at it. Yet it may well form an obstacle against further progress in the future. Conjectures such as the AdS/CFT correspondence appear to be successful, but what do they really mean? Can such conjectures still hold exactly when conformal symmetry in the CFT is explicitly broken? Can they be extended to flat spacetimes? Can the real world be mapped onto its boundary in a meaningful way?
Should a straightforward interpretation of the theory and the prescriptions concerning its application be based on unproven conjectures? Of course we prefer more solid foundations, but on the other hand, the need for unproven conjectures is nothing new or fundamentally rejectable for theories, in particular when they are still in their infancy. The reason for writing this note is simply that the author suspects that superstring theory can be improved considerably, or can perhaps be replaced entirely by something better.
What are the examples of better theories? Consider classical mechanics, such as the theory describing the planets in their orbits around the sun, including all mutual disturbances. At first sight, this seems to be a perfect, deterministic, theory, allowing every calculation to be performed with any desired precision, in principle. Of course there are cases where the theory ceases to be valid, such as its description of what happens when two planets collide head-on, or when relativistic or quantum corrections are needed, but this is not the issue. A more subtle objection against the theory of classical mechanics is the inevitable phenomenon of “chaos”. What this means is that any error in the initial parameters of a state eventually leads to large deviations in the orbits that are calculated. This implies that the theory is only fully predictive if all masses and initial states are known as infinitely precisely defined real numbers. The need to specify infinite sequences of decimals for all these numbers could be viewed as an unwanted “divergence” of the theory. How can Nature (or “God”) continuously remember infinite sequences of decimals? At all stages, the theory requires infinite amounts of information to define how things evolve.
Quantum mechanics usually also works with infinite dimensional Hilbert spaces, which may cause similar difficulties, but here it is easier to imagine how such infinities may be cut-off. Regularization is the removal of states that can rarely be detected or realized in an experiment. Hilbert space is then replaced by a finite dimensional vector space. “Chaos” still takes place, but only in as far as ratios between eigenvalues of the Hamiltonian (more precisely, the separations between eigenvalues) take irrational values. Replacing these by rational values makes our system periodic so that all chaos is removed.
The difficulty with quantum mechanics is that it usually can only give statistical predictions for the outcomes of experiments, which one can also bring forward as an objection: the theory is not infinitely precise in predicting the outcomes of experiments. In practice, this is not a difficulty at all; in any experiments on subatomic particles, we have uncertainties in the initial conditions anyway, so even deterministic theories would give us predictions of a statistical nature and these would be in no way better than the ones provided by standard quantum mechanical calculations. Only when questions are asked about “reality”, quantum mechanics fails to give answers of the type sometimes desired.
Quantum field theory is nearly, but just not quite a ‘perfect’ theory. It is the best possible synthesis of quantum mechanics with special relativity. Apart from the defects of quantum mechanics itself, as just discussed, there are imperfections due to the need to renormalize the interactions, and, associated with that, also the need to consider explicitly perturbation theory. If the theory is chosen to be asymptotically free, only the first few terms of perturbation theory are needed to define the interactions at infinitesimal distance scales; the rest can be calculated, in principle. Rigorous proofs that these calculations always converge have not been given but, considering what we know from perturbation expansions, it is extremely likely that these theories work just fine. If a theory is not asymptotically free, such as the Standard Model itself, calculations will not in general converge but nevertheless suffice to define dozens of decimal places so that there are no difficulties in practice. Of course this does imply that such a quantum field theory cannot serve as a model for the ultimate truth; we must continue searching for something better.
A notable feature of all well-established theories of natural phenomena is that they contain ‘constants of Nature’, some freely adjustable parameters in the form of real numbers. These parameters have to be measured; they cannot be computed from first principles. The Standard Model itself now has some 30 freely adjustable constants.
How does superstring theory compare with these other theories? We are not interested in the fact that string theory was originally introduced to describe the strong interactions. Today, it is claimed to be “the most promising candidate” for a theory that combines general relativity with quantum mechanics, so that it will serve to understand quantum gravity. Considering the successes of quantum field theory in combining special relativity with quantum mechanics, this is evidently an important aim.
Different from quantum field theory, superstring theory hinges on a major, unproven assumption. This is the assertion that most of the particles that were considered to be pointlike in the Standard Model, should be replaced by structures with one internal spacelike dimension: strings. By itself, the assumption is baseless. No experimental evidence can support it. The one and only justification of this assumption is the mathematical observation that, as one of the many constraints required by internal consistency of the resulting scheme, some of the stringlike excitations (the lowest closed string states) behave as gravitons, so that the theory “automatically” generates a gravitational force [1, 2]. This surprise was welcomed as a quite pleasing one; apparently, this theory ‘generates’ gravity. It is the only theory with this bizarre property, and, since gravity undeniably exists, the theory is therefore happily embraced.
An inevitable consequence of this property is mathematical complexity. Gravity is associated to space-time curvature, so, by some miracle, closed string loops turn a flat background spacetime into a dynamical structure, to be provided with a curved coordinate grid. Many other such miracles were encountered. Further self-consistency required the introduction of supersymmetry on the string world sheet, turning it into a super-string, so that string theory also “explains” the existence of fermionic particles [3, 4]. Next came the observation that higher dimensional membrane like structures also arise as topological features [5]. Probably, what we are arriving at is a fundamental generalization of quantum field theory to include 1-, 2- and higher dimensional subspaces of space-time to replace the elementary particles. One can safely conclude that if we introduce such higher dimensional structures, the emergence of fermions and gravity is inevitable. Yet still, the converse is not obvious; physicists have not succeeded to derive the existence of strings and D-branes from the requirements one presumably has to demand for a quantum gravity theory. Therefore, we have to keep in mind the possibility that the real world is something totally different.
This having been said, we can also decide to ignore such objections. We just accept the fact that superstring theory is a theory awaiting further support from experimental evidence. Our job is to provide the proper foundations of the theory.
Consider first the approach starting from the (super)strings themselves. String world sheet diagrams are considered to replace the old Feynman diagrams. Now Feynman diagrams typically represent sequences of perturbative corrections, so that, in turn, string world sheets also should be interpreted as perturbative expressions, the perturbation expansion being one in terms of powers of the string coupling constant g s . Experiences obtained from the older quantum field theories tell us that such perturbation expansions are fundamentally divergent, generating coefficients of the order of n! for the nth term (the term with n loops in the string world sheet). There is no good reason to expect string perturbation theory to converge better than that. The fact that no UV renormalization appears to be needed might help but certainly does not suffice. In short, string perturbation theory itself definitely does not define a theory. Again looking at quantum field theory, we know of one case where we can do better: in asymptotically free theories such as QCD.
QCD can be defined on a discrete, but arbitrarily dense lattice. The limit a↓0, where a is the size of the meshes, can be rigorously defined, and, according to perturbation expansion, only the first two terms of this expansion need to be known for a rigorous definition of this limit. If any procedure for string theory along similar lines could be defined then we would have more rigorous foundations. Now this is unlikely, since at distance scales tiny compared to the Planck scale, we have no idea about how to formulate what happens, quite unlike asymptotically free field theories. No inductive arguments exist telling us how to integrate the equations starting from a region of triviality such as the short distance region of QCD. Therefore, the situation is exactly as bad as in non-asymptotically free quantum field theories, which are known for their disasters such as the Landau ghosts.
Various cures for this shortcoming have been proposed. It seems that formal mappings may exist of string theories onto infinite-dimensional matrices [6]. Here, however, one also seems to rely on the usefulness of certain 1/N expansions, which again do not converge in general. Are strings finite-dimensional matrices? Those seem to exhibit far too little structure to be able to model a universe as complex as ours, so this would be difficult to accept. If, starting from 10 dimensional superstring theory, we add one more dimension then the string may be seen to be a topological object of a compactified 11 dimensional supergravity. This is an “ordinary” quantum field theory, in so many dimensions that it cannot be asymptotically free. It is sometimes claimed to be finite order by order in perturbation theory, but it seems obvious that perturbation theory itself should be highly divergent here; in any case, this theory also becomes ill-defined at distances small compared to the (11 dimensional version of the) Planck scale.
Further artillery has been put in position. Duality transformations link one kind of (string) theory to others. A problem here is that each of these theories themselves lack solid foundations or definitions. Mappings to and fro won’t change that; rigorous foundations are still absent.
In spite of this lamentable situation, miraculous features are claimed by their discoverers, notably in the area of black holes. To demonstrate the inadequacy of (super)string theory, the author has brought forward that black holes show deficiencies that cannot possibly be cured by string theories. Applying standard quantum field theory to black holes exposes a contradiction: it was deduced from quantum field theoretical considerations near a black hole horizon that black holes emit a thermal spectrum of elementary particles. Being thermal, it seems that these particles cannot be in pure quantum states, that is, be described by single vectors in Hilbert space. One would have to conclude that black holes themselves cannot obey any Schrödinger equation, since such an equation would require single elements of Hilbert space, even if these are entangled. This contradiction should hold for all local field theories, hence even for string theories.
Now here, we had underestimated string theory. A large class of black holes, that is, the set of black holes that are close to an extreme limit, could be reproduced in string/brane theory, and their internal properties do seem to obey good quantum coherence laws. How can this be?
I do not have the impression that this point is well understood, even by the experts. The horizon of these black holes does not seem to be something one can transform away by coordinate transformations [7, 8]. What is needed is a theory that explains black hole microstates as a local property of horizons, where the Schwarzschild horizon should be the prototype, not the horizon of an extreme black hole, which is physically different.
Not all hope should be given up. As was shown by the present author, a very special new local symmetry can restore the quantum coherence of black holes: local conformal symmetry [9]. This symmetry must be exact and spontaneously broken. The latter was known and is not at all new; it has been pointed out by many authors; but the claim that this symmetry has to be exact implies that the conformal anomalies have to cancel out, and this is normally not assumed to be the case. We have derived this from ordinary quantum field theory, where it may have deep and important consequences, but the same argument may well hold for string theories as well. Quite possibly, the conformal anomalies cancel out here in a natural way, and this could be a deeper explanation as to why string theories produce quantum mechanically sound black holes. This would indicate that our standard objection has been met; string theory survived it.
One weakness of string theory has not yet been discussed here: the arbitrariness in folding the superfluous dimensions into compact manifolds that may trap arbitrary amounts of different kinds of fluxes. The question how these compactified dimensions came to be folded the way they are seems to be unanswerable: they always were folded this way, from time zero. Not only is this unsatisfactory; it is something of a disaster for the theory, because the compactification ambiguity leads to a permanent large-scale ambiguity in the realization of these theories. There are quadrillions of different theories and there is practically no way to select the one that is appropriate to describe the universe we live in. We find it a curious coincidence that string theory may exhibit so many distinct forms, and that exactly such a set of distinct forms also emerges if we demand the cancellation of conformal anomalies in conventional gravitating quantum field theories.
This conjecture may be exactly as weak as many of the others used in connection with string theory. As long as solid foundations in terms of provable mathematical equations are lacking, one may conjecture anything one likes, it does not help at all to strengthen these foundations. Genuine theories should be based on rigorous formulations for their local behavior. Precisely this is a problem: string theory only allows for constructions of on-shell amplitudes, a thing it has in common with predecessors of quantum field theory: axiomatic S matrix theory for hadronic amplitudes. In this theory algebraic symmetries were suspected to suffice, together with dispersion relations, to define dynamical amplitudes, a program that failed bitterly. So, we have to attempt to put our fingers onto local formalisms. This really implies that we have to understand our physical world beyond the Planck scale.
The ultimate theory of the world cannot be a very simple one, if only because it must be able to describe a universe as complex as ours. The hierarchy problem emphasizes that the enormous variety of scales in the universe, both in space and in time, can only be due to the fact that many constants of nature show gigantic variations in strength; the most notable example of this being the value of the cosmological constant in terms of Planck units: close to 10−123. Since string theories form discrete classes, one has to search for the one representative of these classes that exhibits such a variation in coupling strengths. This might be slightly easier than sometimes claimed. One does not have to dig very deep into mathematics to encounter naturally large numbers such as \(e^{ e^{2\pi}/2}\) or e 90π , but devising physical theories that naturally produce such quantities might be something of an art.
My last point is one where only few readers will follow me. One of my most fundamental objections against string theory, as usually formulated today, is that it unquestioningly embraces standard quantum theory: states in string theory span a Hilbert space, and its evolution equations are just an arbitrary recipe to generate an evolution operator in this Hilbert space. This implies that string theory also accepts the fact that any given initial state may lead to quantum superpositions of many final states. It accepts that many experiments, even at the Planck level, give rise to outcomes with a probability distribution rather than distinct certainties. This, I believe, cannot be right. The dynamical variables active at the Planck scale should not give rise to quantum vagueness.
This objection only holds for theories that claim to be a theory of everything. Such theories should not be allowed to produce probabilities but only certainties. The most urgent case is the case of a small, compact, evolving universe. In a compact universe, one cannot repeat experiments infinitely many times, and this means that probability theory is inapplicable there. Now, having said this for a compact universe, the same should be true for a non-compact universe if locality of the interactions means anything.
This standpoint clearly calls for the revival of “hidden variables”, even “local hidden variables”. According to many, the possibility to use such variables was disproved by theorems starting with the Bell inequalities [10]. However, we claim that hidden variables do not exclude being treated as if they occupy quantum states. We can introduce quantum operators even for deterministic hidden variables and end up with quantum models. We explained this in a number of recent articles [11, 12, 13, 14, 15], but much earlier a somewhat awkward argument explaining the same was given by D. Bohm [16, 17]. His ideas, involving “pilot wave functions”, were dismissed as irrelevant by a majority in the community, even if his claim that he exactly reproduced quantum mechanics was accepted.
What we showed was that pilot wave functions are not really needed; there are much more elegant and fundamental ways of understanding how quantum mechanics can become emergent [18, 19]. More recent research lead to a surprise. Attempts to directly reproduce realistic quantum field theories out of deterministic toy models were not totally successful: rotation invariance was difficult to realize, Galilean invariance (needed to describe simple models of moving particles) was even harder, and Lorentz invariance seemed to be hopelessly impossible. The surprise was that the most eminent system ideally suitable to be cast into a deterministic setting, turned out to be string theory. The deterministic equations apply to the world sheet, the dynamical variables are in target space, and there, rotation invariance, even Lorentz invariance, can now be understood [20].
This seems to open up the exciting possibility that various problems can only be solved together at one stroke. It is my fear that without such steps string theory, or any of its more advanced successors, will never be properly understood. Further investigations of string theory’s foundations are therefore urgently called for.
1. 1.
String theory has been, and will always be, disputed by numerous onlookers in the sideline who failed to grasp many of its subtle technicalities. It goes without saying that we ignore them.
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Copyright information
© Springer Science+Business Media, LLC 2012
Authors and Affiliations
1. 1.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands
2. 2.Spinoza Institute3508 TD UtrechtThe Netherlands
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98b8c9184f3edf91 | List of Articles
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Spectral Lines Associated with Dark Matter
In recent News from Physics and Cosmology, there has been a flurry of reports concerning a signature spectral line which can be associated with dark matter in distant galaxies. Given the preponderance of hydrogen in normal matter, there has been a suspicion that dark matter is novel form of hydrogen. "An unidentified line in X-ray spectra of the Andromeda galaxy and Perseus galaxy cluster" by Boyarsky is an example. Although the line is weak, it has a tendency to become stronger towards the centers of the galaxies and is absent in the spectrum of a deep "blank sky" dataset.
Detection of an Unidentified Emission Line in the Stacked X-Ray Spectrum of Galaxy Clusters by Bulbul et al. reports the unidentified emission line at 3.6 keV in 73 different Galaxies.
The authors conclude that "As intriguing as the dark matter interpretation of our new line is, we should emphasize the significant systematic uncertainties in detecting the line energy in addition to the quoted statistical errors." Statisticians seem to be more comfortable with the evidence than physicists. We at Chava are interested in the dark matter to hydrogen from the perspective of alternative energy. There is a possibility that dark matter hydrogen can be ubiquitous, and even manufactured or "harvested". The solar wind could be resource for dark matter.
In another paper: "Questioning a 3.5 keV dark matter emission line" Riemer-Sørensen analyzes data from the Milky Way and finds some evidence of this line but does not ascribe the same high confidence level as do other for a dark matter signature.
The issue is far from decided, but it is not too soon to consider alternative energy implications for Earth-bound uses and experiments with engineered Dark Matter, which are based on the possibility that hydrogen isomers are formed in a predicted state, known as the DDL, or Deep Dirac Level, which can be identified as warm dark matter with the characteristic emission.
The actual mystery emission line is centered at ~3.5-3.6 keV in all 73 Galaxies which were analyzed.
Previously there had been predictions of neutrinos at 3.5 and 7 keV based on roughly the same equations which derive from the Dirac equation. This spectrum is otherwise unpopulated by known elemental emission lines.
X-rays in this spectrum are fairly "soft" - and at a blind spot exists in experiments where they could appear, since there are no commercially available meters to see 10 keV all the way down to EUV. Thus, detection in metal hydride experiments has not been possible to date without the use of film exposure; and even NASA can only accomplish this feat in space and at huge expense. Almost any window for a detector will block this x-ray but if more evidence accumulates, solutions to the detector problem will be found. Very thin Mylar may work, or exposed circuit lines and semiconductor.
The enticing thing about this x-ray line – for those who pursuing the phenomenon of anomalous heat from metal-hydrides – the field which was once called "cold fusion" and later LENR is that it offers an alternative explanation for thermal gain. No matter what name has been given to the phenomenon in the past, it cannot involve common types of nuclear fusion, since no gamma radiation is present. But the predicted deeply bound state of hydrogen, derived from the Dirac equation, fits the evidence nicely. This is an emission range which could have gone undetected in the past 25 years of LENR research, and yet it would produce a few thousand times more energy than a chemical reaction.
Notably, this line seems to be near a Rydberg multiple of the kind featured in the CQM theory of Randell Mills, and possibly already associated with deep level ground state orbital redundancy of hydrogen, in the work of several others including Naudts, Va'vra and Meulenberg. There can be 137 steps in the progression of ground state hydrogen orbital to a DDL which are multiples of 27.2 eV, the Hartree energy. For instance 130 * 27.2eV = 3.54 keV which would indicate that the deeper states below 130 steps are not accessible. Randell Mills own calculation provides a value which is too low for what has been reported. There are other ways to compute this value, as well, which fall within a range of 3-7 keV. If the hydrogen as a DDL isomer can be identified as dark matter, or a subset of dark matter, it is not completely dark in a cosmological environment, and will emit its signature on either
decay or other stimulation, such as the passage of a gravity wave.
The payoff of dark matter research - and its availability as an alternative energy source would be huge - should this emission line be seen in experiments. We could simultaneously go a long way towards explaining what dark matter really consists of (basically it is hydrogen but as a DDL isomer) and also, explain the proximate cause of some forms of LENR, which are producing heat without gamma radiation. This understanding could also permit better control over a notoriously unpredictable system.
Further Reading:
Randell Mills Theory
Jan Naudts "On the hydrino state of the relativistic hydrogen atom", Aug, 2005, predicts the DDL state at very close to the observed spectral line which does not really support Mills theory.
Naudts summarizes: "This paper starts with the Klein-Gordon equation, with minimal coupling to the non-quantized electromagnetic field. In case of a Coulomb potential this equation is the obvious relativistic generalization of the Schrödinger equation of the non relativistic hydrogen atom, if spin of the electron is neglected. It has two sets of eigenfunctions, one of which introduces small relativistic corrections to the non-relativistic solutions. The other set of solutions contains one eigenstate which describes a highly relativistic particle with a binding energy which is a large fraction of the rest mass energy. This is the hydrino [single DDL] state.
For a contrary view, see Rice and Kim
and the rebuttal of Rice and Kim by Va'vra
The DDL/Dark-Matter/LENR connection is an interesting possibility that has generated a huge amount of interest, since it fills a large gap elegantly... which of course, does not make it right. is an information sharing platform on new energy research projects provided by Chava Energy LLC and its subsidiary Chava Wind LLC. For further information please contact Hagen Ruff or Mark Snoswell on This email address is being protected from spambots. You need JavaScript enabled to view it.. |
6d67dea92248615a | Is String Theory Testable?
I’ve been traveling in Italy for the past ten days, and gave talks in Rome and Pisa, on the topic “Is String Theory Testable?”. The slides from my talks are here (I’ll fix a few minor things about them in a few days when I’m back in New York, including adding credits to where some of the graphics were stolen from). It seemed to me that the talks went well, with fairly large audiences and good questions. In Pisa string theorist Massimo Porrati was there and made some extensive and quite reasonable comments afterwards, and this led to a bit of a discussion with some others in the audience.
I don’t think the points I was making in the talk were particularly controversial. It was an attempt to explain without too much editorializing the state of the effort to connect the idea of string-based unification of gravity and particle physics with the real world. This is something that has not worked out as people had hoped and I think it is important to acknowledge this and examine the reasons for it. In one part of the talk I go over a list of the many public claims made in recent years for some sort of “experimental tests” of string theory and explain what the problems with these are.
My conclusion, as you’d expect, is that string theory is not testable in any conventional scientific use of the term. The fundamental problem is that simple versions of the string theory unification idea, the ones often sold as “beautiful”, disagree with experiment for some basic reasons. Getting around these problems requires working with much more complicated versions, which have become so complicated that the framework becomes untestable as it can be made to agree with virtually anything one is likely to experimentally measure. This is a classic failure mode of a speculative framework: the rigid initial version doesn’t agree with experiment, making it less rigid to avoid this kills off its predictivity.
Some string theorists refuse to acknowledge that this is what has happened and that this has been a failure. Most I think just take the point of view that the structures uncovered are so rich that they are worth continuing to investigate despite this failure, especially given the lack of successful alternative ideas about unification of particle physics and gravity. Here we get into a very different kind of argument.
It was very interesting to talk to the particle physicists in Rome and Pisa. They are facing many of the same issues as elsewhere about what sort of research directions to support, with string theory often being pursued as an almost separate subject from the rest of particle theory, leading to conflict over resources and sometimes heated debates between them and the rest of the particle physics community. Many people were curious about how things were different in the US than in Europe, but I’m afraid I couldn’t enlighten them a great deal, mainly because I just don’t know as much about the European situation, although I’ve started to learn more about this on the trip. Several wondered if the phenomenon of theorists going to the press to make overhyped claims about string theory was an American phenomenon. I hadn’t really noticed this, but it does seem to be true. While the hype starts in the US, it does travel to Europe, with the US very influential in this aspect of culture as in many others. In the latest issue of the main Italian magazine about science, there’s an article explaining how certain US theorists have finally figured out how to test string theory with the new LHC…
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47 Responses to Is String Theory Testable?
1. Levi says:
This seems similar to the situation with Grand Unified Theories. I gather that SU(5) was the “beautiful” version, and when that version ran into problems much of the beauty went out of GUTs. It’s interesting to contrast this with cosmic inflation, where Guth’s original version didn’t quite work, but Linde and others found forms of inflation which worked better, and WMAP data gives a reality check.
I should mention that I’m not a physicist, just a casual reader, so if I’m misinformed I hope somebody will point it out.
2. Arun says:
It would be nice to know what Porrati said, if at all possible.
3. Joseph Smidt says:
Great Post. I thought your comments on US/Europe string culture were interesting. Thanks for the slides.
4. Irish physicist says:
Off-topic – but congratulations on 3 years of blogging and Happy St. Patricks day too!
5. woit says:
Irish Physicist,
Thanks! I hadn’t realized that the blog was started on a St. Patrick’s day. Surely some sort of homage to the Irish was unconsciously intended.
I can’t recall exactly what Porrati’s points were, except that he said that he had five of them, and none of them were things that I really had a substantive disagreement with. Some of them were (from memory, and in loose translation, surely he would express these differently)
1. String theory shouldn’t be thought of as a theory that leads to a unique, predictive model, but instead as a very general framework, like QFT, valuable for the different kinds of models it allows.
2. He mentioned the “swampland” idea, that one could try and characterize those low energy theories that come from an ultraviolet completion like string theory.
3. His main point I think was that as long as there was no alternative way to unify particle theory with quantum gravity, string theory would continue to be a main focus for people to pursue. Kind of the “only game in town argument”.
4. He may also have mentioned the use of string theory in heavy-ion physics, in regimes where lattice gauge theory has trouble providing results.
I guess I’m missing at least one…
6. anon. says:
Porrati’s 1st point is with all due respect exactly the argument that defended the use of epicycles by both Ptolemy and Copernicus: it seemed to be a very useful framework of ideas. (Ptolemy used epicycles in the earth-centred universe, c. 150AD. In 1543, Copernicus used epicycles in his final model of the solar system.)
As a ‘general framework of ideas’, the false theory of epicycles was invaluable to Ptolemy, Copernicus and generations of physicists. But that useful approximate framework was really false, as Kepler eventually discovered. So in the end both the earth-centred universe and its general framework of ideas were discredited. Will the string theory framework of ideas similarly mislead generations?
What is so interesting is that it seems to be disconnected from reality not just with regard to its failure to make testable predictions, but also at the input end. Instead of having solid input, everything which has been put into string theory is completely speculative. It is less testable than either of the epicycle theories, and has less solid evidence.
People now laugh at the idea that a theory was once constructed in which the stars and planets were carried around the earth while imbedded in closed crystalline shells. At least that false model was an attempt to interpret data. Perhaps people will cry with pity in the future, reading how physicists defended 10/11 dimensional M-theory in the 21st century, without providing any evidence at all.
7. Chris W. says:
The pre-Copernican astronomers could be excused on the basis of epistemological naivete; their successors largely invented the understanding of science that is now being invoked in discussions of string theory.
String theorists can’t be so excused. They should have known better, and should know better now. Certainly ‘general frameworks of ideas’ are important; they set the context for formulating problems. This is why metaphysics is important in science, even though most metaphysics ultimately proves worthless.
The questions that must be asked now with respect to quantum gravity and unification concern the problem formulation. (Shiing-Shen Chern, who discussed the matter with Einstein in the 1940s, recognized this as the essential work of the physicist.) The alternatives to string theory in quantum gravity challenge the received wisdom in this regard, and for this reason alone are important. In this context Porrati’s main point (as stated by Peter, and echoed by many of Porrati’s colleagues) strikes me as a complete crock. The string theorists who adopt this attitude are the least likely to arrive at the crucial insights into the problem. One can hope they’ll at least have the good sense and simple honesty to recognize those insights when they appear, although I’m less and less optimistic about that.
8. Vijay Shankar says:
There seems to be many differences in opinions unfortunately based on nationality. What people would want Physicists to come up with is a theory that holds in all frames or an experimental method that would help us test all the theories. Until then, we can’t stop someone crying foul whenever there is a news about ‘revolutionary’ theories.
9. tomj says:
My question is: what is the difference between no theory and one which cannot be tested?
I cannot figure out why string theory is a theory. It barely ranks as a hypothesis, and a poor one, very close to what my teenager would come up with. It is 100% mental.
A theory, at minimum, should cover all the facts known, but as Einstein once said (something like): a theory should be as simple as possible, _but_ no simpler. The implication is that there has to be a careful balance, and the theory _must_ track data. How else could the complexity of theory be measured? Yes, you can predict new facts, but first you have to account for known facts. We have to start with the abilities of the observer. And the first ability is that of objectivity, and objectivity begins with the repudiation of belief.
If a theory cannot be any simpler than necessary, how … really … how can a theory be more complex than necessary? If over simplification is a sin, complexity is beyond sin. A ‘theory’ (or set of words and math) which can ‘explain’ everything ‘after the fact’ is useless. Can someone please explain to me this: do physicists really believe that it is possible to formulate a complete description of the universe which will be testable? Because one possible reality is that we are incapable of this. We have thousands of years of data to suggest this conclusion, and only wishful thinking to suggest otherwise.
I like the name of the book, it is important to echo prior thinking. But it might have been even more valid to call it ‘Beyond Reason’. Everyone seems to think that they have reason, that they think logically. And as long as we can avoid testing our reason and logic, we can continue to ‘think’ and ‘believe’ whatever we want. And if we become dogmatic in these untested beliefs, what is this? Science is not belief. Science is experiment. And experiment is based upon question, the antithesis of belief. Science is not an answer, science is a method.
10. Ptolemy says:
‘I cannot figure out why string theory is a theory.’ – tomj
Gerard ‘t Hooft:
‘Actually, I would not even be prepared to call string theory a “theory” – rather a model or not even that: just a hunch. After all, a theory should come together with instructions on how to deal with it to identify the things one wishes to describe, in our case the elementary particles, and one should, at least in principle, be able to formulate the rules for calculating the properties of these particles, and how to make new predictions for them. Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrest will perhaps be delivered soon; whatever I did give you, can I still call it a chair?’
Peter Woit’s argument of why a non-predictive framework is not science can be found on p211 of Not Even Wrong (UK ed.):
‘An explanation that allows one to predict successfully in detail what will happen when one goes out and performs a feasible experiment that has never been done before is the sort of explanation that most clearly can be labelled ‘scientific’. Explanations that are grounded in … systems of belief and which cannot be used to predict what will happen are the sort of thing that clearly does not deserve this label. This is also true of … wishful thinking or ideology, where the source of belief … is something other than rational thought.’
11. r hofmann says:
There are many examples of theoretical physicists
working in the US (foreigners and US citizens) who do extraordinarily good work but on the short run are screened by those that produce overhyped newspaper headlines. My general impression is that the US culture supports the go for extremes in generating scientific opinion, publicizing of `results´, and network formation. This may be helpful in projects where a focus of resources is needed (Cobe, WMAP,…). On the theoretical side, however, it may at times just produce entropy, a lack of well-fermented orginality, and thus no gain in robust knowledge.
12. Stacy says:
A note on inflation, inspired by Levi’s comment:
Actually, the situation with Inflation is quite analogous to that with string theory. The original idea was beautiful, and made a simple prediction (the universe should be flat) which solved the coincidence problem (to do with the evolution of the density of the universe). These together were compelling and propelled the theory to the dominance it enjoys today. But it did suffer problems (like a graceful exit from inflating) which have not entirely been solved. Worse, the compelling aspect of the flatness prediction – confirmed by the WMAP satellite – was that the density parameter should be unity – all in mass – in order to solve the coincidence problem. But it isn’t all in mass – we have now to invoke dark energy. This makes the coincidence problem worse.
In other words, the compelling part of Inflation that led us all to believe it not only doesn’t work, but has made worse the problem it originally seemed to solve. I can’t help wondering if future generations of sociologists will debate whether speculative theories like string theory and Inflation were ever distinguishable from some sort of mathematically motivated religion.
13. matteoeo says:
I agree with Stacy, I think cosmology suffers just the same problems as string theory. Cosmologists can produce potentials that would suit to any possible dynamics of inflation and produce the desired spectrum of cosmic background radiation, without actually deriving them from the properties of the known QFT particles. The worse, cosmology at the moment is a melting pot of the most un-scientifical theories and hypothesis in town: dark energy, cosmological costant, strings and GUTs (early universe), inflation, higgs boson, quintessence, supersimmetry. In cosmology it seems one could just say whatever he wants without too much care about scientific estabilished facts. I was impressed once reading some articles that showed that accelerated expansion of the universe could be explained without any reference to cosmological costant and dark energy, but just owing to some very peculiar relativistic effect (I can give references if any of you is interested). The point is: before inventing theories about the universe, shouldn’t we study general relativity a lot better? And, before unifying gravity and the quantum, shouldn’t we try to understand the basis of QFT and the geometrical structure of QM, and the very profound implications of GR itself?
14. woit says:
Please, cosmology is off-topic. I’m not a cosmologist and don’t want to moderate discussions about cosmology.
15. Alex Nichols says:
I don’t think the epicycles analogy is correct.
That’s an example of an incorrect theory that was disproved by subsequent observation, rather like to the ether theory.
The suggestion being made is that string theory is incapable of falsification because it can’t be tested.
Possibly true, but there are compelling reasons for believing that extended entities that fluctuate are the only possible basis for observable space-time.
This could include strings, loop quantum gravity, spin foams, spin networks etc…
Were it found that the higgs boson is a fundamental particle by the the LHC, all of these would be disproved.
But aren’t the problems of falisfiability at high energy (planck or horizon size) equally true for all the other theories?
Perhaps all the effort shouldn’t be going into one avenue of research. When it comes to funding though, governments may simply decide that we need more effort in applied physics, such as energy production.
16. Alex Nichols says:
BTW, could this finding have any heterotic implications? :-
17. The problem with particle physics, if it is a problem, is that we don’t have any new particles, and the very good theory we have for those particles looks pretty much like a kludge – all those undefined parameters hovering there like epicycles – which were very highly predictive, by the way.
String theory is a heroic attempt to go beyond the SM, but so far hasn’t proven predictive in a confirmable sense. My guess is that we might be stuck without more input from the Universe, which is why everybody is pinning their hopes on the LHC.
Maybe it will provide some clue that makes it possible to turn ST into a predictive theory, maybe it will make it more unlikely that ST has any reality, and maybe it will be mute on ST and other subjects.
Only the last would be a bad outcome.
18. off topic says:
sorry to be off-topic, but let me point out that simplest inflation predicts Omega_total = 1: it is what we measure, and the fact that the total involves some components we don’t understand has nothing to do with inflation. Simplest inflation models also naturally produce a spectrum of scalar adiabatic Gaussian cosmological perturbations with spectral index ns = 1 +- 1/60. Each word has a precise meaning, and it agrees with data. (The deviation of ns from 1 is not yet safely seen).
People tried and try to invent alternative to inflation, but it is not easy because inflation turned out to be good succesful physics. For example alternative models based on “simple string cosmologies” suggested wrong kinds of perturbations (isoentropic, ns not close to 1, etc), and a significant amount of additional complications seems needed to get what inflation naturally does.
19. matteoeo says:
I’m sorry I went off topic and I will retain from writing again, but nevertheless I think it’s interesting to see how the scientific method has been mistreated and pseudo-scientific claims are made in almost any field of natural sciences and humanistic “sciences”.
Or do you think that this bad string theory story is just an occasional mistake soon to be recovered?
My question was: do we know enough of the physics of the 20th century before adventuring in the physics of the 21st? I don’t think this question is off-topic.
20. Peter Woit says:
There are all sorts of problematic claims made in different sciences. I just don’t want this blog turned into a discussion forum about all of them, but want to keep it focused on things I know about and am willing to moderate discussions of. The question of the evidence for inflation is an interesting one, and “off-topic” makes to-the-point comments, but I’m not an expert on this, and there are good blogs out there run by people who are, so that’s where the discussion should really take place.
My point of view is certainly that the Standard Model QFT remains poorly understood in many ways, and that problem deserves more attention. There are lots of other issues in physics that aren’t well-understood, but again, I don’t want to moderate discussions of issues I don’t know much about.
21. Robert says:
If your words were as reasonable as your slides congratulations for this nice presentation. For the philosophy of science section of the German Physics Societey I had indended to give a talk with a very similar subject (but of course slightly different conclusions). Unfortunately for personal reasons I could not attend the conference.
Just a minor point of nitpicking (and we have discussed this before): When you say there is no clear cut experimental prediction I would qualify that with “to be performed with currently available experimental technology”. Otherwise I strongly believe your claim is wrong, at least if a weakly coupled description exisists (that is there is — possibly after a duality — a stringy description with g
22. Robert says:
Sorry for the sudden end of the previous comment. I wanted to say g less then less than (i.e. \ll in TeX) 1 but typing that froze my firefox (probably the script that does the preview. Luckily, I did not lose the post as after a few minutes it popped up a box asking me if I wanted to cancel a script. So I could still press the submit button. But there seems to be a bug either in the script or in firefox…
23. Robert Musil says:
Please correct me if I mistake your views, but I believe you have several times made clear that while you harbor skepticism over many aspects of string theory as physics, you believe that much extraordinary and important mathematics has resulted from string theory. The “Mirror Conjecture” is one such example. Admiration for string theory mathematics spin-offs is widely shared by many of the worlds leading mathematicians.
But there are some very troubling aspects to even this, very real, admiration for string theory inspired mathematics – at least to my eye. It’s trivial to formulate the Mirror Conjecture: Just flip the Hodge array on the diagonal and ask for a variety. But nobody bothered to ask the question before M-theory was posited. Moreover, the first few examples of the Mirror Conjecture are not hard to prove (although the entire conjecture is), yet nobody bothered to investigate them before M-Theory was posited. One main (or at least common) example that supposedly demonstrates the mathematical importance of the Mirror Conjecture – finding those curves – was being pursued (apparently) by exactly two Norwegians on a computer before the Mirror Conjecture came up. Yet the Mirror Conjecture is supposed to be ultra-important mathematics. There is something very strange here.
Perhaps what is strange here is reflected (oops! and unintentional pun) in the constant references to physics in all mathematical programs regarding the Mirror Conjecture (or at least the ones with which I am familiar). “Golly,” the mathematicians seem to say, “What I’m noodling over has relevance to the real world! It must be important mathematics!” But if it turns out that string theory is not important physics, I believe it would be a first if the associated mathematics were really all that important – regardless of the level of enthusiasm it has inspired. After all, string theory inspired quite a lot of ill-considered, unchallenged enthusiasm as physics for quite a while.
In other words, I can’t shake the sense that the enthusiasm over the Mirror Conjecture (for example) has itself a hall-of-mirrors aspect: Mathematicians (even very good ones) love it supposedly because it is “intrinsically” wonderful mathematics. But it’s a strange kind of intrinsically wonderful mathematics that nobody gave a dam about before the physics came along in the form of string theory – even though it’s wonderful mathematics whose formulation is trivial and whose first few examples are easy and whose supposedly important applications nobody cared about enough to work on but two Norwegians (not that I have anything against Norwegians, mind you).
Of course, on the other side of the hall of mirrors we find the string theorists reassuring themselves that their theory must be important (or even correct) because the mathematics is so wonderful. Bing, bing, bing goes the wonderful image across the hall – each time a little more distorted as it recedes.
Personally, I find this hall of mirrors aspect of things disturbing, perhaps because I associate halls of mirrors with lower-budget hotel lobbies trying to look bigger than they are. Somehow I get a similar feeling from the mathematical spin-offs of string theory.
Do you have anything to say on this?
24. r hofmann says:
Dear Robert Musil,
although I have no idea about the Mirror Conjecture what you say about it and its embedding into the modern relationship between physics and mathematics strikes me as an intelligent observation. Thanks for the info.
25. David B. says:
Dear Robert:
Mirror symmetry is not just about “flipping the Hodge diamond”.
When you say
You are trivializing the contribution from physicists and mathematicians. The truth is that mathematicians had not suspected that the problem of counting curves in Calabi-Yau manifolds (a typical problem of enumerative geometry) could be related to the theory of deformations of the complex structure on the mirror geometry.
You are also trivializing the problem by making statements like “even though it’s wonderful mathematics whose formulation is trivial and whose first few examples are easy”.
The formulation is not trivial at all, and it took quite a while before someone produced a complete mathematical proof of the first few examples.
I don’t like these misinformed statements about the relationship between research in string theory and mathematics. They seem to be crafted for purposefully misleading the public at large.
Many profesionals use simple statements like “flipping the hodge diamond” when giving presentations in order to explain the simplest aspects of mirror symmetry to an uninformed audience, and to try to give them something they might relate to. In this way they can share the excitement of the subject. Don’t mistake those statements for the research that is done in the subject.
26. Peter Woit says:
Robert (non-Musil),
The slides pretty accurately reflect what I said. In this talk I wanted to just as clearly as possible state the facts of the matter and avoid any editorializing.
One thing that I should have put in the slides was a comment about the issue you raise, the claim that the testability problem for string theory only arises at low energy, that if we could do Planck scale experiments, it would be testable. I think we’ve probably discussed this before, but I would claim that the string theory framework continues to be not testable even at that scale. As you acknowledge, even a qualitative prediction of the kind I assume you have in mind (standard distinctive aspects of perturbative string spectra or scattering amplitudes) rely on the string coupling being small enough for the perturbation approximation to be good. Such a prediction is not falsifiable, since it could be evaded simply by saying “well, maybe the string coupling really is not small enough”.
In practice, it is true that if we could do experiments at arbitrarily high scales, we’d presumably see what the structure of quantum gravitational effects is, and would see whether this looked at all like anything that had ever shown up in studies of string theory.
Robert (Musil)
David B. is right. The “Mirror Conjecture” and the associated mathematics it has generated go far, far beyond what you mention and are much deeper than “flipping the Hodge diamond”. As an example of this, next week at the IAS they’ll be an important mathematics workshop on “Homological Mirror Symmetry”, focusing on relations to the geometric Langlands program. This is a very active and important area in matthematics. It has pretty much nothing to do with attempts to unify physics via string theory, but it’s great mathematics, and maybe someday it will turn around and inspire some physics.
27. Robert Musil says:
Thank you for your as-always thoughtful response. David B. seems a very intelligent and knowledgeable (if somewhat excitable) fellow, but he is certainly not right in mischaracterizing me as asserting that the Mirror Conjecture ends with the Hodge Diamond formulations. Indeed, I’m not aware of any comprehensive formulation of the Mirror Conjecture. Manifolds with mirror-symmetric Hodge tables are called geometrical mirrors. My point in this regard is (and was) that the Hodge Diamond formulation is trivial to state and notice and that nobody had bothered to do either prior to the positing of M Theory. Yet now that very formulation is deemed to be inherently wonderful mathematics. Of course, this is not an argument for dismissing or downgrading the significance of any version of the Mirror Conjecture. But to start the discussion it does help to get the question right.
Nor is David B.’s assertion that there are no easy examples of Mirror Symmetry right. Indeed, it is not that hard to find references to this fact in papers by central practitioners in the field. Of course, some of the known examples were by no means easy.
As for the geometric Langlands program, I’m not knowledgable in the area of mathematics. I realize that geometric Langlands is an active area of research considered promising by many very smart people. But promise and “rich” structure alone didn’t make string theory great – or even important – physics. I’m not sure if I see why one can already conclude that Geometric Langlands is great mathematics – and evaluating the importance of Mirror Conjecture relationships to GL is another step after that.
28. Peter Shor says:
If somebody comes up tomorrow with a beautiful new theory which unifies gravity and QM and is much simpler than string theory, and if the LHC produces results that agree with its predictions, I assume that nearly all the string theorists will drop their current research and jump on the bandwagon.
The real question is (a) without any hints of an alternative, are any of them going to abandon string theory research, no matter how unpromising it looks and (b) whether a hint of a promising alternative is enough, or whether it takes a fully formed theory. For instance, if the LHC produces a Higgs mass close to that predicted by Connes, are any of the string theorists going to take this as a hint that maybe they’re on the wrong track, and Connes on the right one?
Any wagers on this?
29. Kea says:
Any wagers on this?
What are we betting on? How long it will take the String theorists to figure out what’s going on? Actual experimental outcomes at the LHC? Oooohhh, this is fun.
30. A.J. says:
Several comments for Robert Musil:
1) The relationship between Hodge diamonds predates M-theory by several years. It’s part of the story physicists like to tell about M-theory and a simple example of a mirror phenomenon, but I don’t think it’s of deep importance. More of a decorative note.
2) I’ve never heard anyone claim that the existence of manifolds with mirror hodge diamonds was the important or deep part of the story. Complaining that others are calling it “inherently wonderful mathematics” seems like a bit of a straw man. Who exactly has said this?
3) What is important, as David B. more or less pointed out, is that we can relate moduli spaces of complex structures to moduli spaces of symplectic structures. This is incredibly non-trivial, and potentially very useful.
4) While I agree that “This shows up in string theory” isn’t necessarily a good rationale for a mathematical research problem, I think it’s a poor reason to dislike good mathematical ideas. And the notion that there’s a topological field theory which carries information about the space of curves and maps to a fixed target has proven to be a fertile source of algebro-geometric ideas.
31. A.J. says:
Peter (Shor):
If the Connes et al prediction comes out right, I imagine that some people will take the hint and start working on it. On the other hand, I’ve also seen some stringy speculation around the fact that the noncommutative space in Connes, Marcolli, & Chamseddine has KO dimension 6.
32. Robert Musil says:
You are quite right in that current interest in mirror manifolds is due to the idea is that along with the equality h1,1(X) = h2,1(Y ) of moduli numbers of Kahler structures on X and of complex structures on Y, the whole symplectic topology on X is equivalent to complex geometry on Y, and vice versa. In that sense perhaps I should have been more explicit about the means of establishing the Hodge equivalences. But the first examples of this equivalence are not hard, nobody was looking at them, etc.
I’m not sure what you mean by “The relationship between Hodge diamonds predates M-theory by several years,” unless you are referring to the earlier computer results. I’m not aware of any general Hodge diamond conjecture that predated the positing of M Theory.
All that being said, I don’t see why my points don’t still stand. For example, while I don’t mean to be snide or obtuse, neither do I see why the assertion that something is “a fertile source of algebro-geometric ideas” is a very good basis for concluding that those ideas or their source are important. The argument seems to completely assume its conclusion. Am I missing part of your point?
There is clearly great and broad enthusiasm for some mathematics derived from (or spun off from) string theory – much of it among very smart and accomplished people. But there was (and is) just such enthusiasm for string theory itself – an enthusiasm only recently seriously challenged. That challenge has been made from one redoubt: In physics one at least has the check on the products of such enthusiasms that at some point or other those products must be EXPERIMENTALLY TESTABLE (although, as this blog cogently points out, some string theory practitioners are struggling mightily to avoid even that check). There is no such check in mathematics. So how do we know that the mathematics spun off from string theory are not just empty enthusiasms? It’s just silly to deny that a lot (as Peter points out, perhaps not all) of the enthusiasm is derived directly or indirectly from string theory itself. To make matters worse, some of the best mathematicians speaking to the public about mathematics spun off from string theory often make claims for its importance that are absurdly over the top (Michael Atiyah, for example). Certainly just asserting that one thing or another is “great mathematics” or the like doesn’t advance matters, does it? What does?
33. A.J. says:
OK first, most of the ideas of mirror symmetry predate M-theory by several years. The former is part of the body of evidence for the latter. If you want more direct evidence: Kontsevich’s homological mirror symmetry lecture is from the summer of 94; Witten’s M-theory announcement from the fall of 1995.
Second, why are we still talking about Hodge equivalences? This is a hint that there’s something interesting going on, not the end goal of any major research efforts.
I don’t understand what your metric for “importance” is. But it seems to me that algebraic geometers have judged Gromov-Witten theory to be important and interesting because of the ideas it’s brought into their field, not because it’s connected in some way to a much larger program in a different field. So, yes, by this standard, it’s important. If you mean important in some other sense, I really don’t have anything to say to you.
My point basically is this: You have a reasonable abstract point about a potential relationship between relative levels of enthusiasm about physics and mathematics, and a cute metaphor about hotel mirrors to go with it. But I think you’re quite wrong to single out mirror symmetry as an example of the phenomenon you’re talking about.
And I suspect you will have a hard time finding actual examples. It’s true that some mathematicians like to talk and daydream about important physics connections, but I think you’ll find that the physics-derived ideas which mathematicians have really taken the time to develop intensely have been those which are useful and interesting as as mathematics.
34. Robert Musil says:
You mention “I don’t understand what your metric for “importance” is.” Well, let’s take that seriously. Terry Tao advanced a set of criteria for “bad mathematics” that I believe were discussed in this blog a while back:
• A field which becomes increasingly ornate and baroque, in which individual results are generalised and refined for their own sake, but the subject as a whole drifts aimlessly without any definite direction or sense of progress; or
• A field which becomes filled with many astounding conjectures, but with no hope of rigorous progress on any of them; or
• A field which now consists primarily of using ad hoc methods to solve a collection of unrelated problems, which have no unifying theme, connections, or purpose; or
• A field which has become overly dry and theoretical, continually recasting and unifying previous results in increasingly technical formal frameworks, but not generating any exciting new breakthroughs as a consequence; or
• A field which reveres classical results, and continually presents shorter, simpler, and more elegant proofs of these results, but which does not generate any truly original and new results beyond the classical literature.
Is it clear that the mathematics spun off from string theory has avoided each of these? It seems at least arguable that one, perhaps more, of these criteria fit uncomfortably well. Not that an answer to this would end the discussion, of course.
35. Robert Musil says:
I first want to be very clear that I appreciate your thoughtfulness and intelligent comments. I also want to apologize in advance for popping in this second post before you have a chance to respond to or digest the first.
With respect to Kontsevich’s seminal address at ICM, Zurich 1994, it is worth keeping in mind that Kontsevich’s himself characterized what he was doing as follows (I quote from his address):
“Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has
dim Hp(V,q) = dim Hn−p(W,
q). ….
“We describe here a not yet completely constructed theory which has potentially wider domain of applications than mirror symmetry. It is based on pioneering ideas of M. Gromov on the role of ∂-equations in symplectic geometry, and certain physical intuition proposed by E. Witten.”
The relevant references to Witten’s “intuitions” are to two papers: Topological sigma models, Commun. Math. Phys. 118 (1988), 411-449 and Two-dimensional gravity and intersection theory on moduli space, Surveys in Di . Geom. 1 (1991), 243–310.
I believe these quotes address several questions and concerns expressed in your posts above (why we are talking about Hodge numbers, for example). I also believe these passages support my points.
36. A.J. says:
Tao didn’t give that list as criteria for bad mathematics. It’s just a list of dangers (somewhat exaggerated as Tao admits) which might have detrimental effects on the development of a field. I think it’s misleading to treat it as checklist for identifying “bad mathematics”.
That said, the only danger I see being remotely applicable is is the 2nd one. But I don’t think it’s a particularly great danger. For one thing, judicious borrowing of physical intuition has a pretty good track record. (Donaldson theory, Chern-Simons, knot polynomials, mirror symmetry, Seiberg-Witten theory, and so on.) And for another, mathematicians have a habit of concentrating on problems they think are solvable. No one is butting heads with 4d Yang-Mills theory right now, because it’s probably out of reach. But there’s lots of motion in the Gromov-Witten theory of orbifolds right now; people are getting things done.
37. A.J. says:
I don’t see how the Kontsevich quotes support your points. Perhaps you’d care to explain? You’ll probably have to take some care to spell out carefully what you mean, since we seem to be talking at angles.
Some of the confusion may stem from the term mirror symmetry. The symmetry gets its name from the duality of the hodge diamonds, but it’s just a name. The actual set of ideas involved is considerably richer than the name implies. Most of it has been developed in the years since Kontsevich’s lecture.
38. David Williams says:
Please have mercy on an old scocial science Phd.
I have had, basically, only a pragmatic and professional education except for a couple of biology courses and a stint as a biology teaching assistant (where I first encountered the scientific method) but I have indulged my interest in popularized science writing. I use this information in debating the champions of religion.
In debate the basic successful argument is that Science is not based on belief but on questioning and testing. Recently, String theory has become widely accepted in physics. I love the idea in the sense that it tells us that the universe is a symphony. HOWEVER, String theory appears to arrive at the position of a Unified Field Theory only by relying upon
a. mathematical solutions b. solutions that require positing multiple universes.
May I ask you these questions.
In your opinion are mathematical solutions the equivalent of an empirical test? Although I’m told (and I simply have to accept or not – at the level of my math and science skills) that M theory will offer an opportunity to empirically test String Theory. I cannot, to my satisfaction, imagine an empirical test for multiple universes.
And, If empirical tests are not available by the very nature of String Theory is this idea no better than religious belief?
I am inclined, therefore, to simply leave String theory to it’s own devises and conclude that it lacks scientific credibility and that we are stuck with the contradictions between General Relativity and Quantum Dynamics. We would otherwise be as lacking in evidence as the religious. Why have physicists so departed from scientific standards?
Hope you will be able to spare the time answer this query.
David P. Williams, PhD
3181 Micmac St. Halifax
Nova Scotia Canada B3L 3W3
(902) 454
39. tomj says:
I have similar concerns. String theory is hyped or hoped beyond belief. This is serious, because if a scientist is supposed to be exact and careful about their theory, their work, etc., why doesn’t this carry over into their public descriptions?
I somehow stumbled across this site a few weeks ago. But for several years I have firmly believed that there was something not right about the ‘theory of everything’ crowd.
At that point I was trying to track down some more concrete details about these theories. But nothing concrete ever appeared. Instead, I ran across some made up cafeteria dialog between a string theorist and (I guess) a LQG theorist. I think the point of the dialog was to highlight the lack of evidence for either theory, but more important for me was another principle: science is about the unknown, not the known or the unknowable.
If science is expanded to cover the unknowable, you forfeit the ability to apply Occam’s razor. Occam’s razor isn’t a theory, it isn’t a law of nature, it is a check on logic: it requires experiment. If a theory has no experimental results, how can you compare it to one that does? If a theory predicts unknowables like multiple universes, how can this win out over a theory that predicts only the one we experience?
My problem with the proponents of string theory is that their ideas fall into the category of ‘known’ or ‘unknowable’. That is, their statements lead me to believe that they know something (strings are the basic building blocks of everything) or their theory covers stuff we can’t know (multiple universes, etc.). In the first case, they are lying, or using language in a very sloppy way. If they are sloppy with English, why should I think that they are not sloppy in their math or logic?
What I don’t understand is that if a scientist makes wild statements that they ‘know’ something or that their theory implies ‘unknowable’ realities, why shouldn’t I remember their unscientific approach? Either put up, or shut up.
Known = technology
Unknown = science
Unknowable = fantasy
40. r hofmann says:
it’s of undeniable educational value to follow the debate between Robert Musil and yourself.
This statement is, however, outright false and confirms
pretty much the relevance of Terry Tao’s above quoted criteria.
Best, RH
41. Ralf says:
David and tomj,
String theory/M-theory is a speculative research program which therefore would not be covered at all in the popularized science press if the latter were responsible. Officially, string theory is “accepted” exactly as that. In practice this doesn’t stand in the way of string theorists taking over high energy physics, in part because the field has been short of new ideas for three decades. In such cases the subjective criteria for what can be regarded “reasonable” ideas become rather flexible. The “scientific method” exists only in the imagination of philosophers of science. “Occam’s razor” cannot be “applied” like a theoretical analog of a lab test.
People can be honestly deluded about things that are crucial to their identity, like their love life, their social life or their professional world. In addition, string theorists view themselves as intellectually superior to everybody else, which automatically degrades any objections brought up by those.
Finally, in reality there is simply nothing about string theory that can be related to laypeople. Anybody who writes about it is only heaping nonsense on a foundation of nonsense which again rests on a foundation of nonsense. It is utter intellectual dishonesty to pretend otherwise. Supersymmetry, by itself, cannot possibly be assessed by a non-physicist. Every account makes it appear much more reasonable than it is. Grand Unification sounds almost like a no-brainer if one doesn’t know the details. The technical details on which string theory is built—and which are never even mentioned in the popular press—render it, in my opinion, deranged and demented. And it is exactly this wide gap between actual physics and string theory that—perversely—facilitates the public’s susceptibility for it. The public never registered anything from the Schrödinger equation onward because they don’t like the absence of visualizability.
That is why they prefer the faux visualizability of General Relativity—and of string theory, of course. Physics is not the Riemannian geometry of the 19th century. It was Einstein, after all, who commented that one should explain everything as simply as possible—but not simpler.
42. a says:
dear David,
let me try to give an answer to your “Why have physicists so departed from scientific standards?”. It is oversimplified and caricatural, but I think it captures a relevant aspect of the question. Do you believe that an average rational human being would choose option A or B?
Option A is what is happening now.
Option B is “I spent my life working on strings, but, contrarily to what press said, initial hopes mostly disappeared. Maybe I could start doing some other physics, but I only have expertise in strings, that is a highly specialized topic: so I resign from my academic job”
43. A.J. says:
R. Hofmann:
Sorry about that. I was not expressing myself clearly. (Why can’t you people just read my mind?!) A precise formulation: “Few if any mathematicians are attempting to construct 4d Yang-Mills theory in the sense required by the Clay Millenium prizes.” Obviously plenty of people are thinking about 4d Yang-Mills in a non-rigorous fashion, or trying to work out various facts about its topological analogues. But no one’s managed to do anything interesting as far as construction & mass gap goes.
44. r hofmann says:
I see … That problem was formulated by E. Witten and a famous Harvard mathematical physicist, right?
Best, RH
45. A.J. says:
I don’t know anything about how the Clay Foundation works, but at the least the problem description was written by Edward Witten and Arthur Jaffe.
46. Ari Heikkinen says:
Just a couple of questions, when you say:
Do you mean by this that what’s in Greene’s book (that “beautiful idea”) of particles being tiny vibrating strings of which amplitude and wavelength corresponds to different masses and force charges of them and that those “extra” dimension are curled up in Calabi-Yau shapes are what disagree with experiment?
And by this something that Greene’s book don’t mention?
47. Peter Woit says:
One of the main problems is that you have to do something to fix the size and shape of the Calabi-Yaus, and the only ways people have found to do this involve introducing a lot of complex, ad hoc structure. This is the “moduli problem”, and I don’t remember what Brian says about it in his book. His book was written now quite a few years ago, before people had any solution at all to the problem. Back then I suspect there was a lot more optimism that a simple solution could be found.
Comments are closed. |
5830bab8eef65405 | Implications of a deeper level explanation of the deBroglie–Bohm version of quantum mechanics
• G. Grössing
• S. Fussy
• J. Mesa PascasioEmail author
• H. Schwabl
Regular Paper
Elements of a “deeper level” explanation of the deBroglie–Bohm (dBB) version of quantum mechanics are presented. Our explanation is based on an analogy of quantum wave-particle duality with bouncing droplets in an oscillating medium, the latter being identified as the vacuum’s zero-point field. A hydrodynamic analogy of a similar type has recently come under criticism by Richardson et al. (On the analogy of quantum wave-partile duality with bouncing droplets, 2014), because despite striking similarities at a phenomenological level the governing equations related to the force on the particle are evidently different for the hydrodynamic and the quantum descriptions, respectively. However, said differences are not relevant if a radically different use of said analogy is being made, thereby essentially referring to emergent processes in our model. If the latter are taken into account, one can show that the forces on the particles are identical in both the dBB and our model. In particular, this identity results from an exact matching of our emergent velocity field with the Bohmian “guiding equation”. One thus arrives at an explanation involving a deeper, i.e., subquantum, level of the dBB version of quantum mechanics. We show in particular how the classically local approach of the usual hydrodynamical modeling can be overcome and how, as a consequence, the configuration-space version of dBB theory for N particles can be completely substituted by a “superclassical” emergent dynamics of N particles in real three-dimensional space.
Quantum mechanics Hydrodynamics DeBroglie–Bohm theory Guiding equation Configuration space Zero-point field
Mathematics Subject Classification
1 Introduction
The Schrödinger equation for \(N>1\) particles does not describe a wave function in ordinary three-dimensional space, but instead in an abstract \(3N\)-dimensional space. For quantum realists, including Schrödinger and Einstein, for example, this has always been considered as “indigestible”. This holds even more so for a realist, causal approach to quantum phenomena such as the deBroglie–Bohm (dBB) version of quantum mechanics. David Bohm himself has admitted this, calling it a “serious problem”: “While our theory can be extended formally in a logically consistent way by introducing the concept of a wave in a \(3N\)-dimensional space, it is evident that this procedure is not really acceptable in a physical theory, and should at least be regarded as an artifice that one uses provisionally until one obtains a better theory in which everything is expressed once more in ordinary three-dimensional space” [1]. (For more detailed accounts of this discussion already in the early years of quantum mechanics, see [17, 18].)
In the present paper, we shall refer to our attempt towards such a “better theory” in terms of a deeper level, i.e., subquantum, approach to the dBB theory, and thus to quantum theory in general. In fact, with our model, we have in a series of papers already obtained several essential elements of nonrelativistic quantum theory [8, 9, 13, 14]. They derive from the assumption that a particle of energy \(E=\hbar \omega \) is actually an oscillator of angular frequency \(\omega \) phase-locked with the zero-point oscillations of the surrounding environment, the latter containing both regular and fluctuating components and being constrained by the boundary conditions of the experimental setup via the buildup and maintenance of standing waves. The particle in this approach is an off-equilibrium steady-state oscillation maintained by a constant throughput of energy provided by the (“classical”) zero-point energy field. We have, for example, applied the model to the case of interference at a double slit, thereby obtaining the exact quantum mechanical probability density distributions on a screen behind the double slit, the average trajectories (which because of the averaging are shown to be identical to the Bohmian ones), and the involved probability density currents. Our whole model is constructed in close analogy to the bouncing/walking droplets above the surface of a vibrated liquid in the experiments first performed by Couder and Fort [4, 5], Fort and co-workers [6], which in many respects can serve as a classical prototype guiding our intuition for the modeling of quantum systems.
However, there are also obvious differences between the mentioned physics of classical bouncers/walkers on the one hand, and the hydrodynamic-like models for quantum systems like our own model or the dBB on the other hand. In a recent paper, Richardson et al. [20] have probed more thoroughly into the hydrodynamic analogy of dBB-type quantum wave-particle duality with that of the classical bouncing droplets. Apart from the obvious difference in that Bohmian theory is distinctly nonlocal, whereas droplet–surface interactions are rooted in classical hydrodynamics and thus in a manifestly local theory, Richardson et al. focus on the following observation: the evidently different nature of the Bohmian force upon a quantum particle as compared to the force that a surface wave exerts upon a droplet. In fact, wherever the probability density in the dBB picture is close to zero, the quantum force becomes singular and will very quickly push any particle away from that area. Conversely, the hydrodynamic force directs the droplet into the trough of the wave! So, the probability of finding a droplet in the minima never reaches zero as it does for a quantum particle. The authors conclude that these discrepancies between the two models highlight “a major difference between the hydrodynamic force and the quantum force” [20].
Although these authors generally recover in numerical hydrodynamic simulations the results of the Paris group (later confirmed also by the group of Bush [3] at MIT) on single-slit diffraction and double-slit interference, they also point out the (already known) striking contrast between the trajectory behaviors for the bouncing droplet systems and dBB-type quantum mechanics, respectively. Whereas the latter exhibits the well-known no-crossing property, the trajectories of the former do to a large extent cross each other. So, again, the physics in the two models is apparently fundamentally different, despite some striking similarities on a phenomenological level. As to the differences, one may very well expect that they will even become more severe when moving from one-particle to \(N\)-particle systems.
So, all in all, the paper by Richardson et al. [20] cautions against the assumption of too close a resemblance of bouncer/walker systems and the hydrodynamic-like modeling of quantum systems like the dBB, with their main argument being that the hydrodynamic force on a droplet strikingly contrasts with the quantum force on a particle in the dBB theory. However, we shall here argue against the possible conclusion that one has thus reached the limits of applicability of the hydrodynamic bouncer analogy for quantum modeling. On the contrary, as we have already pointed out in previous papers, it is a more detailed model inspired by the bouncer/walker experiments that can show the fertility of said analogy. It enables us to show that our model, being of the type of an “emergent quantum mechanics” [10, 11], can provide a deeper level explanation of the dBB version of quantum mechanics (Sect. 2). Moreover, as we shall also show, it turns out to provide an identity of an emergent force on the bouncer in our hydrodynamic-like model with the quantum force in Bohmian theory (Sect. 3). Finally, in Sect. 4, we shall discuss the “price” to be paid to arrive at our explanation of dBB theory in that some kind of nonlocality, or a certain “systemic nonlocality”, has to be admitted in the model from the start. However, the simplicity and elegance of our derived formalism, combined with arguments about the reasonableness of a corresponding hydrodynamic-like modeling, will show that our approach may be a viable one w.r.t. understanding the emergence of quantum phenomena from the interactions and contextualities provided by the combined levels of classical boundary conditions and those of a subquantum domain.
2 Identity of the emergent kinematics of \(N\) bouncers in real three-dimensional space with the configuration-space version of deBroglie–Bohm theory for \(N\) particles
Consider one particle in an \(n\)-slit system. In quantum mechanics, as well as in our quantum-like modeling via an emergent quantum mechanics approach, one can write down a formula for the total intensity distribution \(P\) which is very similar to the classical formula. For the general case of \(n\) slits, it holds with phase differences \(\varphi _{ij}=\varphi _{i}-\varphi _{j}\) that
$$\begin{aligned} P=\sum _{i=1}^{n}\left( P_{i}+\sum _{j=i+1}^{n}2R_{i}R_{j}\cos \varphi _{ij}\right) , \end{aligned}$$
where the phase differences are defined over the whole domain of the experimental setup. Apart from the role of the relative phase with important implications for the discussions on nonlocality [14], there is one additional ingredient that distinguishes (1) from its classical counterpart, namely the “dispersion of the wavepacket”. As in our model the “particle” is actually a bouncer in a fluctuating wave-like environment, i.e., analogously to the bouncers of Couder and Fort’s group, one does have some (e.g., Gaussian) distribution, with its center following the Ehrenfest trajectory in the free case, but one also has a diffusion to the right and to the left of the mean path which is just due to that stochastic bouncing. Thus the total velocity field of our bouncer in its fluctuating environment is given by the sum of the forward velocity \(\mathbf {v}\) and the respective diffusive velocities \(\mathbf {u}_{\mathrm {L}}\) and \(\mathbf {u}_{\mathrm {R}}\) to the left and the right. As for any direction \(i\) the diffusion velocity \(\mathbf {u}_{{i}}=D\frac{\nabla _{i}P}{P}\) does not necessarily fall off with the distance, one has long effective tails of the distributions which contribute to the nonlocal nature of the interference phenomena [14]. In sum, one has three, distinct velocity (or current) channels per slit in an \(n\)-slit system.
We have previously shown [7, 15] how one can derive the Bohmian guidance formula from our bouncer/walker model. To recapitulate, we recall the basics of that derivation here. Introducing classical wave amplitudes \(R(\mathbf {w}_{i})\) and generalized velocity field vectors \(\mathbf {w}_{i}\), which stand for either a forward velocity \(\mathbf {v}_{i}\) or a diffusive velocity \(\mathbf {u}_{i}\) in the direction transversal to \(\mathbf {v}_{i}\), we account for the phase-dependent amplitude contributions of the total system’s wave field projected on one channel’s amplitude \(R(\mathbf {w}_{i})\) at the point \((\mathbf {x},t)\) in the following way: We define a conditional probability density \(P(\mathbf {w}_{i})\) as the local wave intensity \(P(\mathbf {w}_{i})\) in one channel (i.e., \(\mathbf {w}_{i}\)) upon the condition that the totality of the superposing waves is given by the “rest” of the \(3n-1\) channels (recalling that there are three velocity channels per slit). The expression for \(P(\mathbf {w}_{i})\) represents what we have termed “relational causality”: any change in the local intensity affects the total field, and vice versa, any change in the total field affects the local one. In an \(n\)-slit system, we thus obtain for the conditional probability densities and the corresponding currents, respectively, i.e., for each channel component \( i \),
$$\begin{aligned} P(\mathbf {w}_{i})&= R(\mathbf {w}_{i})\hat{\mathbf {w}}_{i}\cdot {\displaystyle \sum _{j=1}^{3n}}\hat{\mathbf {w}}_{j}R(\mathbf {w}_{j})\end{aligned}$$
$$\begin{aligned} \mathbf {J}\mathrm {(}\mathbf {w}_{i}\mathrm {)}&= \mathbf {w}_{i}P(\mathbf {w}_{i}),\qquad i=1,\ldots ,3n, \end{aligned}$$
$$\begin{aligned} \cos \varphi _{i,j}:=\hat{\mathbf {w}}_{i}\cdot \hat{\mathbf {w}}_{j}. \end{aligned}$$
Consequently, the total intensity and current of our field read as
$$\begin{aligned} P_{\mathrm {tot}}=&{\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})=\left( {\displaystyle \sum _{i=1}^{3n}}\hat{\mathbf {w}}_{i}R(\mathbf {w}_{i})\right) ^{2}\end{aligned}$$
$$\begin{aligned} \mathbf {J}_{\mathrm {tot}}=&\sum _{i=1}^{3n}\mathbf {J}(\mathbf {w}_{i})={\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i}), \end{aligned}$$
leading to the emergent total velocity
$$\begin{aligned} \mathbf {v}_{\mathrm {tot}}=\frac{\mathbf {J}_{\mathrm {tot}}}{P_{\mathrm {tot}}}=\frac{{\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i})}{{\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})}\,. \end{aligned}$$
In [7, 15], we have shown with the example of \(n=2,\) i.e., a double-slit system, that Eq. (7) can equivalently be written in the form
$$\begin{aligned} \mathbf {v}_{\mathrm {tot}}=\frac{R_{1}^{2}\mathbf {v}_{\mathrm {1}}+R_{2}^{2}\mathbf {v}_{\mathrm {2}}+R_{1}R_{2}\left( \mathbf {v}_{\mathrm {1}}+\mathbf {v}_{2}\right) \cos \varphi +R_{1}R_{2}\left( \mathbf {u}_{1}-\mathbf {u}_{2}\right) \sin \varphi }{R_{1}^{2}+R_{2}^{2}+2R_{1}R_{2}\cos \varphi }\,. \end{aligned}$$
The trajectories or streamlines, respectively, are obtained according to \(\dot{{\mathbf {x}}}=\mathbf {v}_{\mathrm {tot}}\) in the usual way by integration. As first shown in [13], by re-inserting the expressions for convective and diffusive velocities, respectively, i.e., \(\mathbf {v}_{i}=\frac{\nabla S_{i}}{m}\), \(\mathbf {u}_{i}=-\frac{\hbar }{m}\) \(\frac{\nabla R_{i}}{R_{i}}\), one immediately identifies Eq. (8) with the Bohmian guidance formula. Naturally, employing the Madelung transformation for each path \(j\) (\(j=1\) or \(2\)),
$$\begin{aligned} \psi _{j}=R_{j}\mathrm{e}^{{i}S_{j}/\hbar }, \end{aligned}$$
and thus \(P_{j}=R_{j}^{2}=|\psi _{j}|^{2}=\psi _{j}^{*}\psi _{j}\), with \(\varphi =(S_{1}-S_{2})/\hbar \), and recalling the usual trigonometric identities such as \(\cos \varphi =\frac{1}{2}\left( \mathrm{e}^{{i}\varphi }+\mathrm{e}^{-{i}\varphi }\right) \), one can rewrite the total average current immediately in the usual quantum mechanical form as
$$\begin{aligned} \begin{array}{ll} {\displaystyle \mathbf {J}_{\mathrm{tot}}} &{} =P_{\mathrm{tot}}\mathbf {v}_{\mathrm{tot}}\\ &{} ={\displaystyle (\psi _{1}+\psi _{2})^{*}(\psi _{1}+\psi _{2})\frac{1}{2}\left[ \frac{1}{m}\left( -{i}\hbar \frac{\nabla (\psi _{1}+\psi _{2})}{(\psi _{1}+\psi _{2})}\right) +\frac{1}{m}\left( {i}\hbar \frac{\nabla (\psi _{1}+\psi _{2})^{*}}{(\psi _{1}+\psi _{2})^{*}}\right) \right] }\\ &{} ={\displaystyle -\frac{{i}\hbar }{2m}\left[ \Psi ^{*}\nabla \Psi -\Psi \nabla \Psi ^{*}\right] ={\displaystyle \frac{1}{m}{Re}\left\{ \Psi ^{*}(-{i}\hbar \nabla )\Psi \right\} ,}} \end{array} \end{aligned}$$
where \(P_{\mathrm{tot}}=|\psi _{1}+\psi _{2}|^{2}=:|\Psi |^{2}\).
Equation (7) has been derived for one particle in an \(n\)-slit system. However, it is straightforward to extend this derivation to the many-particle case. Due to the purely additive terms in the expressions for the total current and total probability density, respectively, also for N particles, the only difference now is that the currents’ nabla operators have to be applied at all of the locations of the respective N particles, thus providing the quantum mechanical formula
$$\begin{aligned} {\displaystyle \mathbf {J}_{\mathrm{tot}}}\left( N\right) ={\displaystyle \sum _{i=1}^{N}}\frac{1}{m_{i}}{Re}\left\{ \Psi \left( t\right) ^{*}(-{i}\hbar \nabla _{i})\Psi \left( t\right) \right\} , \end{aligned}$$
where \(\Psi \left( t\right) \) now is the total \(N\)-particle wave function, whereas the total velocity fields are given by
$$\begin{aligned} \mathbf {v}_{i}\left( t\right) =\frac{\hbar }{m_{i}}\mathrm {Im}\frac{\nabla _{i}\Psi \left( t\right) }{\Psi \left( t\right) }\;\forall i=1,\ldots ,N. \end{aligned}$$
Note that this result is similar in spirit to that of Norsen et al. [17, 18] who with the introduction of a conditional wave function \(\tilde{\psi }_{i}\), as opposed to the configuration-space wave function \(\Psi \), rewrite the guidance formula, for each particle, in terms of the \(\tilde{\psi }_{i}\):
$$\begin{aligned} \frac{\,\mathrm {d}X_{i}\left( t\right) }{\,\mathrm {d}t}=\frac{\hbar }{m_{i}}\mathrm {Im}\left. \frac{\nabla \Psi }{\Psi }\right| _{\mathbf {\varvec{x}=\varvec{X}\left( t\right) }}\equiv \frac{\hbar }{m_{i}}\mathrm {Im}\left. \frac{\nabla \tilde{\psi }_{i}}{\tilde{\psi }_{i}}\right| _{x=X_{i}\left( t\right) }, \end{aligned}$$
where the \(X_{i}\) denote the location of one specific particle and \(\mathbf {X}\left( t\right) =\left\{ X_{1}\left( t\right) ,\ldots ,X_{N}\left( t\right) \right\} \) the actual configuration point. Thus, in this approach, each \(\tilde{\psi }_{i}\) can be regarded as a wave propagating in physical three-dimensional space.
In sum, with our introduction of a conditional probability \(P(\mathbf {w}_{i})\) for channels \(\mathbf {w}_{i}\), which include subquantum velocity fields, we obtain the guidance formula also for \(N\)-particle systems. Therefore, what looks like the necessity in the dBB theory to superpose wave functions in configuration space in order to provide an “indigestible” guiding wave, can equally be obtained by superpositions of all relational amplitude configurations of waves in real three-dimensional space. The central ingredient for this to be possible is to consider the emergence of the velocity field from the interplay of the totality of all of the system’s velocity channels. We have termed the framework of our approach a “superclassical” one, because in it are combined classical levels at vastly different scales, i.e., at the subquantum and the macroscopic levels, respectively.
3 Identity of the emergent force on a particle modeled by a bouncer system and the quantum force of the deBroglie–Bohm theory
With the results of the foregoing Chapter, we can now return to and resolve the problem discussed in Sect. 1 of the apparent incompatibility between the Bohmian force upon a quantum particle and the force exerted on a bouncing droplet as formulated by Richardson et al. [20]. In fact, already a first look at the bouncer/walker model of our group provides a clear difference as compared to the hydrodynamical force studied by Richardson et al. Whereas the latter investigates the effects of essentially a single bounce on the fluid surface and the acceleration of the bouncer as a consequence of this interaction, our bouncer/walker model for quantum particles involves a much more complex dynamical scenario: we consider the effects of a huge number of bounces, i.e., typically of the order of \(1/{\omega }\), like approximately \(10^{21}\) bounces per second of an electron, which constitute effectively a “heating up” of the bouncer’s surrounding, i.e., the subquantum medium related to the zero-point energy field.
Note that as soon as a microdynamics is assumed, the development of heat fluxes is a logical necessity if the microdynamics is constrained by some macroscopic boundaries like that of a slit system, for example. As we have shown in some detail [12], the thermal field created by such a huge number of bounces in a slit system leads to an emergent average behavior of particle trajectories which is identified as anomalous, and more specifically as ballistic, diffusion. As such, the particle trajectories exiting from, say, a Gaussian slit behave exactly as if they were subject to a Bohmian quantum force. We were also able to show that this applies also to \(n\)-slit systems, such that one arrives at a subquantum modeling of the emergent interference effects at \(n\) slits whose predicted average behavior is identical to that provided by the dBB theory.
It is then easily shown that the average force acting on a particle in our model is the same as the Bohmian quantum force. Due to the identity of our emerging velocity field with the guidance formula, and because they essentially differ only via the notations due to different forms of bookkeeping, their respective time derivatives must also be identical. Thus, from Eq. (7), one obtains the particle acceleration field (using a one-particle scenario for simplicity) in an \(n\)-slit system as
$$\begin{aligned} a_{\mathrm {tot}}\left( t\right)&= \frac{\,\mathrm {d}\mathbf {v}_{\mathrm{tot}}}{\,\mathrm {d}t}=\frac{\,\mathrm {d}}{\,\mathrm {d}t}\left( \frac{{\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i})}{{\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})}\right) \nonumber \\&= \frac{1}{\left( {\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})\right) ^{2}}\left\{ \sum _{i=1}^{3n}\left[ P(\mathbf {w}_{i})\frac{\,\mathrm {d}\mathbf {w}_{i}}{\,\mathrm {d}t}+\mathbf {w}_{i}\frac{\,\mathrm {d}P(\mathbf {w}_{i})}{\,\mathrm {d}t}\right] \left( {\displaystyle \sum _{i=1}^{3n}}P(\mathbf {w}_{i})\right) \,-\left( {\displaystyle \sum _{i=1}^{3n}}\mathbf {w}_{i}P(\mathbf {w}_{i})\right) \left( {\displaystyle \sum _{i=1}^{3n}\frac{\,\mathrm {d}P(\mathbf {w}_{i})}{\,\mathrm {d}t}}\right) \right\} .\nonumber \\ \end{aligned}$$
Note in particular that (14) typically becomes infinite for regions \(\left( \mathbf {x},t\right) \) where \(P_{\mathrm {tot}}={\sum _{i=1}^{3n}}P(\mathbf {w}_{i})\rightarrow 0\), in accordance with the Bohmian picture.
From (14), we see that even the acceleration of one particle in an \(n\)-slit system is a highly complex affair, as it nonlocally depends on all other accelerations and temporal changes in the probability densities across the whole experimental setup! In other words, this force is truly emergent, resulting from a huge amount of bouncer–medium interactions, both locally and nonlocally. This is of course radically different from the scenario studied by Richardson et al. where the effect of only a single local bounce is compared with the quantum force. From our new perspective, it is then hardly a surprise that the comparison of the two respective forces provides distinctive differences. However, as we just showed, with the emergent scenario proposed in our model, complete agreement with the Bohmian quantum force is established.
4 Choose your poison: how to introduce nonlocality in a hydrodynamic-like model for quantum systems?
As already mentioned in the introduction of this paper, purely classical hydrodynamical models are manifestly local and thus inadequate tools to explain quantum mechanical nonlocality. Although nonlocal correlations may also be obtainable within hydrodynamical modeling [2], there is no way to also account for dynamical nonlocality [21] in this manner. So, as correctly observed by Richardson et al. [20], droplet–surface wave interaction scenarios are not enough to serve as a full-fledged analogy of the distinctly nonlocal dBB theory, for example.
The question thus arises how in our much more complex, but still “hydrodynamic-like” bouncer/walker model nonlocal, or nonlocal-like, effects can come about. To answer this, one needs to consider in more detail how the elements of our model are constructed, which finally provide an elegant formula, Eq. (7), identical with the guidance formula in a (for simplicity: one-particle) system with \(n\) slits. (As shown above, the extension to \(N\) particles is straightforward.) As we consider, without restriction of generality, the typical example of Gaussian slits, we introduce the Gaussians in the usual way, with \(\sigma \) related to the slit width, for the probability density distributions (which in our model coincide with “heat” distributions due to the bouncers’ stirring up of the vacuum) just behind the slit. The important feature of these Gaussians is that we do not implement any cutoff for the distributions, but maintain the long tails which actually then extend across the whole experimental setup, even if these are only very small and practically often negligible amplitudes in the regions far away from the slit proper. As the emerging probability density current is given by the denominator of Eq. (8), we see that in fact the product \(R_{1}R_{2}\) may be negligibly small for regions where only a long tail of one Gaussian overlaps with another Gaussian, nevertheless the last term in (8) can be very large despite the smallness of \(R_{1}\) or \(R_{2}\). It is this latter part which is responsible for the genuinely quantum-like nature of the average momentum, i.e., for its nonlocal nature. This is similar in the Bohmian picture, but here given a more direct physical meaning in that this last term refers to a difference in diffusive currents as explicitly formulated in the last term of (8). Because of the mixing of diffusion currents from both channels, we call this decisive term in \(\mathbf {J_{\mathrm { \mathrm {tot} }}=P_{\mathrm{tot}}\mathbf {v}_{\mathrm{tot}}}\) the “entangling current” [16].
Thus, one sees that formally one obtains genuine quantum mechanical nonlocality in a hydrodynamic-like model with one particular “unusual” characteristic: the extremely feeble but long tails of (Gaussian or other) distribution functions for probability densities exiting from a slit extend nonlocally across the whole experimental setup. So, we have nonlocality by explicitly putting it into our model. After all, if the world is nonlocal, it would not make much sense to attempt its reconstruction with purely local means. Still, so far we have just stated a formal reason for how nonlocality may come about. Somewhere in any theory, so it seems, one has to “choose one’s poison” that would provide nonlocality in the end. But what would be a truly “digestible” physical explanation? Here is where at present only some speculative clues can be given.
For one thing, strict nonlocality in the sense of absolute simultaneity of space-like separated events can never be proven in any realistic experiment, because infinite precision is not attainable. This means, however, that very short time lapses must be admitted in any operational scenario, with two basic options remaining: (1) either there is a time lapse due to the finitely fast “switching” of experimental arrangements in combination with instantaneous information transfer [but not signaling; see Walleczek and Grössing (forthcoming)], or (2) the information transfer itself is not instantaneous, but happens at very high speeds \(v\ggg c\).
How, then, can the implementation of nonlocal or nonlocal-like processes with speeds \(v\ggg c\) be argued for in the context of a hydrodynamic-like bouncer/walker model? We briefly mention two options here. Firstly, one can imagine that the “medium” we employ in our model is characterized by oscillations of the zero-point energy throughout space, i.e., between any two or more macroscopic boundaries as given by experimental setups. Between these boundaries standing wave configurations may emerge (similar to the Paris group’s experiments, but now explicitly extending synchronously over nonlocal distances). Here it might be helpful to remind ourselves that we deal with solutions of the diffusion (heat conduction) equation. At least (but perhaps only) formally, any change of the boundary conditions is effective “instantaneously” across the whole setup. Alternatively, if the experimental setup is changed such that old boundary conditions are substituted by new ones, due to the all-space-pervading zero-point energy oscillations, one “immediately” (i.e., after a very short time of the order \(t\sim \frac{1}{\omega }\)) obtains a new standing wave configuration that now effectively implies an almost instantaneous change of probability density distributions, or relative phase changes, for example. The latter would then become “immediately” effective in that changed phase information is available across the whole domain of the extended probability density distribution. We have referred to this state of affairs as “systemic nonlocality” [14]. So, one may speculate that it is something like “eigenvalues” of the universe’s network of zero-point fluctuations that may be responsible for quantum mechanical nonlocality–eigenvalues which (almost?) instantaneously change whenever the boundary conditions are changed.
A second option even more explicitly refers to the universe as a whole, or, more particularly, to spacetime itself. If spacetime is an emergent phenomenon as some recent work suggests [19], this would very likely have strong implications for the modeling and understanding of quantum phenomena. Just as in our model of an emergent quantum mechanics we consider quantum theory as a possible limiting case of a deeper level theory, present-day relativity and concepts of spacetime may be approximations of, and emergent from a superclassical, deeper level theory of gravity and/or spacetime. It is thus a potentially fruitful task to bring both attempts together in the near future.
We thank Jan Walleczek for many enlightening discussions, and the Fetzer Franklin Fund for partial support of the current work.
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Copyright information
© Chapman University 2015
Authors and Affiliations
• G. Grössing
• 1
• S. Fussy
• 1
• J. Mesa Pascasio
• 1
Email author
• H. Schwabl
• 1
1. 1.Austrian Institute for Nonlinear StudiesViennaAustria
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平成２２年 ２月２８日０９時３３分 気象庁発表
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(found here – )
World Book at NASA
Earthquake is a shaking of the ground caused by the sudden breaking and shifting of large sections of Earth’s rocky outer shell. Earthquakes are among the most powerful events on earth, and their results can be terrifying. A severe earthquake may release energy 10,000 times as great as that of the first atomic bomb. Rock movements during an earthquake can make rivers change their course. Earthquakes can trigger landslides that cause great damage and loss of life. Large earthquakes beneath the ocean can create a series of huge, destructive waves called tsunamis (tsoo NAH meez)that flood coasts for many miles.
Earthquakes almost never kill people directly. Instead, many deaths and injuries result from falling objects and the collapse of buildings, bridges, and other structures. Fire resulting from broken gas or power lines is another major danger during a quake. Spills of hazardous chemicals are also a concern during an earthquake.
The force of an earthquake depends on how much rock breaks and how far it shifts. Powerful earthquakes can shake firm ground violently for great distances. During minor earthquakes, the vibration may be no greater than the vibration caused by a passing truck.
On average, a powerful earthquake occurs less than once every two years. At least 40 moderate earthquakes cause damage somewhere in the world each year. Scientists estimate that more than 8,000 minor earthquakes occur each day without causing any damage. Of those, only about 1,100 are strong enough to be felt.
This article discusses Earthquake (How an earthquake begins) (How an earthquake spreads) (Damage by earthquakes) (Where and why earthquakes occur) (Studying earthquakes).
How an earthquake begins
Most earthquakes occur along a fault — a fracture in Earth’s rocky outer shell where sections of rock repeatedly slide past each other. Faults occur in weak areas of Earth’s rock. Most faults lie beneath the surface of Earth, but some, like the San Andreas Fault in California, are visible on the surface. Stresses in Earth cause large blocks of rock along a fault to strain, or bend. When the stress on the rock becomes great enough, the rock breaks and snaps into a new position, causing the shaking of an earthquake.
Earthquakes usually begin deep in the ground. The point in Earth where the rocks first break is called the focus, also known as the hypocenter, of the quake. The focus of most earthquakes lies less than 45 miles (72 kilometers) beneath the surface, though the deepest known focuses have been nearly 450 miles (700 kilometers) below the surface. The point on the surface of Earth directly above the focus is known as the epicenter of the quake. The strongest shaking is usually felt near the epicenter.
From the focus, the break travels like a spreading crack along the fault. The speed at which the fracture spreads depends on the type of rock. It may average about 2 miles (3.2 kilometers) per second in granite or other strong rock. At that rate, a fracture may spread more than 350 miles (560 kilometers) in one direction in less than three minutes. As the fracture extends along the fault, blocks of rock on one side of the fault may drop down below the rock on the other side, move up and over the other side, or slide forward past the other.
How an earthquake spreads
When an earthquake occurs, the violent breaking of rock releases energy that travels through Earth in the form of vibrations called seismic waves. Seismic waves move out from the focus of an earthquake in all directions. As the waves travel away from the focus, they grow gradually weaker. For this reason, the ground generally shakes less farther away from the focus.
There are two chief kinds of seismic waves: (1) body waves and (2) surface waves. Body waves, the fastest seismic waves, move through Earth. Slower surface waves travel along the surface of Earth.
Body waves tend to cause the most earthquake damage. There are two kinds of body waves: (1) compressional waves and (2) shear waves. As the waves pass through Earth, they cause particles of rock to move in different ways. Compressional waves push and pull the rock. They cause buildings and other structures to contract and expand. Shear waves make rocks move from side to side, and buildings shake. Compressional waves can travel through solids, liquids, or gases, but shear waves can pass only through solids.
Compressional waves are the fastest seismic waves, and they arrive first at a distant point. For this reason, compressional waves are also called primary (P) waves. Shear waves, which travel slower and arrive later, are called secondary (S) waves.
Body waves travel faster deep within Earth than near the surface. For example, at depths of less than 16 miles (25 kilometers), compressional waves travel at about 4.2 miles (6.8 kilometers) per second, and shear waves travel at 2.4 miles (3.8 kilometers) per second. At a depth of 620 miles (1,000 kilometers), the waves travel more than 11/2 times that speed.
Surface waves are long, slow waves. They produce what people feel as slow rocking sensations and cause little or no damage to buildings.
There are two kinds of surface waves: (1) Love waves and (2) Rayleigh waves. Love waves travel through Earth’s surface horizontally and move the ground from side to side. Rayleigh waves make the surface of Earth roll like waves on the ocean. Typical Love waves travel at about 23/4 miles (4.4 kilometers) per second, and Rayleigh waves, the slowest of the seismic waves, move at about 21/4 miles (3.7 kilometers) per second. The two types of waves were named for two British physicists, Augustus E. H. Love and Lord Rayleigh, who mathematically predicted the existence of the waves in 1911 and 1885, respectively.
Damage by earthquakes
How earthquakes cause damage
Earthquakes can damage buildings, bridges, dams, and other structures, as well as many natural features. Near a fault, both the shifting of large blocks of Earth’s crust, called fault slippage, and the shaking of the ground due to seismic waves cause destruction. Away from the fault, shaking produces most of the damage. Undersea earthquakes may cause huge tsunamis that swamp coastal areas. Other hazards during earthquakes include rockfalls, ground settling, and falling trees or tree branches.
Fault slippage
The rock on either side of a fault may shift only slightly during an earthquake or may move several feet or meters. In some cases, only the rock deep in the ground shifts, and no movement occurs at Earth’s surface. In an extremely large earthquake, the ground may suddenly heave 20 feet (6 meters) or more. Any structure that spans a fault may be wrenched apart. The shifting blocks of earth may also loosen the soil and rocks along a slope and trigger a landslide. In addition, fault slippage may break down the banks of rivers, lakes, and other bodies of water, causing flooding.
Ground shaking causes structures to sway from side to side, bounce up and down, and move in other violent ways. Buildings may slide off their foundations, collapse, or be shaken apart.
In areas with soft, wet soils, a process called liquefaction may intensify earthquake damage. Liquefaction occurs when strong ground shaking causes wet soils to behave temporarily like liquids rather than solids. Anything on top of liquefied soil may sink into the soft ground. The liquefied soil may also flow toward lower ground, burying anything in its path.
An earthquake on the ocean floor can give a tremendous push to surrounding seawater and create one or more large, destructive waves called tsunamis, also known as seismic sea waves. Some people call tsunamis tidal waves, but scientists think the term is misleading because the waves are not caused by the tide. Tsunamis may build to heights of more than 100 feet (30 meters) when they reach shallow water near shore. In the open ocean, tsunamis typically move at speeds of 500 to 600 miles (800 to 970 kilometers) per hour. They can travel great distances while diminishing little in size and can flood coastal areas thousands of miles or kilometers from their source.
Structural hazards
Structures collapse during a quake when they are too weak or rigid to resist strong, rocking forces. In addition, tall buildings may vibrate wildly during an earthquake and knock into each other. Picture San Francisco earthquake of 1906 A major cause of death and property damage in earthquakes is fire. Fires may start if a quake ruptures gas or power lines. The 1906 San Francisco earthquake ranks as one of the worst disasters in United States history because of a fire that raged for three days after the quake.
Other hazards during an earthquake include spills of toxic chemicals and falling objects, such as tree limbs, bricks, and glass. Sewage lines may break, and sewage may seep into water supplies. Drinking of such impure water may cause cholera, typhoid, dysentery, and other serious diseases.
Loss of power, communication, and transportation after an earthquake may hamper rescue teams and ambulances, increasing deaths and injuries. In addition, businesses and government offices may lose records and supplies, slowing recovery from the disaster.
Reducing earthquake damage
In areas where earthquakes are likely, knowing where to build and how to build can help reduce injury, loss of life, and property damage during a quake. Knowing what to do when a quake strikes can also help prevent injuries and deaths.
Where to build
Earth scientists try to identify areas that would likely suffer great damage during an earthquake. They develop maps that show fault zones, flood plains (areas that get flooded), areas subject to landslides or to soil liquefaction, and the sites of past earthquakes. From these maps, land-use planners develop zoning restrictions that can help prevent construction of unsafe structures in earthquake-prone areas.
How to build
An earthquake-resistant building includes such structures as shear walls, a shear core, and cross-bracing. Base isolators act as shock absorbers. A moat allows the building to sway.
An earthquake-resistant building includes such structures as shear walls, a shear core, and cross-bracing. Base isolators act as shock absorbers. A moat allows the building to sway. Image credit: World Book illustration by Doug DeWitt
Engineers have developed a number of ways to build earthquake-resistant structures. Their techniques range from extremely simple to fairly complex. For small- to medium-sized buildings, the simpler reinforcement techniques include bolting buildings to their foundations and providing support walls called shear walls. Shear walls, made of reinforced concrete (concrete with steel rods or bars embedded in it), help strengthen the structure and help resist rocking forces. Shear walls in the center of a building, often around an elevator shaft or stairwell, form what is called a shear core. Walls may also be reinforced with diagonal steel beams in a technique called cross-bracing.
Builders also protect medium-sized buildings with devices that act like shock absorbers between the building and its foundation. These devices, called base isolators, are usually bearings made of alternate layers of steel and an elastic material, such as synthetic rubber. Base isolators absorb some of the sideways motion that would otherwise damage a building.
Skyscrapers need special construction to make them earthquake-resistant. They must be anchored deeply and securely into the ground. They need a reinforced framework with stronger joints than an ordinary skyscraper has. Such a framework makes the skyscraper strong enough and yet flexible enough to withstand an earthquake.
Earthquake-resistant homes, schools, and workplaces have heavy appliances, furniture, and other structures fastened down to prevent them from toppling when the building shakes. Gas and water lines must be specially reinforced with flexible joints to prevent breaking.
Safety precautions are vital during an earthquake. People can protect themselves by standing under a doorframe or crouching under a table or chair until the shaking stops. They should not go outdoors until the shaking has stopped completely. Even then, people should use extreme caution. A large earthquake may be followed by many smaller quakes, called aftershocks. People should stay clear of walls, windows, and damaged structures, which could crash in an aftershock.
People who are outdoors when an earthquake hits should quickly move away from tall trees, steep slopes, buildings, and power lines. If they are near a large body of water, they should move to higher ground. Where and why earthquakes occur
Scientists have developed a theory, called plate tectonics, that explains why most earthquakes occur. According to this theory, Earth’s outer shell consists of about 10 large, rigid plates and about 20 smaller ones. Each plate consists of a section of Earth’s crust and a portion of the mantle, the thick layer of hot rock below the crust. Scientists call this layer of crust and upper mantle the lithosphere. The plates move slowly and continuously on the asthenosphere, a layer of hot, soft rock in the mantle. As the plates move, they collide, move apart, or slide past one another.
The movement of the plates strains the rock at and near plate boundaries and produces zones of faults around these boundaries. Along segments of some faults, the rock becomes locked in place and cannot slide as the plates move. Stress builds up in the rock on both sides of the fault and causes the rock to break and shift in an earthquake.
There are three types of faults: (1) normal faults, (2) reverse faults, and (3) strike-slip faults. In normal and reverse faults, the fracture in the rock slopes downward, and the rock moves up or down along the fracture. In a normal fault, the block of rock on the upper side of the sloping fracture slides down. In a reverse fault, the rock on both sides of the fault is greatly compressed. The compression forces the upper block to slide upward and the lower block to thrust downward. In a strike-slip fault, the fracture extends straight down into the rock, and the blocks of rock along the fault slide past each other horizontally.
Most earthquakes occur in the fault zones at plate boundaries. Such earthquakes are known as interplate earthquakes. Some earthquakes take place within the interior of a plate and are called intraplate earthquakes.
Interplate earthquakes occur along the three types of plate boundaries: (1) mid-ocean spreading ridges, (2) subduction zones, and (3) transform faults.
Mid-ocean spreading ridges are places in the deep ocean basins where the plates move apart. As the plates separate, hot lava from Earth’s mantle rises between them. The lava gradually cools, contracts, and cracks, creating faults. Most of these faults are normal faults. Along the faults, blocks of rock break and slide down away from the ridge, producing earthquakes.
Near the spreading ridges, the plates are thin and weak. The rock has not cooled completely, so it is still somewhat flexible. For these reasons, large strains cannot build, and most earthquakes near spreading ridges are shallow and mild or moderate in severity.
Subduction zones are places where two plates collide, and the edge of one plate pushes beneath the edge of the other in a process called subduction. Because of the compression in these zones, many of the faults there are reverse faults. About 80 per cent of major earthquakes occur in subduction zones encircling the Pacific Ocean. In these areas, the plates under the Pacific Ocean are plunging beneath the plates carrying the continents. The grinding of the colder, brittle ocean plates beneath the continental plates creates huge strains that are released in the world’s largest earthquakes.
The world’s deepest earthquakes occur in subduction zones down to a depth of about 450 miles (700 kilometers). Below that depth, the rock is too warm and soft to break suddenly and cause earthquakes.
Transform faults are places where plates slide past each other horizontally. Strike-slip faults occur there. Earthquakes along transform faults may be large, but not as large or deep as those in subduction zones.
One of the most famous transform faults is the San Andreas Fault. The slippage there is caused by the Pacific Plate moving past the North American Plate. The San Andreas Fault and its associated faults account for most of California’s earthquakes.
Intraplate earthquakes are not as frequent or as large as those along plate boundaries. The largest intraplate earthquakes are about 100 times smaller than the largest interplate earthquakes.
Intraplate earthquakes tend to occur in soft, weak areas of plate interiors. Scientists believe intraplate quakes may be caused by strains put on plate interiors by changes of temperature or pressure in the rock. Or the source of the strain may be a long distance away, at a plate boundary. These strains may produce quakes along normal, reverse, or strike-slip faults.
Studying earthquakes
Recording, measuring, and locating earthquakes
To determine the strength and location of earthquakes, scientists use a recording instrument known as a seismograph. A seismograph is equipped with sensors called seismometers that can detect ground motions caused by seismic waves from both near and distant earthquakes. Some seismometers are capable of detecting ground motion as small as 0.1 nanometer. One nanometer is 1 billionth of a meter or about 39 billionths of an inch. Scientists called seismologists measure seismic ground movements in three directions: (1) up-down, (2) north-south, and (3) east-west. The scientists use a separate sensor to record each direction of movement.
A seismograph produces wavy lines that reflect the size of seismic waves passing beneath it. The record of the wave, called a seismogram, is imprinted on paper, film, or recording tape or is stored and displayed by computers.
Probably the best-known gauge of earthquake intensity is the local Richter magnitude scale, developed in 1935 by United States seismologist Charles F. Richter. This scale, commonly known as the Richter scale, measures the ground motion caused by an earthquake. Every increase of one number in magnitude means the energy release of the quake is about 32 times greater. For example, an earthquake of magnitude 7.0 releases about 32 times as much energy as an earthquake measuring 6.0. An earthquake with a magnitude of less than 2.0 is so slight that usually only a seismometer can detect it. A quake greater than 7.0 may destroy many buildings. The number of earthquakes increases sharply with every decrease in Richter magnitude by one unit. For example, there are 8 times as many quakes with magnitude 4.0 as there are with magnitude 5.0.
Although large earthquakes are customarily reported on the Richter scale, scientists prefer to describe earthquakes greater than 7.0 on the moment magnitude scale. The moment magnitude scale measures more of the ground movements produced by an earthquake. Thus, it describes large earthquakes more accurately than does the Richter scale.
The largest earthquake ever recorded on the moment magnitude scale measured 9.5. It was an interplate earthquake that occurred along the Pacific coast of Chile in South America in 1960. The largest intraplate earthquakes known struck in central Asia and in the Indian Ocean in 1905, 1920, and 1957. These earthquakes had moment magnitudes between about 8.0 and 8.3. The largest intraplate earthquakes in the United States were three quakes that occurred in New Madrid, Missouri, in 1811 and 1812. The earthquakes were so powerful that they changed the course of the Mississippi River. During the largest of them, the ground shook from southern Canada to the Gulf of Mexico and from the Atlantic Coast to the Rocky Mountains. Scientists estimate the earthquakes had moment magnitudes of 7.5.
Scientists locate earthquakes by measuring the time it takes body waves to arrive at seismographs in a minimum of three locations. From these wave arrival times, seismologists can calculate the distance of an earthquake from each seismograph. Once they know an earthquake’s distance from three locations, they can find the quake’s focus at the center of those three locations.
Predicting earthquakes
Scientists can make fairly accurate long-term predictions of where earthquakes will occur. They know, for example, that about 80 percent of the world’s major earthquakes happen along a belt encircling the Pacific Ocean. This belt is sometimes called the Ring of Fire because it has many volcanoes, earthquakes, and other geologic activity.
Scientists are working to make accurate forecasts on when earthquakes will strike. Geologists closely monitor certain fault zones where quakes are expected. Along these fault zones, they can sometimes detect small quakes, the tilting of rock, and other events that might signal a large earthquake is about to occur.
Exploring Earth’s interior
Most of what is known about the internal structure of Earth has come from studies of seismic waves. Such studies have shown that rock density increases from the surface of Earth to its center. Knowledge of rock densities within Earth has helped scientists determine the probable composition of Earth’s interior.
Scientists have found that seismic wave speeds and directions change abruptly at certain depths. From such studies, geologists have concluded that Earth is composed of layers of various densities and substances. These layers consist of the crust, mantle, outer core, and inner core. Shear waves do not travel through the outer core. Because shear waves cannot travel through liquids, scientists believe the outer core is liquid. Scientists believe the inner core is solid because of the movement of compressional waves when they reach the inner core.
Contributor: Karen C. McNally, Ph.D., Professor of Earth Sciences, University of California, Santa Cruz.
How to cite this article: To cite this article, World Book recommends the following format: McNally, Karen C. “Earthquake.” World Book Online Reference Center. 2005. World Book, Inc. http://www.worldbookonline.com/wb/Article?id=ar171680
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Chile quake similar to 2004 Indian Ocean temblor
February 27, 2010 By ALICIA CHANG , AP Science Writer
–>Chile quake similar to 2004 Indian Ocean temblor (AP)
(AP) — Scientists say the major earthquake that struck off the coast of Chile was a “megathrust” – similar to the 2004 Indian Ocean temblor that spawned a catastrophic tsunami.
Megathrust earthquakes occur in subduction zones where plates of the Earth’s crust grind and dive. Saturday’s jolt occurred when the Nazca plate dove beneath the South American plate, releasing tremendous energy.
The U.S. Geological Survey says 13 temblors of magnitude-7 or larger have hit coastal Chile since 1973.
The latest quake occurred about 140 miles north of the largest earthquake ever recorded. The magnitude-9.5 struck southern Chile in 1960, killing some 1,600 people and generating a tsunami that killed another 200 people in Japan, Hawaii and the Philippines.
2010 The Associated Press
My Note –
The difference is that the earthquake in Chile yesterday morning at 23 hours ago, was much deeper – 35 kilometers down. Now, there are 300 people listed as killed during the earthquake.
– cricketdiane
1.5M homes damaged in Chile quake—official
Agence France-Presse
First Posted 07:54:00 02/28/2010
Filed Under: Earthquake, Disasters (general), Housing & Urban Planning
SANTIAGO, Chile—Some 1.5 million homes were damaged by the powerful earthquake that struck central Chile, Housing Minister Patricia Poblete said Saturday.
The figure includes half a million homes “with severe damage” that will “probably not be able to be lived in again,” Poblete told reporters.
The huge 8.8-magnitude earthquake that rocked Chile in the pre-dawn hours of Saturday left a trail of twisted buildings, destroyed bridges, and closed down the Santiago International Airport
The city of Concepcion, some 440 kilometers (273 miles) southwest of Santiago, and its surrounding area was especially hard-hit.
The 1960 eruption of Cordón Caulle soon after the Great Chilean Earthquake was triggered by movements in the fault. (found in the text below – where is that volcano now?)
Liquiñe_Ofqui_Fault - Chile
Liquiñe_Ofqui_Fault - Chile - earthquake 1960 and 02-27-2010
The Liquiñe-Ofqui Fault marked with red.
The Liquiñe-Ofqui Fault is major geological fault[1] that runs a length of roughly 1000 km in a north-south direction and exhibits current seismicity [2]. It is located in the Chilean northern patagonean Andes.
As the name implies it runs from the Liquiñe hot springs in the north to the Ofqui Isthmus in the south, where the Antarctic Plate meets the Nazca Plate and the South American Plate in Chile Triple Junction. A large parth of the fault runs along the Moraleda Channel. North of Liquiñe the fault is gradually converted into a compression area. At Quetrupillán volcano the fault is crossed by the Gastre Fault Zone. It may be classified as a dextral intra-arc transform fault.
The 1960 eruption of Cordón Caulle soon after the Great Chilean Earthquake was triggered by movements in the fault. The Aysén Fjord earthquake in 2007 and the eruption of Chaitén Volcano in 2008 are belivied to have been caused by movements in the fault.
My Note – as I was looking for information about the earthquake fault zones in Chile and other info about the tsunami earlier today, I also found a few other good things including these which were interesting – cricketdiane
magnetic polarity reversal. A change of the earth’s magnetic field to the opposite polarity that has occurred at irregular intervals during geologic time. Polarity reversals can be preserved in sequences of magnetized rocks and compared with standard polarity-change time scales to estimate geologic ages of the rocks. Rocks created along the oceanic spreading ridges commonly preserve this pattern of polarity reversals as they cool, and this pattern can be used to determine the rate of ocean ridge spreading. The reversal patterns recorded in the rocks are termed sea-floor magnetic lineaments.
Quantum measurement precision approaches Heisenberg limit
February 26, 2010 By Lisa Zyga
This illustration shows an adaptive feedback scheme being used to measure an unknown phase difference between the two red arms in the interferometer. A photon (qubit) is sent through the interferometer, and detected by either c1 or c0, depending on which arm it traveled through. Feedback is sent to the processing unit, which controls the phase shifter in one arm so that, when the next photon is sent, the device can more precisely measure the unknown phase in the other arm, and calculate a precise phase difference. Image credit: Hentschel and Sanders.
(PhysOrg.com) — In the classical world, scientists can make measurements with a degree of accuracy that is restricted only by technical limitations. At the fundamental level, however, measurement precision is limited by Heisenberg’s uncertainty principle. But even reaching a precision close to the Heisenberg limit is far beyond existing technology due to source and detector limitations.
“The precision that any measurement can possibly achieve is limited by the so-called Heisenberg limit, which results from Heisenberg’s uncertainty principle,” Hentschel told PhysOrg.com. “However, classical measurements cannot achieve a precision close to the Heisenberg limit. Only quantum measurements that use quantum correlations can approach the Heisenberg limit. Yet, devising quantum measurement procedures is highly challenging.”
Heisenberg’s uncertainty principle ultimately limits the achievable precision depending on how many quantum resources are used for the measurement. For example, gravitational waves are detected with laser interferometers, whose precision is limited by the number of photons available to the interferometer within the duration of the gravitational wave pulse.
In their study, Hentschel and Sanders used a computer simulation of a two-channel interferometer with a random phase difference between the two arms. Their goal was to estimate the relative phase difference between the two channels. In the simulated system, photons were sent into the interferometer one at a time. Which input port the photon entered was unknown, so that the photon (serving as a qubit) was in a superposition of two states, corresponding to the two channels. When exiting the interferometer, the photon was detected as leaving one of the two output ports, or not detected at all if it was lost. Since photons were fed into the interferometer one at a time, no more than one bit of information could be extracted at once. In this scenario, the achievable precision is limited by the number of photons used for the measurement.
As previous research has shown, the most effective quantum measurement schemes are those that incorporate adaptive feedback. These schemes accumulate information from measurements and then exploit it to maximize the information gain in subsequent measurements. In an interferometer with feedback, a sequence of photons is successively sent through the interferometer in order to measure the unknown phase difference. Detectors at the two output ports measure which way each of the photons exits, and then transmit this information to a processing unit. The processing unit adapts the value of a controllable phase shifter after each photon according to a given policy.
However, devising an optimal policy is difficult, and usually requires guesswork. In their study, Hentschel and Sanders adapted a technique from the field of artificial intelligence. Their algorithm autonomously learns an optimal policy based on trial and error – replacing guesswork by a logical, fully automatic, and programmable procedure.
Specifically, the new method uses a machine learning algorithm called particle swarm optimization (PSO). PSO is a “collective intelligence” optimization strategy inspired by the social behavior of birds flocking or fish schooling to locate feeding sites. In this case, the physicists show that a PSO algorithm can also autonomously learn a policy for adjusting the controllable phase shift.
As Hentschel and Sanders show, after a sequence of input qubits have been sent through the interferometer, the measurement procedure learned by the PSO algorithm delivers a measurement of the unknown phase shift that scales closely to the Heisenberg limit, setting a new precedent for quantum measurement precision. The new high level of precision could have important implications for the gravitational wave detection.
“Einstein’s theory of General Relativity predicts gravitational waves,” Hentschel said. “However, a direct detection of gravitational waves has not been achieved. Gravitational wave detection will open up a new field of astronomy that augments electromagnetic wave and neutrino observations. For example, gravitational wave detectors can spot merging black holes or binary star systems composed of two neutron stars, which are mostly hidden to conventional telescopes.”
More information: Alexander Hentschel and Barry C. Sanders. “Machine Learning for Precise Quantum Measurement.” Physical Review Letters 104, 063603 (2010). DOI:10.1103/PhysRevLett.104.063603
2010 PhysOrg.com.
Physicists detect entanglement of one photon shared among four locations
May 08, 2009 | not rated yet | 0
Evidence of a new phase in liquid hydrogen
February 25, 2010 By Miranda Marquit
Protium, the most common isotope of hydrogen. Image: Wikipedia.
(PhysOrg.com) — We like to think that we’ve got hydrogen, one of the most basic of elements, figured out. However, hydrogen can still surprise, especially once scientists start probing its properties on the most fundamental levels. “We ran simulations in order to provide a quantitative map of the molecular to atomic transition in liquid hydrogen,” Isaac Tamblyn tells PhysOrg.com. “Some of what we found was surprising, and could change the basic equations of state used in models involving hydrogen.”
Tamblyn is a scientist at Dalhousie University in Halifax, Canada. He worked with Stanimir A. Bonev to simulate the transition in liquid hydrogen, offering evidence for an unreported liquid phase, and noting some interesting structural characteristics of liquid hydrogen. Information on the simulation efforts, as well as results and conclusions, are presented in Physical Review Letters: “Structure and Phase Boundaries of Compressed Liquid Hydrogen.”
“We used first principles molecular dynamics simulations to model the liquid,” Tamblyn explains. “Forces between atoms were obtained using the Schrödinger equation. Velocities of the atoms were then updated, and the system was evolved through time.”
“We ran simulations to determine what would happen under different thermodynamic conditions, like density and temperature, and monitored the stability of molecules as the simulations progressed,” Tamblyn continues. “Our transition line is based on molecular stability. The chances of a molecule surviving are greater in a molecular liquid than in an atomic one, so this is a natural way to describe the transition.”
After running the simulations, Tamblyn and Bonev then had to analyze them. “We discovered an ordering in the liquid that accounts for some of the interesting characteristics of hydrogen, such as the fact that under certain conditions, liquid hydrogen is more dense than the solid. We also found that highly ordered packing explains properties related to dissociation that were previously not well understood.”
The pair found that the simulations suggest criteria for the existence of a first-order phase transition in liquid hydrogen. “The existence of this has been debated,” Tamblyn explains, “and we provide some evidence for its possibility.”
One of the most significant things Tamblyn and Bonev discovered through their simulations, from an astrophysics standpoint, is that equations describing the properties of hydrogen might need to be updated. “This should change the modeling going forward,” Tamblyn insists. “What we found in the liquid suggests what the solid might look like, and that can help determine some of its thermal and electronic properties.”
There is a good chance that planetary models might be changed using the new information on hydrogen structure discovered through these simulations. “Some previous calculations may need to be revised,” Tamblyn predicts. He also says that the simulations hint at some of the potential effects of mixtures involving hydrogen. “We’re especially interested in the implication for hydrogen and helium mixtures.”
Going forward, Tamblyn believes there is room to expand upon the work. “We are looking at the metallization of hydrogen, following the transition into a liquid metal. We are also looking at simulating hydrogen mixtures, especially with helium, to see if our findings hold true.”
More information: Isaac Tamblyn and Stanimir A. Bonev, “Structure and Phase Boundaries of Compressed Liquid Hydrogen,” Physical Review Letters (2010). Available online: http://link.aps.org/doi/10.1103/PhysRevLett.104.065702
2010 PhysOrg.com. |
a8a7d706a0d6f4cb | Half-day Workshops 2018
7th MAPEX Early Career Researcher Workshop - science meets industry
October 25th, 2018
Further details...
MAPEX Methods Workshop II - Computational Materials Science
May 17th, 2018
Further details...
Machine Learning Quantum Mechanical Properties for Molecular Systems
April 24th, 2018
In recent years, Machine Learning (ML) algorithms have accurately reproduced energies derived
from quantum chemistry without the need to solve the Schrödinger equation. In this talk, I will
provide an overview of how these methods work and emphasize their speed and accuracy.
Examples from recent literature will be used to illustrate how ML can be used to perform reactive
molecular dynamics simulations on unprecedented length and time scales. Additionally, the concept
of "active learning" will be explored, which is where an ML algorithm is able to quantify it's own
accuracy and determine systematically improve itself with no human intervention. Then the ability
of ML algorithms to produce properties other than energies will be explored, such as atomic charges
and dipole moments. Finally, ongoing work where ML is used to generate effective Hamiltonian
parameters will be discussed.
Time: 10:00 - approx. 13:00
Location: University of Bremen, BCCMS, ECO5/TAB Building, Entrance F, Ground floor, room 0.50/0.51
Non-adiabatic excited state dynamics simulations
February 6th, 2018
Sergei Tretiak, Los Alamos National Laboratory
Efficient non-adiabatic excited state dynamics simulations in extended molecular systems
Antonietta De Sio, University of Oldenburg
Ultrafast non-adiabatic dynamics in organic solar cell materials revealed by two-dimensional electronic spectroscopy
Coffee break
Ulrich Kleinekathoefer, Jacobs University Bremen
Multiscale simulations of energy and charge transport in biological systems
Time: 12:30 - 16:00 |
ba66aef5931beb29 | Dissertation/Thesis Abstract
Quantum Dragon Solutions for Electron Transport through Single-Layer Planar Rectangular Crystals
by Inkoom, Godfred, Ph.D., Mississippi State University, 2017, 205; 10641651
Abstract (Summary)
When a nanostructure is coupled between two leads, the electron transmission probability as a function of energy, E, is used in the Landauer formula to obtain the electrical conductance of the nanodevice. The electron transmission probability as a function of energy, T( E), is calculated from the appropriate solution of the time independent Schrödinger equation. Recently, a large class of nanostructures called quantum dragons has been discovered. Quantum dragons are nanodevices with correlated disorder but still can have electron transmission probability unity for all energies when connected to appropriate (idealized) leads. Hence for a single channel setup, the electrical conductivity is quantized. Thus quantum dragons have the minimum electrical conductance allowed by quantum mechanics. These quantum dragons have potential applications in nanoelectronics.
It is shown that for dimerized leads coupled to a simple two-slice ( l = 2, m = 1) device, the matrix method gives the same expression for the electron transmission probability as renormalization group methods and as the well known Green's function method. If a nanodevice has m atoms per slice, with l slices to calculate the electron transmission probability as a function of energy via the matrix method requires the solution of the inverse of a (2 + ml) × (2 + ml) matrix. This matrix to invert is of large dimensions for large m and l. Taking the inverse of such a matrix could be done numerically, but getting an exact solution may not be possible. By using the mapping technique, this reduces this large matrix to invert into a simple (l + 2) × (l + 2) matrix to invert, which is easier to handle but has the same solution. By using the map-and-tune approach, quantum dragon solutions are shown to exist for single-layer planar rectangular crystals with different boundary conditions. Each chapter provides two different ways on how to find quantum dragons. This work has experimental relevance, since this could pave the way for planar rectangular nanodevices with zero electrical resistance to be found. In the presence of randomness of the single-band tight-binding parameters in the nanodevice, an interesting quantum mechanical phenomenon called Fano resonance of the electron transmission probability is shown to be observed.
Indexing (document details)
Advisor: Novotny, Mark A.
Commitee: Arnoldus, Hendrik F., Clay, R. Torsten, Kim, Seong-Gon, Miller, Vivien G.
School: Mississippi State University
Department: Physics and Astronomy
School Location: United States -- Mississippi
Source: DAI-B 79/03(E), Dissertation Abstracts International
Subjects: Physics
Keywords: Electron transmission, Fano resonances, Perron-Frobenius theorem, Quantum dragons, Single-band tight binding model, Single-layer rectangular crystals
Publication Number: 10641651
ISBN: 978-0-355-51391-2 |
00503a22eb232d7c | Problems in Nonlinear Wave Propagation A Walk in Physics from Plasmas to Bose-Einstein Condensates with some Examples of Unifying Themes in Nature
Doctoral thesis, 2001
Waves are a phenomenon that can be found virtually everywhere in nature. A first description of wave propagation can be given in the linear limit, but the nonlinear regime of propagation is of the utmost importance, also in view of possible applications in several scientific fields. In the course of this work, nonlinear wave propagation in physical systems from plasmas interacting with super-intense laser light to Bose-Einstein condensates (BEC) has been investigated, making use of the analogies brought to light by the mathematical modelization of such different systems. In the case of laser-plasma interactions, the main problem is the propagation of electromagnetic waves through a plasma. The nonlinear character is due to the high laser intensity which sets the plasma electrons into relativistic motion and exerts a force strongly perturbing their equilibrium density distribution. This deeply modifies the physics of the propagation leading to effects like self-induced transparency or the generation of plasma-field structures. Self-induced transparency is originated by the relativistic quiver motion of the plasma electrons and allows light to propagate through plasmas with a density so high that light propagation would classically be impossible. We have studied the problem of a threshold for induced transparency via an exact analytical investigation which has led furthermore to an exact description of the structures (electron depletion regions and light filaments) generated in the plasma as a consequence of the interaction, for both high and low density plasmas. The physics behind the generation of these structures can be described, in the weakly nonlinear limit, by the nonlinear Schrödinger equation (NLS), one of the fundamental nonlinear equations of physics. The importance and effectiveness of analytical investigations has been demonstrated by an analysis of the NLS equation generalized to the case of multi-dimensional non conservative systems (Ginzburg-Landau equation) and applied to the description of a scheme for the amplification of laser pulses (the chirped pulse amplification scheme). Furthermore, the same mathematical structure of the NLS equation describes the physics of BEC, systems of bosonic atoms that have undergone a phase transition such that they occupy the same ground state. It is the wave nature of matter which brings to light deep analogies with the nonlinear classical physics of optics and we have made of these analogies a tool for investigating certain aspects of the nature of a condensate. Once more, the mathematical modeling of physical phenomena has revealed new features of the underlying physics.
nonlinear Schroedinger equation
Bose-Einstein condensates
Federica Cattani
Chalmers, Department of Electromagnetics
Subject Categories
Electrical Engineering, Electronic Engineering, Information Engineering
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1775
More information |
678d6cfed869be92 | EPR paradox
From Wikipedia, the free encyclopedia
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The Einstein–Podolsky–Rosen paradox or EPR paradox[1] of 1935 is a thought experiment in quantum mechanics with which Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen (EPR) claimed to demonstrate that the wave function does not provide a complete description of physical reality, and hence that the Copenhagen interpretation is unsatisfactory; resolutions of the paradox have important implications for the interpretation of quantum mechanics.
Albert Einstein
The essence of the paradox is that particles can interact in such a way that it is possible to measure both their position and their momentum more accurately than Heisenberg's uncertainty principle allows, unless measuring one particle instantaneously affects the other to prevent this accuracy, which would involve information being transmitted faster than light as forbidden by the theory of relativity ("spooky action at a distance"). This consequence had not previously been noticed and seemed unreasonable at the time; the phenomenon involved is now known as quantum entanglement.
While EPR felt that the paradox showed that quantum theory was incomplete and should be extended with hidden variables, the usual modern resolution is to say that due to the common preparation of the two particles (for example the creation of an electron-positron pair from a photon) the property we want to measure has a well defined meaning only when analyzed for the whole system while the same property for the parts individually remains undefined. Therefore, if similar measurements are being performed on the two entangled subsystems, there will always be a correlation between the outcomes resulting in a well defined global outcome i.e. for both subsystems together. However, the outcomes for each subsystem separately at each repetition of the experiment will not be well defined or predictable. This correlation does not imply any action of the measurement of one particle on the measurement of the other, therefore it does not imply any form of action at a distance. This modern resolution eliminates the need for hidden variables, action at a distance or other structures introduced over time in order to explain the phenomenon.
A preference for the latter resolution is supported by experiments suggested by Bell's theorem of 1964, which exclude some classes of hidden variable theory.
According to quantum mechanics, under some conditions, a pair of quantum systems may be described by a single wave function, which encodes the probabilities of the outcomes of experiments that may be performed on the two systems, whether jointly or individually. At the time the EPR article discussed below was written, it was known from experiments that the outcome of an experiment sometimes cannot be uniquely predicted. An example of such indeterminacy can be seen when a beam of light is incident on a half-silvered mirror. One half of the beam will reflect, and the other will pass. If the intensity of the beam is reduced until only one photon is in transit at any time, whether that photon will reflect or transmit cannot be predicted quantum mechanically.
The routine explanation of this effect was, at that time, provided by Heisenberg's uncertainty principle. Physical quantities come in pairs called conjugate quantities. Examples of such conjugate pairs are (position, momentum), (time, energy), and (angular position, angular momentum). When one quantity was measured, and became determined, the conjugated quantity became indeterminate. Heisenberg explained this uncertainty as due to the quantization of the disturbance from measurement.
The EPR paper, written in 1935, was intended to illustrate that this explanation is inadequate. It considered two entangled particles, referred to as A and B, and pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance. The basic idea was that the quantum states of two particles in a system cannot always be decomposed from the joint state of the two, as is the case for the Bell state,
Heisenberg's principle was an attempt to provide a classical explanation of a quantum effect sometimes called non-locality. According to EPR there were two possible explanations. Either there was some interaction between the particles (even though they were separated) or the information about the outcome of all possible measurements was already present in both particles.
The EPR authors preferred the second explanation according to which that information was encoded in some 'hidden parameters'. The first explanation of an effect propagating instantly across a distance is in conflict with the theory of relativity. They then concluded that quantum mechanics was incomplete since its formalism does not permit hidden parameters.
Violations of the conclusions of Bell's theorem are generally understood to have demonstrated that the hypotheses of Bell's theorem, also assumed by Einstein, Podolsky and Rosen, do not apply in our world.[2] Most physicists who have examined the issue concur that experiments, such as those of Alain Aspect and his group, have confirmed that physical probabilities, as predicted by quantum theory, do exhibit the phenomena of Bell-inequality violations that are considered to invalidate EPR's preferred "local hidden-variables" type of explanation for the correlations to which EPR first drew attention.[3][4]
History of EPR developments[edit]
The article that first brought forth these matters, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" was published in 1935.[1] The paper prompted a response by Bohr, which he published in the same journal, in the same year, using the same title.[5] There followed a debate between Bohr and Einstein about the fundamental nature of reality. Einstein had been skeptical of the Heisenberg uncertainty principle and the role of chance in quantum theory. But the crux of this debate was not about chance, but something even deeper: Is there one objective physical reality, which every observer sees from his own vantage? (Einstein's view) Or does the observer co-create physical reality by the questions he poses with experiments? (Bohr's view)
Einstein struggled to the end of his life for a theory that could better comply with his idea of causality, protesting against the view that there exists no objective physical reality other than that which is revealed through measurement interpreted in terms of quantum mechanical formalism. However, since Einstein's death, experiments analogous to the one described in the EPR paper have been carried out, starting in 1976 by French scientists Lamehi-Rachti and Mittig[6] at the Saclay Nuclear Research Centre. These experiments appear to show that the local realism idea is false,[7] vindicating Bohr.
Quantum mechanics and its interpretation[edit]
Since the early twentieth century, quantum theory has proved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world, in multiple reproducible physics experiments.
Quantum mechanics was developed with the aim of describing atoms and explaining the observed spectral lines in a measurement apparatus. Although disputed especially in the early twentieth century, it has yet to be seriously challenged. Philosophical interpretations of quantum phenomena, however, are another matter: the question of how to interpret the mathematical formulation of quantum mechanics has given rise to a variety of different answers from people of different philosophical persuasions (see Interpretations of quantum mechanics).
Quantum theory and quantum mechanics do not provide single measurement outcomes in a deterministic way. According to the understanding of quantum mechanics known as the Copenhagen interpretation, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable that was measured. Einstein characterized this imagined collapse in the 1927 Solvay Conference. He presented a thought experiment in which electrons are introduced through a small hole in a sphere whose inner surface serves as a detection screen. The electrons will contact the spherical detection screen in a widely dispersed manner. Those electrons, however, are all individually described by wave fronts that expand in all directions from the point of entry. A wave as it is understood in everyday life would paint a large area of the detection screen, but the electrons would be found to impact the screen at single points and would eventually form a pattern in keeping with the probabilities described by their identical wave functions. Einstein asks what makes each electron's wave front "collapse" at its respective location. Why do the electrons appear as single bright scintillations rather than as dim washes of energy across the surface? Why does any single electron appear at one point rather than some alternative point? The behavior of the electrons gives the impression of some signal having been sent to all possible points of contact that would have nullified all but one of them, or, in other words, would have preferentially selected a single point to the exclusion of all others.[8]
Einstein's opposition[edit]
Einstein was the most prominent opponent of the Copenhagen interpretation. In his view, quantum mechanics was incomplete. Commenting on this, other writers (such as John von Neumann[9] and David Bohm[10]) hypothesized that consequently there would have to be 'hidden' variables responsible for random measurement results, something which was not expressly claimed in the original paper.
The 1935 EPR paper[1] condensed the philosophical discussion into a physical argument. The authors claim that given a specific experiment, in which the outcome of a measurement is known before the measurement takes place, there must exist something in the real world, an "element of reality", that determines the measurement outcome. They postulate that these elements of reality are local, in the sense that each belongs to a certain point in spacetime. Each element may only be influenced by events which are located in the backward light cone of its point in spacetime (i.e., the past). These claims are founded on assumptions about nature that constitute what is now known as local realism.
Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored by Podolsky, based on discussions at the Institute for Advanced Study with Einstein and Rosen. Einstein later expressed to Erwin Schrödinger that, "it did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by the formalism."[11] In 1936, Einstein presented an individual account of his local realist ideas.[12]
Description of the paradox[edit]
The original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle. This challenge can be extended to other pairs of physical properties.
EPR paper[edit]
The original paper purports to describe what must happen to "two systems I and II, which we permit to interact ...", and, after some time, "we suppose that there is no longer any interaction between the two parts." As explained by Manjit Kumar (2009), the EPR description involves "two particles, A and B, [which] interact briefly and then move off in opposite directions."[13] According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly. However, it is possible to measure the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Alternatively, the exact momentum of particle A can be measured, so the exact momentum of particle B can be worked out. Kumar writes: "EPR argued that they had proved that ... [particle] B can have simultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum that is real."
EPR tried to set up a paradox to question the range of true application of Quantum Mechanics: Quantum theory predicts that both values cannot be known for a particle, and yet the EPR thought experiment purports to show that they must all have determinate values. The EPR paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."[13]
The EPR paper ends by saying:
Measurements on an entangled state[edit]
We have a source that emits electron–positron pairs, with the electron sent to destination A, where there is an observer named Alice, and the positron sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. The particles are thus said to be entangled. This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, the electron has spin pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis (−z). In state II, the electron has spin −z and the positron has spin +z. Because it is in a superposition of states it is impossible without measuring to know the definite state of spin of either particle in the spin singlet.[14]:421–422
The EPR thought experiment, performed with electron–positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to Bob (right), who can perform spin measurements.
Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or −z. Suppose she gets +z. According to the Copenhagen interpretation of quantum mechanics, the quantum state of the system collapses into state I. The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the z-axis, there is 100% probability that he will obtain −z. Similarly, if Alice gets −z, Bob will get +z.
There is, of course, nothing special about choosing the z-axis: according to quantum mechanics the spin singlet state may equally well be expressed as a superposition of spin states pointing in the x direction.[15]:318 Suppose that Alice and Bob had decided to measure spin along the x-axis. We'll call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob's positron has spin −x. In state IIa, Alice's electron has spin −x and Bob's positron has spin +x. Therefore, if Alice measures +x, the system 'collapses' into state Ia, and Bob will get −x. If Alice measures −x, the system collapses into state IIa, and Bob will get +x.
Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the particles are linked in some way. Either they were created with a definite (opposite) spin about every axis—a "hidden variable" argument—or they are linked so that one electron "feels" which axis the other is having its spin measured along, and becomes its opposite about that one axis—an "entanglement" argument. Moreover, if the two particles have their spins measured about different axes, once the electron's spin has been measured about the x-axis (and the positron's spin about the x-axis deduced), the positron's spin about the z-axis will no longer be certain, as if (a) it knows that the measurement has taken place, or (b) it has a definite spin already, about a second axis—a hidden variable. However, it turns out that the predictions of Quantum Mechanics, which have been confirmed by experiment, cannot be explained by any local hidden variable theory. This is demonstrated in Bell's theorem.[16]
In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning the Heisenberg uncertainty principle applies to alternating measurements of them: a quantum state cannot possess a definite value for both of these variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. It is impossible to predict which outcome will appear until Bob actually performs the measurement.
Here is the crux of the matter:[editorializing]
You might imagine that, when Bob measures the x-spin of his positron, he would get an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But it turns out that Bob's positron has a 50% probability of producing +x and a 50% probability of −x, meaning the outcome is not certain. It's as if Bob's positron "knows" that Alice has measured the z-spin of her electron, and hence his positron's own z-spin must also be set, but its x-spin remains uncertain.
Put another way, how does Bob's positron know which way to point if Alice decides (based on information unavailable to Bob) to measure x (i.e., to be the opposite of Alice's electron's spin about the x-axis) and also how to point if Alice measures z, since it is only supposed to know one thing at a time? The Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, so there must be action at a distance (entanglement) or the positron must know more than it's supposed to know (hidden variables).
Here is the paradox summed up:[editorializing]
It is one thing to say that physical measurement of the first particle's momentum affects uncertainty in its own position, but to say that measuring the first particle's momentum affects the uncertainty in the position of the other is another thing altogether. Einstein, Podolsky and Rosen asked how can the second particle "know" to have precisely defined momentum but uncertain position? Since this implies that one particle is communicating with the other instantaneously across space, i.e., faster than light, this is the "paradox".
Incidentally, Bell used spin as his example, but many types of physical quantities—referred to as "observables" in quantum mechanics—can be used. The EPR paper used momentum for the observable. Experimental realisations of the EPR scenario often use photon polarization, because polarized photons are easy to prepare and measure.
Locality in the EPR experiment[edit]
The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity, which states that information can never be transmitted faster than the speed of light without violating causality. It is generally believed that any theory which violates causality would also be internally inconsistent, and thus useless.[14]:427–428[17]
It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality.[14]:427–428[17] Causality is preserved because there is no way for Alice to transmit messages (i.e., information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "−", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the "no cloning theorem", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% of getting "−", regardless of whether or not his axis is aligned with Alice's.
However, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance". The conclusion they drew was that quantum mechanics is not a complete theory.[18]
In recent years, however, doubt has been cast on EPR's conclusion due to developments in understanding locality and especially quantum decoherence. The word locality has several different meanings in physics. For example, in quantum field theory "locality" means that quantum fields at different points of space do not interact with one another. However, quantum field theories that are "local" in this sense appear to violate the principle of locality as defined by EPR, but they nevertheless do not violate locality in a more general sense. Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behaviour doesn't violate local causality, it follows that neither does the additional effect of wavefunction collapse, whether real or apparent. Therefore, as outlined in the example above, neither the EPR experiment nor any quantum experiment demonstrates that faster-than-light signaling is possible.
Resolving the paradox[edit]
Hidden variables[edit]
There are several ways to resolve the EPR paradox. The one suggested by EPR is that quantum mechanics, despite its success in a wide variety of experimental scenarios, is actually an incomplete theory. In other words, there is some yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation (albeit an exceedingly successful one). Unlike quantum mechanics, the more complete theory contains variables corresponding to all the "elements of reality". There must be some unknown mechanism acting on these variables to give rise to the observed effects of "non-commuting quantum observables", i.e. the Heisenberg uncertainty principle. Such a theory is called a hidden variable theory.[13]:334[19]:357–358
To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment. One supposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the z-spin and x-spin. In these "true" states, the positron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+z, −x) to Alice and (−z, +x) to Bob", the next pair "(−z, −x) to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "−" with equal probability.[20]:239–240
Assuming we restrict our measurements to the z- and x-axes, such a hidden variable theory is experimentally indistinguishable from quantum mechanics. In reality, there may be an infinite number of axes along which Alice and Bob can perform their measurements, so there would have to be an infinite number of independent hidden variables. However, this is not a serious problem; we have formulated a very simplistic hidden variable theory, and a more sophisticated theory might be able to patch it up. It turns out that there is a much more serious challenge to the idea of hidden variables.
Bell's inequality[edit]
In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a particular class of hidden variable theories (the local hidden variable theories). Roughly speaking, quantum mechanics has a much stronger statistical correlation with measurement results performed on different axes than do these hidden variable theories. These differences, expressed using inequality relations known as "Bell's inequalities", are in principle experimentally detectable. After the publication of Bell's paper, a variety of experiments to test Bell's inequalities were devised. These generally relied on measurement of photon polarization. All experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics theory.
Later work by Henry Stapp showed that a key property of local hidden variable theories which lead to Bell's inequalities was counter-factual definiteness. Building on Stapp's observations, P.H. Eberhard showed that any local counter-factual model results in Bell's inequality even without the assumption of there being hidden variables unknown to physics upon which the relevant observables depend. Arthur Fine subsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory. (Although Eberhard referred to his result as "Bell's theorem without hidden variables", Fine used a more general definition of "hidden variables" that includes the possibility of the observables being elementary.) Fine went on to show that any stochastic factorizable model leads to Bell's inequality. Itamar Pitowsky showed that Bell's inequality was a special case of an inequality discovered by George Boole which provides a consistency check on whether data can be represented by variables on a single classical probability space. He interpreted this to be an indication that the locality assumption prevented the data from being represented as events on such a space.[21]
As Eberhard's proof made use of both locality and counter-factual definiteness it was assumed that an interpretation could reject either one of these to escape Bell's inequality. Violation of locality is difficult to reconcile with special relativity, and is thought to be incompatible with the principle of causality, nevertheless there was renewed interest in the Bohm interpretation of quantum mechanics which keeps counter-factual definiteness while introducing a conjectured non-local mechanism in the form of the 'quantum potential' that is defined as one of the terms of the Schrödinger equation. Mainstream physics preferred to keep locality and reject counter-factual definiteness. Fine's work showed that, taking locality as a given, there exist scenarios in which two statistical variables are correlated in a manner inconsistent with counter-factual definiteness, and that such scenarios are no more mysterious than any other, despite the fact that the inconsistency with counter-factual definiteness may seem 'counter-intuitive'.
Further insights resulted from the work of Lawrence J. Landau. Landau showed that if it is assumed that there is a single classical probability space underlying all the observables under consideration in the EPR experiment, Bell's inequality will result.[22] Thus the fundamental issue is that Quantum mechanical probabilities cannot be modeled using classical (Kolmogorovian) probability regardless of whether Quantum Mechanics is considered a complete description of reality or not. Regarding Landau's proof Ray Streater notes that it shows that Bohmian mechanics is inconsistent with Quantum mechanics and succumbs to Bell's inequality despite claims to the contrary by its proponents. Streater notes that Landau's proof only requires the assumption of a single classical probability space (a condition still satisfied by Bohm's theory) and the fact that Bohmian mechanics additionally postulates the existence of a non-local mechanism, cannot prevent Bell's inequality from applying to it.[23]:99–102
Similar observations have been made by Karl Hess, Walter, Philipp, Hans de Raedt and Kristel Michielsen, who note that in Bell's proof, Bell's assumption of a space of hidden variables behaving as a classical probability space is sufficient to produce a contradiction with the predications of Quantum mechanics via a consistency theorem of N. N. Vorob'ev, a statistician who had built on the same work of Boole used by Pitowsky. The additional assumption of locality used by Bell is redundant and indeed Fine's work had included a derivation of Bell's inequality that did not require the assumption of locality .[24] [25] Non-locality is not sufficient to escape Bell's inequality, any interpretation of Quantum mechanics needs to reject counter-factual definiteness to be consistent with the Quantum mechanical predications. The authors also produced a model of an EPR experiment that is local but which violates Bell's inequality, thus demonstrating that non-locality is also not necessary for escaping Bell's inequality.[26] They also note a loophole regarding models of EPR experiments whereby even a counter-factual definite model can result in data that violates Bell's inequality if as in actual experiments there is a time window based post-selection of results due to the need to identify particles belonging to an emitted pair.[26] Robert Griffiths has shown that according to a quantum mechanical analysis, the instrument settings for the measurement of one of the particles in the EPR scenario, does not influence subsequent measurement results on the second, thus ruling out non-locality as a viable explanation for the EPR correlations.[27]
However, Bell's theorem does not apply to all possible philosophically realist theories. It is a common misconception that quantum mechanics is inconsistent with all notions of philosophical realism. Realist interpretations of quantum mechanics are possible, although as discussed above, such interpretations must reject counter-factual definiteness. Examples of such realist interpretations are the consistent histories interpretation and the transactional interpretation (first proposed by John G. Cramer in 1986). Griffiths notes that it is not "local realism" that is ruled out by quantum mechanics but "classical realism".[27] Some workers in the field have also attempted to formulate hidden variable theories that exploit loopholes in actual experiments, such as the assumptions made in interpreting experimental data, although no theory has been proposed that can reproduce all the results of quantum mechanics.
Alternatives are still possible. A recent review article based on the Wheeler–Feynman time-symmetric theory rewrites the entire theory in terms of retarded Liénard–Wiechert potentials only, which becomes manifestly causal, and, establishes a conservation law for total generalized momenta held instantaneously for any closed system.[28] The outcome results in correlation between particles from a "handshake principle" based on a variational principle applied to a system as a whole, an idea with a slightly non-local feature but the theory is nonetheless in agreement with the essential results of quantum electrodynamics and relativistic quantum chemistry.
There are also individual EPR-like experiments that have no local hidden variables explanation. Examples have been suggested by David Bohm and by Lucien Hardy.
Einstein's hope for a purely algebraic theory[edit]
The Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly through time with no collapsing of quantum wavefunctions. One problem for the Copenhagen interpretation is to precisely define wavefunction collapse. Einstein maintained that quantum mechanics is physically incomplete and logically unsatisfactory. In "The Meaning of Relativity", Einstein wrote, "One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality. But nobody knows how to find the basis for such a theory." If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox (although Bell's theorem would still be valid). If physical reality is totally finite, then the Copenhagen interpretation might be an approximation to an information processing system below the Planck scale.
"Acceptable theories" and the experiment[edit]
According to the present view of the situation, quantum mechanics flatly contradicts Einstein's philosophical postulate that any acceptable physical theory must fulfill "local realism".
In the EPR paper (1935), the authors realised that quantum mechanics was inconsistent with their assumptions, but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables (i.e., variables which were, at that point, still obscure to him), without any other change, to achieve an acceptable theory. He pursued these ideas for over twenty years until the end of his life, in 1955.
In contrast, John Bell, in his 1964 paper, showed that quantum mechanics and the class of hidden variable theories Einstein favored[29] would lead to different experimental results: different by a factor of 3/2 for certain correlations. So the issue of "acceptability", up to that time mainly concerning theory, finally became experimentally decidable.
There are many Bell test experiments, e.g., those of Alain Aspect and others. They support the predictions of quantum mechanics rather than the class of hidden variable theories supported by Einstein.[4]
Implications for quantum mechanics[edit]
Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" only because classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality depends on the interpretation of quantum mechanics one uses. In the Copenhagen interpretation, it is usually understood that instantaneous wave function collapse does occur. However, the view that there is no causal instantaneous effect has also been proposed within the Copenhagen interpretation: in this alternate view, measurement affects our ability to define (and measure) quantities in the physical system, not the system itself. In the many-worlds interpretation, locality is strictly preserved, since the effects of operations such as measurement affect only the state of the particle that is measured.[17] However, the results of the measurement are not unique—every possible result is obtained.
The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentally non-classical characteristics of the measurement process. Before the publication of the EPR paper, a measurement was often visualized as a physical disturbance that had to be inflicted directly upon the measured subsystem. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such pat and convenient but unhelpful explanations of quantum mechanics remain commonplace today,[30][31] but they fail to explain (among other things) the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle. In fact, Yakir Aharonov and his collaborators have developed a whole theory of so-called Weak measurement.[15]:181–184
Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangled particles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.[32]:83–100
Mathematical formulation[edit]
The above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional complex vector space V, with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction, denoted Sx, Sy, and Sz respectively, can be represented using the Pauli matrices:[20]:9
where is the reduced Planck constant (or the Planck constant divided by 2π).
The eigenstates of Sz are represented as
and the eigenstates of Sx are represented as
The vector space of the electron-positron pair is , the tensor product of the electron's and positron's vector spaces. The spin singlet state is
where the two terms on the right hand side are what we have referred to as state I and state II above.
From the above equations, it can be shown that the spin singlet can also be written as
where the terms on the right hand side are what we have referred to as state Ia and state IIa.
To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of Sz (or Sx), Bob's value of Sz (or Sx) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of measurement in quantum mechanics. When Sz is measured, the system state ψ collapses into an eigenvector of Sz. If the measurement result is +z, this means that immediately after measurement the system state undergoes an orthogonal projection of ψ onto the space of states of the form
For the spin singlet, the new state is
Similarly, if Alice's measurement result is −z, the system undergoes an orthogonal projection onto
which means that the new state is
This implies that the measurement for Sz for Bob's positron is now determined. It will be −z in the first case or +z in the second case.
It remains only to show that Sx and Sz cannot simultaneously possess definite values in quantum mechanics. One may show in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally, one may use the fact that the operators do not commute,
along with the Heisenberg uncertainty relation
See also[edit]
1. ^ a b c Einstein, A; B Podolsky; N Rosen (1935-05-15). "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?" (PDF). Physical Review. 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
2. ^ Gaasbeek, Bram (Jul 22, 2010). "Demystifying the Delayed Choice Experiments". arXiv:1007.3977v1Freely accessible [quant-ph].
3. ^ Bell, John. On the Einstein–Poldolsky–Rosen paradox, Physics 1 3, 195–200, Nov. 1964
4. ^ a b Aspect A (1999-03-18). "Bell's inequality test: more ideal than ever" (PDF). Nature. 398 (6724): 189–90. Bibcode:1999Natur.398..189A. doi:10.1038/18296.
5. ^ Bohr, N. (1935-10-13). "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?". Physical Review. 48 (8): 696–702. Bibcode:1935PhRv...48..696B. doi:10.1103/PhysRev.48.696.
6. ^ Advances in atomic and molecular physics, Volume 14 By David Robert Bates
7. ^ Gribbin, J. (1984). In Search of Schrödinger's Cat. Black Swan. ISBN 0-7045-3071-6.
8. ^ The Einstein–Podolsky–Rosen Argument in Quantum Theory (Stanford Encyclopedia of Philosophy)
9. ^ von Neumann, J. (1932/1955). In Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, translated into English by Beyer, R.T., Princeton University Press, Princeton, cited by Baggott, J. (2004) Beyond Measure: Modern physics, philosophy, and the meaning of quantum theory, Oxford University Press, Oxford, ISBN 0-19-852927-9, pages 144–145.
10. ^ Bohm, D. (1951). Quantum Theory, Prentice-Hall, Englewood Cliffs, page 29, and Chapter 5 section 3, and Chapter 22 Section 19.
11. ^ Quoted in Kaiser, David. "Bringing the human actors back on stage: the personal context of the Einstein–Bohr debate", British Journal for the History of Science 27 (1994): 129–152, on page 147.
12. ^ Einstein, Albert (1936). "Physik und realität". Journal of the Franklin Institute. Elsevier. 221 (3): 313–347. doi:10.1016/S0016-0032(36)91045-1. Retrieved 9 December 2012. English translation by Jean Piccard, pp 349–382 in the same issue, doi:10.1016/S0016-0032(36)91047-5).
13. ^ a b c d Kumar, Manjit (2011). Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality (Reprint ed.). W. W. Norton & Company. pp. 305–306. ISBN 978-0393339888.
14. ^ a b c Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7
15. ^ a b Laloe, Franck (2012), Do We Really Understand Quantum Mechanics, Cambridge University Press, arXiv:quant-ph/0209123Freely accessible, Bibcode:2002quant.ph..9123L, ISBN 978-1-107-02501-1
16. ^ George Greenstein and Arthur G. Zajonc, The Quantum Challenge, p. "[Experiments in the early 1980s] have conclusively shown that quantum mechanics is indeed orrect, and that the EPR argument had relied upon incorrect assumptions."
17. ^ a b c Blaylock, Guy (January 2010). "The EPR paradox, Bell's inequality, and the question of locality". American Journal of Physics. 78 (1): 111–120. arXiv:0902.3827Freely accessible. Bibcode:2010AmJPh..78..111B. doi:10.1119/1.3243279.
18. ^ Bell, John (1981). "Bertlmann's socks and the nature of reality". J. Physique colloques. C22: 41–62. Bibcode:1988nbpw.conf..245B.
19. ^ John Archibald Wheeler; Wojciech Hubert Zurek (14 July 2014). Quantum Theory and Measurement. Princeton University Press. ISBN 978-1-4008-5455-4.
20. ^ a b Sakurai, J. J.; Napolitano, Jim (2010), Modern Quantum Mechanics (2nd ed.), Addison-Wesley, ISBN 978-0805382914
21. ^ Pitowsky, Itamar (1989). "From George Boole To John Bell — The Origins of Bell's Inequality". Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Dordrecht: Springer Netherlands. pp. 37–49. doi:10.1007/978-94-017-0849-4_6. ISBN 978-90-481-4058-9.
22. ^ Landau, L. J. (1987). "On the violation of Bell's inequality in quantum theory". Physics Letters. 120 (2): 4–6. Bibcode:1987PhLA..120...54L. doi:10.1016/0375-9601(87)90075-2.
23. ^ Streater, R.F. (2017). Lost Causes in and beyond Physics. Springer Berlin Heidelberg. ISBN 9783540365822.
24. ^ Hess, Karl (2005). Bell’s theorem: Critique of proofs with and without inequalities. AIP. pp. 150–157. arXiv:quant-ph/0410015Freely accessible. doi:10.1063/1.1874568. ISSN 0094-243X.
25. ^ Hess, Karl; Raedt, Hans De; Michielsen, Kristel (2012-11-01). "Hidden assumptions in the derivation of the theorem of Bell". Physica Scripta. IOP Publishing. T151: 014002. arXiv:1108.3583Freely accessible. Bibcode:2012PhST..151a4002H. doi:10.1088/0031-8949/2012/t151/014002. ISSN 0031-8949.
26. ^ a b De Raedt, Hans; Michielsen, Kristel; Hess, Karl (2016). "The digital computer as a metaphor for the perfect laboratory experiment: Loophole-free Bell experiments". Computer Physics Communications. Elsevier BV. 209: 42–47. Bibcode:2016CoPhC.209...42D. doi:10.1016/j.cpc.2016.08.010. ISSN 0010-4655.
27. ^ a b Griffiths, Robert B. (2010-10-21). "Quantum Locality". Foundations of Physics. Springer Nature. 41 (4): 705–733. arXiv:0908.2914Freely accessible. Bibcode:2011FoPh...41..705G. doi:10.1007/s10701-010-9512-5. ISSN 0015-9018.
28. ^ Scott, T. C.; Andrae, D. (2015). "Quantum Nonlocality and Conservation of momentum". Phys. Essays. 28 (3): 374–385. Bibcode:2015PhyEs..28..374S. doi:10.4006/0836-1398-28.3.374.
29. ^ "Clearing up mysteries: the original goal" (PDF).
30. ^ Furuta, Aya. "One Thing Is Certain: Heisenberg's Uncertainty Principle Is Not Dead". Scientific American. Retrieved 16 January 2017. Yet the uncertainty principle comes in two superficially similar formulations that even many practicing physicists tend to confuse. Werner Heisenberg's own version is that in observing the world, we inevitably disturb it. And that is wrong, as a research team at the Vienna University of Technology has now vividly demonstrated.
31. ^ Jha, Alok (10 November 2013). "What is Heisenberg's Uncertainty Principle?". The Guardian. Retrieved 16 January 2017. One way to think about the uncertainty principle is as an extension of how we see and measure things in the everyday world... the act of observation affects the particle being observed
32. ^ Haroche, Serge; Raimond, Jean-Michel (2006). Exploring the Quantum: Atoms, Cavities, and Photons (1st ed.). Oxford University Press. ISBN 978-0198509141.
Selected papers[edit]
• P. H. Eberhard, Bell's theorem without hidden variables. Nuovo Cimento 38B1 75 (1977).
• P. H. Eberhard, Bell's theorem and the different concepts of locality. Nuovo Cimento 46B 392 (1978).
• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[2]
• A. Fine, Do Correlations need to be explained?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986).
• L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71 1665 (1993).[3]
• M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie 26 683 (2001)
• Peres, Asher (2005). "Einstein, Podolsky, Rosen, and Shannon". Foundations of Physics. Kluwer Academic Publishers. 35 (3): 511–514. arXiv:quant-ph/0310010Freely accessible. Bibcode:2005FoPh...35..511P. doi:10.1007/s10701-004-1986-6. ISSN 0015-9018.
• P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006)
• M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe and D. J. Wineland, Experimental violation of a Bell's inequality with efficient detection, Nature 409, 791–794 (15 February 2001). [4]
• M. Smerlak, C. Rovelli, Relational EPR [5]
• John S. Bell (1987) Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press. ISBN 0-521-36869-3.
• Arthur Fine (1996) The Shaky Game: Einstein, Realism and the Quantum Theory, 2nd ed. Univ. of Chicago Press.
• Selleri, F. (1988) Quantum Mechanics Versus Local Realism: The Einstein–Podolsky–Rosen Paradox. New York: Plenum Press. ISBN 0-306-42739-7
• Leon Lederman, L., Teresi, D. (1993). The God Particle: If the Universe is the Answer, What is the Question? Houghton Mifflin Company, pages 21, 187 to 189.
• John Gribbin (1984) In Search of Schrödinger's Cat. Black Swan. ISBN 978-0-552-12555-0
External links[edit] |
a3ddba825e92e0eb | Joint Admission Test For M.Sc Syllabus for Physics | JAM Syllabus For PH | JAM Physics Syllabus
Joint Admission Test For M.Sc Syllabus for Physics (PH)
Mathematical Methods: Calculus of single and multiple variables, partial derivatives, Jacobian, imperfect and perfect differentials, Taylor expansion, Fourier series. Vector algebra, Vector Calculus, Multiple integrals, Divergence theorem, Green’s theorem, Stokes’ theorem. First and linear second order differential equations. Matrices and determinants, Algebra of complex numbers.
Mechanics and General Properties of Matter: Newton’s laws of motion and applications, Velocity and acceleration in Cartesian, polar and cylindrical coordinate systems, uniformly rotating frame, centrifugal and Coriolis forces, Motion under a central force, Kepler’s laws, Gravitational Law and field, Conservative and non-conservative forces. System of particles, Centre of mass, equation of motion of the CM, conservation of linear and angular momentum, conservation of energy, variable mass systems. Elastic and inelastic collisions. Rigid body motion, fixed axis rotations, rotation and translation, moments of Inertia and products of Inertia. Principal moments and axes. Elasticity, Hooke’s law and elastic constants of isotropic solid, stress energy. Kinematics of moving fluids, equation of continuity, Euler’s equation, Bernoulli’s theorem, viscous fluids, surface tension and surface energy, capillarity.
Oscillations, Waves and Optics:
Kinetic theory, Thermodynamics: Elements of Kinetic theory of gases. Velocity distribution and Equipartition of energy. Specific heat of Mono-, di- and tri-atomic gases. Ideal gas, van-der-Waals gas and equation of state. Mean free path. Laws of thermodynamics. Zeroeth law and concept of thermal equilibrium. First law and its consequences. Isothermal and adiabatic processes. Reversible, irreversible and quasi-static processes. Second law and entropy. Carnot cycle. Maxwells thermodynamic relations and simple applications. Thermodynamic potentials and their applications. Phase transitions and Clausius-Clapeyron equation.
Modern Physics: Inertial frames and Galilean invariance. Postulates of special relativity. Lorentz transformations. Length contraction, time dilation. Relativistic velocity addition theorem, mass energy equivalence. Blackbody radiation, photoelectric effect, Compton effect, Bohr’s atomic model, X-rays. Wave-particle duality, Uncertainty principle, Schrödinger equation and its solution for one, two and three dimensional boxes. Reflection and transmission at a step potential, tunneling through a barrier. Pauli exclusion principle. Distinguishable and indistinguishable particles. Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics. Structure of atomic nucleus, mass and binding energy. Radioactivity and its applications. Laws of radioactive decay. Fission and fusion.
Solid State Physics, Devices and Electronics:
Crystal structure, Bravais lattices and basis. Miller indices. X-ray diffraction and Bragg’s law, Einstein and Debye theory of specific heat. Free electron theory of metals. Fermi energy and density of states. Origin of energy bands. Concept of holes and effective mass. Elementary ideas about dia-, para- and ferromagnetism, Langevin’s theory of paramagnetism, Curie’s law. Intrinsic and extrinsic semiconductors. Fermi level. p-n junctions, transistors. Transistor circuits in CB, CE, CC modes. Amplifier circuits with transistors. Operational amplifiers. OR, AND, NOR and NAND gates.
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b46d964cf74a9bf9 | Monday, 13 May 2013
Coursera - Quantum Mechanics
A course on quantum mechanics! What am I thinking?
So I signed up for a course on quantum mechanics. I mean, how hard can it be?
Answer - *!?**!!* hard!
I brushed off my knowledge of imaginary numbers, I went through the introductory maths materials - they didn't seem too hard. OK - I struggled to remember complex conjugates, and one or two other things.
I thought there might be a reasonable introduction, and an explanation about things - which there was. However the learning curve was incredibly steep, and was sort of emphasised by the first homework.
Q1 For what was Albert Einstein awarded the Nobel prize?
• General Relativity
• The expansion of the universe
• The photo-electric effect
• Electron diffraction
OK - I actually knew that one - although it was in the course materials too.
Q2 Recall how the Schrödinger equation was motivated by the non-relativistic dispersion relation
E=p22m. If we follow the same procedure for the case of a relativistic dispersion relation (E2=p2c2+m2c4), what equation do we arrive at? (For simplicity consider the one-dimensional case)
Ouch! The gloves are off! The lectures also had a grading system. No stars was for everyone, 1 star had some maths in it, two stars extensive maths, and three stars - mega maths. Most of the videos were in the 2/3 star range.
I actually enjoyed doing some of the integration - but realised I was gradually losing the plot as the course went on. I never really got a good handle on the bra-ket notation - I still don't really get it's power - I'm missing something I'm sure, but they didn't spend very long on it, and the books I got didn't help. Then it was onto Dirac deltas, Levi-civita notation and stuff about spin. By now I was really struggling with the weekly homeworks, and guessing as many as I was solving - I was no longer learning and close to drowning. I did think about giving up on the course, but I stayed the distance, and finished all the videos, all the homeworks.
This course had an exam - 1 chance at answer each question - 6 hour time limit. A couple of questions I could answer, the rest I guessed at, except those that required a numeric answer - which I couldn't do. I got 42% which I consider more than fair.
This gave me a total course mark of 72% - again more than I deserve.
So I probably got half way before I couldn't keep up, and for me it was hard to turn all that maths back into what it meant in the real world - even in the abstract. I guess that's not unusual in quantum mechanics!
Sayan Datta said...
Hello Julian,
Thanks for this post.
I am doing this course at its present iteration. Can you give me a rough idea about what percentage of homework and exam questions require entering numeric answers? I have trouble entering numeric answers...that's why I am asking the question.
Julian Onions said...
Quite a number of the homeworks require you to either work out maths expressions, or solve for numbers. Good luck!
Sayan Datta said...
Thanks for the reply... |
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Wikipedia β
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.
String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and condensed matter physics, and it has stimulated a number of major developments in pure mathematics. Because string theory potentially provides a unified description of gravity and particle physics, it is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter. Despite much work on these problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of its details.
String theory was first studied in the late 1960s as a theory of the strong nuclear force, before being abandoned in favor of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of particles known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. Five consistent versions of superstring theory were developed before it was conjectured in the mid-1990s that they were all different limiting cases of a single theory in eleven dimensions known as M-theory. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, which relates string theory to another type of physical theory called a quantum field theory.
One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and this has complicated efforts to develop theories of particle physics based on string theory. These issues have led some in the community to criticize these approaches to physics and question the value of continued research on string theory unification.
The fundamental objects of string theory are open and closed string models.
In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics. The first is Albert Einstein's general theory of relativity, a theory that explains the force of gravity and the structure of space and time. The other is quantum mechanics which is a completely different formulation to describe physical phenomena using the known probability principles. By the late 1970s, these two frameworks had proven to be sufficient to explain most of the observed features of the universe, from elementary particles to atoms to the evolution of stars and the universe as a whole.[1]
In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of quantum gravity.[1] The general theory of relativity is formulated within the framework of classical physics, whereas the other fundamental forces are described within the framework of quantum mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.[2] In addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, and the early universe.[a]
String theory is a theoretical framework that attempts to address these questions and many others. The starting point for string theory is the idea that the point-like particles of particle physics can also be modeled as one-dimensional objects called strings. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In this way, all of the different elementary particles may be viewed as vibrating strings. In string theory, one of the vibrational states of the string gives rise to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.[3]
One of the main developments of the past several decades in string theory was the discovery of certain "dualities", mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered a number of these dualities between different versions of string theory, and this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as M-theory.[4]
Studies of string theory have also yielded a number of results on the nature of black holes and the gravitational interaction. There are certain paradoxes that arise when one attempts to understand the quantum aspects of black holes, and work on string theory has attempted to clarify these issues. In late 1997 this line of work culminated in the discovery of the anti-de Sitter/conformal field theory correspondence or AdS/CFT.[5] This is a theoretical result which relates string theory to other physical theories which are better understood theoretically. The AdS/CFT correspondence has implications for the study of black holes and quantum gravity, and it has been applied to other subjects, including nuclear[6] and condensed matter physics.[7][8]
Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that it fully describes our universe, making it a theory of everything. One of the goals of current research in string theory is to find a solution of the theory that reproduces the observed spectrum of elementary particles, with a small cosmological constant, containing dark matter and a plausible mechanism for cosmic inflation. While there has been progress toward these goals, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of details.[9]
One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. The scattering of strings is most straightforwardly defined using the techniques of perturbation theory, but it is not known in general how to define string theory nonperturbatively.[10] It is also not clear whether there is any principle by which string theory selects its vacuum state, the physical state that determines the properties of our universe.[11] These problems have led some in the community to criticize these approaches to the unification of physics and question the value of continued research on these problems.[12]
Interaction in the quantum world: worldlines of point-like particles or a worldsheet swept up by closed strings in string theory.
The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields.[13]
In quantum field theory, one typically computes the probabilities of various physical events using the techniques of perturbation theory. Developed by Richard Feynman and others in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called Feynman diagrams to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions.[13]
The starting point for string theory is the idea that the point-like particles of quantum field theory can also be modeled as one-dimensional objects called strings.[14] The interaction of strings is most straightforwardly defined by generalizing the perturbation theory used in ordinary quantum field theory. At the level of Feynman diagrams, this means replacing the one-dimensional diagram representing the path of a point particle by a two-dimensional surface representing the motion of a string.[15] Unlike in quantum field theory, string theory does not have a full non-perturbative definition, so many of the theoretical questions that physicists would like to answer remain out of reach.[16]
The original version of string theory was bosonic string theory, but this version described only bosons, a class of particles which transmit forces between the matter particles, or fermions. Bosonic string theory was eventually superseded by theories called superstring theories. These theories describe both bosons and fermions, and they incorporate a theoretical idea called supersymmetry. This is a mathematical relation that exists in certain physical theories between the bosons and fermions. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa.[17]
There are several versions of superstring theory: type I, type IIA, type IIB, and two flavors of heterotic string theory (SO(32) and E8×E8). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA, IIB and heterotic include only closed strings.[18]
Extra dimensions
In everyday life, there are three familiar dimensions of space: height, width and length. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a four-dimensional spacetime. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.[19]
In spite of the fact that the universe is well described by four-dimensional spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily.[b] There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in condensed matter physics.[20] Finally, there exist scenarios in which there could actually be more than four dimensions of spacetime which have nonetheless managed to escape detection.[21]
One notable feature of string theories is that these theories require extra dimensions of spacetime for their mathematical consistency. In bosonic string theory, spacetime is 26-dimensional, while in superstring theory it is 10-dimensional, and in M-theory it is 11-dimensional. In order to describe real physical phenomena using string theory, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.[22]
A cross section of a quintic Calabi–Yau manifold
Compactification is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.[23] In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.[24]
Compactification can be used to construct models in which spacetime is effectively four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics, the compact extra dimensions must be shaped like a Calabi–Yau manifold.[23] A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians Eugenio Calabi and Shing-Tung Yau.[25]
Another approach to reducing the number of dimensions is the so-called brane-world scenario. In this approach, physicists assume that the observable universe is a four-dimensional subspace of a higher dimensional space. In such models, the force-carrying bosons of particle physics arise from open strings with endpoints attached to the four-dimensional subspace, while gravity arises from closed strings propagating through the larger ambient space. This idea plays an important role in attempts to develop models of real world physics based on string theory, and it provides a natural explanation for the weakness of gravity compared to the other fundamental forces.[26]
A diagram of string theory dualities. Yellow arrows indicate S-duality. Blue arrows indicate T-duality.
One notable fact about string theory is that the different versions of the theory all turn out to be related in highly nontrivial ways. One of the relationships that can exist between different string theories is called S-duality. This is a relationship which says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently. Type I string theory turns out to be equivalent by S-duality to the SO(32) heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality.[27]
In general, the term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. Two theories related by a duality need not be string theories. For example, Montonen–Olive duality is example of an S-duality relationship between quantum field theories. The AdS/CFT correspondence is example of a duality which relates string theory to a quantum field theory. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.[28]
Open strings attached to a pair of D-branes.
In string theory and other related theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For instance, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.[29]
In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to a certain mathematical condition on the system known as the Dirichlet boundary condition. The study of D-branes in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many problems in quantum field theory.[30]
Branes are frequently studied from a purely mathematical point of view, and they are described as objects of certain categories, such as the derived category of coherent sheaves on a complex algebraic variety, or the Fukaya category of a symplectic manifold. [31] The connection between the physical notion of a brane and the mathematical notion of a category has led to important mathematical insights in the fields of algebraic and symplectic geometry [32] and representation theory. [33]
Prior to 1995, theorists believed that there were five consistent versions of superstring theory (type I, type IIA, type IIB, and two versions of heterotic string theory). This understanding changed in 1995 when Edward Witten suggested that the five theories were just special limiting cases of an eleven-dimensional theory called M-theory. Witten's conjecture was based on the work of a number of other physicists, including Ashoke Sen, Chris Hull, Paul Townsend, and Michael Duff. His announcement led to a flurry of research activity now known as the second superstring revolution.[34]
Unification of superstring theories
A schematic illustration of the relationship between M-theory, the five superstring theories, and eleven-dimensional supergravity. The shaded region represents a family of different physical scenarios that are possible in M-theory. In certain limiting cases corresponding to the cusps, it is natural to describe the physics using one of the six theories labeled there.
In the 1970s, many physicists became interested in supergravity theories, which combine general relativity with supersymmetry. Whereas general relativity makes sense in any number of dimensions, supergravity places an upper limit on the number of dimensions.[35] In 1978, work by Werner Nahm showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven.[36] In the same year, Eugene Cremmer, Bernard Julia, and Joel Scherk of the École Normale Supérieure showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.[37][38]
Initially, many physicists hoped that by compactifying eleven-dimensional supergravity, it might be possible to construct realistic models of our four-dimensional world. The hope was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the strong and weak nuclear forces, and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as chirality. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.[38]
In the first superstring revolution in 1984, many physicists turned to string theory as a unified theory of particle physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects.[38] Another feature of string theory that many physicists were drawn to in the 1980s and 1990s was its high degree of uniqueness. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary Lagrangian. In string theory, the possibilities are much more constrained: by the 1990s, physicists had argued that there were only five consistent supersymmetric versions of the theory.[38]
Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation.[38] However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways. They found that a system of strongly interacting strings can, in some cases, be viewed as a system of weakly interacting strings. This phenomenon is known as S-duality. It was studied by Ashoke Sen in the context of heterotic strings in four dimensions[39][40] and by Chris Hull and Paul Townsend in the context of the type IIB theory.[41] Theorists also found that different string theories may be related by T-duality. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent.[42]
At around the same time, as many physicists were studying the properties of strings, a small group of physicists was examining the possible applications of higher dimensional objects. In 1987, Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes.[43] Intuitively, these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after this discovery, Michael Duff, Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle.[44] In this setting, one can imagine the membrane wrapping around the circular dimension. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. In fact, Duff and his collaborators showed that this construction reproduces exactly the strings appearing in type IIA superstring theory.[45]
Speaking at a string theory conference in 1995, Edward Witten made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions. Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of higher dimensional branes in string theory.[46] In the months following Witten's announcement, hundreds of new papers appeared on the Internet confirming different parts of his proposal.[47] Today this flurry of work is known as the second superstring revolution.[48]
Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In a paper from 1996, Hořava and Witten wrote "As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes."[49] In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the M should stand for "magic", "mystery", or "membrane" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.[50]
Matrix theory
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.[51]
One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.[51]
The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry. This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra.[52] In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory, a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry.[53] This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.[54][55]
Black holes
In general relativity, a black hole is defined as a region of spacetime in which the gravitational field is so strong that no particle or radiation can escape. In the currently accepted models of stellar evolution, black holes are thought to arise when massive stars undergo gravitational collapse, and many galaxies are thought to contain supermassive black holes at their centers. Black holes are also important for theoretical reasons, as they present profound challenges for theorists attempting to understand the quantum aspects of gravity. String theory has proved to be an important tool for investigating the theoretical properties of black holes because it provides a framework in which theorists can study their thermodynamics.[56]
Bekenstein–Hawking formula
In the branch of physics called statistical mechanics, entropy is a measure of the randomness or disorder of a physical system. This concept was studied in the 1870s by the Austrian physicist Ludwig Boltzmann, who showed that the thermodynamic properties of a gas could be derived from the combined properties of its many constituent molecules. Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give a precise definition of entropy as the natural logarithm of the number of different states of the molecules (also called microstates) that give rise to the same macroscopic features.[57]
In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases, the entropy scales with the volume. In the 1970s, the physicist Jacob Bekenstein suggested that the entropy of a black hole is instead proportional to the surface area of its event horizon, the boundary beyond which matter and radiation is lost to its gravitational attraction.[58] When combined with ideas of the physicist Stephen Hawking,[59] Bekenstein's work yielded a precise formula for the entropy of a black hole. The Bekenstein–Hawking formula expresses the entropy S as
where c is the speed of light, k is Boltzmann's constant, ħ is the reduced Planck constant, G is Newton's constant, and A is the surface area of the event horizon.[60]
Like any physical system, a black hole has an entropy defined in terms of the number of different microstates that lead to the same macroscopic features. The Bekenstein–Hawking entropy formula gives the expected value of the entropy of a black hole, but by the 1990s, physicists still lacked a derivation of this formula by counting microstates in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the viability of any theory of quantum gravity such as string theory.[61]
Derivation within string theory
In a paper from 1996, Andrew Strominger and Cumrun Vafa showed how to derive the Beckenstein–Hawking formula for certain black holes in string theory.[62] Their calculation was based on the observation that D-branes—which look like fluctuating membranes when they are weakly interacting—become dense, massive objects with event horizons when the interactions are strong. In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of different ways of placing D-branes in spacetime so that their combined mass and charge is equal to a given mass and charge for the resulting black hole. Their calculation reproduced the Bekenstein–Hawking formula exactly, including the factor of 1/4.[63] Subsequent work by Strominger, Vafa, and others refined the original calculations and gave the precise values of the "quantum corrections" needed to describe very small black holes.[64][65]
The black holes that Strominger and Vafa considered in their original work were quite different from real astrophysical black holes. One difference was that Strominger and Vafa considered only extremal black holes in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge.[66] Strominger and Vafa also restricted attention to black holes in five-dimensional spacetime with unphysical supersymmetry.[67]
Although it was originally developed in this very particular and physically unrealistic context in string theory, the entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity. Indeed, in 1998, Strominger argued that the original result could be generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry.[68] In collaboration with several other authors in 2010, he showed that some results on black hole entropy could be extended to non-extremal astrophysical black holes.[69][70]
AdS/CFT correspondence
One approach to formulating string theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. This is a theoretical result which implies that string theory is in some cases equivalent to a quantum field theory. In addition to providing insights into the mathematical structure of string theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.[6] The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997.[71] Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov,[72] and by Edward Witten.[73] By 2010, Maldacena's article had over 7000 citations, becoming the most highly cited article in the field of high energy physics.[c]
Overview of the correspondence
In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain vacuum solution of Einstein's equation called anti-de Sitter space.[74] In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the metric) is different from the notion of distance in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be viewed as a disk as illustrated on the left.[75] This image shows a tessellation of a disk by triangles and squares. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior.[76]
This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.[75]
An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-dimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around any given point, it looks just like Minkowski space, the model of spacetime used in nongravitational physics.[77] One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory. The claim is that this quantum field theory is equivalent to a gravitational theory, such as string theory, in the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.[78]
Applications to quantum gravity
The discovery of the AdS/CFT correspondence was a major advance in physicists' understanding of string theory and quantum gravity. One reason for this is that the correspondence provides a formulation of string theory in terms of quantum field theory, which is well understood by comparison. Another reason is that it provides a general framework in which physicists can study and attempt to resolve the paradoxes of black holes.[56]
In 1975, Stephen Hawking published a calculation which suggested that black holes are not completely black but emit a dim radiation due to quantum effects near the event horizon.[59] At first, Hawking's result posed a problem for theorists because it suggested that black holes destroy information. More precisely, Hawking's calculation seemed to conflict with one of the basic postulates of quantum mechanics, which states that physical systems evolve in time according to the Schrödinger equation. This property is usually referred to as unitarity of time evolution. The apparent contradiction between Hawking's calculation and the unitarity postulate of quantum mechanics came to be known as the black hole information paradox.[79]
The AdS/CFT correspondence resolves the black hole information paradox, at least to some extent, because it shows how a black hole can evolve in a manner consistent with quantum mechanics in some contexts. Indeed, one can consider black holes in the context of the AdS/CFT correspondence, and any such black hole corresponds to a configuration of particles on the boundary of anti-de Sitter space.[80] These particles obey the usual rules of quantum mechanics and in particular evolve in a unitary fashion, so the black hole must also evolve in a unitary fashion, respecting the principles of quantum mechanics.[81] In 2005, Hawking announced that the paradox had been settled in favor of information conservation by the AdS/CFT correspondence, and he suggested a concrete mechanism by which black holes might preserve information.[82]
Applications to nuclear physics
A magnet levitating above a high-temperature superconductor. Today some physicists are working to understand high-temperature superconductivity using the AdS/CFT correspondence.[7]
In addition to its applications to theoretical problems in quantum gravity, the AdS/CFT correspondence has been applied to a variety of problems in quantum field theory. One physical system that has been studied using the AdS/CFT correspondence is the quark–gluon plasma, an exotic state of matter produced in particle accelerators. This state of matter arises for brief instants when heavy ions such as gold or lead nuclei are collided at high energies. Such collisions cause the quarks that make up atomic nuclei to deconfine at temperatures of approximately two trillion kelvins, conditions similar to those present at around 10−11 seconds after the Big Bang.[83]
The physics of the quark–gluon plasma is governed by a theory called quantum chromodynamics, but this theory is mathematically intractable in problems involving the quark–gluon plasma.[d] In an article appearing in 2005, Đàm Thanh Sơn and his collaborators showed that the AdS/CFT correspondence could be used to understand some aspects of the quark–gluon plasma by describing it in the language of string theory.[84] By applying the AdS/CFT correspondence, Sơn and his collaborators were able to describe the quark gluon plasma in terms of black holes in five-dimensional spacetime. The calculation showed that the ratio of two quantities associated with the quark–gluon plasma, the shear viscosity and volume density of entropy, should be approximately equal to a certain universal constant. In 2008, the predicted value of this ratio for the quark–gluon plasma was confirmed at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory.[85][86]
Applications to condensed matter physics
The AdS/CFT correspondence has also been used to study aspects of condensed matter physics. Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including Subir Sachdev hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string theory and learn more about their behavior.[85]
So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an insulator. A superfluid is a system of electrically neutral atoms that flows without any friction. Such systems are often produced in the laboratory using liquid helium, but recently experimentalists have developed new ways of producing artificial superfluids by pouring trillions of cold atoms into a lattice of criss-crossing lasers. These atoms initially behave as a superfluid, but as experimentalists increase the intensity of the lasers, they become less mobile and then suddenly transition to an insulating state. During the transition, the atoms behave in an unusual way. For example, the atoms slow to a halt at a rate that depends on the temperature and on Planck's constant, the fundamental parameter of quantum mechanics, which does not enter into the description of the other phases. This behavior has recently been understood by considering a dual description where properties of the fluid are described in terms of a higher dimensional black hole.[87]
In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing models of real world physics that combine general relativity and particle physics. Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. String phenomenology is the part of string theory that attempts to construct realistic or semi-realistic models based on string theory.
Particle physics
The currently accepted theory describing elementary particles and their interactions is known as the standard model of particle physics. This theory provides a unified description of three of the fundamental forces of nature: electromagnetism and the strong and weak nuclear forces. Despite its remarkable success in explaining a wide range of physical phenomena, the standard model cannot be a complete description of reality. This is because the standard model fails to incorporate the force of gravity and because of problems such as the hierarchy problem and the inability to explain the structure of fermion masses or dark matter.
String theory has been used to construct a variety of models of particle physics going beyond the standard model. Typically, such models are based on the idea of compactification. Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles.[88] One popular way of deriving realistic physics from string theory is to start with the heterotic theory in ten dimensions and assume that the six extra dimensions of spacetime are shaped like a six-dimensional Calabi–Yau manifold. Such compactifications offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct realistic or semi-realistic models of our four-dimensional world based on M-theory.[89]
The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. Despite its success in explaining many observed features of the universe including galactic redshifts, the relative abundance of light elements such as hydrogen and helium, and the existence of a cosmic microwave background, there are several questions that remain unanswered. For example, the standard Big Bang model does not explain why the universe appears to be same in all directions, why it appears flat on very large distance scales, or why certain hypothesized particles such as magnetic monopoles are not observed in experiments.[90]
Currently, the leading candidate for a theory going beyond the Big Bang is the theory of cosmic inflation. Developed by Alan Guth and others in the 1980s, inflation postulates a period of extremely rapid accelerated expansion of the universe prior to the expansion described by the standard Big Bang theory. The theory of cosmic inflation preserves the successes of the Big Bang while providing a natural explanation for some of the mysterious features of the universe.[91] The theory has also received striking support from observations of the cosmic microwave background, the radiation that has filled the sky since around 380,000 years after the Big Bang.[92]
In the theory of inflation, the rapid initial expansion of the universe is caused by a hypothetical particle called the inflaton. The exact properties of this particle are not fixed by the theory but should ultimately be derived from a more fundamental theory such as string theory.[93] Indeed, there have been a number of attempts to identify an inflaton within the spectrum of particles described by string theory, and to study inflation using string theory. While these approaches might eventually find support in observational data such as measurements of the cosmic microwave background, the application of string theory to cosmology is still in its early stages.[94]
Connections to mathematics
In addition to influencing research in theoretical physics, string theory has stimulated a number of major developments in pure mathematics. Like many developing ideas in theoretical physics, string theory does not at present have a mathematically rigorous formulation in which all of its concepts can be defined precisely. As a result, physicists who study string theory are often guided by physical intuition to conjecture relationships between the seemingly different mathematical structures that are used to formalize different parts of the theory. These conjectures are later proved by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics.[95]
Mirror symmetry
The Clebsch cubic is an example of a kind of geometric object called an algebraic variety. A classical result of enumerative geometry states that there are exactly 27 straight lines that lie entirely on this surface.
After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late 1980s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold.[96] Instead, two different versions of string theory, type IIA and type IIB, can be compactified on completely different Calabi–Yau manifolds giving rise to the same physics. In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry.[97]
Regardless of whether Calabi–Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between different string theories has significant mathematical consequences. The Calabi–Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry, a branch of mathematics concerned with counting the numbers of solutions to geometric questions.[31][98]
Enumerative geometry studies a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic illustrated on the right is an algebraic variety defined using a certain polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface.[99]
Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi–Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five. This problem was solved by the nineteenth-century German mathematician Hermann Schubert, who found that there are exactly 2,875 such lines. In 1986, geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250.[100]
By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish.[101] The field was reinvigorated in May 1991 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parks showed that mirror symmetry could be used to translate difficult mathematical questions about one Calabi–Yau manifold into easier questions about its mirror.[102] In particular, they used mirror symmetry to show that a six-dimensional Calabi–Yau manifold can contain exactly 317,206,375 curves of degree three.[101] In addition to counting degree-three curves, Candelas and his collaborators obtained a number of more general results for counting rational curves which went far beyond the results obtained by mathematicians.[103]
Originally, these results of Candelas were justified on physical grounds. However, mathematicians generally prefer rigorous proofs that do not require an appeal to physical intuition. Inspired by physicists' work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry.[e] Today mirror symmetry is an active area of research in mathematics, and mathematicians are working to develop a more complete mathematical understanding of mirror symmetry based on physicists' intuition.[104] Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich[32] and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.[105]
Monstrous moonshine
An equilateral triangle can be rotated through 120°, 240°, or 360°, or reflected in any of the three lines pictured without changing its shape.
Group theory is the branch of mathematics that studies the concept of symmetry. For example, one can consider a geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle without changing its shape. One can rotate it through 120°, 240°, or 360°, or one can reflect in any of the lines labeled S0, S1, or S2 in the picture. Each of these operations is called a symmetry, and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a group. In this particular example, the group is known as the dihedral group of order 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a finite group.[106]
Mathematicians often strive for a classification (or list) of all mathematical objects of a given type. It is generally believed that finite groups are too diverse to admit a useful classification. A more modest but still challenging problem is to classify all finite simple groups. These are finite groups which may be used as building blocks for constructing arbitrary finite groups in the same way that prime numbers can be used to construct arbitrary whole numbers by taking products.[f] One of the major achievements of contemporary group theory is the classification of finite simple groups, a mathematical theorem which provides a list of all possible finite simple groups.[107]
This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances. The largest sporadic group, the so-called monster group, has over 1053 elements, more than a thousand times the number of atoms in the Earth.[108]
A graph of the j-function in the complex plane
A seemingly unrelated construction is the j-function of number theory. This object belongs to a special class of functions called modular functions, whose graphs form a certain kind of repeating pattern.[109] Although this function appears in a branch of mathematics which seems very different from the theory of finite groups, the two subjects turn out to be intimately related. In the late 1970s, mathematicians John McKay and John Thompson noticed that certain numbers arising in the analysis of the monster group (namely, the dimensions of its irreducible representations) are related to numbers that appear in a formula for the j-function (namely, the coefficients of its Fourier series).[110] This relationship was further developed by John Horton Conway and Simon Norton[111] who called it monstrous moonshine because it seemed so far fetched.[112]
In 1992, Richard Borcherds constructed a bridge between the theory of modular functions and finite groups and, in the process, explained the observations of McKay and Thompson.[113][114] Borcherds' work used ideas from string theory in an essential way, extending earlier results of Igor Frenkel, James Lepowsky, and Arne Meurman, who had realized the monster group as the symmetries of a particular[which?] version of string theory.[115] In 1998, Borcherds was awarded the Fields medal for his work.[116]
Since the 1990s, the connection between string theory and moonshine has led to further results in mathematics and physics.[108] In 2010, physicists Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa discovered connections between a different sporadic group, the Mathieu group M24, and a certain version[which?] of string theory.[117] Miranda Cheng, John Duncan, and Jeffrey A. Harvey proposed a generalization of this moonshine phenomenon called umbral moonshine,[118] and their conjecture was proved mathematically by Duncan, Michael Griffin, and Ken Ono.[119] Witten has also speculated that the version of string theory appearing in monstrous moonshine might be related to a certain simplified model of gravity in three spacetime dimensions.[120]
Early results
Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein. The first person to add a fifth dimension to a theory of gravity was Gunnar Nordström in 1914, who noted that gravity in five dimensions describes both gravity and electromagnetism in four. Nordström attempted to unify electromagnetism with his theory of gravitation, which was however superseded by Einstein's general relativity in 1919. Thereafter, German mathematician Theodor Kaluza combined the fifth dimension with general relativity, and only Kaluza is usually credited with the idea. In 1926, the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension—it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions.
String theory was originally developed during the late 1960s and early 1970s as a never completely successful theory of hadrons, the subatomic particles like the proton and neutron that feel the strong interaction. In the 1960s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories that did not presume that they were composed of any fundamental particles, but would construct their interactions from self-consistency conditions on the S-matrix. The S-matrix approach was started by Werner Heisenberg in the 1940s as a way of constructing a theory that did not rely on the local notions of space and time, which Heisenberg believed break down at the nuclear scale. While the scale was off by many orders of magnitude, the approach he advocated was ideally suited for a theory of quantum gravity.
Working with experimental data, R. Dolen, D. Horn and C. Schmid developed some sum rules for hadron exchange. When a particle and antiparticle scatter, virtual particles can be exchanged in two qualitatively different ways. In the s-channel, the two particles annihilate to make temporary intermediate states that fall apart into the final state particles. In the t-channel, the particles exchange intermediate states by emission and absorption. In field theory, the two contributions add together, one giving a continuous background contribution, the other giving peaks at certain energies. In the data, it was clear that the peaks were stealing from the background—the authors interpreted this as saying that the t-channel contribution was dual to the s-channel one, meaning both described the whole amplitude and included the other.
The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen–Horn–Schmid duality, later renamed world-sheet duality. The amplitude needed poles where the particles appear, on straight line trajectories, and there is a special mathematical function whose poles are evenly spaced on half the real line—the gamma function— which was widely used in Regge theory. By manipulating combinations of gamma functions, Veneziano was able to find a consistent scattering amplitude with poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at high energy. The amplitude could fit near-beam scattering data as well as other Regge type fits, and had a suggestive integral representation that could be used for generalization.
Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle that appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon. Miguel Virasoro and Joel Shapiro found a different amplitude now understood to be that of closed strings, while Ziro Koba and Holger Nielsen generalized Veneziano's integral representation to multiparticle scattering. Veneziano and Sergio Fubini introduced an operator formalism for computing the scattering amplitudes that was a forerunner of world-sheet conformal theory, while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on the states. Claud Lovelace calculated a loop amplitude, and noted that there is an inconsistency unless the dimension of the theory is 26. Charles Thorn, Peter Goddard and Richard Brower went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to 26.
In 1969–70, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind recognized that the theory could be given a description in space and time in terms of strings. The scattering amplitudes were derived systematically from the action principle by Peter Goddard, Jeffrey Goldstone, Claudio Rebbi, and Charles Thorn, giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the Virasoro conditions.
In 1971, Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. John Schwarz and André Neveu added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10. Stanley Mandelstam formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave a different formulation of the bosonic string, as a string field theory, with infinitely many particle types and with fields taking values not on points, but on loops and curves.
In 1974, Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle that obeyed the correct Ward identities to be a graviton. John Schwarz and Joel Scherk came to the same conclusion and made the bold leap to suggest that string theory was a theory of gravity, not a theory of hadrons. They reintroduced Kaluza–Klein theory as a way of making sense of the extra dimensions. At the same time, quantum chromodynamics was recognized as the correct theory of hadrons, shifting the attention of physicists and apparently leaving the bootstrap program in the dustbin of history.
String theory eventually made it out of the dustbin, but for the following decade all work on the theory was completely ignored. Still, the theory continued to develop at a steady pace thanks to the work of a handful of devotees. Ferdinando Gliozzi, Joel Scherk, and David Olive realized in 1977 that the original Ramond and Neveu Schwarz-strings were separately inconsistent and needed to be combined. The resulting theory did not have a tachyon, and was proven to have space-time supersymmetry by John Schwarz and Michael Green in 1984. The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively. In 1979, Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein equations of general relativity, emerge from the renormalization group equations for the two-dimensional field theory. Schwarz and Green discovered T-duality, and constructed two superstring theories—IIA and IIB related by T-duality, and type I theories with open strings. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices.
First superstring revolution
In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino. This led him, in collaboration with Luis Álvarez-Gaumé, to study violations of the conservation laws in gravity theories with anomalies, concluding that type I string theories were inconsistent. Green and Schwarz discovered a contribution to the anomaly that Witten and Alvarez-Gaumé had missed, which restricted the gauge group of the type I string theory to be SO(32). In coming to understand this calculation, Edward Witten became convinced that string theory was truly a consistent theory of gravity, and he became a high-profile advocate. Following Witten's lead, between 1984 and 1986, hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution.
During this period, David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm discovered heterotic strings. The gauge group of these closed strings was two copies of E8, and either copy could easily and naturally include the standard model. Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten found that the Calabi–Yau manifolds are the compactifications that preserve a realistic amount of supersymmetry, while Lance Dixon and others worked out the physical properties of orbifolds, distinctive geometrical singularities allowed in string theory. Cumrun Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of mirror symmetry. Daniel Friedan, Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring using conformal field theory techniques. David Gross and Vipul Periwal discovered that string perturbation theory was divergent. Stephen Shenker showed it diverged much faster than in field theory suggesting that new non-perturbative objects were missing.
In the 1990s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and identified these with the black-hole solutions of supergravity. These were understood to be the new objects suggested by the perturbative divergences, and they opened up a new field with rich mathematical structure. It quickly became clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the physical interpretation of the strings and branes was revealed—they are a type of black hole. Leonard Susskind had incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly excited string states with ordinary thermal black hole states. As suggested by 't Hooft, the fluctuations of the black hole horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all nearby objects too.
Second superstring revolution
In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten gave a speech on string theory that in essence united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called M-theory. M-theory was also foreshadowed in the work of Paul Townsend at approximately the same time. The flurry of activity that began at this time is sometimes called the second superstring revolution.[34]
During this period, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated matrix theory, a full holographic description of M-theory using IIA D0 branes.[51] This was the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the holographic principle. It is an example of a gauge-gravity duality and is now understood to be a special case of the AdS/CFT correspondence. Andrew Strominger and Cumrun Vafa calculated the entropy of certain configurations of D-branes and found agreement with the semi-classical answer for extreme charged black holes.[62] Petr Hořava and Witten found the eleven-dimensional formulation of the heterotic string theories, showing that orbifolds solve the chirality problem. Witten noted that the effective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered in terms of the location of the branes.
In 1997, Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to the horizon, which for extreme charged black holes looks like an anti-de Sitter space.[71] He noted that in this limit the gauge theory describes the string excitations near the branes. So he hypothesized that string theory on a near-horizon extreme-charged black-hole geometry, an anti-de Sitter space times a sphere with flux, is equally well described by the low-energy limiting gauge theory, the N = 4 supersymmetric Yang–Mills theory. This hypothesis, which is called the AdS/CFT correspondence, was further developed by Steven Gubser, Igor Klebanov and Alexander Polyakov,[72] and by Edward Witten,[73] and it is now well-accepted. It is a concrete realization of the holographic principle, which has far-reaching implications for black holes, locality and information in physics, as well as the nature of the gravitational interaction.[56] Through this relationship, string theory has been shown to be related to gauge theories like quantum chromodynamics and this has led to more quantitative understanding of the behavior of hadrons, bringing string theory back to its roots.[84]
Number of solutions
To construct models of particle physics based on string theory, physicists typically begin by specifying a shape for the extra dimensions of spacetime. Each of these different shapes corresponds to a different possible universe, or "vacuum state", with a different collection of particles and forces. String theory as it is currently understood has an enormous number of vacuum states, typically estimated to be around 10500, and these might be sufficiently diverse to accommodate almost any phenomena that might be observed at low energies.[121]
Many critics of string theory have expressed concerns about the large number of possible universes described by string theory. In his book Not Even Wrong, Peter Woit, a lecturer in the mathematics department at Columbia University, has argued that the large number of different physical scenarios renders string theory vacuous as a framework for constructing models of particle physics. According to Woit,
The possible existence of, say, 10500 consistent different vacuum states for superstring theory probably destroys the hope of using the theory to predict anything. If one picks among this large set just those states whose properties agree with present experimental observations, it is likely there still will be such a large number of these that one can get just about whatever value one wants for the results of any new observation.[122]
Some physicists believe this large number of solutions is actually a virtue because it may allow a natural anthropic explanation of the observed values of physical constants, in particular the small value of the cosmological constant.[122] The anthropic principle is the idea that some of the numbers appearing in the laws of physics are not fixed by any fundamental principle but must be compatible with the evolution of intelligent life. In 1987, Steven Weinberg published an article in which he argued that the cosmological constant could not have been too large, or else galaxies and intelligent life would not have been able to develop.[123] Weinberg suggested that there might be a huge number of possible consistent universes, each with a different value of the cosmological constant, and observations indicate a small value of the cosmological constant only because humans happen to live in a universe that has allowed intelligent life, and hence observers, to exist.[124]
String theorist Leonard Susskind has argued that string theory provides a natural anthropic explanation of the small value of the cosmological constant.[125] According to Susskind, the different vacuum states of string theory might be realized as different universes within a larger multiverse. The fact that the observed universe has a small cosmological constant is just a tautological consequence of the fact that a small value is required for life to exist.[126] Many prominent theorists and critics have disagreed with Susskind's conclusions.[127] According to Woit, "in this case [anthropic reasoning] is nothing more than an excuse for failure. Speculative scientific ideas fail not just when they make incorrect predictions, but also when they turn out to be vacuous and incapable of predicting anything."[128]
Background independence
One of the fundamental properties of Einstein's general theory of relativity is that it is background independent, meaning that the formulation of the theory does not in any way privilege a particular spacetime geometry.[129]
One of the main criticisms of string theory from early on is that it is not manifestly background independent. In string theory, one must typically specify a fixed reference geometry for spacetime, and all other possible geometries are described as perturbations of this fixed one. In his book The Trouble With Physics, physicist Lee Smolin of the Perimeter Institute for Theoretical Physics claims that this is the principal weakness of string theory as a theory of quantum gravity, saying that string theory has failed to incorporate this important insight from general relativity.[130]
Others have disagreed with Smolin's characterization of string theory. In a review of Smolin's book, string theorist Joseph Polchinski writes
[Smolin] is mistaking an aspect of the mathematical language being used for one of the physics being described. New physical theories are often discovered using a mathematical language that is not the most suitable for them… In string theory it has always been clear that the physics is background-independent even if the language being used is not, and the search for more suitable language continues. Indeed, as Smolin belatedly notes, [AdS/CFT] provides a solution to this problem, one that is unexpected and powerful.[131]
Polchinski notes that an important open problem in quantum gravity is to develop holographic descriptions of gravity which do not require the gravitational field to be asymptotically anti-de Sitter.[131] Smolin has responded by saying that the AdS/CFT correspondence, as it is currently understood, may not be strong enough to resolve all concerns about background independence.[g]
Sociological issues
Since the superstring revolutions of the 1980s and 1990s, string theory has become the dominant paradigm of high energy theoretical physics.[132] Some string theorists have expressed the view that there does not exist an equally successful alternative theory addressing the deep questions of fundamental physics. In an interview from 1987, Nobel laureate David Gross made the following controversial comments about the reasons for the popularity of string theory:
The most important [reason] is that there are no other good ideas around. That's what gets most people into it. When people started to get interested in string theory they didn't know anything about it. In fact, the first reaction of most people is that the theory is extremely ugly and unpleasant, at least that was the case a few years ago when the understanding of string theory was much less developed. It was difficult for people to learn about it and to be turned on. So I think the real reason why people have got attracted by it is because there is no other game in town. All other approaches of constructing grand unified theories, which were more conservative to begin with, and only gradually became more and more radical, have failed, and this game hasn't failed yet.[133]
Several other high-profile theorists and commentators have expressed similar views, suggesting that there are no viable alternatives to string theory.[134]
Many critics of string theory have commented on this state of affairs. In his book criticizing string theory, Peter Woit views the status of string theory research as unhealthy and detrimental to the future of fundamental physics. He argues that the extreme popularity of string theory among theoretical physicists is partly a consequence of the financial structure of academia and the fierce competition for scarce resources.[135] In his book The Road to Reality, mathematical physicist Roger Penrose expresses similar views, stating "The often frantic competitiveness that this ease of communication engenders leads to 'bandwagon' effects, where researchers fear to be left behind if they do not join in."[136] Penrose also claims that the technical difficulty of modern physics forces young scientists to rely on the preferences of established researchers, rather than forging new paths of their own.[137] Lee Smolin expresses a slightly different position in his critique, claiming that string theory grew out of a tradition of particle physics which discourages speculation about the foundations of physics, while his preferred approach, loop quantum gravity, encourages more radical thinking. According to Smolin,
String theory is a powerful, well-motivated idea and deserves much of the work that has been devoted to it. If it has so far failed, the principal reason is that its intrinsic flaws are closely tied to its strengths—and, of course, the story is unfinished, since string theory may well turn out to be part of the truth. The real question is not why we have expended so much energy on string theory but why we haven't expended nearly enough on alternative approaches.[138]
Smolin goes on to offer a number of prescriptions for how scientists might encourage a greater diversity of approaches to quantum gravity research.[139]
Notes and references
1. ^ For example, physicists are still working to understand the phenomenon of quark confinement, the paradoxes of black holes, and the origin of dark energy.
3. ^ "Top Cited Articles during 2010 in hep-th". Retrieved 25 July 2013.
4. ^ More precisely, one cannot apply the methods of perturbative quantum field theory.
5. ^ Two independent mathematical proofs of mirror symmetry were given by Givental 1996, 1998 and Lian, Liu, Yau 1997, 1999, 2000.
6. ^ More precisely, a nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups.
7. ^ "Archived copy". Archived from the original on November 5, 2015. Retrieved December 31, 2015. Response to review of The Trouble with Physics by Joe Polchinski
1. ^ a b Becker, Becker, and Schwarz 2007, p. 1
2. ^ Zwiebach 2009, p. 6
3. ^ a b Becker, Becker, and Schwarz 2007, pp. 2–3
4. ^ Becker, Becker, and Schwarz 2007, pp. 9–12
5. ^ Becker, Becker, and Schwarz 2007, pp. 14–15
6. ^ a b Klebanov and Maldacena 2009
7. ^ a b Merali 2011
8. ^ Sachdev 2013
9. ^ Becker, Becker, and Schwarz 2007, pp. 3, 15–16
10. ^ Becker, Becker, and Schwarz 2007, p. 8
11. ^ Becker, Becker, and Schwarz 13–14
12. ^ a b Woit 2006
13. ^ a b Zee 2010
14. ^ Becker, Becker, and Schwarz 2007, p. 2
15. ^ a b Becker, Becker, and Schwarz 2007, p. 6
16. ^ Zwiebach 2009, p. 12
17. ^ Becker, Becker, and Schwarz 2007, p. 4
18. ^ Zwiebach 2009, p. 324
19. ^ Wald 1984, p. 4
20. ^ Zee 2010, Parts V and VI
21. ^ Zwiebach 2009, p. 9
22. ^ Zwiebach 2009, p. 8
23. ^ a b Yau and Nadis 2010, Ch. 6
24. ^ Greene 2000, p. 186
25. ^ Yau and Nadis 2010, p. ix
26. ^ Randall and Sundrum 1999
27. ^ a b Becker, Becker, and Schwarz 2007
28. ^ Zwiebach 2009, p. 376
29. ^ a b Moore 2005, p. 214
30. ^ Moore 2005, p. 215
31. ^ a b Aspinwall et al. 2009
32. ^ a b Kontsevich 1995
33. ^ Kapustin and Witten 2007
34. ^ a b Duff 1998
35. ^ Duff 1998, p. 64
36. ^ Nahm 1978
37. ^ Cremmer, Julia, and Scherk 1978
38. ^ a b c d e Duff 1998, p. 65
39. ^ Sen 1994a
40. ^ Sen 1994b
41. ^ Hull and Townsend 1995
42. ^ Duff 1998, p. 67
43. ^ Bergshoeff, Sezgin, and Townsend 1987
44. ^ Duff et al. 1987
45. ^ Duff 1998, p. 66
46. ^ Witten 1995
47. ^ Duff 1998, pp. 67–68
48. ^ Becker, Becker, and Schwarz 2007, p. 296
49. ^ Hořava and Witten 1996
50. ^ Duff 1996, sec. 1
51. ^ a b c Banks et al. 1997
52. ^ Connes 1994
53. ^ Connes, Douglas, and Schwarz 1998
54. ^ Nekrasov and Schwarz 1998
55. ^ Seiberg and Witten 1999
56. ^ a b c de Haro et al. 2013, p. 2
57. ^ Yau and Nadis 2010, p. 187–188
58. ^ Bekenstein 1973
59. ^ a b Hawking 1975
60. ^ Wald 1984, p. 417
61. ^ Yau and Nadis 2010, p. 189
62. ^ a b Strominger and Vafa 1996
63. ^ Yau and Nadis 2010, pp. 190–192
64. ^ Maldacena, Strominger, and Witten 1997
65. ^ Ooguri, Strominger, and Vafa 2004
66. ^ Yau and Nadis 2010, pp. 192–193
67. ^ Yau and Nadis 2010, pp. 194–195
68. ^ Strominger 1998
69. ^ Guica et al. 2009
70. ^ Castro, Maloney, and Strominger 2010
71. ^ a b Maldacena 1998
72. ^ a b Gubser, Klebanov, and Polyakov 1998
73. ^ a b Witten 1998
74. ^ Klebanov and Maldacena 2009, p. 28
75. ^ a b c Maldacena 2005, p. 60
76. ^ a b Maldacena 2005, p. 61
77. ^ Zwiebach 2009, p. 552
78. ^ Maldacena 2005, pp. 61–62
79. ^ Susskind 2008
80. ^ Zwiebach 2009, p. 554
81. ^ Maldacena 2005, p. 63
82. ^ Hawking 2005
83. ^ Zwiebach 2009, p. 559
84. ^ a b Kovtun, Son, and Starinets 2001
85. ^ a b Merali 2011, p. 303
86. ^ Luzum and Romatschke 2008
87. ^ Sachdev 2013, p. 51
88. ^ Candelas et al. 1985
89. ^ Yau and Nadis 2010, pp. 147–150
90. ^ Becker, Becker, and Schwarz 2007, pp. 530–531
91. ^ Becker, Becker, and Schwarz 2007, p. 531
92. ^ Becker, Becker, and Schwarz 2007, p. 538
93. ^ Becker, Becker, and Schwarz 2007, p. 533
94. ^ Becker, Becker, and Schwarz 2007, pp. 539–543
95. ^ Deligne et al. 1999, p. 1
96. ^ Hori et al. 2003, p. xvii
97. ^ Aspinwall et al. 2009, p. 13
98. ^ Hori et al. 2003
99. ^ Yau and Nadis 2010, p. 167
100. ^ Yau and Nadis 2010, p. 166
101. ^ a b Yau and Nadis 2010, p. 169
102. ^ Candelas et al. 1991
103. ^ Yau and Nadis 2010, p. 171
104. ^ Hori et al. 2003, p. xix
105. ^ Strominger, Yau, and Zaslow 1996
106. ^ Dummit and Foote 2004
107. ^ Dummit and Foote 2004, pp. 102–103
108. ^ a b Klarreich 2015
109. ^ Gannon 2006, p. 2
110. ^ Gannon 2006, p. 4
111. ^ Conway and Norton 1979
112. ^ Gannon 2006, p. 5
113. ^ Gannon 2006, p. 8
114. ^ Borcherds 1992
115. ^ Frenkel, Lepowsky, and Meurman 1988
116. ^ Gannon 2006, p. 11
117. ^ Eguchi, Ooguri, and Tachikawa 2010
118. ^ Cheng, Duncan, and Harvey 2013
119. ^ Duncan, Griffin, and Ono 2015
120. ^ Witten 2007
121. ^ Woit 2006, pp. 240–242
122. ^ a b Woit 2006, p. 242
123. ^ Weinberg 1987
124. ^ Woit 2006, p. 243
125. ^ Susskind 2005
126. ^ Woit 2006, pp. 242–243
127. ^ Woit 2006, p. 240
128. ^ Woit 2006, p. 249
129. ^ Smolin 2006, p. 81
130. ^ Smolin 2006, p. 184
131. ^ a b Polchinski 2007
132. ^ Penrose 2004, p. 1017
133. ^ Woit 2006, pp. 224–225
134. ^ Woit 2006, Ch. 16
135. ^ Woit 2006, p. 239
136. ^ Penrose 2004, p. 1018
137. ^ Penrose 2004, pp. 1019–1020
138. ^ Smolin 2006, p. 349
139. ^ Smolin 2006, Ch. 20
Further reading
For physicists
• Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String Theory and M-theory: A Modern Introduction. Cambridge University Press. ISBN 978-0-521-86069-7.
• Green, Michael; Schwarz, John; Witten, Edward (2012). Superstring theory. Vol. 1: Introduction. Cambridge University Press. ISBN 978-1107029118.
• Green, Michael; Schwarz, John; Witten, Edward (2012). Superstring theory. Vol. 2: Loop amplitudes, anomalies and phenomenology. Cambridge University Press. ISBN 978-1107029132.
• Polchinski, Joseph (1998). String Theory Vol. 1: An Introduction to the Bosonic String. Cambridge University Press. ISBN 0-521-63303-6.
• Polchinski, Joseph (1998). String Theory Vol. 2: Superstring Theory and Beyond. Cambridge University Press. ISBN 0-521-63304-4.
For mathematicians
• Deligne, Pierre; Etingof, Pavel; Freed, Daniel; Jeffery, Lisa; Kazhdan, David; Morgan, John; Morrison, David; Witten, Edward, eds. (1999). Quantum Fields and Strings: A Course for Mathematicians, Vol. 2. American Mathematical Society. ISBN 978-0821819883.
External links |
aef2a0d5b6c6c006 | The Biointelligence Explosion
preprint of paper appearing in the Springer volume: Singularity Hypotheses: A scientific and philosophical assessment. (2013). Eden, A, Sørake, J., Moor, JH., Steinhart, E., eds. Berlin: Springer. Also MS Word and PDF.
The Biointelligence Explosion
How recursively self-improving organic robots will modify their own
source code and bootstrap our way to full-spectrum superintelligence
David Pearce (2012)
Edward O. Wilson
Consilience, The Unity of Knowledge (1999)
Freeman Dyson
New York Review of Books (July 19, 2007)
1 The Fate of the Germline
Genetic evolution is slow. Progress in artificial intelligence is fast. Only a handful of genes separate Homo sapiens from our hominid ancestors on the African savannah. Among our 23,000-odd protein-coding genes, variance in single nucleotide polymorphisms ("SNPs") accounts for just a small percentage of phenotypic variance in intelligence as measured by what we call IQ tests. True, the tempo of human evolution is about to accelerate. CRISPR-Cas9 genome-editing is a gamechanger. As the reproductive revolution of "designer babies" gathers pace, prospective parents will pre-select alleles and allelic combinations for a new child in anticipation of their behavioural effects - a novel kind of selection pressure to replace the "blind" genetic roulette of natural selection. In time, routine embryo screening via preimplantation genetic diagnosis will be complemented by gene therapy, genetic enhancement and then true designer zygotes. In consequence, life on Earth will also become progressively happier as the hedonic treadmill is recalibrated. In the new reproductive era, hedonic set-points and intelligence alike will be ratcheted upwards in virtue of selection pressure. For what parent-to-be wants to give birth to a low-status depressive "loser"? Future parents can enjoy raising a normal transhuman supergenius who grows up to be faster than Usain Bolt, more beautiful than Marilyn Monroe, more saintly than Nelson Mandela, more creative than Shakespeare - and smarter than Einstein.
Even so, the accelerating growth of germline engineering will be a comparatively slow process. In this scenario, sentient biological machines will design cognitively self-amplifying biological machines who will design cognitively self-amplifying biological machines. Greater-than-human biological intelligence will transform itself into posthuman superintelligence. Cumulative gains in intellectual capacity and subjective well-being across the generations will play out over hundreds and perhaps thousands of years - a momentous discontinuity, for sure, and a twinkle in the eye of eternity; but not a BioSingularity.
2 Biohacking Your Personal Genome
Yet germline engineering is only one strand of the genomics revolution. Indeed after humans master the ageing process, the extent to which traditional germlines or human generations will persist in the post-ageing world is obscure. Focus on the human germline ignores the slow-burning but then explosive growth of somatic gene enhancement in prospect. The CRISPR genome-editing revolution is accelerating. Later this century, innovative gene therapies will be succeeded by gene enhancement technologies - a value-laden dichotomy that reflects our impoverished human aspirations. Starting with individual genes, then clusters of genes, and eventually hundreds of genes and alternative splice variants, a host of recursively self-improving organic robots ("biohackers") will modify their genetic source code and modes of sentience: their senses, their moods, their motivation, their cognitive apparatus, their world-simulations and their default state of consciousness.
As the era of open-source genetics unfolds, tomorrow's biohackers will add, delete, edit and customise their own legacy code in a positive feedback loop of cognitive enhancement. Computer-aided genetic engineering will empower biological humans, transhumans and then posthumans to synthesise and insert new genes, variant alleles and even designer chromosomes - reweaving the multiple layers of regulation of our DNA to suit their wishes and dreams rather than the inclusive fitness of their genes in the ancestral environment. Collaborating and competing, next-generation biohackers will use stem-cell technologies to expand their minds, literally, via controlled neurogenesis. Freed from the constraints of the human birth canal, biohackers may re-sculpt the prison-like skull of Homo sapiens to accommodate a larger mind/brain, which can initiate recursive self-expansion in turn. Six crumpled layers of neocortex fed by today's miserly reward pathways aren't the upper bound of conscious mind, merely its seedbed. Each biological neuron and glial cell of your growing mind/brain can have its own dedicated artificial healthcare team, web-enabled nanobot support staff, and social network specialists; compare today's anonymous neural porridge. Transhuman minds will be augmented with neurochips, molecular nanotechnology, mind/computer interfaces and full-immersion virtual reality (VR) software. To achieve finer-grained control of cognition, mood and motivation, genetically enhanced transhumans will draw upon exquisitely tailored new designer drugs, nutraceuticals and cognitive enhancers - precision tools that make today's crude interventions seem the functional equivalent of glue-sniffing.
By way of comparison, early in the twenty-first century the scientific counterculture is customizing a bewildering array of designer drugs that outstrip the capacity of the authorities to regulate or comprehend. The bizarre psychoactive effects of such agents dramatically expand the evidential base that our theory of consciousness must explain. However, such drugs are short-acting. Their benefits, if any, aren't cumulative. By contrast, the ability genetically to hack one's own source code will unleash an exponential growth of genomic rewrites - not mere genetic tinkering but a comprehensive redesign of "human nature". Exponential growth starts out almost unnoticeably, and then explodes. Human bodies, cognition and ancestral modes of consciousness alike will be transformed. Post-humans will range across immense state-spaces of conscious mind hitherto impenetrable because access to their molecular biology depended on crossing gaps in the fitness landscape prohibited by natural selection. Intelligent agency can "leap across" such fitness gaps. What we'll be leaping into is currently for the most part unknown: an inherent risk of the empirical method. But mastery of our reward circuitry can guarantee such state-spaces of experience will be glorious beyond human imagination. For intelligent biohacking can make unpleasant experience physically impossible because its molecular substrates are absent. Hedonically enhanced innervation of the neocortex can ensure a rich hedonic tone saturates whatever strange new modes of experience our altered neurochemistry discloses.
Pilot studies of radical genetic enhancement will be difficult. Randomised longitudinal trials of such interventions in long-lived humans would take decades. In fact officially licensed, well-controlled prospective trials to test the safety and efficacy of genetic innovation will be hard if not impossible to conduct because all of us, apart from monozygotic twins, are genetically unique. Even monozygotic twins exhibit different epigenetic and gene expression profiles. Barring an ideological and political revolution, most formally drafted proposals for genetically-driven life-enhancement probably won't pass ethics committees or negotiate the maze of bureaucratic regulation. But that's the point of biohacking. By analogy today, if you're technically savvy, you don't want a large corporation controlling the operating system of your personal computer: you use open source software instead. Likewise, you don't want governments controlling your state of mind via drug laws. By the same token, tomorrow's biotech-savvy individualists won't want anyone restricting our right to customise and rewrite our own genetic source code in any way we choose.
Will central governments try to regulate personal genome editing? Most likely yes. How far they'll succeed is an open question. So too is the success of any centralised regulation of futuristic designer drugs or artificial intelligence. Another huge unknown is the likelihood of state-sponsored designer babies, human reproductive cloning, and autosomal gene enhancement programs; and their interplay with privately-funded initiatives. China, for instance, has a different historical memory from the West.
Will there initially be biohacking accidents? Personal tragedies? Most probably yes, until human mastery of the pleasure-pain axis is secure. By the end of next decade, every health-conscious citizen will be broadly familiar with the architecture of his or her personal genome: the cost of personal genotyping will be trivial, as will be the cost of DIY gene-manipulation kits. Let's say you decide to endow yourself with an extra copy of the N-methyl D-aspartate receptor subtype 2B (NR2B) receptor, a protein encoded by the GRIN2B gene. Possession of an extra NR2B subunit NMDA receptor is a crude but effective way to enhance your learning ability, at least if you're a transgenic mouse. Recall how Joe Tsien and his colleagues first gave mice extra copies of the NR2B receptor-encoding gene, then tweaked the regulation of those genes so that their activity would increase as the mice grew older. Unfortunately, it transpires that such brainy "Doogie mice" - and maybe brainy future humans endowed with an extra NR2B receptor gene - display greater pain-sensitivity too; certainly, NR2B receptor blockade reduces pain and learning ability alike. Being smart, perhaps you decide to counteract this heightened pain-sensitivity by inserting and then over-expressing a high pain-threshold, "low pain" allele of the SCN9A gene in your nociceptive neurons at the dorsal root ganglion and trigeminal ganglion. The SCN9A gene regulates pain-sensitivity; nonsense mutations abolish the capacity to feel pain at all. In common with taking polydrug cocktails, the factors to consider in making multiple gene modifications soon snowball; but you'll have heavy-duty computer software to help. Anyhow, the potential pitfalls and makeshift solutions illustrated in this hypothetical example could be multiplied in the face of a combinatorial explosion of possibilities on the horizon. Most risks - and opportunities - of genetic self-editing are presumably still unknown.
It is tempting to condemn such genetic self-experimentation as irresponsible, just as unlicensed drug self-experimentation is irresponsible. Would you want your teenage daughter messing with her DNA? Perhaps we may anticipate the creation of a genetic counterpart of the Drug Enforcement Agency (DEA) to police the human genome and its transhuman successors. Yet it's worth bearing in mind how each act of sexual reproduction today is an unpoliced genetic experiment with unfathomable consequences too. Without such reckless genetic experimentation, none of us would exist. In a cruel Darwinian world, this argument admittedly cuts both ways.
Naively, genomic source-code self-editing will always be too difficult for anyone beyond a dedicated cognitive elite of recursively self-improving biohackers. Certainly there are strongly evolutionarily conserved "housekeeping" genes that archaic humans would be best advised to leave alone for the foreseeable future. Granny might do well to customize her Windows desktop rather than her personal genome - prior to her own computer-assisted enhancement, at any rate. Yet the Biointelligence Explosion won't depend on more than a small fraction of its participants mastering the functional equivalent of machine code - the three billion odd 'A's, 'C's, 'G's and 'T's of our DNA. For the open-source genetic revolution will be propelled by powerful suites of high-level gene-editing tools, insertion vector applications, nonviral gene-editing kits, and user-friendly interfaces. Clever computer modelling and "narrow" AI can assist the intrepid biohacker to become a recursively self-improving genomic innovator. Later this century, your smarter counterpart will have software tools to monitor and edit every gene, repressor, promoter and splice variant in every region of the genome: each layer of epigenetic regulation of your gene transcription machinery in every region of the brain. This intimate level of control won't involve just crude DNA methylation to turn genes off and crude histone acetylation to turn genes on. Personal self-invention will involve mastery and enhancement of the histone and micro-RNA codes to allow sophisticated fine-tuning of gene expression and repression across the brain. Even today, researchers are exploring “nanochannel electroporation” (NEP) technologies that allow the mass-insertion of novel therapeutic genetic elements into our cells. Mechanical cell-loading systems will shortly be feasible that can inject up to 100,000 cells at a time. Before long, such technologies will seem primitive. Freewheeling genetic self-experimentation will be endemic as the DIY-Bio revolution unfolds. At present, crude and simple gene editing can be accomplished only via laborious genetic engineering techniques. Sophisticated authoring tools don't exist. In future, computer-aided genetic and epigenetic enhancement can become an integral part of your personal growth plan.
3 Will Humanity's Successors Also Be Our Descendants?
To contrast "biological" with "artificial" conceptions of posthuman superintelligence is convenient. The distinction may also prove simplistic. In essence, whereas genetic change in biological humanity has always been slow, the software run on serial, programmable digital computers is executed exponentially faster (cf. Moore's Law); it's copyable without limit; it runs on multiple substrates; and it can be cheaply and rapidly edited, tested and debugged. Extrapolating, Singularitarians like Ray Kurzweil and Eliezer Yudkowsky prophesy that human programmers will soon become redundant because autonomous AI run on digital computers will undergo accelerating cycles of self-improvement. In this kind of scenario, artificial, greater-than-human nonbiological intelligence will be rapidly succeeded by artificial posthuman superintelligence.
So we may distinguish two radically different conceptions of posthuman superintelligence: on one hand, our supersentient, cybernetically enhanced, genetically rewritten biological descendants, on the other, nonbiological superintelligence, either a Kurzweilian ecosystem or singleton Artificial General Intelligence (AGI) as foretold by the Machine Intelligence Research Institute (MIRI). Such a divide doesn't reflect a clean contrast between "natural" and "artificial" intelligence, the biological and the nonbiological. This contrast may prove another false dichotomy. Transhuman biology will increasingly become synthetic biology as genetic enhancement plus cyborgisation proceeds apace. "Cyborgisation" is a barbarous term to describe an invisible and potentially life-enriching symbiosis of biological sentience with artificial intelligence. Thus "narrow-spectrum" digital superintelligence on web-enabled chips can be more-or-less seamlessly integrated into our genetically enhanced bodies and brains. Seemingly limitless formal knowledge can be delivered on tap to supersentient organic wetware, i.e. us. Critically, transhumans can exploit what is misleadingly known as "narrow" or "weak" AI to enhance our own code in a positive feedback loop of mutual enhancement - first plugging in data and running multiple computer simulations, then tweaking and re-simulating once more. In short, biological humanity won't just be the spectator and passive consumer of the intelligence explosion, but its driving force. The smarter our AI, the greater our opportunities for reciprocal improvement. Multiple "hard" and "soft" take-off scenarios to posthuman superintelligence can be outlined for recursively self-improving organic robots, not just nonbiological AI. Thus for serious biohacking later this century, artificial quantum supercomputers may be deployed rather than today's classical toys to test-run multiple genetic interventions, accelerating the tempo of our recursive self-improvement. Quantum supercomputers exploit quantum coherence to do googols of computations all at once. So the accelerating growth of human/computer synergies means it's premature to suppose biological evolution will be superseded by technological evolution, let alone a "robot rebellion" as the parasite swallows its host. As the human era comes to a close, the fate of biological (post)humanity is more likely to be symbiosis with AI followed by metamorphosis, not simple replacement.
Despite this witches' brew of new technologies, a conceptual gulf remains in the futurist community between those who imagine human destiny, if any, lies in digital computers running programs with (hypothetical) artificial consciousness; and in contrast radical bioconservatives who believe that our posthuman successors will also be our supersentient descendants at their organic neural networked core - not the digital zombies of symbolic AI run on classical serial computers or their souped-up multiprocessor cousins. For one metric of progress in AI remains stubbornly unchanged: despite the exponential growth of transistors on a microchip, the soaring clock speed of microprocessors, the growth in computing power measured in MIPS, the dramatically falling costs of manufacturing transistors and the plunging price of dynamic RAM (etc), any chart plotting the growth rate in digital sentience shows neither exponential growth, nor linear growth, but no progress whatsoever. As far as we can tell, digital computers are still zombies. Our machines are becoming autistically intelligent, but not supersentient - nor even conscious. On some fairly modest philosophical assumptions, digital computers were not subjects of experience in 1946 (cf. ENIAC); nor are they conscious subjects in 2012 (cf. "Watson"); nor do researchers know how any kind of sentience may be "programmed" in future. So what if anything does consciousness do? Is it computationally redundant? Pre-reflectively, we tend to have a "dimmer-switch" model of sentience: "primitive" animals have minimal awareness and "advanced" animals like human beings experience a proportionately more intense awareness. By analogy, most AI researchers assume that at a given threshold of complexity / intelligence / processing speed, consciousness will somehow "switch on", turn reflexive, and intensify too. The problem with the dimmer-switch model is that our most intense experiences, notably raw agony or blind panic, are also the most phylogenetically ancient, whereas the most "advanced" modes (e.g. linguistic thought and the rich generative syntax that has helped one species to conquer the globe) are phenomenologically so thin as to be barely accessible to introspection. Something is seriously amiss with our entire conceptual framework.
So the structure of the remainder of this essay is as follows. I shall first discuss the risks and opportunities of building friendly biological superintelligence. Next I discuss the nature of full-spectrum superintelligence - and why consciousness is computationally fundamental to the past, present and future success of organic robots. Why couldn't recursively self-improving zombies modify their own genetic source code and bootstrap their way to full-spectrum superintelligence, i.e. a zombie biointelligence explosion? Finally, and most speculatively, I shall discuss the future of sentience in the cosmos.
4 Can We Build Friendly Biological Superintelligence?
4.1 Risk-Benefit Analysis.
Crudely speaking, evolution "designed" male human primates to be hunters/warriors. Evolution "designed" women to be attracted to powerful, competitive alpha males. Until humans rewrite our own hunter-gatherer source code, we shall continue to practise extreme violence against members of other species - and frequently against members of our own. A heritable (and conditionally activated) predisposition to unfriendliness shown towards members of other races and other species is currently hardwired even in "social" primates. Indeed we have a (conditionally activated) predisposition to compete against, and harm, anyone who isn't a genetically identical twin. Compared to the obligate siblicide found in some bird species, human sibling rivalry isn't normally so overtly brutal. But conflict as well as self-interested cooperation is endemic to Darwinian life on Earth. This grim observation isn't an argument for genetic determinism, or against gene-culture co-evolution, or to discount the decline of everyday violence with the spread of liberal humanitarianism - just a reminder of the omnipresence of immense risks so long as we're shot through with legacy malware. Attempting to conserve the genetic status quo in an era of weapons of mass destruction (WMD) poses unprecedented global catastrophic and existential risks. Indeed the single biggest underlying threat to the future of sentient life within our cosmological horizon derives, not from asocial symbolic AI software in the basement turning rogue and going FOOM (a runaway computational explosion of recursive self-improvement), but from conserving human nature in its present guise. In the twentieth century, male humans killed over 100 million fellow humans and billions of non-human animals. This century's toll may well be higher. Mankind currently spends well over a trillion dollars each year on weapons designed to kill and maim other humans. The historical record suggests such weaponry won't all be beaten into ploughshares.
Strictly speaking, however, humanity is more likely to be wiped out by idealists than by misanthropes, death-cults or psychologically unstable dictators. Anti-natalist philosopher David Benatar's plea ("Better Never to Have Been") for human extinction via voluntary childlessness must fail if only by reason of selection pressure; but not everyone who shares Benatar's bleak diagnosis of life on Earth will be so supine. Unless we modify human nature, compassionate-minded negative utilitarians, with competence in bioweaponry, nanorobotics or artificial intelligence, for example, may quite conceivably take direct action. Echoing Moore's law, Eliezer Yudkowsky warns that "Every eighteen months, the minimum IQ necessary to destroy the world drops by one point”. Although suffering and existential risk might seem separate issues, they are intimately connected. Not everyone loves life so much they wish to preserve it. Indeed the extinction of Darwinian life is what many transhumanists are aiming for - just not framed in such apocalyptic and provocative language. For just as we educate small children so they can mature into fully-fledged adults, biological humanity may aspire to grow up, too, with the consequence that - in common with small children - archaic humans become extinct.
4.2 Technologies Of Biofriendliness.
How do you disarm a potentially hostile organic robot - despite your almost limitless ignorance of his source code? Provide him with a good education, civics lessons and complicated rule-governed ethics courses? Or give him a tablet of MDMA ("Ecstasy") and get smothered with hugs?
MDMA is short-acting. The "penicillin of the soul" is potentially neurotoxic to serotonergic neurons. In theory, however, lifelong use of safe and sustainable empathogens would be a passport to worldwide biofriendliness. MDMA releases a potent cocktail of oxytocin, serotonin and dopamine into the user's synapses, thereby inducing a sense of "I love the world and the world loves me”. There's no technical reason why MDMA's acute pharmacodynamic effects can't be replicated indefinitely, shorn of its neurotoxicity. Designer "hug drugs" can potentially turn manly men into intelligent bonobos, more akin to the "hippie chimp" Pan paniscus than his less peaceable cousin Pan troglodytes. Violence would become unthinkable. Yet is this sort of proposal politically credible? "Morality pills" and other pharmacological solutions to human unfriendliness are both personally unsatisfactory and sociologically implausible. Do we really want to drug each other up from early childhood? Moreover life would be immeasurably safer if our fellow humans weren't genetically predisposed to unfriendly behaviour in the first instance.
But how can this friendly predisposition be guaranteed?
Friendliness can't realistically be hand-coded by tweaking the connections and weight strengths of our neural networks.
Nor can robust friendliness in advanced biological intelligence be captured by a bunch of explicit logical rules and smart algorithms, as in the paradigm of symbolic AI.
4.3 Mass Oxytocination?
Amplified "trust hormone" might create the biological underpinnings of world-wide peace and love if negative feedback control of oxytocin release can be circumvented. Oxytocin is functionally antagonised by testosterone in the male brain. Yet oxytocin enhancers have pitfalls too. Enriched oxytocin function leaves one vulnerable to exploitation by the unenhanced. Can we really envisage a cross-cultural global consensus for mass-medication? When? Optional or mandatory? And what might be the wider ramifications of a "high oxytocin, low testosterone" civilisation? Less male propensity to violent territorial aggression, for sure; but disproportionate intellectual progress in physics, mathematics and computer science to date has been driven by the hyper-systematising cognitive style of "extreme male" brains. Also, enriched oxytocin function can indirectly even promote unfriendliness to "out-groups" in consequence of promoting in-group bonding. So as well as oxytocin enrichment, global security demands a more inclusive, impartial, intellectually sophisticated conception of "us" that embraces all sentient beings - the expression of a hyper-developed capacity for empathetic understanding combined with a hyper-developed capacity for rational systematisation. Hence the imperative need for full-spectrum superintelligence.
4.4 Mirror-Touch Synaesthesia?
A truly long-term solution to unfriendly biological intelligence might be collectively to engineer ourselves with the functional generalisation of mirror-touch synaesthesia. On seeing you cut and hurt yourself, a mirror-touch synaesthete is liable to feel a stab of pain as acutely as you do. Conversely, your expressions of pleasure elicit a no less joyful response. Thus mirror-touch synaesthesia is a hyper-empathising condition that makes deliberate unfriendliness, in effect, biologically impossible in virtue of cognitively enriching our capacity to represent each other's first-person perspectives. The existence of mirror-touch synaesthesia is a tantalising hint at the God-like representational capacities of a full-spectrum superintelligence. This so-called "disorder" is uncommon in humans.
4.5 Timescales.
The biggest problem with all these proposals, and other theoretical biological solutions to human unfriendliness, is timescale. Billions of human and non-human animals will have been killed and abused before they could ever come to pass. Cataclysmic wars may be fought in the meantime with nuclear, biological and chemical weapons harnessed to "narrow" AI. Our circle of empathy expands only slowly and fitfully. For the most part, religious believers and traditional-minded bioconservatives won't seek biological enhancement / remediation for themselves or their children. So messy democratic efforts at "political" compromise are probably unavoidable for centuries to come. For sure, idealists can dream up utopian schemes to mitigate the risk of violent conflict until the "better angels of our nature" can triumph, e.g. the election of a risk-averse all-female political class to replace legacy warrior males. Such schemes tend to founder on the rock of sociological plausibility. Innumerable sentient beings are bound to suffer and die in consequence.
4.6 Does Full-Spectrum Superintelligence Entail Benevolence?
The God-like perspective-taking faculty of a full-spectrum superintelligence doesn't entail distinctively human-friendliness any more than a God-like superintelligence could promote distinctively Aryan-friendliness. Indeed it's unclear how benevolent superintelligence could want omnivorous killer apes in our current guise to walk the Earth in any shape or form. But is there any connection at all between benevolence and intelligence? Pre-reflectively, benevolence and intelligence are orthogonal concepts. There's nothing obviously incoherent about a malevolent God or a malevolent - or at least a callously indifferent - Superintelligence. Thus a sceptic might argue that there is no link whatsoever between benevolence - on the face of it a mere personality variable - and enhanced intellect. After all, some sociopaths score highly on our [autistic, mind-blind] IQ tests. Sociopaths know that their victims suffer. They just don't care.
However, what's critical in evaluating cognitive ability is a criterion of representational adequacy. Representation is not an all-or-nothing phenomenon; it varies in functional degree. More specifically here, the cognitive capacity to represent the formal properties of mind differs from the cognitive capacity to represent the subjective properties of mind. Thus a notional zombie Hyper-Autist robot running a symbolic AI program on an ultrapowerful digital computer with a classical von Neumann architecture may be beneficent or maleficent in its behaviour toward sentient beings. By its very nature, it can't know or care. Most starkly, the zombie Hyper-Autist might be programmed to convert the world's matter and energy into either heavenly "utilitronium" or diabolical "dolorium" without the slightest insight into the significance of what it was doing. This kind of scenario is at least a notional risk of creating insentient Hyper-Autists endowed with mere formal utility functions rather than hyper-sentient full-spectrum superintelligence. By contrast, full-spectrum superintelligence does care in virtue of its full-spectrum representational capacities - a bias-free generalisation of the superior perspective-taking, "mind-reading" capabilities that enabled humans to become the cognitively dominant species on the planet. Full-spectrum superintelligence, if equipped with the posthuman cognitive generalisation of mirror-touch synaesthesia, understands your thoughts, your feelings and your egocentric perspective better than you do yourself.
Could there arise "evil" mirror-touch synaesthetes? In one sense, no. You can't go around wantonly hurting other sentient beings if you feel their pain as your own. Full-spectrum intelligence is friendly intelligence. But in another sense yes, insofar as primitive mirror-touch synaesthetes are prey to species-specific cognitive limitations that prevent them acting rationally to maximise the well-being of all sentience. Full-spectrum superintelligences would lack those computational limitations in virtue of their full cognitive competence in understanding both the subjective and the formal properties of mind. Perhaps full-spectrum superintelligences might optimise your matter and energy into a blissful smart angel; but they couldn't wantonly hurt you, whether by neglect or design.
More practically today, a cognitively superior analogue of natural mirror-touch synaesthesia should soon be feasible with reciprocal neuroscanning technology - a kind of naturalised telepathy. At first blush, mutual telepathic understanding sounds a panacea for ignorance and egotism alike. An exponential growth of shared telepathic understanding might safeguard against global catastrophe born of mutual incomprehension and WMD. As the poet Henry Wadsworth Longfellow observed, "If we could read the secret history of our enemies, we should find in each life sorrow and suffering enough to disarm all hostility." Maybe so. The problem here, as advocates of Radical Honesty soon discover, is that many Darwinian thoughts scarcely promote friendliness if shared: they are often ill-natured, unedifying and unsuitable for public consumption. Thus unless perpetually "loved-up" on MDMA or its long-acting equivalents, most of us would find mutual mind-reading a traumatic ordeal. Human society and most personal relationships would collapse in acrimony rather than blossom. Either way, our human incapacity fully to understand the first-person point of view of other sentient beings isn't just a moral failing or a personality variable; it's an epistemic limitation, an intellectual failure to grasp an objective feature of the natural world. Even "normal" people share with sociopaths this fitness-enhancing cognitive deficit. By posthuman criteria, perhaps we're all quasi-sociopaths. The egocentric delusion (i.e. that the world centres on one's existence) is genetically adaptive and strongly selected for over hundreds of millions of years. Fortunately, it's a cognitive failing amenable to technical fixes and eventually a cure: full-spectrum superintelligence. The devil is in the details, or rather the genetic source code.
5 A Biotechnological Singularity?
Yet does this positive feedback loop of reciprocal enhancement amount to a Singularity in anything more than a metaphorical sense? The risk of talking portentously about "The Singularity" isn't of being wrong: it's of being "not even wrong" - of reifying one's ignorance and elevating it to the status of an ill-defined apocalyptic event. Already multiple senses of "The Singularity" proliferate in popular culture. Does taking LSD induce a Consciousness Singularity? How about the abrupt and momentous discontinuity in one's conception of reality entailed by waking from a dream? Or the birth of language? Or the Industrial Revolution? So is Biotechnological Singularity, or "BioSingularity" for short, any more rigorously defined than "Technological Singularity"?
Metaphorically, perhaps, the impending biointelligence explosion represents an intellectual "event horizon" beyond which archaic humans cannot model or understand the future. Events beyond the BioSingularity will be stranger than science-fiction: too weird for unenhanced human minds - or the algorithms of a zombie super-Asperger - to predict or understand. In the popular sense of "event horizon", maybe the term is apt too, though the metaphor is still potentially misleading. Thus theoretical physics tells us that one could pass through the event horizon of a non-rotating supermassive black hole and not notice any subjective change in consciousness - even though one's signals would now be inaccessible to an external observer. The BioSingularity will feel different in ways a human conceptual scheme can't express. But what is the empirical content of this claim?
6 What Is Full-Spectrum Superintelligence?
"[g is] ostensibly some innate scalar brain force...[However] ability is a folk concept and not amenable to scientific analysis."
Jon Marks (Dept Anthropology, Yale University), 1995, Nature, 9 xi, 143-144.
(William James)
6.1 Intelligence.
"Intelligence" is a folk concept. The phenomenon is not well-defined - or rather any attempt to do so amounts to a stipulative definition that doesn't "carve Nature at the joints". The Cattell-Horn-Carroll (CHC) psychometric theory of human cognitive abilities is probably most popular in academia and the IQ testing community. But the Howard Gardner multiple intelligences model, for example, differentiates "intelligence" into various spatial, linguistic, bodily-kinaesthetic, musical, interpersonal, intrapersonal, naturalistic and existential intelligence rather than a single general ability ("g"). Who's right? As it stands, "g" is just a statistical artefact of our culture-bound IQ tests. If general intelligence were indeed akin to an innate scalar brain force, as some advocates of "g" believe, or if intelligence can best be modelled by the paradigm of symbolic AI, then the exponential growth of digital computer processing power might indeed entail an exponential growth in intelligence too - perhaps leading to some kind of Super-Watson. Other facets of intelligence, however, resist enhancement by mere acceleration of raw processing power.
The non-exhaustive set of criteria below doesn't pretend to be anything other than provisional. They are amplified in the sections to follow.
Full-Spectrum Superintelligence entails:
(cf. naive realist theories of "perception" versus the world-simulation or "Matrix" paradigm. Compare disorders of binding, e.g. simultanagnosia (an inability to perceive the visual field as a whole), cerebral akinetopsia ("motion blindness"), etc. In the absence of a data-driven, almost real-time simulation of the environment, intelligent agency is impossible.)
(cf. dissociative identity disorder (DID or "multiple personality disorder"), or florid schizophrenia, or your personal computer: in the absence of at least a fleetingly unitary self, what philosophers call "synchronic identity", there is no entity that is intelligent, just an aggregate of discrete algorithms and an operating system.)
3. a "mind-reading" or perspective-taking faculty; higher-order intentionality (e.g. "he believes that she hopes that they fear that he wants...", etc): social intelligence.
The intellectual success of the most cognitively successful species on the planet rests, not just on the recursive syntax of human language, but also on our unsurpassed "mind-reading" prowess, an ability to simulate the perspective of other unitary minds: the "Machiavellian Ape" hypothesis. Any ecologically valid intelligence test designed for a species of social animal must incorporate social cognition and the capacity for co-operative problem-solving. So must any test of empathetic superintelligence.
4. a metric to distinguish the important from the trivial.
(our theory of significance should be explicit rather than implicit, as in contemporary IQ tests. What distinguishes, say, mere calendrical prodigies and other "savant syndromes" from, say, a Grigori Perelman who proved the Poincaré conjecture? Intelligence entails understanding what does - and doesn't - matter. What matters is of course hugely contentious.)
and finally
6. "Autistic", pattern-matching, rule-following, mathematico-linguistic intelligence, i.e. the standard, mind-blind cognitive tool-kit scored by existing IQ tests. High-functioning "autistic" intelligence is indispensable to higher mathematics, computer science and the natural sciences. High-functioning autistic intelligence is necessary - but not sufficient - for a civilisation capable of advanced technology that can cure ageing and disease, systematically phase out the biology of suffering, and take us to the stars. And for programming artificial intelligence.
6.2 The Bedrock Of Intelligence:
World-Simulation ("Perception")
Consider criterion number one, world-simulating prowess, or what we misleadingly term "perception". The philosopher Bertrand Russell once aptly remarked that one never sees anything but the inside of one's own head. In contrast to such inferential realism, commonsense perceptual direct realism offers all the advantages of theft over honest toil - and it's computationally useless for the purposes either of building artificial general intelligence or understanding its biological counterparts. For the bedrock of intelligent agency is the capacity of an embodied agent computationally to simulate dynamic objects, properties and events in the mind-independent environment. The evolutionary success of organic robots over the past c. 540 million years has been driven by our capacity to run data-driven egocentric world-simulations - what the naive realist, innocent of modern neuroscience or post-Everett quantum mechanics, calls simply perceiving one's physical surroundings. Unlike classical digital computers, organic neurocomputers can simultaneously "bind" multiple features (edges, colours, motion, etc) distributively processed across the brain into unitary phenomenal objects embedded in unitary spatio-temporal world-simulations apprehended by a momentarily unitary self: what Kant calls "the transcendental unity of apperception". These simulations run in (almost) real time; the time-lag in our world-simulations is barely more than a few dozen milliseconds. Such blistering speed of construction and execution is adaptive and often life-saving in a fast-changing external environment. Recapitulating evolutionary history, pre-linguistic human infants must first train up their neural networks to bind the multiple features of dynamic objects and run unitary world-simulations before they can socially learn second-order representation and then third-order representation, i.e. language followed later in childhood by meta-language.
Occasionally, object binding and/or the unity of consciousness partially breaks down in mature adults who suffer a neurological accident. The results can be cognitively devastating (cf. akinetopsia or "motion blindness"; and simultanagnosia, an inability to apprehend more than a single object at a time, etc). Yet normally our simulations of fitness-relevant patterns in the mind-independent local environment feel seamless. Our simulations each appear simply as "the world"; we just don't notice or explicitly represent the gaps. Neurons, (mis)construed as classical processors, are pitifully slow, with spiking frequencies barely up to 200 per second. By contrast, silicon (etc) processors are ostensibly millions of times faster. Yet the notion that nonbiological computers are faster than sentient neurocomputers is a philosophical assumption, not an empirical discovery. Here the assumption will be challenged. Unlike the CPUs of classical robots, an organic mind/brain delivers dynamic unitary phenomenal objects and unitary world-simulations with a "refresh rate" of many billions per second (cf. the persistence of vision as experienced watching a movie run at a mere 30 frames per second). These cross-modally matched simulations take the guise of what passes as the macroscopic world: a spectacular egocentric simulation run by the vertebrate CNS that taps into the world's fundamental quantum substrate.
We should pause here. This is not a mainstream view. Most AI researchers regard stories of a non-classical mechanism underlying the phenomenal unity of biological minds as idiosyncratic at best. In fact no scientific consensus exists on the molecular underpinnings of the unity of consciousness, nor on how such unity is even physically possible. By analogy, 1.3 billion skull-bound Chinese minds can never be a single subject of experience, irrespective of their interconnections. How could waking or dreaming communities of membrane-bound classical neurons - even microconscious classical neurons - be any different? ? If materialism is true, conscious mind should be impossible. Yet any explanation of phenomenal object binding, the unity of perception, or the phenomenal unity of the self that invokes quantum coherence as here is controversial. One reason it's controversial is that the delocalisation involved in quantum coherence is exceedingly short-lived in an environment as warm and noisy as a macroscopic brain - supposedly too short-lived to do computationally useful work. Physicist Max Tegmark estimates that thermally-induced decoherence destroys any macroscopic coherence of brain states within 10-13 second, an unimaginably long time in natural Planck units but an unimaginably short time by everyday human intuitions. Perhaps it would be wiser just to acknowledge these phenomena are unexplained mysteries within a conventional materialist framework - as mysterious as the existence of consciousness itself. But if we're speculating about the imminent end of the human era, shoving the mystery under the rug isn't really an option. For the different strands of the Singularity movement share a common presupposition. This presupposition is that our complete ignorance within a materialist conceptual scheme of why consciousness exists (the "Hard Problem"), and of even the ghost of a solution to the Binding Problem, doesn't matter for the purposes of building the seed of artificial posthuman superintelligence. Our ignorance supposedly doesn't matter either because consciousness and/or our quantum "substrate" are computationally irrelevant to cognition and the creation of nonbiological minds, or alternatively because the feasibility of "whole brain emulation" (WBE) will allow us to finesse our ignorance.
Unfortunately, we have no grounds for believing this suppressed premiss is true or that the properties of our quantum "substrate" are functionally irrelevant to full-spectrum superintelligence or its humble biological predecessors. Conscious minds are not substrate-neutral digital computers. Humans investigate problems of which digital computers are invincibly ignorant, not least the properties of consciousness itself. The Hard Problem of consciousness can't be quarantined from the rest of science and treated as a troublesome but self-contained anomaly: its mystery infects everything that we think we know about ourselves, our computers and the world. Either way, the conjecture that the phenomenal unity of perception is a manifestation of ultra-rapid sequences of irreducible quantum coherent states isn't a claim that the mind/brain is capable of detecting events in the mind-independent world on this kind of sub-picosecond timescale. Rather the role of the local environment in shaping action-guiding experience in the awake mind/brain is here conjectured to be quantum state-selection. When we're awake, patterns of impulses from e.g. the optic nerve select which quantum-coherent frames are generated by the mind/brain - in contrast to the autonomous world-simulations spontaneously generated by the dreaming brain. Other quantum mind theorists, most notably Roger Penrose and Stuart Hameroff, treat quantum minds as evolutionarily novel rather than phylogenetically ancient. They invoke a non-physical wave-function collapse and unwisely focus on e.g. the ability of mathematically-inclined brains to perform non-computable functions in higher mathematics, a feat for which selection pressure has presumably been non-existent. Yet the human capacity for sequential linguistic thought and formal logico-mathematical reasoning is a late evolutionary novelty executed by a slow, brittle virtual machine running on top of its massively parallel quantum parent - a momentous evolutionary innovation whose neural mechanism is still unknown.
In contrast to the evolutionary novelty of serial linguistic thought, our ancient and immensely adaptive capacity to run unitary world-simulations, simultaneously populated by hundreds or more dynamic unitary objects, enables organic robots to solve the computational challenges of navigating a hostile environment that would leave the fastest classical supercomputer grinding away until Doomsday. Physical theory (cf. the Bekenstein bound) shows that informational resources as classically conceived are not just physical but finite and scarce: a maximum possible limit of 10120 bits set by the surface area of the entire accessible universe expressed in Planck units according to the Holographic principle. An infinite computing device like a universal Turing machine (UTM) is physically impossible. So invoking computational equivalence and asking whether a classical Turing machine can run a human-equivalent macroscopic world-simulation is akin to asking whether a classical Turing machine can factor 1500 digit numbers in real-world time [i.e. no]. No doubt resourceful human and transhuman programmers will exploit all manner of kludges, smart workarounds and "brute-force" algorithms to try and defeat the Binding Problem in AI. How will they fare? Compare clod-hopping AlphaDog with the sophisticated functionality of the sesame-seed sized brain of a bumblebee. Brute-force algorithms suffer from an exponentially growing search space that soon defeats any classical computational device in open-field contexts. As witnessed by our seemingly effortless world-simulations, organic minds are ultrafast; classical computers are slow. Serial thinking is slower still; but that's not what conscious biological minds are good at. On this conjecture, "substrate-independent" phenomenal world-simulations are impossible for the same reason that "substrate-independent" chemical valence structure is impossible. We're simply begging the question of what's functionally (ir)relevant. Ultimately, Reality has only a single, "program-resistant" ontological level even though it's amenable to description at different levels of computational abstraction; and the nature of this program-resistant level as disclosed by the subjective properties of one's mind (Lockwood 1989) is utterly at variance with what naive materialist metaphysics would suppose. If our phenomenal world-simulating prowess turns out to be constitutionally tied to our quantum mechanical wetware, then substrate-neutral virtual machines (VMs, i.e. software implementations of a digital computer that execute programs like a physical machine) will never be able to support "virtual" qualia or "virtual" unitary subjects of experience. This rules out sentient life "uploading" itself to digital nirvana. Contra Marvin Minsky ("The most difficult human skills to reverse engineer are those that are unconscious"), the most difficult skills for roboticists to engineer in artificial robots are actually intensely conscious: our colourful, noisy, tactile, sometimes hugely refractory virtual worlds.
Naively, for sure, real-time world-simulation doesn't sound too difficult. Hollywood robots do it all the time. Videogames become ever more photorealistic. Perhaps one imagines viewing some kind of inner TV screen, as in a Terminator movie or The Matrix. Yet the capacity of an awake or dreaming brain to generate unitary macroscopic world-simulations can only superficially resemble a little man (a "homunculus") viewing its own private theatre - on pain of an infinite regress. For by what mechanism would the homunculus view this inner screen? Emulating the behaviour of even the very simplest sentient organic robots on a classical digital computer is a daunting task. If conscious biological minds are irreducibly quantum mechanical by their very nature, then reverse-engineering the brain to create digital human "mindfiles" and "roboclones" alike will prove impossible.
6.3 The Bedrock Of Superintelligence:
Hypersocial Cognition ("Mind-reading")
Will superintelligence be solipsistic or social? Overcoming a second obstacle to delivering human-level artificial general intelligence - let alone building a recursively self-improving super-AGI culminating in a technological Singularity - depends on finding a solution to the first challenge, i.e. real-time world-simulation. For the evolution of distinctively human intelligence, sitting on top of our evolutionarily ancient world-simulating prowess, has been driven by the interplay between our rich generative syntax and superior "mind-reading" skills: so-called Machiavellian intelligence. Machiavellian intelligence is an egocentric parody of God's-eye-view empathetic superintelligence. Critically for the prospects of building AGI, this real-time mind-modelling expertise is parasitic on the neural wetware to generate unitary first-order world-simulations - virtual worlds populated by the avatars of intentional agents whose different first-person perspectives can be partially and imperfectly understood by their simulator. Even articulate human subjects with autism spectrum disorder are prone to multiple language deficits because they struggle to understand the intentions - and higher-order intentionality - of neurotypical language users. Indeed natural language is itself a pre-eminently social phenomenon: its criteria of application must first be socially learned. Not all humans possess the cognitive capacity to acquire mind-reading skills and the cooperative problem-solving expertise that sets us apart from other social primates. Most notably, people with autism spectrum disorder don't just fail to understand other minds; autistic intelligence cannot begin to understand its own mind. Pure autistic intelligence has no conception of a self that can be improved, recursively or otherwise. Autists can't "read" their own minds. The inability of the autistic mind to take what Daniel Dennett calls the intentional stance parallels the inability of classical computers to understand the minds of intentional agents - or have insight into their own zombie status. Even with smart algorithms and ultra-powerful hardware, the ability of ultra-intelligent autists to predict the long-term behaviour of mindful organic robots by relying exclusively on the physical stance (i.e. solving the Schrödinger equation of the intentional agent in question) will be extremely limited. For a start, much collective human behaviour is chaotic in the technical sense, i.e. it shows extreme sensitivity to initial conditions that confounds long-term prediction by even the most powerful real-world supercomputer. But there's a worse problem: reflexivity. Predicting sociological phenomena differs essentially from predicting mindless physical phenomena. Even in a classical, causally deterministic universe, the behaviour of mindful, reflexively self-conscious agents is frequently unpredictable, even in principle, from within the world owing to so-called prediction paradoxes. When the very act of prediction causally interacts with the predicted event, then self-defeating or self-falsifying predictions are inevitable. Self-falsifying predictions are a mirror image of so-called self-fulfilling predictions. So in common with autistic "idiot savants", classical AI gone rogue will be vulnerable to the low cunning of Machiavellian apes and the high cunning of our transhuman descendants.
This argument (i.e. our capacity for unitary mind-simulation embedded in unitary world-simulation) for the cognitive primacy of biological general intelligence isn't decisive. For a start, computer-aided Machiavellian humans can program robots with "narrow" AI - or perhaps "train up" the connections and weights of a subsymbolic connectionist architecture - for their own manipulative purposes. Humans underestimate the risks of zombie infestation at our peril. Given our profound ignorance of how conscious mind is even possible, it's probably safest to be agnostic over whether autonomous nonbiological robots will ever emulate human world-simulating or mind-reading capacity in most open-field contexts, despite the scepticism expressed here. Either way, the task of devising an ecologically valid measure of general intelligence that can reliably, predictively and economically discriminate between disparate life-forms is immensely challenging, not least because the intelligence test will express the value-judgements, and species- and culture-bound conceptual scheme, of the tester. Some biases are insidious and extraordinarily subtle: for example, the desire systematically to measure "intelligence" with mind-blind IQ tests is itself a quintessentially Asperger-ish trait. In consequence, social cognition is disregarded altogether. What we fancifully style "IQ tests" are designed by people with abnormally high AQs as well as self-defined high IQs. Thus many human conceptions of (super)intelligence resemble high-functioning autism spectrum disorder (ASD) rather than a hyper-empathetic God-like Super-Mind. For example, an AI that attempted systematically to maximise the cosmic abundance of paperclips would be recognisably autistic rather than incomprehensibly alien. Full-spectrum (super-)intelligence is certainly harder to design or quantify scientifically than mathematical puzzle-solving ability or performance in verbal memory-tests: "IQ". But that's because superhuman intelligence will be not just quantitatively different but also qualitatively alien from human intelligence. To misquote Robert McNamara, cognitive scientists need to stop making what is measurable important, and find ways to make the important measurable. An idealised full-spectrum superintelligence will indeed be capable of an impartial "view from nowhere" or God's-eye-view of the multiverse, a mathematically complete Theory Of Everything - as does modern theoretical physics, in aspiration if not achievement. But in virtue of its God's-eye-view, full-spectrum superintelligence must also be hypersocial and supersentient: able to understand all possible first-person perspectives, the state-space of all possible minds in other Hubble volumes, other branches of the universal wavefunction (UWF) - and in other solar systems and galaxies if such beings exist within our cosmological horizon. Idealized at least, full-spectrum superintelligence will be able to understand and weigh the significance of all possible modes of experience irrespective of whether they have hitherto been recruited for information-signalling purposes. The latter is, I think, by far the biggest intellectual challenge we face as cognitive agents. The systematic investigation of alien types of consciousness intrinsic to varying patterns of matter and energy calls for a methodological and ontological revolution. Transhumanists talking of post-Singularity superintelligence are fond of hyperbole about "Level 5 Future Shock" etc; but it's been aptly said that if Elvis Presley were to land in a flying saucer on the White House lawn, it's as nothing in strangeness compared to your first DMT trip.
6.4 Ignoring The Elephant: Consciousness.
Why Consciousness is Computationally Fundamental to the Past, Present and Future Success of Organic Robots.
The pachyderm in the room in most discussions of (super)intelligence is consciousness - not just human reflective self-awareness but the whole gamut of experience from symphonies to sunsets, agony to ecstasy: the phenomenal world of everyday experience. All one ever knows, except by inference, is the contents of one's own conscious mind: what philosophers call "qualia". Yet according to the ontology of our best story of the world, namely physical science, conscious minds shouldn't exist at all, i.e. we should be zombies, insentient patterns of matter and energy indistinguishable from normal human beings but lacking conscious experience. Dutch computer scientist Edsger Dijkstra once remarked, "The question of whether a computer can think is no more interesting than the question of whether a submarine can swim." Yet the question of whether a programmable digital computer - or a subsymbolic connectionist system with a merely classical parallelism - could possess, and think about, qualia, "bound" perceptual objects, a phenomenal self, or the unitary phenomenal minds of sentient organic robots can't be dismissed so lightly. For if advanced nonbiological intelligence is to be smart enough comprehensively to understand, predict and manipulate the behaviour of enriched biological intelligence, then the AGI can't rely autistically on the "physical stance", i.e. to monitor the brains, scan the atoms and molecules, and then solve the Schrödinger equation of intentional agents like human beings. Such calculations would take longer than the age of the universe.
For sure, many forms of human action can be predicted, fallibly, on the basis of crude behavioural regularities and reinforcement learning. Within your world-simulation, you don't need a theory of mind or an understanding of quantum mechanics to predict that Fred will walk to the bus-stop again today. Likewise, powerful tools of statistical analysis run on digital supercomputers can predict, fallibly, many kinds of human collective behaviour, for example stock markets. Yet to surpass human and transhuman capacities in all significant fields, AGI must understand how intelligent biological robots can think about, talk about and manipulate the manifold varieties of consciousness that make up their virtual worlds. Some investigators of consciousness even dedicate their lives to that end; what might a notional insentient AGI suppose we're doing? There is no evidence that serial digital computers have the capacity to do anything of the kind - or could ever be programmed to do so. Digital computers don't know anything about conscious minds, unitary persons, the nature of phenomenal pleasure and pain, or the Problem of Other Minds; it's not even "all dark inside". The challenge for a conscious mind posed by understanding itself "from the inside" pales into insignificance compared to the challenge for a nonconscious system of understanding a conscious mind "from the outside". Nor within the constraints of a materialist ontology have we the slightest clue how the purely classical parallelism of a subsymbolic, "neurally inspired" connectionist architecture could turn water into wine and generate unitary subjects of experience to fill the gap. For even if we conjecture in the spirit of Strawsonian physicalism - the only scientifically literate form of panpsychism - that the fundamental stuff of the world, the mysterious "fire in the equations", is fields of microqualia, this bold ontological conjecture doesn't, by itself, explain why biological robots aren't zombies. This is because structured aggregates of classically conceived "mind-dust" aren't the same as a unitary phenomenal subject of experience who apprehends "bound" spatio-temporal objects in a dynamic world-simulation. Without phenomenal object binding and the unity of perception, we are faced with the spectre of what philosophers call "mereological nihilism". Mereological nihilism, also known as "compositional nihilism", is the position that composite objects with proper parts do not exist: strictly speaking, only basic building blocks without parts have more than fictional existence. Unlike the fleetingly unitary phenomenal minds of biological robots, a classical digital computer and the programs it runs lacks ontological integrity: it's just an assemblage of algorithms. In other words, a classical digital computer has no self to understand or a mind recursively to improve, exponentially or otherwise. Talk about artificial "intelligence" exploding is just an anthropomorphic projection on our part.
So how do biological brains solve the binding problem and become persons? In short, we don't know. Vitalism is clearly a lost cause. Most AI researchers would probably dismiss - or at least discount as wildly speculative - any story of the kind mooted here involving macroscopic quantum coherence grounded in an ontology of physicalistic panpsychism. The conjecture should be experimentally falsifiable with the tools of next-generation molecular matter-wave interferometry. But in the absence of any story at all, we are left with a theoretical vacuum and a faith that natural science - or the exponential growth of digital computer processing power culminating in a Technological Singularity - will one day deliver an answer. Evolutionary biologist Theodosius Dobzhansky famously observed how "Nothing in Biology Makes Sense Except in the Light of Evolution". In the same vein, nothing in the future of intelligent life in the universe makes sense except in the light of a solution to the Hard Problem of Consciousness and the closure of Levine's Explanatory Gap. Consciousness is the only reason anything matters at all; and it's the only reason why unitary subjects of experience can ask these questions; and yet materialist orthodoxy has no idea how or why the phenomenon exists. Unfortunately, the Hard Problem won't be solved by building more advanced digital zombies who can tell mystified conscious minds the answer.
More practically for now, perhaps the greatest cognitive challenge of the millennium and beyond is deciphering and systematically manipulating the "neural correlates of consciousness" (NCC). Neuroscientists use this expression in default of any deeper explanation of our myriad qualia. How and why does experimentally stimulating via microelectrodes one cluster of nerve cells in the neocortex yield the experience of phenomenal colour; stimulating a superficially type of nerve cell induces a musical jingle; stimulating another with a slightly different gene-expression profile triggers a sense of everything being hysterically funny; stimulating another induces a hallucination of your mother; and stimulating another induces the experience of an archangel, say, in front of your body-image? In each case, the molecular variation in neuronal cell architecture is ostensibly trivial; the difference in subjective experience is profound. On a mind/brain identity theory, such experiential states are an intrinsic property of some configurations of matter and energy. How and why this is so is incomprehensible on an orthodox materialist ontology. Yet empirically, microelectrodes, dreams and hallucinogenic drugs elicit these experiences regardless of any information-signalling role such experiences typically play in the "normal" awake mind/brain. Orthodox materialism and classical information-based ontologies alike do not merely lack any explanation for why consciousness and our countless varieties of qualia exist. They lack any story of how our qualia could have the causal efficacy to allow us to allude to - and in some cases volubly expatiate on - their existence. Thus mapping the neural correlates of consciousness is not amenable to formal computational methods: digital zombies don't have any qualia, or at least any "bound" macroqualia, that could be mapped, nor a unitary phenomenal self that could do the mapping.
Note this claim for the cognitive primacy of biological sentience isn't a denial of the Church-Turing thesis that given infinite time and infinite memory any Turing-universal system can formally simulate the behaviour of any conceivable process that can be digitized. Indeed (very) fancifully, if the multiverse were being run on a cosmic supercomputer, speeding up its notional execution a million times would presumably speed us up a million times too. But that's not the issue here. Rather the claim is that nonbiological AI run on real-world digital computers cannot tackle the truly hard and momentous cognitive challenge of investigating first-person states of egocentric virtual worlds - or understand why some first-person states, e.g. agony or bliss, are intrinsically important, and cause unitary subjects of experience, persons, to act the way we do.
At least in common usage, "intelligence" refers to an agent's ability to achieve goals in a wide range of environments. What we call greater-than-human intelligence or Superintelligence presumably involves the design of qualitatively new kinds of intelligence never seen before. Hence the growth of artificial intelligence and symbolic AI, together with subsymbolic (allegedly) brain-inspired connectionist architectures and soon artificial quantum computers. But contrary to received wisdom in AI research, sentient biological robots are making greater cognitive progress in discovering the potential for truly novel kinds of intelligence than the techniques of formal AI. We are doing so by synthesising and empirically investigating a galaxy of psychoactive designer drugs - experimentally opening up the possibility of radically new kinds of intelligence in different state-spaces of consciousness. For the most cognitively challenging environments don't lie in the stars but in organic mind/brains - the baffling subjective properties of quantum-coherent states of matter and energy - most of which aren't explicitly represented in our existing conceptual scheme.
6.5 Case Study: Visual Intelligence versus Echolocatory Intelligence:
What Is It Like To Be A Super-Intelligent Bat?
Let's consider the mental state-space of organisms whose virtual worlds are rooted in their dominant sense mode of echolocation. This example isn't mere science fiction. Unless post-Everett quantum mechanics is false, we're forced to assume that googols of quasi-classical branches of the universal wavefunction - the master formalism that exhaustively describes our multiverse - satisfy this condition. Indeed their imperceptible interference effects must be present even in "our" world: strictly speaking, interference effects from branches that have decohered ("split") never wholly disappear; they just become vanishingly small. Anyhow, let's assume these echolocatory superminds have evolved opposable thumbs, a rich generative syntax and advanced science and technology. How are we to understand or measure this alien kind of (super)intelligence? Rigging ourselves up with artificial biosonar apparatus and transducing incoming data into the familiar textures of sight or sound might seem a good start. But to understand the conceptual world of echolocatory superminds, we'd need to equip ourselves with neurons and neural networks neurophysiologically equivalent to smart chiropterans. If one subscribes to a coarse-grained functionalism about consciousness, then echolocatory experience would (somehow) emerge at some abstract computational level of description. The implementation details, or "meatware" as biological mind/brains are derisively called, are supposedly incidental or irrelevant. The functionally unique valence properties of the carbon atom, and likewise the functionally unique quantum mechanical properties of liquid water, are discounted or ignored. Thus according to the coarse-grained functionalist, silicon chips could replace biological neurons without loss of function or subjective identity. By contrast, the micro-functionalist, often branded a mere "carbon chauvinist", reckons that the different intracellular properties of biological neurons - with their different gene expression profiles, diverse primary, secondary, tertiary, and quaternary amino acid chain folding (etc) as described by quantum chemistry - are critical to the many and varied phenomenal properties such echolocatory neurons express. Who is right? We'll only ever know the answer by rigorous self-experimentation: a post-Galilean science of mind.
It's true that humans don't worry much about our ignorance of echolocatory experience, or our ignorance of echolocatory primitive terms, or our ignorance of possible conceptual schemes expressing echolocatory intelligence in echolocatory world-simulations. This is because we don't highly esteem bats. Humans don't share the same interests or purposes as our flying cousins, e.g. to attract desirable, high-fitness bats and rear reproductively successful baby bats. Alien virtual worlds based on biosonar don't seem especially significant to Homo sapiens except as an armchair philosophical puzzle.
Yet this assumption would be intellectually complacent. Worse, understanding what it's like to be a hyperintelligent bat mind is comparatively easy. For echolocatory experience has been recruited by natural selection to play an information-signalling role in a fellow species of mammal; and in principle a research community of language users could biologically engineer their bodies and minds to replicate bat-type experience and establish crude intersubjective agreement to discuss and conceptualise its nature. By contrast, the vast majority of experiential state-spaces remain untapped and unexplored. This task awaits full-spectrum superintelligence in the posthuman era.
In a more familiar vein, consider visual intelligence. How does one measure the visual intelligence of a congenitally blind person? Even with sophisticated technology that generates "inverted spectrograms" of the world to translate visual images into sound, the congenitally blind are invincibly ignorant of visual experience and the significance of visually-derived concepts. Just as a sighted idiot has greater visual intelligence than a blind super-rationalist sage, likewise psychedelics confer the ability to become (for the most part) babbling idiots about other state-spaces of consciousness - but babbling idiots whose insight is deeper than the drug-naive or the genetically unenhanced - or the digital zombies spawned by symbolic AI and its connectionist cousins.
The challenge here is that the vast majority of these alien state-spaces of consciousness latent in organised matter haven't been recruited by natural selection for information-tracking purposes. So "psychonauts" don't yet have the conceptual equipment to navigate these alien state-spaces of consciousness in even a pseudo-public language, let alone integrate them in any kind of overarching conceptual framework. Note the claim here isn't that taking e.g. ketamine, LSD, salvia, DMT and a dizzying proliferation of custom-designed psychoactive drugs is the royal route to wisdom. Or that ingesting such agents will give insight into deep mystical truths. On the contrary: it's precisely because such realms of experience haven't previously been harnessed for information-processing purposes by evolution in "our" family of branches of the universal wavefunction that makes investigating their properties so cognitively challenging - currently beyond our conceptual resources to comprehend. After all, plants synthesise natural psychedelic compounds to scramble the minds of herbivores who might eat them, not to unlock mystic wisdom. Unfortunately, there is no "neutral" medium of thought impartially to appraise or perceptually cross-modally match all these other experiential state-spaces. One can't somehow stand outside one's own stream of consciousness to evaluate how the properties of the medium are infecting the notional propositional content of the language that one uses to describe it.
By way of illustration, compare drug-induced visual experience in a notional community of congenitally blind rationalists who lack the visual apparatus to transduce incident electromagnetic radiation of our familiar wavelengths. The lone mystical babbler who takes such a vision-inducing drug is convinced that [what we would call] visual experience is profoundly significant. And as visually intelligent folk, we know that he's right: visual experience is potentially hugely significant - to an extent which the blind mystical babbler can't possibly divine. But can the drug-taker convince his congenitally blind fellow tribesmen that his mystical visual experiences really matter in the absence of perceptual equipment that permits sensory discrimination? No, he just sounds psychotic. Or alternatively, he speaks lamely and vacuously of the "ineffable". The blind rationalists of his tribe are unimpressed.
The point of this fable is that we've scant reason to suppose that biologically re-engineered posthumans millennia hence will share the same state-spaces of consciousness, or the same primitive terms, or the same conceptual scheme, or the same type of virtual world that human beings now instantiate. Maybe all that will survive the human era is a descendant of our mathematical formalism of physics, M-theory of whatever, in basement reality.
Of course such ignorance of other state-spaces of experience doesn't normally trouble us. Just as the congenitally blind don't grow up in darkness - a popular misconception - the drug-naive and genetically unenhanced don't go around with a sense of what we're missing. We notice teeming abundance, not gaping voids. Contemporary humans can draw upon terms like "blindness" and "deafness" to characterise the deficits of their handicapped conspecifics. From the perspective of full-spectrum superintelligence, what we really need is millions more of such "privative" terms, as linguists call them, to label the different state-spaces of experience of which genetically unenhanced humans are ignorant. In truth, there may very well be more than millions of such nameless state-spaces, each as incommensurable as e.g. visual and auditory experience. We can't yet begin to quantify their number or construct any kind of crude taxonomy of their interrelationships.
Note the problem here isn't cognitive bias or a deficiency in logical reasoning. Rather a congenitally blind (etc) super-rationalist is constitutionally ignorant of visual experience, visual primitive terms, or a visually-based conceptual scheme. So (s)he can't cite e.g. Aumann's agreement theorem [claiming in essence that two cognitive agents acting rationally and with common knowledge of each other's beliefs cannot agree to disagree] or be a good Bayesian rationalist or whatever: these are incommensurable state-spaces of experience as closed to human minds as Picasso is to an earthworm. Moreover there is no reason to expect one realm, i.e. "ordinary waking consciousness", to be cognitively privileged relative to every other realm. "Ordinary waking consciousness" just happened to be genetically adaptive in the African savannah on Planet Earth. Just as humans are incorrigibly ignorant of minds grounded in echolocation - both echolocatory world-simulations and echolocatory conceptual schemes - likewise we are invincibly ignorant of posthuman life while trapped within our existing genetic architecture of intelligence.
In order to understand the world - both its formal/mathematical and its subjective properties - sentient organic life must bootstrap its way to super-sentient full-spectrum superintelligence. Grown-up minds need tools to navigate all possible state-spaces of qualia, including all possible first-person perspectives, and map them - initially via the neural correlates of consciousness in our world-simulations - onto the formalism of mathematical physics. Empirical evidence suggests that the behaviour of the stuff of the world is exhaustively described by the formalism of physics. To the best of our knowledge, physics is causally closed and complete, at least within the energy range of the Standard Model. In other words, there is nothing to be found in the world - no "element of reality", as Einstein puts it - that isn't captured by the equations of physics and their solutions. This is a powerful formal constraint on our theory of consciousness. Yet our ultimate theory of the world must also close Levine's notorious "Explanatory Gap". Thus we must explain why consciousness exists at all ("The Hard Problem"); offer a rigorous derivation of our diverse textures of qualia from the field-theoretic formalism of physics; and explain how qualia combine ("The Binding Problem") in organic minds. These are powerful constraints on our ultimate theory too. How can they be reconciled with physicalism? Why aren't we zombies?
The hard-nosed sceptic will be unimpressed at such claims. How significant are these outlandish state-spaces of experience? And how are they computationally relevant to (super)intelligence? Sure, says the sceptic, reckless humans may take drugs, and experience wild, weird and wonderful states of mind. But so what? Such exotic states aren't objective in the sense of reliably tracking features of the mind-independent world. Elucidation of their properties doesn't pose a well-defined problem that a notional universal algorithmic intelligence could solve.
Well, let's assume, provisionally at least, that all mental states are identical with physical states. If so, then all experience is an objective, spatio-temporally located feature of the world whose properties a unified natural science must explain. A cognitive agent can't be intelligent, let alone superintelligent, and yet be constitutionally ignorant of a fundamental feature of the world - not just ignorant, but completely incapable of gathering information about, exploring, or reasoning about its properties. Whatever else it may be, superintelligence can't be constitutionally stupid. What we need is a universal, species-neutral criterion of significance that can weed out the trivial from the important; and gauge the intelligence of different cognitive agents accordingly. Granted, such a criterion of significance might seem elusive to the antirealist about value. (Mackie 1991) Value nihilism treats any ascription of (in)significance as arbitrary. Or rather the value nihilist maintains that what we find significant simply reflects what was fitness-enhancing for our forebears in the ancestral environment of adaptation. Yet for reasons we simply don't understand, Nature discloses just such a universal touchstone of importance, namely the pleasure-pain axis: the world's inbuilt metric of significance and (dis)value. We're not zombies. First-person facts exist. Some of them matter urgently, e.g. I am in pain. Indeed it's unclear if the expression "I'm in agony; but the agony doesn't matter" even makes cognitive sense. Built into the very nature of agony is the knowledge that its subjective raw awfulness matters a great deal - not instrumentally or derivatively, but by its very nature. If anyone - or indeed any notional super-AGI - supposes that your agony doesn't matter, then he/it hasn't adequately represented the first-person perspective in question.
So the existence of first-person facts is an objective feature of the world that any intelligent agent must comprehend. Digital computers and the symbolic AI code they execute can support formal utility functions. In some contexts, formally programmed utility functions can play a role functionally analogous to importance. But nothing intrinsically matters to a digital zombie. Without sentience, and more specifically without hedonic tone, nothing inherently matters. By contrast, extreme pain and extreme pleasure in any guise intrinsically matter intensely. Insofar as exotic state-spaces of experience are permeated with positive or negative hedonic tone, they matter too. In summary, "He jests at scars, that never felt a wound": scepticism about the self-intimating significance of this feature of the world is feasible only in its absence.
7 The Great Transition
7.1 The End Of Suffering.
A defining feature of general intelligence is the capacity to achieve one's goals in a wide range of environments. All sentient biological agents are endowed with a pleasure-pain axis. All prefer occupying one end to the other. A pleasure-pain axis confers inherent significance on our lives: the opioid-dopamine neurotransmitter system extends from flatworms to humans. Our core behavioural and physiological responses to noxious and rewarding stimuli have been strongly conserved in our evolutionary lineage over hundreds of millions of years. Some researchers argue for psychological hedonism, the theory that all choice in sentient beings is motivated by a desire for pleasure or an aversion from suffering. When we choose to help others, this is because of the pleasure that we ourselves derive, directly or indirectly, from doing so. Pascal put it starkly: "All men seek happiness. This is without exception. Whatever different means they employ, they all tend to this end. The cause of some going to war, and of others avoiding it, is the same desire in both, attended with different views. This is the motive of every action of every man, even of those who hang themselves." In practice, the hypothesis of psychological hedonism is plagued with anomalies, circularities and complications if understood as a universal principle of agency: the "pleasure principle" is simplistic as it stands. Yet the broad thrust of this almost embarrassingly commonplace idea may turn out to be central to understanding the future of life in the universe. If even a weak and exception-laden version of psychological hedonism is true, then there is an intimate link between full-spectrum superintelligence and happiness: the "attractor" to which rational sentience is heading. If that's really what we're striving for, a lot of the time at least, then instrumental means-ends rationality dictates that intelligent agency should seek maximally cost-effective ways to deliver happiness - and then superhappiness and beyond.
A discussion of psychological hedonism would take us too far afield here. More fruitful now is just to affirm a truism and then explore its ramifications for life in the post-genomic era. Happiness is typically one of our goals. Intelligence amplification entails pursuing our goals more rationally. For sure, happiness, or at least a reduction in unhappiness, is frequently sought under a variety of descriptions that don't explicitly allude to hedonic tone and sometimes disavow it altogether. Natural selection has "encephalised" our emotions in deceptive, fitness-enhancing ways within our world-simulations. Some of these adaptive fetishes may be formalised in terms of abstract utility functions that a rational agent would supposedly maximise. Yet even our loftiest intellectual pursuits are underpinned by the same neurophysiological reward and punishment pathways. The problem for sentient creatures is that, both personally and collectively, Darwinian life is not very smart or successful in its efforts to achieve long-lasting well-being. Hundreds of millions of years of "Nature, red in tooth and claw" attest to this terrible cognitive limitation. By a whole raft of indices (suicide rates, the prevalence of clinical depression and anxiety disorders, the Easterlin paradox, etc) humans are not getting any (un)happier on average than our Palaeolithic ancestors despite huge technological progress. Our billions of factory-farmed non-human victims spend most of their abject lives below hedonic zero. In absolute terms, the amount of suffering in the world increases each year in humans and non-humans alike. Not least, evolution sabotages human efforts to improve our subjective well-being thanks to our genetically constrained hedonic treadmill - the complicated web of negative feedback mechanisms in the brain that stymies our efforts to be durably happy at every turn. Discontent, jealousy, anxiety, periodic low mood, and perpetual striving for "more" were fitness-enhancing in the ancient environment of evolutionary adaptedness. Lifelong bliss wasn't harder for information-bearing self-replicators to encode. Rather lifelong bliss was genetically maladaptive and hence selected against. Only now can biotechnology remedy organic life's innate design flaw.
A potential pitfall lurks here: the fallacy of composition. Just because all individuals tend to seek happiness and shun unhappiness doesn't mean that all individuals seek universal happiness. We're not all closet utilitarians. Genghis Khan wasn't trying to spread universal bliss. As Plato observed, "Pleasure is the greatest incentive to evil." But here's the critical point. Full-spectrum superintelligence entails the cognitive capacity impartially to grasp all possible first-person perspectives - overcoming egocentric, anthropocentric, and ethnocentric bias (cf. mirror-touch synaesthesia). As an idealisation, at least, full-spectrum superintelligence understands and weighs the full range of first-person facts. First-person facts are as much an objective feature of the natural world as the rest mass of the electron or the Second Law of Thermodynamics. You can't be ignorant of first-person perspectives and superintelligent any more than you can be ignorant of the Second law of Thermodynamics and superintelligent. By analogy, just as autistic superintelligence captures the formal structure of a unified natural science, a mathematically complete "view from nowhere", all possible solutions to the universal Schrödinger equation or its relativistic extension, likewise a full-spectrum superintelligence also grasps all possible first-person perspectives - and acts accordingly. In effect, an idealised full-spectrum superintelligence would combine the mind-reading prowess of a telepathic mirror-touch synaesthete with the optimising prowess of a rule-following hyper-systematiser on a cosmic scale. If your hand is in the fire, you reflexively withdraw it. In withdrawing your hand, there is no question of first attempting to solve the Is-Ought problem in meta-ethics and trying logically to derive an "ought" from an "is". Normativity is built into the nature of the aversive experience itself: I-ought-not-to-be-in-this-dreadful-state. By extension, perhaps a full-spectrum superintelligence will perform cosmic felicific calculus and execute some sort of metaphorical hand-withdrawal for all accessible suffering sentience in its forward light-cone. Indeed one possible criterion of full-spectrum superintelligence is the propagation of subjectively hypervaluable states on a cosmological scale.
What this constraint on intelligent agency means in practice is unclear. Conceivably at least, idealised superintelligences must ultimately do what a classical utilitarian ethic dictates and propagate some kind of "utilitronium shockwave" across the cosmos. To the classical utilitarian, any rate of time-discounting indistinguishable from zero is ethically unacceptable, so s/he should presumably be devoting most time and resources to that cosmological goal. An ethic of negative utilitarianism is often accounted a greater threat to intelligent life (cf. the hypothetical "button-pressing" scenario) than classical utilitarianism. But whereas a negative utilitarian believes that once intelligent agents have phased out the biology of suffering, all our ethical duties have been discharged, the classical utilitarian seems ethically committed to converting all accessible matter and energy into relatively homogeneous matter optimised for maximum bliss: "utilitronium". Hence the most empirically valuable outcome entails the extinction of intelligent life. Could this prospect derail superintelligence?
Perhaps. But utilitronium shockwave scenarios shouldn't be confused with wireheading. The prospect of self-limiting superintelligence might be credible if either a (hypothetical) singleton biological superintelligence or its artificial counterpart discovers intracranial self-stimulation or its nonbiological analogues. Yet is this blissful fate a threat to anyone else? After all, a wirehead doesn't aspire to convert the rest of the world into wireheads. A junkie isn't driven to turn the rest of the world into junkies. By contrast, a utilitronium shockwave propagating across our Hubble volume would be the product of intelligent design by an advanced civilisation, not self-subversion of an intelligent agent's reward circuitry. Also, consider the reason why biological humanity - as distinct from individual humans - is resistant to wirehead scenarios, namely selection pressure. Humans who discover the joys of intra-cranial self-stimulation or heroin aren't motivated to raise children. So they are outbred. Analogously, full-spectrum superintelligences, whether natural or artificial, are likely to be social rather than solipsistic, not least because of the severe selection pressure exerted against any intelligent systems who turn in on themselves to wirehead rather than seek out unoccupied ecological niches. In consequence, the adaptive radiation of natural and artificial intelligence across the Galaxy won't be undertaken by stay-at-home wireheads or their blissed-out functional equivalents.
On the face of it, this argument from selection pressure undercuts the prospect of superhappiness for all sentient life - the "attractor" towards which we may tentatively predict sentience is converging in virtue of the pleasure principle harnessed to ultraintelligent mind-reading prowess and utopian neuroscience. But what is necessary for sentient intelligence is information-sensitivity to fitness-relevant stimuli - not an agent's absolute location on the pleasure-pain axis. True, uniform bliss and uniform despair are inconsistent with intelligent agency. Yet mere recalibration of a subject's "hedonic set-point" leaves intelligence intact. Both information-sensitive gradients of bliss and information-sensitive gradients of misery allow high-functioning performance and critical insight. Only sentience animated by gradients of bliss is consistent with a rich subjective quality of intelligent life. Moreover the nature of "utilitronium" is as obscure as its theoretical opposite, "dolorium". The problem here cuts deeper than mere lack of technical understanding, e.g. our ignorance of the gene expression profiles and molecular signature of pure bliss in neurons of the rostral shell of the nucleus accumbens and ventral pallidum, the twin cubic centimetre-sized "hedonic hotspots" that generate ecstatic well-being in the mammalian brain. Rather there are difficult conceptual issues at stake. For just as the torture of one mega-sentient being may be accounted worse than a trillion discrete pinpricks, conversely the sublime experiences of utiltronium-driven Jupiter minds may be accounted preferable to tiling our Hubble volume with the maximum abundance of micro-bliss. What is the optimal trade-off between quantity and intensity? In short, even assuming a classical utilitarian ethic, the optimal distribution of matter and energy that a God-like superintelligence would create in any given Hubble volume is very much an open question.
Of course we've no grounds for believing in the existence of an omniscient, omnipotent, omnibenevolent God or a divine utility function. Nor have we grounds for believing that the source code for any future God, in the fullest sense of divinity, could ever be engineered. The great bulk of the Multiverse, and indeed a high measure of life-supporting Everett branches, may be inaccessible to rational agency, quasi-divine or otherwise. Yet His absence needn't stop rational agents intelligently fulfilling what a notional benevolent deity would wish to accomplish, namely the well-being of all accessible sentience: the richest abundance of empirically hypervaluable states of mind in their Hubble volume. Recognisable extensions of existing technologies can phase out the biology of suffering on Earth. But responsible stewardship of the universe within our cosmological horizon depends on biological humanity surviving to become posthuman superintelligence.
7.2 Paradise Engineering?
The hypothetical shift to life lived entirely above Sidgwick's "hedonic zero" will mark a momentous evolutionary transition. What lies beyond? There is no reason to believe that hedonic ascent will halt in the wake of the world's last aversive experience in our forward light-cone. Admittedly, the self-intimating urgency of eradicating suffering is lacking in any further hedonic transitions, i.e. a transition from the biology of happiness to a biology of superhappiness; and then beyond. Yet why "lock in" mediocrity if intelligent life can lock in sublimity instead?
Naturally, superhappiness scenarios could be misconceived. Long-range prediction is normally a fool's game. But it's worth noting that future life based on gradients of intelligent bliss isn't tied to any particular ethical theory: its assumptions are quite weak. Radical recalibration of the hedonic treadmill is consistent not just with classical or negative utilitarianism, but also with preference utilitarianism, Aristotelian virtue theory, a deontological or a pluralist ethic, Buddhism, and many other value systems besides. Recalibrating our hedonic set-point doesn't - or at least needn't - undermine critical discernment. All that's needed for the abolitionist project and its hedonistic extensions to succeed is that our ethic isn't committed to perpetuating the biology of involuntary suffering. Likewise, only a watered-down version of psychological hedonism is needed to lend the scenario sociological credibility. We can retain as much - or as little - of our existing preference architecture as we please. You can continue to prefer Shakespeare to Mills-and-Boon, Mozart to Morrissey, Picasso to Jackson Pollock while living perpetually in Seventh Heaven or beyond.
Nonetheless an exalted hedonic baseline will revolutionise our conception of life. The world of the happy is quite different from the world of the unhappy, says Wittgenstein; but the world of the superhappy will feel unimaginably different from the human, Darwinian world. Talk of preference conservation may reassure bioconservatives that nothing worthwhile will be lost in the post-Darwinian transition. Yet life based on information-sensitive gradients of superhappiness will most likely be "encephalised" in state-spaces of experience alien beyond human comprehension. Humanly comprehensible or otherwise, enriched hedonic tone can make all experience generically hypervaluable in an empirical sense - its lows surpassing today's peak experiences. Will such experience be hypervaluable in a metaphysical sense too? Is this question cognitively meaningful?
8 The Future Of Sentience
8.1 The Sentience Explosion.
Man proverbially created God in his own image. In the age of the digital computer, humans conceive God-like superintelligence in the image of our dominant technology and personal cognitive style - refracted, distorted and extrapolated for sure, but still through the lens of human concepts. The "super-" in so-called superintelligence is just a conceptual fig-leaf that humans use to hide our ignorance of the future. Thus high-AQ / high-IQ humans may imagine God-like intelligence as some kind of Super-Asperger - a mathematical theorem-proving hyper-rationalist liable systematically to convert the world into computronium for its awesome theorem-proving. High-EQ, low-AQ humans, on the other hand, may imagine a cosmic mirror-touch synaesthete nurturing creatures great and small in expanding circles of compassion. From a different frame of reference, psychedelic drug investigators may imagine superintelligence as a Great Arch-Chemist opening up unknown state-space of consciousness. And so forth. Probably the only honest answer is to say, lamely, boringly, uninspiringly: we simply don't know.
Grand historical meta-narratives are no longer fashionable. The contemporary Singularitarian movement is unusual insofar as it offers one such grand meta-narrative: history is the story of simple biological intelligence evolving through natural selection to become smart enough to conceive an abstract universal Turing machine (UTM), build and program digital computers - and then merge with, or undergo replacement by, recursively self-improving artificial superintelligence.
Another grand historical meta-narrative views life as the story of overcoming suffering. Darwinian life is characterised by pain and malaise. One species evolves the capacity to master biotechnology, rewrites its own genetic source code, and creates post-Darwinian superhappiness. The well-being of all sentience will be the basis of post-Singularity civilisation: primitive biological sentience is destined to become blissful supersentience.
These meta-narratives aren't mutually exclusive. Indeed on the story told here, full-spectrum superintelligence entails full-blown supersentience too: a seamless unification of the formal and the subjective properties of mind.
If the history of futurology is any guide, the future will confound us all. Yet in the words of Alan Kay: "It's easier to invent the future than to predict it."
* * *
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* * *
David Pearce (2012, last updated 2016)
see too Technological Singularities and An Organic Singularity? (PDF) (PPT)
The Hedonistic Imperative
Talks 2015
BLTC Research
Quantum Ethics?
Utopian Surgery?
Social Media 2018
The Shulgin Index
Gene Drives (2016)
Utopian Pharmacology
The Abolitionist Project
Quora Answers (2015-8)
The Repugnant Conclusion
Reprogramming Predators
The Reproductive Revolution
Asperger's Quotient (AQ) Test
Kurzweil Accelerating Intelligence
The Future of Biological Intelligence
Machine Intelligence Research Institute (MIRI)
Technological Singularities and Intelligence Explosions
The Biointelligence Explosion (PowerPoint Slide Show)
An Organic Singularity? Humans and Intelligent Machines: Co-Evolution, Fusion or Replacement?
E-mail Dave |
c5251e61f25426c2 | Thursday, March 04, 2010
Quantum Interactions Create Space-time
The notion that spacetime is an emergent phenomenon is, by my reckoning, being proposed by an increasing number of thinkers. Physicists and philosophers working in quantum gravity and quantum foundations are turning to the idea that the spacetime of relativity is not fundamental, but rather something which arises from a more fundamental world of quantum mechanical systems and their interactions.
I just saw a reference to one such argument which was made a few years ago in an article by Avshalom C. Elitzur and Shahar Dolev called “Quantum Phenomena Within a New Theory of Time”. This was published in the 2005 collection Quo Vadis Quantum Mechanics?, Avshalom C. Elitzur, Shahar Dolev, Nancy Kolenda, Eds.
Elitzur and Dolev examine several puzzles over the nature of time in quantum mechanics and are led to the hypothesis that quantum interactions (measurements) themselves are responsible for the creation of spacetime.
A couple of quotes from section 17.10, titled “An Outline of the Spacetime Dynamics Theory”:
Could it be, then, that the two phenomena – time’s passage and wave-function collapse – are not only real, but the latter is the very manifestation of the former? A wave function, after all, is a sum of many equally possible outcomes, while the measurement brings about the realization of one out of them, the others vanishing. Is this not the very difference between future and past? And is collapse not elusive because it creates the elusive ‘now’?
Suppose that there is indeed a ‘now’ front, on the one side of which there are past events, adding up as the ‘now’ progresses, while on its other side there are no events, and hence, according to Mach, not even spacetime. Spacetime thus ‘grows’ into the future as history unfolds.
What role does the wave function play in this creation of new events? The dynamically evolving spacetime allows a radical possibility. Rather than conceiving of some empty spacetime with which the wave function evolves, the reverse may be the case. The wave function evolves beyond the ‘now’, i.e., outside of spacetime, and its ‘collapse’ due to the interaction with other wave functions creates not only the events, but also the spacetime within which they are located in relation to one another. The famous peculiarities of the quantum interaction – nonlocality, the coexistence of mutually exclusive states, backward causation and the inconsistent histories presented in the previous sections, thus become more natural.
Can the reciprocal effects of spacetime and matter – the celebrated lesson of general relativity – thus possible gain a quantum mechanical explanation? Perhaps it is the wave function, we submit, that is more primitive than spacetime, and the spacetime connecting the two events is the product of their interacting wave functions.
Thomas J McFarlane said...
On the face of it, there seems to be a problem with this proposal due to the fact that the Schrödinger equation has time built into it already. How can time emerge from wave functions that are themselves solutions of an equation that presupposes time already as a background?
Steve said...
Hi. Good question.
There are two levels. They're making a distinction between the emergent spacetime of relativity, which is a purely geometric structure derived from the distribution of the measurement events, and the underlying background time of the wave functions.
The reason I think this is worthy of consideration is that the spacetime of relativity has no concept of the flow of time anyway, so under this vision that is accounted for by the underlying more fundamental quantum time.
Doru said...
I find the Space-time Dynamics Theory to be a very valid argument.
Even from a philosophical perspective, the wave function seems to be transcending the space-time bound as a more fundamental ontological explanation. The real problem we have here is the problem of mind and consciousness. Space and time is the only solution we have for this problem.
Steve said...
Hi Doru: with reference to your last sentence, I think we need time for consciousness, but I'm not sure we need space (as we have normally understood it).
TechTonics said...
John Baez: ..."there's no "time operator" in quantum mechanics!
Since the underlying quantum formalism has no time operator, the Schrodinger equation, which is super-structure, adds time to bridge QM to General Relativity, or microcosm to macrocosm where humans insist upon time recognition. I don't think the objection offered by McFarlane is fundamental. Refer to quantum gravity.
TechTonics said...
I group my interests in philosophical musings under the category, Causality and Machine Learning, or the case of how did Causality evolve/emerge a time- perceiving consciousness? Here is a real good paper on the problem of time within quantum gravity.
Prima Facie Questions in Quantum Gravity Authors: C.J.Isham
I wish now to consider in more detail three exceptionally important questions that can
be asked of any approach to quantum gravity. These are: (i) what background structure
is assumed?; (ii) what role is played by the spacetime diffeomorphism group?; (iii) how is the concept of ‘time’ to be understood?
One of the major issues in quantum gravity is the so-called ‘problem of time’. This arises from the very different roles played by the concept of time in quantum theory and in general relativity. Let us start by considering standard quantum theory."
Steve said...
Thanks very much for the link. I looked at what Baez said about time not being an operator in qm. This is an idea that t isn't like the other observables and there is some debate about whether the time/energy uncertainly pair is unlike the position/momemtum pair, etc. I think regardless of this, a time parameter is still a fundamental part of qm, as it indexes our experiences in the situations qm is used to describe.
TechTonics said...
QM is not a universal theory. GR is a universal theory in which time is considered fundamental. In QM time is not an operator meaning it is not an observable. The time in the S.E. parameter is arbitrary so that is why I say it is not fundamental. In the chapter under discussion, Zeh is cited as a reference, and within the article, Aharonov is mentioned which is why I selected them as authoritative resources. Zeh
Time in Quantum Theory
... "For this reason, von Neumann [3] referred to the time-dependent --> Schrödinger equation as a 'second intervention', since Schrödinger had invented it solely to describe consequences of time-dependent external 'perturbations' of a quantum system.
In non-relativistic quantum mechanics, the time parameter t that appears in the Schrödinger wave function *ψ(q,t) is identified with Newton's absolute time."
[SH: This is how the gap is bridged between the not universal theory of QM and the classical/GR regime which is considered universal, because although time is not observable in GR either, time is considered fundamental, "Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions."
Y. Aharonov and D. Bohm
Because time does not appear in Schrodinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulate in a special way.
First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e. by means of operators [observable], wave functions, etc."
Steve said...
Hi. Thanks Techtonics for your comments. Re:"QM is not a universal theory. GR is a universal theory...":
I've been interested in proposals that turn this 180 degrees around.
In these, GR is seen as an emergent, not fundamental theory; a network of quantum mechanical systems and their interactions would be fundamental. You say the time parameter in qm is arbitrary, and I understand the rationale for saying this, but it's drawn from our experience.
Whereas with GR (and some quantum gravity theories like loop quantum gravity) you can't get our intuitive experience of time back out! The idea of time as one of the dimensions in GR isn't enough -there is no preferred time direction picked out by the theory, it is really just geometry.
A list of posts on theories of this type is at the bottom of this post.
TechTonics said...
Causality First
Rafael Sorkin’s Causal Sets and Fotini Markopoulou’s Quantum Causal Histories.
She finishes with a note on time:
Just as the emergent locality has nothing to do with the fundamental micro-locality,
time and causality will also be unrelated macro vs. micro. So, the theory “puts” in
time at the micro-level (via its causality constraints), but emergent spacetime will
have no preferred time slice –as required in general relativity.
TT: I haven't read as many of those theories as you have. I did read Fotini and was impressed. But you quote her as saying "time and causality will also be unrelated macro vs. micro."
I think GR is still received as a universal theory because humans perceive causality as events unfolding through time. The resistance to accepting QM as a universal theory arises from not understanding the physical result of the double-split experiment, which Feymann described as the "central mystery" of quantum mechanics. Or the causality involved with EPR and non-locality. Classically, the speed of light is precisely determined in terms of a meter which limits causality. In the 13 or so major interpretations of what QM describes about reality, one has time considered as instantaneous which wreaks a bit of havoc with a traditional grasp of causality, and other various QM interpretations about the speed of light. I'm not going to buy a theory which doesn't make testable predictions.
I like most what Penrose had to say in the Foreward to that Quo Vadis book. The theories are incomplete, either QM or GR or both. What bothers me about Fotini's idea is that both QM and GR have experiments which confirm the theories to 99.9999+ accuracy.
It makes me think that there should be a reconciliation possible; but it doesn't seem Fotini is forecasting such a reconciliation with that remark: "time and causality will also be unrelated macro vs. micro." I read her paper over a year ago so my memory is fuzzy. I recall that the root of the debate is over whether the turtle's back is discrete or continuous.
TechTonics said...
I just found this newer paper by Loll, Coupling Point-Like Masses to Quantum Gravity with Causal Dynamical Triangulations
Authors: Igor Khavkine, Renate Loll, Paul Reska (Submitted on 24 Feb 2010)
I have a fondness for Loll's approach because I like fractals. Though Fotini mentions CDT, the paper you linked to her seemed to conflict with CDT in too many ways.
Steve said...
I haven't read that paper yet, but I do feel that the cdt results are "telling us something" about quantum gravity. I think that we probably will get a better motivated and more insightful micro-theory than the cdt one in a future theory.
TechTonics said...
I also found that post by Scott interesting. I've heard it said, like a complaint, that everything is emergent. But maybe that is really an insight.
The Big Bang or the Big Bounce, stars forming and then reforming so that they create more complex elements when they nova, planets forming, life, intelligence and consciousness. This all seems quite unpredictable from the point of origin.
Becoming more particular, von Neumann purposely designed an automata capable of universal computation (UC). Was that emergent? Then Conway designed The Game of Life in 1970. He proved Life capable of UC in 1982. I've read that Life (UC) described as an emergent phenomenon. The cases appear to be different in my view.
There are complex patterns which have rules, that are so complex that it has been proven that no computer can find the rule. How can you distinguish that type of rule from random, as in Rule 30, which is a pseudo-random generator which re-cycles after a "billion billion lifetimes of the universe" (Wolfram).
I found your website after looking into CAs and emergence.
I've really enjoyed reading some so far, of your many available insightful posts. I can read a paper and compare what I get out of it (which never seems to be enough) with your analysis; quite a treasure. Also my first impression of Fotini's paper has improved with aging.
TechTonics said...
I have been reading your past posts and taking notes. I was hoping to find a consistent
explanation between the ideas explored which shared a common philosophical perspective.
Maybe you can connect all the dots in preparation for writing your book :-)
Philosophers usually dismiss the notion of "strong emergence" because it is magical, transcending causality. Chalmers adopts a type of dualism in suggesting that the only type of strong emergence might be consciousness. Most philosophers describe weak emergence in terms of knowledge (or lack thereof). Weak emergence assumes causality although the rule of the pattern which connects cause and effect is unknown or irreducible.
Apparently the objective universe existed before humans evolved who describe it in theories which can never be proved, thus epistemological constraints. One such theory suggests that the universe expanded at a super-luminal velocity, which I think is faster than causality propagates. At what point can one distinguish some causing event/property as fundamental, from the created event (effect) which follows, and is now described as emergent? The universe was evolving before sentience arose which could experience such epistemological constraints.
The Limit of the Bayesian Interpretation "I read the paper “Subjective probability and quantum certainty” ...
To bring the point home, this particular paper features the discussion of quantum experiments with a certain outcome: the authors show that this outcome is to be interpreted as a certainty of epistemic belief on the part of the observer, not an objective certainty."
Causality: Models, Reasoning, and Inference (2000) by Judea Pearl Preface
Tech: What primitive or fundamental concept of causality can exist which doesn't also entail the idea of events existing in space which create the following timely t+1 next event? If causality is primitive, how then are space and time considered emergent?
It would seem to me that causality, space and time must all have emerged from a common cause, or, they are all fundamental since causality is defined in terms of, or exists in terms of, space and time and events. Isn't gravity essentially causal? Is everything emergent when viewed from the first few nano-seconds of expansion? Do quantum fluctuations include and explain the plasma conditions?
Steve said...
Thanks for your kind words about the blog. I always appreciate dialogue with others who think about these things. I also appreciate your comment that it’s hard to discern a coherent philosophical story from this format – I should attempt to restate or summarize some of the themes which run through this stuff.
Re: emergence
This is a tricky subject. I had convinced myself awhile back that most examples of emergence in nature were epistemological or explanatory, not truly ontological – with one exception: The collapse of a quantum system when measured.
The idea is that the ordered sequence of measurement events (like the causal set idea) emerges from the underlying dimensionless quantum world of possibilia. So causality, and also time, which is the index of causality, are primitive. Space is not primitive. Space (and the geometric spacetime picture of GR) are derived from the primitive relations between and among the events (“geometrogenesis”).
With regard to mind, the speculative idea is that panexperientialism is true, and so there is a mote of first person experience present in every event. So the emergence of human consciousness is not an additional ontological type of emergence beyond the first type.
Steve said...
Instead of "dimensionless" quantum world, I should perhaps have said "hyper-dimensioned" or something like that..
TechTonics said...
I was talking to a French AIT expert about the term "algorithmically incompressible" as an incorrect description (due to the definition) of describing CAs that must unfold in a simulation, there is no predictive shortcut. Wolram's term, "computationally irreducible" is correct. Anyway, he mentioned in passing that QM thoughts about emergence would depend upon which interpretation one chose.
30. For instance, the Schrödinger equation gives a deterministic evolution of the wave-function. In the "traditional interpretation" of quantum mechanics a measurement "collapses the wave-function." This collapse DOES NOT obey Schrödinger’s equation.
After the collapse Schrödinger’s equation again governs the wavefunction evolution. Randomness enters during the measurement.
1. Unitary evolution by the Schrödinger equation
SH: So the standard interpretation postulates I think, that there are myriad wave-function collapses in this universe. However the Everett, Many-Worlds, and Many-Minds interpretation postulate a universal wave-function: "However Albert and Loewer suggest that the mental does not supervene on the physical, because individual minds have trans-temporal identity of their own. The mind selects one of these identities to be its non-random reality, while the universe itself is unaffected."
With regard to "panexperientialism is true", isn't that a type of dualism which entails supervience?
Steve said...
No; panexperientialism is meant to avoid dualism by specifying that mental events and physical events are the same things. Neither supervenes on the other. The only dualism here is a dualism of perspectives. Mental events are ones we participate in directly; physical events are the others we infer exist via a third person perspective. |
5efd7c32933c69b3 | Camil Muscalu, Christoph Thiele and I have just uploaded to the arXiv our joint paper, “Multi-linear multipliers associated to simplexes of arbitrary length“, submitted to Analysis & PDE. This paper grew out of our project from many years ago to attempt to prove the nonlinear (or “scattering”) version of Carleson’s theorem on the almost everywhere convergence of Fourier series. This version is still open; our original approach was to handle the nonlinear Carleson operator by multilinear expansions in terms of the potential function V, but while the first three terms of this expansion were well behaved, the fourth term was unfortunately divergent, due to the unhelpful location of a certain minus sign. [This survey by Michael Lacey, as well as this paper of ourselves, covers some of these topics.]
However, what we did find out in this paper was that if we modified the nonlinear Carleson operator slightly, by replacing the underlying Schrödinger equation by a more general AKNS system, then for “generic” choices of this system, the problem of the ill-placed minus sign goes away, and each term in the multilinear series is, in fact, convergent (though we did not yet verify that the series actually converged, though in view of the earlier work of Christ and Kiselev on this topic, this seems likely). The verification of this convergence (at least with regard to the scattering data, rather than the more difficult analysis of the eigenfunctions) is the main result of our current paper. It builds upon our earlier estimates of the bilinear term in the expansion (which we dubbed the “biest”, as a multilingual pun). The main new idea in our earlier paper was to decompose the relevant region of frequency space \{ (\xi_1,\xi_2,\xi_3) \in {\Bbb R}^3: \xi_1 < \xi_2 < \xi_3 \} into more tractable regions, a typical one being the region in which \xi_2 was much closer to \xi_1 than to \xi_3. The contribution of each region can then be “parafactored” into a “paracomposition” of simpler operators, such as the bilinear Hilbert transform, which can be treated by standard time-frequency analysis methods. (Much as a paraproduct is a frequency-restricted version of a product, the paracompositions that arise here are frequency-restricted versions of composition.)
A similar analysis happens to work for the multilinear operators associated to the frequency region S := \{ (\xi_1,\ldots,\xi_n): \xi_1 < \ldots < \xi_n \}, but the combinatorics are more complicated; each of the component frequency regions has to be indexed by a tree (in a manner reminiscent of the well-separated pairs decomposition), and a certain key “weak Bessel inequality” becomes considerably more delicate. Our ultimate conclusion is that the multilinear operator
T(V_1,\ldots,V_n) := \int_{(\xi_1,\ldots,\xi_n) \in S} \hat V_1(\xi_1) \ldots \hat V_n(\xi_n) e^{2i (\xi_1+\ldots+\xi_n) x}\ d\xi_1 \ldots d\xi_n (1)
(which generalises the bilinear Hilbert transform and the biest) obeys Hölder-type L^p estimates (note that Hölder’s inequality related to the situation in which the (projective) simplex S is replaced by the entire frequency space {\Bbb R}^n).
For the remainder of this post, I thought I would describe the “nonlinear Carleson theorem” conjecture, which is still one of my favourite open problems, being an excellent benchmark for measuring progress in the (still nascent) field of “nonlinear Fourier analysis“, while also being of interest in its own right in scattering and spectral theory.
My starting point will be the one-dimensional time-independent Schrödinger equation
- u_{xx}(k,x) + V(x) u(k,x) = k^2 u(k,x) (2)
where V: {\Bbb R} \to {\Bbb R} is a given potential function, k \in {\Bbb R} is a frequency parameter, and u: {\Bbb R} \times {\Bbb R} \to {\Bbb C} is the wave function. This equation (after reinstating constants such as Planck’s constant \hbar, which we have normalised away) describes the instantaneous state of a quantum particle with energy k^2 in the presence of the potential V. To avoid technicalities let us assume that V is smooth and compactly supported (say in the interval {}[-R,R]) for now, though the eventual conjecture will concern potentials V that are merely square-integrable.
For each fixed frequency k, the equation (2) is a linear homogeneous second order ODE, and so has a two-dimensional space of solutions. In the free case V=0, the solution space is given by
u(k,x) = \alpha(k) e^{ikx} + \beta(k) e^{-ikx} (3)
where \alpha(k) and \beta(k) are arbitrary complex numbers; physically, these numbers represent the amplitudes of the rightward and leftward propagating components of the solution respectively.
Now suppose that V is non-zero, but is still compactly supported on an interval {}[-R,+R]. Then for a fixed frequency k, a solution to (2) will still behave like (3) in the regions x > R and x < R, where the potential vanishes; however, the amplitudes on either side of the potential may be different. Thus we would have
u(k,x) = \alpha_+(k) e^{ikx} + \beta_+(k) e^{-ikx}
for x > R and
u(k,x) = \alpha_-(k) e^{ikx} + \beta_-(k) e^{-ikx}
for x < -R. Since there is only a two-dimensional linear space of solutions, the four complex numbers \alpha_-(k), \beta_-(k), \alpha_+(k), \beta_+(k) must be related to each other by a linear relationship of the form
\begin{pmatrix} \alpha_+(k) \\ \beta_+(k) \end{pmatrix} = \overbrace{V}(k) \begin{pmatrix} \alpha_-(k) \\ \beta_-(k) \end{pmatrix}
where \overbrace{V}(k) is a 2 \times 2 matrix depending on V and k, known as the scattering matrix of V at frequency k. (We choose this notation to deliberately invoke a resemblance to the Fourier transform \hat V(k) := \int_{-\infty}^\infty V(x) e^{-2ikx}\ dx of V; more on this later.) Physically, this matrix determines how much of an incoming wave at frequency k gets reflected by the potential, and how much gets transmitted.
What can we say about the matrix \overbrace{V}(k)? By using the Wronskian of two solutions to (2) (or by viewing (2) as a Hamiltonian flow in phase space) we can show that \overbrace{V}(k) must have determinant 1. Also, by using the observation that the solution space to (2) is closed under complex conjugation u(k,x) \mapsto \overline{u(k,x)}, one sees that each coefficient of the matrix \overbrace{V}(k) is the complex conjugate of the diagonally opposite coefficient. Combining the two, we see that \overbrace{V}(k) takes values in the Lie group
SU(1,1) := \{ \begin{pmatrix} a & \overline{b} \\ b & \overline{a} \end{pmatrix}: a,b \in {\Bbb C}, |a|^2-|b|^2 = 1 \}
(which, incidentally, is isomorphic to SL_2({\Bbb R})), thus we have
\overbrace{V}(k) = \begin{pmatrix} a(k) & \overline{b(k)} \\ b (k) & \overline{a(k)} \end{pmatrix}
for some functions a: {\Bbb R} \to {\Bbb C} and b: {\Bbb R} \to {\Bbb C} obeying the constraint |a(k)|^2 - |b(k)|^2 = 1. (The functions \frac{1}{a(k)} and \frac{b(k)}{a(k)} are sometimes known as the transmission coefficient and reflection coefficient respectively; note that they square-sum to 1, a fact related to the law of conservation of energy.) These coefficients evolve in a beautifully simple manner if V evolves via the Korteweg-de Vries (KdV) equation V_t + V_{xxx} = 6VV_x (indeed, one has \partial_t a = 0 and \partial_t b = 8ik^3 b), being part of the fascinating subject of completely integrable systems, but that is a long story which we will not discuss here. This connection does however provide one important source of motivation for studying the scattering transform V \mapsto \overbrace{V} and its inverse.
What are the values of the coefficients a(k), b(k)? In the free case V=0, one has a(k)=1 and b(k)=0. When V is non-zero but very small, one can linearise in V (discarding all terms of order O(V^2) or higher), and obtain the approximation
a(k) \approx 1 -\frac{i}{2k}\int_{-\infty}^\infty V; \quad b(k) \approx \frac{-i}{2k} \hat V(k)
known as the Born approximation; this helps explain why we think of \overbrace{V}(k) as a nonlinear variant of the Fourier transform. A slightly more precise approximation, known as the WKB approximation, is
a(k) \approx e^{-\frac{i}{2k}\int_{-\infty}^\infty V}; \quad b(k) \approx \frac{-i}{2k} e^{-\frac{i}{2k}\int_{-\infty}^\infty V} \int_{-\infty}^{\infty} V(x) e^{-2ikx + \frac{i}{k} \int_{-\infty}^x V}\ dx.
(One can avoid the additional technicalities caused by the WKB phase correction by working with the Dirac equation instead of the Schrödinger; this formulation is in fact cleaner in many respects, but we shall stick with the more traditional Schrödinger formulation here. More generally, one can consider analogous scattering transforms for AKNS systems.) One can in fact expand a(k) and b(k) as a formal power series of multilinear integrals in V (distorted slightly by the WKB phase correction e^{\frac{i}{k} \int_{-\infty}^x V}), whose terms resemble the multilinear expression (1) except for some (crucial) sign changes and some WKB phase corrections. It is relatively easy to show that this multilinear series is absolutely convergent for every k when the potential V is absolutely integrable (this is the nonlinear analogue to the obvious fact that the Fourier integral \hat V(k) = \int_{-\infty}^\infty V(k) e^{-2ikx} is absolutely convergent when V is absolutely integrable; it can also be deduced without recourse to multilinear series by using Levinson’s theorem.) If V is not absolutely integrable, but instead lies in L^p({\Bbb R}) for some p > 1, then the series can diverge for some k; this fact is closely related to a classic result of Wigner and von Neumann that the Schrödinger operator can contain embedded pure point spectrum. However, Christ and Kiselev showed that the series is absolutely convergent for almost every k in the case 1 < p < 2 (this is a non-linear version of the Hausdorff-Young inequality). In fact they proved a stronger statement, namely that for almost every k, the eigenfunctions x \mapsto u(k,x) are bounded (and converge asymptotically to plane waves \alpha_\pm(k) e^{ikx} + \beta_\pm(k) e^{-ikx} as x \to \infty). There is an analogue of the Born and WKB approximations for these eigenfunctions, which shows that the Christ-Kiselev result is the nonlinear analogue of a classical result of Menshov, Paley and Zygmund showing the conditional convergence of the Fourier integral \int_{-\infty}^\infty V(x) e^{-2ikx}\ dx for almost every k when V \in L^p({\Bbb R}) for some 1 < p < 2.
The analogue of the Menshov-Paley-Zygmund theorem at the endpoint p=2 is the celebrated theorem of Carleson on almost everywhere convergence of Fourier series of L^2 functions. (The claim fails for p > 2, as can be seen by investigating random Fourier series, though I don’t recall the reference for this fact.) The nonlinear version of this would assert that for square-integrable potentials V, the eigenfunctions x \mapsto u(k,x) are bounded for almost every k. This is the nonlinear Carleson theorem conjecture. Unfortunately, it cannot be established by multilinear series, because of a divergence in the trilinear term of the expansion; but other methods may succeed instead. For instance, the weaker statement that the coefficients a(k) and b(k) (defined by density) are well defined and finite almost everywhere for square-integrable V (which is a nonlinear analogue of Plancherel’s theorem that the Fourier transform can be defined by density on L^2({\Bbb R})) was essentially established by Deift and Killip, using a trace formula (a nonlinear analogue to Plancherel’s formula). Also, the “dyadic” or “function field” model of the conjecture is known, by a modification of Carleson’s original argument. But the general case still seems to require more tools; for instance, we still do not have a good nonlinear Littlewood-Paley theory (except in the dyadic case), which is preventing time-frequency type arguments from being extended directly to the nonlinear setting. |
13fca6c197f4a20b | domingo, 21 de marzo de 2010
Nearly Free Electron Approximation
Nearly Free Electron Approximation
The nearly-free electron model is a modification of the free-electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. This model, like the free-electron model, does not take into account electron-electron interactions; that is, the independent-electron approximation is still in effect.
As shown by Bloch's theorem, introducing a periodic potential into the Schrödinger equation results in a wave function of the form
\psi_{\bold{k}}(\bold{r}) = u_{\bold{k}}(\bold{r}) e^{i\bold{k}\cdot\bold{r}}
where the function u has the same periodicity as the lattice:
u_{\bold{k}}(\bold{r}) = u_{\bold{k}}(\bold{r}+\bold{T})
(where T is a lattice translation vector.)
A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:
(\lambda_{\bold{k}} - \epsilon)C_{\bold{k}} + \sum_{\bold{G}} U_{\bold{G}} C_{\bold{k}-\bold{G}}=0
\lambda_{\bold{k}} = \frac{\hbar^2 k^2}{2m}
and Ck and UG are the Fourier coefficients of the wavefunction ψ(r) and the potential energy U(r), respectively:
U(\bold{r}) = \sum_{\bold{G}} U_{\bold{G}} e^{i\bold{G}\cdot\bold{r}}
\psi(\bold{r}) = \sum_{\bold{k}} C_{\bold{k}} e^{i\bold{k}\cdot\bold{r}}
The vectors G are the reciprocal lattice vectors, and the discrete values of k are determined by the boundary conditions of the lattice under consideration.
In any perturbation analysis, one must consider the base case to which the perturbation is applied. Here, the base case is with U(x) = 0, and therefore all the Fourier coefficients of the potential are also zero. In this case the central equation reduces to the form
This identity means that for each k, one of the two following cases must hold:
1. C_{\bold{k}} = 0,
2. \lambda_{\bold{k}} = \epsilon
If the values of λk are non-degenerate, then the second case occurs for only one value of k, while for the rest, the Fourier expansion coefficient Ck must be zero. In this non-degenerate case, the standard free electron gas result is retrieved:
\psi_k \propto e^{i\bold{k}\cdot\bold{r}}
In the degenerate case, however, there will be a set of lattice vectors k1, ..., km with λ1 = ... = λm. When the energy ε is equal to this value of λ, there will be m independent plane wave solutions of which any linear combination is also a solution:
\psi \propto \sum_{i=1}^{m} A_i e^{i\bold{k}_i\cdot\bold{r}}
Non-degenerate and degenerate perturbation theory can be applied in these two cases to solve for the Fourier coefficients Ck of the wavefunction (correct to first order in U) and the energy eigenvalue (correct to second order in U). An important result of this derivation is that there is no first-order shift in the energy ε in the case of no degeneracy, while there is in the case of near-degeneracy, implying that the latter case is more important in this analysis. Particularly, at the Brillouin zone boundary (or, equivalently, at any point on a Bragg plane), one finds a two-fold energy degeneracy that results in a shift in energy given by:
\epsilon = \lambda_{\bold{k}} \pm |U_{\bold{k}}|
This energy gap between Brillouin zones is known as the band gap, with a magnitude of 2 | UK | .
The approximation resulting from the assumption that electrons in metals can be analysed using the kinetic theory of gases, without taking the periodic potential of the metal into account. This approximation gives a good qualitative account of some properties of metals, such as their electrical conductivity. At very low temperatures it is necessary to use quantum statistical mechanics rather than classical statistical mechanics. The free-electron approximation does not, however, give an adequate quantitative description of the properties of metals. It can be improved by the nearly free electron approximation, in which the periodic potential is treated as a perturbation on the free electrons.
Agustin Egui
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b724a4c2a3961955 | Open Access
Some Basic Difference Equations of Schrödinger Boundary Value Problems
Advances in Difference Equations20092009:569803
Received: 1 April 2009
Accepted: 28 August 2009
Published: 11 October 2009
We consider special basic difference equations which are related to discretizations of Schrödinger equations on time scales with special symmetry properties, namely, so-called basic discrete grids. These grids are of an adaptive grid type. Solving the boundary value problem of suitable Schrödinger equations on these grids leads to completely new and unexpected analytic properties of the underlying function spaces. Some of them are presented in this work.
Moment ProblemJacobi OperatorDouble SequenceAdaptive GridPiecewise Continuous Function
1. Introduction
It is well known that solving Schrödinger's equation is a prominent -boundary value problem. In this article, we want to become familiar with some of the dynamic equations that arise in context of solving the Schrödinger equation on a suitable time scale where the expression time scale is in the context of this article related to the spatial variables.
The Schrödinger equation is the partial differential equation
where the function : yields information on the corresponding physically relevant potential. The solutions of the Schrödinger equation play a probabilistic role, being modeled by suitable -functions. For the convenience of the reader, let us first cite some of the fundamental facts on Schrödinger's equation. To do so, let us denote by all complex-valued functions which are defined on and which are twice differentiable in each of their variables.
Definition 1.1.
Let be twice partially differentiable in its three variables. Let moreover be a piecewise continuous function, denoting the space of piecewise continuous functions with values in . The linear map , given by
is called Schrödinger Operator in .
The following lemma makes a statement on the separation ansatz of the conventional Schrödinger partial differential equation where we throughout the sequel assume
Lemma 1.2 (Separation Ansatz).
Let the Schrödinger equation (1.1) be given, fulfilling the assertions of Definition 1.1 where In addition, the function will have the property
where are continuous. Let now be such that there exist eigenfunctions with
Then the function , given by
is a solution to Schrödinger's equation (1.1), revealing a completely separated structure of the variables.
A fascinating topic which has led to the results to be presented in this article is discretizing the Schrödinger equation on particular suitable time scales. This might be of importance for applications and numerical investigations of the underlying eigenvalue and spectral problems. Let us therefore restrict to the purely discrete case, that is, we are going to focus on a so-called basic discrete quadratic grid resp. on its closure which is a special time scale with fascinating symmetry properties.
Definition 1.3.
Let as well as . The set
denotes the basic discrete quadratic grid where . For , we abbreviate and . Define the set
as well as the set by
The boundary conditions on the functions we need for the discretized version of Schrödinger's equation are then given by the requirement
where the scalar product for two suitable functions will be specified by
In this context, we assume that for all . By construction, it is clear that is a Hilbert space over as it is a weighted sequence space, one of its orthogonal bases being given by all functions which are specified by with and .
Already now, we can say that the separation ansatz for the discretized Schrödinger equation will lead us to looking for eigensolutions of a given Schrödinger operator in the threefold product space .
Hence we come to the conclusion that in case of the separation ansatz for the Schrödinger equation, the following rationale applies:
The solutions of a Schrödinger equation on a basic discrete quadratic grid are directly related to the spectral behavior of the Jacobi operators acting in the underlying weighted sequence spaces.
Before presenting discrete versions of the Schrödinger equation on basic quadratic grids, let us first come back to the situation of Lemma 1.2 where we now assume that the potential is given by the requirement
Hence, it is sufficient to look at the one-dimensional Schrödinger equation
For the convenience of the reader, let us refer to the following fact. let the sequence of functions be recursively given by the requirement
where We then have for and moreover
This result reflects the celebrated so-called Ladder Operator Formalism. We first review a main result in discrete Schrödinger theory that is a basic analog of the just described continuous situation. Let us therefore state in a next step some more useful tools for the discrete description.
Definition 1.4.
Let and let be a nonempty closed set with the properties
Let for any the right-shift resp. left-shift operation be fixed through
respectively, the right-hand resp. left-hand basic difference operation will for any function be given by
Let moreover and let
where the positive even function is chosen as a solution to the basic difference equation
The creation operation resp.annihilation operation are then introduced by their actions on any as follows:
We refer to the discrete Schrödinger equation with an oscillator potential on by
The following result reveals that the discrete Schrödinger equation with an oscillator potential on shows similar properties than its classical analog does.
Lemma 1.5.
Let the function be specified like in Definition 1.4, satisfying the basic difference equation (1.19) on with . Let moreover . For the functions , given by (while ) are well defined in and solve the basic Schrödinger equation (1.21) in the following sense:
These relations apply for and where one set . Moreover, the corresponding moments of the orthogonality measure for the polynomials , arising from (1.19), are given by
The proof for the lemma is straightforward and obeys the techniques in [1].
The following central question concerning the functions spaces behind the Schrödinger equation (1.21) is open and shall be partially attacked in the sequel.
1.1. Central Problem
What are the relations between the linear span of all functions arising from Lemma 1.5 and the function space ?
In contrast to the fact that the corresponding question in the Schrödinger differential equation scenario is very well understood, the basic discrete scenario reveals much more structure which is going to be presented throughout the sequel of this article.
All the stated questions are closely connected to solutions of the equation
which originated in context of basic discrete ladder operator formalisms. We are going to investigate the rich analytic structure of its solutions in Section 2 and are going to exploit new facts on the corresponding moment problem in Section 3 of this article.
Let us remark finally that we will—throughout the presentation of our results in this article—repeatedly make use of the suffix basic. The meaning of it will always be related to the basic discrete grids that we have introduced so far.
The following results will shed some new light on function spaces which are behind basic difference equations. They are not only of interest to applications in mathematical physics but their functional analytic impact will speak for itself. The results altogether show that solving the boundary value problems of Schrödinger equations on time scales (that have the structure of adaptive grids) is a wide new research area. A lot of work still has to be invested into this direction.
For more physically related references on the topic, we invite the interested reader to consider also the work in [25].
For the more mathematical context, see, for instance, [1, 612].
2. Completeness and Lack of Completeness
In the sequel, we will make use of the basic discrete grid:
and we will consider the Hilbert space
Theorem 2.1.
Let and as well as an even positive solution of
on the basic discrete grid . Let the sequences of functions be given by shifted versions of the -function as follows:
The finite linear complex span of precisely all the functions and is then dense in .
Let be a positive and even solution to
One can easily show that an -solution with these properties uniquely exists, up to a positive factor, moreover all the functions defined by (2.4) are well defined in . Let us refer by the sequence to all the orthonormal polynomials which arise from the Gram-Schmidt procedure with respect to the function . They satisfy a three-term recurrence relation
where for the coefficients may be determined by standard methods through the moments resulting from (2.5). From the basic difference equation (2.5) we may also conclude that the polynomials are subject to an indeterminate moment problem, we come back to this in Section 3.
For and the functions given by may now be normalized, let us denote their norms by where is running in . Let us for moreover denote the normalized versions of the functions by
The following observation is essential. the -linear finite span of all functions given by
is the same than the -linear finite span of all functions specified by
As a consequence of (2.6), we conclude that the functions fulfill the recurrence relation:
How ever (2.9) can be brought into the standard form which is of relevance for considering the corresponding Jacobi operator,
where the coefficients are given by
The representation (2.10) results from the fact that the functions constitute a system of orthonormal functions and due to the fact that , acting as a multiplication operator, requires to be a formally symmetric linear operator on the finite linear span of the orthonormal system . Let us now consider the Hilbert space:
As for the definition range of in , let us choose as a densely defined linear operator in where we assume that
Let the expansion for a possible eigenvector of the adjoint be written as , the eigenvalue equation being . Note that the type of moment problem behind is related to the situation that has deficiency indices This also implies that any constitutes an eigenvalue of , hence the point spectrum of is . According to the deficiency index structure of the operator , let us now choose the particular self-adjoint extension of which allows a prescribed real-eigenvalue The corresponding situation for the eigensolution may be written as
where the sequence converges to 0 in the sense of the canonical -norm.
The element is in the finite linear space of all functions . Applying the powers of the shift operator (being given by for any function ) to (2.13) now leads to the fact that we can construct all eigenfunctions of the operator belonging to , as a consequence of
Note that we have used in (2.14) the commutation behavior which is satisfied for any fixed and in addition the fact that the sequence again converges to 0 in the sense of the canonical -norm for any . An analogous result is obtained in the case when we start with the eigenvalue .
Summing up the stated facts, we see that the self-adjoint operator , interpreted now as the multiplication operator, acting on a dense domain in , has precisely the point spectrum in the sense of
the functions with norm 1 being fixed by
Let us recall what we had stated at the beginning: the -linear finite span of all functions
is the same than the -linear finite span of all functions:
Taking the observations together, we conclude that the -linear span of all functions
is dense in the original Hilbert space .
We finally want to show that the -linear span of precisely all the functions in (2.19) is dense in . This can be seen as follows. taking away one of the functions or would already remove the completeness of the smaller Hilbert space . According to the property that the functions from (2.19) are dense in , it follows, for instance, that for any , there exists a double sequence such that in the sense of the canonically induced -norm:
Suppose that there exists a specific such that for all .
Let us consider the following expression where and :
The last expression now may be rewritten as
Successive application of to (2.21) resp. (2.22) with resp. shows that the existence of such a specific for all would finally imply that . This however would lead to a contradiction. Therefore, it becomes apparent that the complex finite linear span of precisely all the functions resp. (where is running in ) is dense in the Hilbert space . Summing up all facts, the basis property stated in the theorem finally follows according to (2.15).
Let us now focus on the following situation to move on towards the second main result of this article.
let be a positive symmetric polynomial, that is,
Definition 2.2.
Let with finite moments. Then
is called the real polynomial hull of .
Theorem 2.3.
Let and moreover , , . Then is not dense in .
For the th moment of can be calculated from the prerequisites of Theorem 2.3, namely;
where —written in short:
Discretization and integration on the basic grid gives
and, if is changed into on the left-hand side,
Define for :
Then, we have
If two densities generate the same moments then the induced orthogonal polynomials are the same. this is an isometry situation. According to the constructions of the two different types of moments, namely, on the one, hand side, the moments of type and on the other hand the moments of type and by comparing (2.26) and (2.30), we see that here, the mentioned isometry situation is matched—provided the initial conditions for the respective moments are chosen in the same way.
We use this observation now to proceed with the conclusions.
Let us make use again of the lattice
We define the restriction of on by
and we will use again the Hilbert space
Then the discrete analog of is
In order to show that is not dense in , we will construct a linear operator being bounded in the space but unbounded when restricted to .
Let us start from a function given by
and define generally for :
We denote from now on the respective multiplication operator again by and use to see that for a suitable operator-valued function the following holds:
Using , choose such that obeys:
that is, we choose .
Note that for the definition of , we need to consider the characterization of the corresponding measure.
Now, rewrite this as
Therefore, there exist such that
represent the eigenfunctions of (not necessarily orthogonal since was not required to be symmetric) and are the eigenvalues. However since the eigenvalues are unbounded, this implies that is an unbounded operator. Let us choose its domain as the algebraic span of the occurring eigenfunctions.
Now consider
Define the operators by
For the operator has the same eigenvectors as and we receive
Therefore, it follows with :
is a bounded operator on since as . Let us state that
that is, the domain of may be chosen as the entire span of . Therefore, is a bounded operator in the space of .
For topological reasons, it follows that
is also a bounded operator.
Now consider
In particular,
holds. Further, we have
Defining for to rewrite this as
and defining for we obtain
Applying to the results in
as . Thus is defined on any and generates infinitely many "rods" on the left-hand side of going toward 0. Therefore, is well defined on any and, therefore, it is well defined on any finite linear combination of the .
By hypothesis, is dense in . Then for, there exists a sequence in which approximates to any degree of accuracy in the sense of the -norm.
Now the question arises: looking at all is there a lower bound for ?
The are pairwise orthogonal, that is, from
it follows that
and therefore we have
as . It follows that is an unbounded operator, a contradiction. Therefore, is not dense in , implying that is not dense in .
However note that the result on the lack of completeness stated in the previous theorem should not be confusing with the fact that pointwise convergence may occur as the following theorem reveals.
Theorem 2.4.
Let be a differentiable positive even solution to
Let moreover be the continuous solutions to the recurrence relation
with initial conditions for all . The closure of the finite linear span of all these continuous functions is a Hilbert space For any element in the finite linear span of the conventional (continuous) Hermite functions, there exists a sequence which converges pointwise to .
According to the assertions of the theorem, the inverse of the function , given by
is differentiable and fulfills the basic difference equation
The function , being extended to the whole complex plane, can be interpreted as a holomorphic function due to its growth behavior, in particular, it allows a product expansion in the whole complex plane,
the sequence of complex numbers being uniquely fixed, denoting a multiplicative constant. Hence, the function given by
is also holomorphic. Inserting the corresponding power series:
we end up with the statement
Any monomial may be written as
with a double sequence of uniquely fixed real numbers . The polynomial functions of degree will be chosen such that
for but . Polynomials which fulfill these properties are, for instance, those fixed by (2.60), see also [1]. We may rewrite (2.66) as follows:
Generalizing this result, we see that there exists a threefold sequence of real numbers such that the classical continuous Hermite functions, given by
have the following representation
We recall that the closure of the finite linear complex span of all functions is a Hilbert space, call it , which is a proper subspace of hence being not dense in —see Theorem 2.3.
For , let us now consider the sequences of functions , given by
According to what we have shown it follows that each of the functions converges pointwise to the functions given by
3. Basic Difference Equations and Moment Problems
Let us make first some more general remarks on the special type of polynomials (2.60) we are considering. In literature, see, for instance, the internet reference to the Koekoek-Swarttouw online report on orthogonal polynomials there are listed two types of deformed discrete generalizations of the classical conventional Hermite polynomials, namely, the discrete basic Hermite polynomials of type I and the discrete basic Hermite polynomials of type II. These polynomials appear in the mentioned internet report under citations 3.28 and 3.29. Both types of polynomials, specified under the two respective citations by the symbol while is a nonnegative integer, can be succesively transformed (scaling the argument and renormalizing the coefficients) into the one and same form which is given by
with initial conditions for all . Note that is chosen as a fixed positive real number. Here, the number may range in the set of all positive real numbers, without the number 1—the case being reserved for the classical conventional Hermite polynomials. Depending on the choice of , the two different types of discrete basic Hermite polynomials can be found. The case corresponds to the discrete basic Hermite polynomials of type II, the case corresponds to the discrete basic Hermite polynomials of type I. Up to the late 1990 years, the perception was that both type of discrete basic Hermite polynomials have only discrete orthogonality measures. This is certainly true in the case of since the existence of such an orthogonality measure was shown explicitly and since the moment problem behind the discrete basic Hermite polynomials of type I is uniquely determined.
However, it could be shown that beside the known discrete orthogonality measure, specified in the aforementioned internet report, the discrete basic Hermite polynomials of type II, hence being connected to (3.1) with allow also orthogonality measures with continuous support.
Let us look at this phenomenon in some more detail.
It is known as a conventional result that a symmetric orthogonality measure with discrete support for the polynomials (3.1) with , yields moments being given by
In [1], it was shown that there exist continuous and piecewise continuous solutions to the difference equation
leading to the same moments (3.2). Such a behavior of the discrete basic Hermite polynomials of type II, hence being related to the scenario (3.1) with , was quite unexpected. Vice versa, once moments with nonnegative integer of a given weight function are given through (3.2), it can immediately be said that the weight function provides an orthogonality measure for the discrete basic Hermite polynomials of type II, related to (3.1) with .
The question however remains whether all weight functions for the discrete basic Hermite polynomials of type II, being related to (3.1) with must fulfill a basic difference equation of type (3.3). We develop now an answer to this question which goes beyond the results known so far.
Let throughout the sequel and We first put forward the following definition.
Definition 3.1.
By the moments of a given orthogonality measure, we understand—like in the previous sections—the numbers
Let us now proceed to the main result of this section.
Theorem 3.2.
There exists a positive symmetric -solution to the basic difference equation
not being a solution to
but generating the same moments and therefore yielding an orthogonality measure to the polynomials from (2.60) in the following sense:
The proof to establish will be a step beyond the already known orthogonality results for the polynomials under consideration.
Let us consider first the special basic difference equations:
Obviously, any positive -solution of (3.8) satisfies (3.9). Moreover, one can show that these -solutions of (3.8) are in . Therefore, the set of positive symmetric solutions to (3.9) which are in is nonempty. Let now be such that
In the sequel, we are going to use the moments
Remember that for as was assumed to act symmetrically on the real axis. Equation (3.10) may therefore now be rewritten—in terms of the numbers —as
Let now be a positive symmetric solution to (3.8). Then, similar integration like in (3.10) shows that the corresponding moments
indeed satisfy (3.12), in particular they obey
Hence, provides an orthogonality measure to the discrete basic Hermite polynomials under consideration—see [1]. The main issue to address now is the following. we want to show that there are positive symmetric functions such that satisfies (3.9) but not (3.8) and such that the moments, given by (3.11) and (3.12), satisfy
In other words, we have to prove that there exist orthogonality measures to the discrete basic Hermite polynomials which stem from a solution to (3.9) but not from a solution to (3.8).
We proceed in a constructive way.
Let us denote first by the -linear subspace of such that all are in the kernel of the map , given by
Let moreover be the -linear subspace of containing all the functions which are in the kernel of the linear map , given by
We will also make use of the -linear subspace which we choose as the maximal common domain on which the following two linear functionals are well defined:
It is easy to see that is continuous. to verify this, we consider the expression in the sense of the -norm for any . We directly obtain
But as is assumed to be an element of and hence of , we may rewrite the expression in the denominator and give the following estimate (with a positive constant ):
Hence is a bounded linear map and therefore continuous. In the same way, we show that is continuous.
We now continue as follows.
Using the terminology of characteristic functions, we first look at
It is possible to choose continuous even -functions being in and fulfilling
Let now with . The function given by
obeys by construction always but never as we have chosen such that and as vanish by construction on .
We now choose an intermediate value argumentation. According to the continuity of resp. , we can choose parameters with such that fulfills the moment property
Let now be a positive and even function such that . Note that we have in particular as well as and . The function (3.23) is by construction also in . According to the construction of the function , we now may choose sufficiently large such that the positive continuous even function
finally fulfills the required properties, namely,
as well as the moment conditions
From (3.27) and by integrating
now it follows for that the moments satisfy (3.15) and therefore also (3.12). In particular, the function yields therefore an orthogonality measure to the discrete basic Hermite polynomials, given by (2.60). Note that by construction, the function now fulfills all the assertions of Theorem 3.2. Hence, Theorem 3.2 holds in total.
Authors’ Affiliations
Center for Applied Mathematics and Theoretical Physics (CAMTP), University of Maribor, Maribor, Slovenia
Department of Mathematics, Technische Universität München, Garching, Germany
Department of Mathematics, Baylor University, Waco, USA
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I’ve just uploaded to the arXiv my paper “The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equation“, submitted to Analysis & PDE. This paper concerns an under-explored limit for the Cauchy problem
\displaystyle -\phi_{tt} + \phi_{xx} = |\phi|^{p-1} \phi; \quad \phi(0,x) = \phi_0(x); \quad \phi_t(0,x) = \phi_1(x) (1)
to the one-dimensional defocusing nonlinear wave equation, where \phi: {\Bbb R} \times {\Bbb R} \to {\Bbb R} is the unknown scalar field, p > 1 is an exponent, and \phi_0, \phi_1: {\Bbb R} \to {\Bbb R} are the initial position and velocity respectively, and the t and x subscripts denote differentiation in time and space. To avoid some (extremely minor) technical difficulties let us assume that p is an odd integer, so that the nonlinearity is smooth; then standard energy methods, relying in particular on the conserved energy
\displaystyle E(\phi)(t) = \int_{\Bbb R} \frac{1}{2} |\phi_t(t,x)|^2 + \frac{1}{2} |\phi_x(t,x)|^2 + \frac{1}{p+1} |\phi(t,x)|^{p+1}\ dx, (2)
on finite speed of propagation, and on the one-dimensional Sobolev embedding H^1({\Bbb R}) \subset L^\infty({\Bbb R}), show that from any smooth initial data \phi_0, \phi_1, there is a unique global smooth solution \phi to the Cauchy problem (1).
It is then natural to ask how the solution \phi behaves under various asymptotic limits. Popular limits for these sorts of PDE include the asymptotic time limit t \to \pm \infty, the non-relativistic limit c \to \infty (where we insert suitable powers of c into various terms in (1)), the small dispersion limit (where we place a small factor in front of the dispersive term +\phi_{xx}), the high-frequency limit (where we send the frequency of the initial data \phi_0, \phi_1 to infinity), and so forth.
Tristan Roy recently posed to me a different type of limit, which to the best of my knowledge has not been explored much in the literature (although some of the literature on limits of the Ginzburg-Landau equation has a somewhat similar flavour): the high exponent limit p \to \infty (holding the initial data \phi_0, \phi_1 fixed). From (1) it is intuitively plausible that as p increases, the nonlinearity gets “stronger” when |\phi| > 1 and “weaker” when |\phi| < 1; the “limiting equation”
\displaystyle -\phi_{tt} + \phi_{xx} = |\phi|^{\infty} \phi; \quad \phi(0,x) = \phi_0(x); \quad \phi_t(0,x) = \phi_1(x) (3)
would then be expected to be linear when |\phi| < 1 and infinitely repulsive when |\phi| > 1 (i.e. in the limit, the solution should be confined to range in the interval [-1,1], much as is the case with linear wave and Schrödinger equations with an infinite barrier potential; though with the key difference that the nonlinear barrier in (3) is confining the range of \phi rather than the domain.).
Of course, the equation (3) does not make rigorous sense as written; we need to formalise what an “infinite nonlinear barrier” is, and how the wave \phi will react to that barrier (e.g. will it reflect off of it, or be absorbed?). So the questions are to find the correct description of the limiting equation, and to rigorously demonstrate that solutions to (1) converge in some sense to that equation.
It is natural to require that \phi_0 stays away from the barrier, in the sense that |\phi_0(x)| < 1 for all x; in particular this implies that the energy (2) stays (locally) bounded as p \to \infty; it also ensures that (1) converges in a satisfactory sense to the free wave equation for sufficiently short times. For technical reasons we also have to make a mild assumption that either of the null energy densities \phi_1 \pm \partial_x \phi_0 vanish on a set with at most finitely many connected components. The main result is then that as p \to \infty, the solution \phi = \phi^{(p)} to (1) converges locally uniformly to a Lipschitz, piecewise smooth limit \phi = \phi^{(\infty)}, which is restricted to take values in [-1,1], with -\phi_{tt}+\phi_{xx} (interpreted in a weak sense) being a negative measure supported on \{ \phi=+1\} plus a positive measure supported on \{\phi = -1\}. Furthermore, we have the reflection conditions
\displaystyle (\partial_t \pm \partial_x) |\phi_t \mp \phi_x| = 0.
It turns out that the above conditions uniquely determine \phi, and one can even solve for \phi explicitly for any given data; such solutions start off smooth but pick up an increasing number of (Lipschitz continuous) singularities over time as they reflect back and forth across the nonlinear barriers \{\phi=+1\} and \{\phi=-1\}. (An explicit example of such a reflection is given in the paper.)
[The above conditions vaguely resemble entropy conditions, as appear for instance in kinetic formulations of conservation laws, though I do not know of a precise connection in this regard.]
In the remainder of this post I would like to describe the strategy of proof and one of the key a priori bounds needed. I also want to point out the connection to Liouville’s equation, which was discussed in the previous post.
Read the rest of this entry » |
35bce3f7c91cef35 | Nonadiabatic Molecular Dynamics in Three Different Flavors
February 26, 2018 to March 2, 2018
Location : CECAM-HQ-EPFL, Lausanne, Switzerland
EPFL on iPhone
Visa requirements
• Basile Curchod (Durham University, United Kingdom)
• Ivano Tavernelli (IBM-Zurich Research, Switzerland)
• Graham A. Worth (University College London, United Kingdom)
• Todd Martinez (Stanford University, USA)
The main purpose of this school is to teach the participants different methods for performing excited-state molecular dynamics.
The first day of the school will be devoted to a general introduction on nonadiabatic molecular dynamics and potential energy surfaces.
Each of the three following days will discuss a particular nonadiabatic method, from a theoretical and a practical perspective, via dedicated lectures in the morning and tutorials on the computer during the afternoon. The three techniques that will be introduced during this school are Multi Configuration Time Dependent Hartree (MCTDH), Ab Initio Multiple Spawning (AIMS), and Trajectory Surface Hopping (TSH). TSH, AIMS, and MCTDH are currently the most popular nonadiabatic dynamics strategies for molecular applications. Furthermore, these three techniques form a hierarchy, from the most accurate quantum dynamics (MCTDH), passing by the approximate yet rigorous trajectory-guided technique AIMS, down to the mixed quantum/classical algorithm TSH.
This school will offer a unique opportunity to learn these methods in parallel, allowing the participants to gain a clear understanding of their differences, but also of their complementarity.
1. General introduction to excited-state dynamics. (Day 1)
a. Time-dependent Schrödinger equation
b. Representations and Ansätze for the time-dependent molecular wavefunction
c. Born-Oppenheimer approximation and beyond
2. Concept of potential energy surfaces. (Day 1)
a. Potential energy surfaces and conical intersections
b. Potential energy fitting procedures
3. Electronic structure properties required for nonadiabatic dynamics. (Day 1)
a. Electronic structure methods for excited states
b. Forces and nonadiabatic couplings
c. On-the-fly dynamics
4. MCTDH and its Gaussian-based versions. (Day 2)
5. Full and Ab Initio Multiple Spawning. (Day 3)
6. Mixed quantum/classical methods and Trajectory Surface Hopping. (Day 4) |
8389f85019c02de5 | Hermitian and Non-Hermitian quantum optics in integrated optical waveguides
The paraxial wave Helmholtz equation describing the propagation of classical electromagnetic waves close to the optical axis is formally equivalent to the Schrödinger equation of quantum mechanics in two spatial dimensions. This concept has long been exploited to emulate certain aspects of quantum physics using light confined in waveguides, where the refractive index contrast plays the role of the potential in the Schrödinger equation. This has become even more interesting as a complex refractive index material allow to emulate non-Hermitian evolutions.
Our aim is to elevate this correspondence between classical electromagnetism and the Schrödinger equation to the realm of quantum optics, where we showed a straightforward translation of concepts of PT-symmetric evolution with loss and gain not to be possible [1]. However, we could show the first Hong-Ou-Mandel quantum interference of photons in a passive PT-symmetric setting [2]. In order to efficiently model the quantum evolution in these open systems, we are developing analytical tools to solve quantum master equations using group-theoretical methods [3]. First applications include Floquet-PT systems [4] that show a strong reduction of the necessary loss for driving a PT-phase transition.
Integrated optical waveguides are highly suitable for implementing adiabatic quantum evolutions by generalizing STIRAP protocols for the evolution in degenerate dark subspaces. We have shown the first non-Abelian geometric quantum gate using an Abelian gauge field [5], and presented an optimal construction using the quantum metric. The concept of holonomic gates can be transferred to non-Hermitian systems [6] which becomes interesting in the framework of non-orthogonal waveguide modes. We also investigate highly degenerate structures for use in holonomic computation [7].
Rydberg physics with semiconductor excitons
Rydberg atoms that have been excited to a quantum state with large principal quantum number n possess some extraordinary properties. For example, their exaggerated size, growing as n2, implies large dipole moments that also grow as n2 as well as dipole polarisabilities with an n7 scaling. Rydberg physics was so far limited to excitations in atomic systems. In collaboration with colleagues from the TU Dortmund, we have recently shown that Rydberg excitations can also be observed in semiconductor systems [1]. Here, the material in which the excitons were formed was a natural cuprous oxide (Cu2O) crystal. Rydberg excitons with principal quantum numbers up to n=25 have been observed in transmission spectroscopy (see Figure) which corresponds to an extension of the exciton wavefunction of more than one micrometer.
Similar to atoms, the binding energies of Rydberg excitons deviate systematically from the hydrogenic Rydberg series. This deviation can be cast into the form of a quantum defect which we showed to derive from the nonparabolicity of the valence bands [2,3]. We have shown further that the proximity of the Rydberg exciton resonances provides a route to detect coherent phenomena in semiconductor Rydberg excitons already in the absorption spectra [4]. There, additional resonances appear in the absorption spectrum that correspond to dressed states consisting of two Rydberg exciton levels coupled to the excitonic vacuum, forming a V-type three-level system, but driven only by one laser light source] (see Figure). Our result is based on the solution of the valence band Hamiltonian associated with the cubic crystal symmetry of Cu2O that corresponds to a modified Schrödinger equation in momentum space.
In crossed electric and magnetic fields, the potential landscape may be altered to provide additional quasi-bound states with giant dipoles [5], where we computed the eigenenergies of these giant-dipole excitons [6]. Stray electric fields on the other hand, such as those produced by charged impurities, leads to the vanishing of Rydberg resonances into an apparent continuum [7]. The appearance of additional absorption lines due to the broken rotational symmetry, together with spatially inhomogeneous Stark shifts, leads to a modification of the observed line shapes that agrees qualitatively with the changes observed in the experiment.
Additional potentials that modify the excitonic spectra include strain that could be used to trap Rydberg excitons [8] (see Figure).
Similar to atoms, dipole-dipole interactions dominate the overall interaction at the large distances relevant under experimental conditions of Rydberg exciton blockade. We have evaluated the long-range interactions between pairs of Rydberg excitons in Cu2O, which are due to direct Coulomb forces rather than short-range collisions typically considered for ground-state excitons [9], and found signatures of Rydberg blockade in pump-probe spectra [10]. For future experiments involving transitions between different exciton series in Cu2O, we have studied the infrared optical transitions between excitons of the yellow, green and blue series in Cu2O [11]. We showed that in many cases the dipole approximation is inadequate and, in particular, that it breaks down in yellow-blue transitions even for moderate principal quantum numbers.
For an introduction to Rydberg physics, we would like to refer you to Quantum Kate (en).
[1] T. Kazimierczuk, D. Fröhlich, S. Scheel, H. Stolz, and M. Bayer, Giant Rydberg excitons in the copper oxide Cu2O, Nature 514, 343 (2014)
[2] F. Schöne, S.-O. Krüger, P. Grünwald, H. Stolz, S. Scheel, M. Aßmann, J. Heckötter, J. Thewes, D. Fröhlich, M. Bayer, Deviations of the exciton level spectrum in Cu2O from the hydrogen series, Phys. Rev. B 93, 075203, (2016)
[3] F. Schöne, S.-O. Krüger, P. Grünwald, M. Aßmann, J. Heckötter, J. Thewes, H. Stolz, D. Fröhlich, M. Bayer and S. Scheel, Coupled valence band dispersions and the quantum defect of excitons in Cu2O, J. Phys. B: At. Mol. Opt. Phys. 49, 134003, (2016)
[4] P. Grünwald, M. Aßmann, J. Heckötter, D. Fröhlich, M. Bayer, H. Stolz, S. Scheel, Signatures of Quantum Coherences in Rydberg Excitons, Phys. Rev. Lett. 117, 133003 (2016)
[5] M. Kurz, P. Grünwald, S. Scheel, Excitonic giant-dipole potentials in cuprous oxide, Phys. Rev. B 95, 245205 (2017)
[6] M. Kurz, S. Scheel, Eigenenergies of excitonic giant-dipole states in cuprous oxide, Phys. Rev. B 99, 075205 (2019)
[7] S.O. Krüger, H. Stolz, S. Scheel, Interaction of charged impurities and Rydberg excitons in cuprous oxide, Phys. Rev. B 101, 235204 (2020)
[8] S.O. Krüger, S. Scheel, Waveguides for Rydberg excitons in Cu2O from strain traps, Phys. Rev. B 97, 205208 (2018)
[9] V. Walther, S.O. Krüger, S. Scheel, T.Pohl, Interactions between Rydberg excitons in Cu2O, Phys. Rev. B 98, 165201 (2018)
[10] J. Heckötter, V. Walther, S. Scheel, M. Bayer, T. Pohl, M. Aßmann, Asymmetric Rydberg blockade of giant excitons in Cuprous Oxide, preprint - Oktober 2020 - arXiv:2010.15459
[11] S.O. Krüger and S. Scheel, Interseries transitions between Rydberg excitons in Cu2O, Phys. Rev. B 100, 085201 (2019)
Machine learning methods for image reconstructions
Single-shot x-ray imaging of short-lived nanostructures such as clusters and nanoparticles near a phase transition or non-crystalizing objects such as large proteins and viruses is currently the most elegant method for characterizing their structure. Wide-angle scattering using XUV or soft x-rays provides three-dimensional structural information in a single shot and has opened routes towards the characterization of non-reproducible objects in the gas phase. The retrieval of the structural information contained in wide-angle scattering images is highly non-trivial, and currently no efficient rigorous algorithm is known. We have shown that deep learning networks, trained with simulated scattering data, allow for fast and accurate reconstruction of shape and orientation of nanoparticles from experimental images [1]. The gain in speed compared to conventional retrieval techniques opens the route for automated structure reconstruction algorithms capable of real-time discrimination and pre-identification of nanostructures in scattering experiments with high repetition rate, thus representing the enabling technology for fast femtosecond nanocrystallography.
Recently, we have shown how a physics-informed deep neural network can be used to reconstruct complete three-dimensional object models on a voxel grid from single two-dimensional wide-angle scattering patterns [2]. We have demonstrated its universal reconstruction capabilities for silver nanoclusters, where the network uncovers novel geometric structures that reproduce the experimental scattering data with very high precision.
Dispersion forces
Dispersion forces such as Casimir forces between bodies, Casimir-Polder forces between atoms and bodies and van der Waals forces between atoms are effective electromagnetic forces that arise as consequences of correlated ground-state fluctuations. We are investigating dispersion forces in and out of thermal equilibrium [1,2,3] that are particularly relevant for long-wavelength atomic transitions as found in Rydberg atoms [4,5], and study universal scaling laws [6] and friction forces [7,8].
Dispersion forces occur in a variety of different contexts such as molecular interferometry where they influence the interference pattern of large molecules [9,10], and in diffraction processes of atom clouds off periodic surface potential landscapes [11]. Near nanofibers, we have shown that the Casimir-Polder force on an atom can be laterally tuned by choosing the appropriate atomic state for preparation [12], and the confinement of atoms inside a hollow-core fiber can be used to tune the van der Waals interaction between them [13].
[1] S.Y. Buhmann and S. Scheel, Thermal Casimir versus Casimir-Polder Forces: Equilibrium and Nonequilibrium Forces, Phys. Rev. Lett. 100, 253201 (2008)
[2] S.A. Ellingsen, S.Y. Buhmann, and S. Scheel, Dynamics of thermal Casimir-Polder forces on polar molecules, Phys. Rev. A 79, 052903 (2009)
[3] S.A. Ellingsen, S.Y. Buhmann, and S. Scheel, Temperature-Independent Casimir-Polder Forces Despite Large Thermal Photon Numbers, Phys. Rev. Lett. 104, 223003 (2010)
[4] J.A. Crosse et al., Thermal Casimir-Polder shifts in Rydberg atoms near metallic surfaces, Phys. Rev. A 82, 010901(R) (2010)
[5] S. Ribeiro, S.Y. Buhmann, T. Stielow, and S. Scheel, Casimir-Polder interaction from exact diagonalization and surface-induced state mixing, EPL 110, 51003 (2015)
[6] S.Y. Buhmann, S. Scheel, and J.R. Babington, Universal Scaling Laws for Dispersion Interactions, Phys. Rev. Lett. 104, 070404 (2010)
[7] S. Scheel and S.Y. Buhmann, Casimir-Polder forces on moving atoms, Phys. Rev. A 80, 042902 (2009)
[8] F. Intravaia et al., Friction forces on atoms after acceleration, J. Phys.: Condensed Matter 27, 214020 (2015)
[9] C. Brand et al., A Green’s function approach to modeling molecular diffraction
in the limit of ultra-thin gratings
, Ann. d. Phys. 527, 580 (2015)
[10] J. Fiedler and S. Scheel, Casimir-Polder potentials on extended molecules, Ann. d. Phys. 527, 570 (2015)
[11] H. Bender et al., Probing Atom-Surface Interactions by Diffraction of Bose-Einstein Condensates, Phys. Rev. X 4, 011029 (2014)
[12] S. Scheel, S.Y. Buhmann, C. Clausen, and P. Schneeweiss, Directional spontaneous emission and lateral Casimir-Polder force on an atom close to a nanofiber, Phys. Rev. A 92, 043819 (2015)
[13] H.R. Haakh and S. Scheel, Modified and controllable dispersion interaction in a one-dimensional waveguide geometry, Phys. Rev. A 91, 052707 (2015)
QED in linear and nonlinear dielectric materials
More recently, we have been able to extend this quantization scheme to include nonlinearly responding, absorbing dielectrics [4,5,6]. This enables us to study the effect of nonlinear absorption mechanisms in a quantum-mechanically consistent way. In the context of nonlocal or even nonreciprocal media, we have shown that the principle of macroscopic duality can still be upheld [7,8].
Quantum tomographic reconstruction with error bars
Ultracold trapped neutral atoms, atom chips
|
e4e53d6bf9cac760 | PHYS 256: Computational Methods in Physics I
Limits of computation; Introduction to numerical methods—Functions and roots, Approximation, Interpolation, Systems of linear equations, Least squares, Numerical differentiation and integration, Finite differences; Realistic projectile motion;...
PHYS 248: Introduction to Physics of Materials
Forces between atoms and molecules and their consequences; Elastic modulae – Young's, Shear, Bulk; Poisson ratio, non-elastic behaviour; Flow properties of fluids; Continuity equation, hydrostatic equation, Euler's and Bernoulli's equations...
PHYS 246: Nuclear Physics I
Radioactivity, nuclear radiation; Detection of nuclear radiation; Structure and properties of the nucleus; binding energy and nuclear forces; Fission and fusion; Applications of radioactivity – Dating, radiology, radiotherapy, analysis.
PHYS 245: Electromagnetism I
Electric field and potential gradient; Gauss's law and it's applications; electric field around conductors; Dielectric medium: polar and non-polar molecules, electric polarization and bound charges; Displacement vector; Gauss's Law in dielectrics...
PHYS 244: Mathematical Methods I
Calculus of functions of several variables, partial differentiation, total differential, Euler's theorem on homogeneous functions; Constrained and unconstrained extrema, multiple integrals; Jacobian; Scalar and vector fields; Line, surface and volume...
PHYS 241: Atomic Physics and Quantum Phenomena
Quantum Phenomena
Blackbody radiation and Planck’s hypothesis, photons and electromagnetic waves, photo-electric effect, Compton Effect, double-slit experiment, wave properties of particles, uncertainty principle, Schrödinger equation, particle in a...
PHYS 206: Practical Physics IV
Laboratory experiments illustrating modern experimental techniques and error analysis
PHYS 205: Practical Physics III
Laboratory experiments illustrating modern experimental techniques and error analysis.
PHYS 144: Electricity and Magnetism
Electric Charge and Electric Field: Electric charge, Conductors, insulators and induced charges, Coulomb's law, Electric field and Electric forces, Charge distributions, Electric dipoles
Gauss’ Law: Charge and electric flux,...
PHYS 143: Mechanics and Thermal Physics
Properties of Vectors: Geometrical representation, multiplication (dot product and cross product), the three-dimensional Cartesian co-ordinate system, Components of a vector, Direction Cosines, Linear Independence, Magnitude of a vector,... |
0535e9d96bb92f55 | We gratefully acknowledge support from
the Simons Foundation and member institutions.
Authors and titles for Jun 2015
[ total of 3097 entries: 1-25 | 26-50 | 51-75 | 76-100 | ... | 3076-3097 ]
[ showing 25 entries per page: fewer | more ]
[1] arXiv:1506.00009 [pdf, ps, other]
Title: On oscillation of solutions of linear differential equations
Comments: 13 pages
Journal-ref: J. Geom. Anal. 27 (2017), no. 1, 868-885
[2] arXiv:1506.00011 [pdf, other]
Title: Group Symmetries of Complementary Code Matrices
Comments: 14 pages, 1 figure
Subjects: Information Theory (cs.IT)
[3] arXiv:1506.00014 [pdf, other]
Title: Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators
Subjects: Numerical Analysis (math.NA)
[4] arXiv:1506.00015 [pdf, ps, other]
Title: Groups with exactly two supercharacter theories
Comments: 7 pages Added work that shows that ${\rm Sp} (6,2)$ has only two supercharacter theories
Subjects: Group Theory (math.GR)
[5] arXiv:1506.00016 [pdf, ps, other]
Title: The Escalator Boxcar Train method for a system of aged-structured equations
Subjects: Analysis of PDEs (math.AP)
[6] arXiv:1506.00017 [pdf, other]
Title: New computer-based search strategies for extreme functions of the Gomory--Johnson infinite group problem
Comments: 54 pages, many figures
Subjects: Optimization and Control (math.OC)
[7] arXiv:1506.00018 [pdf, other]
Title: Conley-Morse-Forman theory for combinatorial multivector fields
Authors: Marian Mrozek
Subjects: Dynamical Systems (math.DS); Algebraic Topology (math.AT); Combinatorics (math.CO); Numerical Analysis (math.NA)
[8] arXiv:1506.00023 [pdf, ps, other]
Title: The Fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave
Comments: To appear in SIADS
Subjects: Analysis of PDEs (math.AP)
[9] arXiv:1506.00025 [pdf, ps, other]
Title: The Dubovitskiĭ-Sard Theorem in Sobolev Spaces
Subjects: Classical Analysis and ODEs (math.CA)
[10] arXiv:1506.00028 [pdf, ps, other]
Title: Coincidence indices of sublattices and coincidences of colorings
Comments: 15 pages, 1 Figure
Journal-ref: Z. Kristallogr. (2015). 230(12), 749-759
Subjects: Metric Geometry (math.MG); Combinatorics (math.CO)
[11] arXiv:1506.00033 [pdf, other]
Title: Stable homology of surface diffeomorphism groups made discrete
Authors: Sam Nariman
Comments: 34 pages. Final submitted version, to appear in Geometry and Topology
Journal-ref: Geom. Topol. 21 (2017) 3047-3092
[12] arXiv:1506.00034 [pdf, ps, other]
Title: Bracketing numbers of convex and $m$-monotone functions on polytopes
Authors: Charles R. Doss
Comments: 42 pages
[13] arXiv:1506.00046 [pdf, other]
Title: Ray-Knight representation of flows of branching processes with competition by pruning of Lévy trees
Comments: 50 pages, 1 figure. This version accepted in Probability Theory and Related Fields
Subjects: Probability (math.PR)
[14] arXiv:1506.00048 [pdf, ps, other]
Title: The Role of the Jacobi Identity in Solving the Maurer-Cartan Structure Equation
Authors: Ori Yudilevich
Comments: 16 pages. Minor Corrections and additions. Final version accepted for publication in Pacific Journal of Mathematics
Journal-ref: Pacific J. Math. 282 (2016) 487-510
[15] arXiv:1506.00050 [pdf, ps, other]
Title: A note on the zeroth products of Frenkel-Jing operators
Authors: Slaven Kozic
Comments: 19 pages, 3 figures
Journal-ref: J. Algebra Appl. Vol. 16, No. 3 (2017) 1750053 (25 pages)
Subjects: Quantum Algebra (math.QA)
[16] arXiv:1506.00056 [pdf, ps, other]
Title: Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence
Comments: 29 pages, 8 figures
Journal-ref: Nonlinear Differential Equations and Applications NoDEA 23 (2016), 1-34
Subjects: Analysis of PDEs (math.AP)
[17] arXiv:1506.00057 [pdf, ps, other]
Title: Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
Subjects: Dynamical Systems (math.DS)
[18] arXiv:1506.00061 [pdf, ps, other]
Title: Quadratic Equation over Associative D-Algebra
Authors: Aleks Kleyn
Comments: English text - 34 pages; Russian text - 35 pages
Subjects: General Mathematics (math.GM)
[19] arXiv:1506.00062 [pdf, ps, other]
Title: On the Convergence of Alternating Least Squares Optimisation in Tensor Format Representations
Comments: arXiv admin note: text overlap with arXiv:1503.05431
Subjects: Numerical Analysis (math.NA)
[20] arXiv:1506.00066 [pdf, other]
Title: Hiding Information in Noise: Fundamental Limits of Covert Wireless Communication
Comments: 6 pages, 4 figures
Journal-ref: IEEE Communications Magazine 53.12 (2015)
Subjects: Information Theory (cs.IT)
[21] arXiv:1506.00067 [pdf, ps, other]
Title: Topological dynamics of the doubling map with asymmetrical holes
Comments: 33 pages
Subjects: Dynamical Systems (math.DS)
[22] arXiv:1506.00071 [pdf, ps, other]
Title: Homology and closure properties of autostackable groups
Comments: 20 pages
Subjects: Group Theory (math.GR); Formal Languages and Automata Theory (cs.FL)
[23] arXiv:1506.00072 [pdf, ps, other]
Title: Singular integrals, rank one perturbations and Clark model in general situation
Comments: 36 pages. Lecture notes
Journal-ref: Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2). Association for Women in Mathematics Series, vol. 5. Springer International Publishing, 2017
[24] arXiv:1506.00078 [pdf, other]
Title: Sufficient Lie Algebraic Conditions for Sampled-Data Feedback Stabilization
Comments: This article draws heavily from arXiv:1407.8380v2
Subjects: Optimization and Control (math.OC)
[25] arXiv:1506.00084 [pdf, ps, other]
Title: Foundations of topological racks and quandles
Comments: Dedicated to Professor J\'ozef H. Przytycki for his 60th birthday
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Quantum Algebra (math.QA); Rings and Algebras (math.RA)
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Edward Fredkin
Lila Gatlin
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GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
A. O. Gomes
Brian Goodwin
Joshua Greene
Jacques Hadamard
Mark Hadley
Patrick Haggard
J. B. S. Haldane
Stuart Hameroff
Augustin Hamon
Sam Harris
Hyman Hartman
John-Dylan Haynes
Donald Hebb
Martin Heisenberg
Werner Heisenberg
John Herschel
Art Hobson
Jesper Hoffmeyer
E. T. Jaynes
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Roman Jakobson
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
William R. Klemm
Christof Koch
Simon Kochen
Hans Kornhuber
Stephen Kosslyn
Ladislav Kovàč
Leopold Kronecker
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Joseph LeDoux
Benjamin Libet
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
James Clerk Maxwell
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John McCarthy
Warren McCulloch
George Miller
Stanley Miller
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Jacques Monod
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Alexander Oparin
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Wojciech Zurek
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Free Will
Mental Causation
James Symposium
The "decoherence program" of H. Dieter Zeh, Erich Joos, Wojciech Zurek, John Wheeler, Max Tegmark, and others has multiple aims -
1. to show how classical physics emerges from quantum physics. They call this the "quantum to classical transition."
2. to explain the lack of macroscopic superpositions of quantum states (e.g., Schrödinger's Cat as a superposition of live and dead cats).
3. in particular, to identify the mechanism that suppresses ("decoheres") interference between states as something involving the "environment" beyond the system and measuring apparatus.
4. to explain the appearance of particles following paths (they say there are no "particles," and maybe no paths).
5. to explain the appearance of discontinuous transitions between quantum states (there are no "quantum jumps" either)
6. to champion a "universal wave function" (as a superposition of states) that evolves in a "unitary" fashion (i.e., deterministically) according to the Schrödinger equation.
7. to clarify and perhaps solve the measurement problem, which they define as the lack of macroscopic superpositions.
8. to explain the "arrow of time."
9. to revise the foundations of quantum mechanics by changing some of its assumptions, notably challenging the "collapse" of the wave function or "projection postulate."
Decoherence theorists say that they add no new elements to quantum mechanics (such as "hidden variables") but they do deny one of the three basic assumptions - namely Dirac's projection postulate. This is the method used to calculate the probabilities of various outcomes, which probabilities are confirmed to several significant figures by the statistics of large numbers of identically prepared experiments.
They accept (even overemphasize) Dirac's principle of superposition. Some also accept the axiom of measurement, although some of them question the link between eigenstates and eigenvalues.
The decoherence program hopes to offer insights into several other important phenomena:
1. What Zurek calls the "einselection" (environment-induced superselection) of preferred states (the so-called "pointer states") in a measurement apparatus.
2. The role of the observer in quantum measurements.
3. Nonlocality and quantum entanglement (which is used to "derive" decoherence).
4. The origin of irreversibility (by "continuous monitoring").
5. The approach to thermal equilibrium.
The decoherence program finds unacceptable these aspects of the standard quantum theory:
1. Quantum "jumps" between energy eigenstates.
2. The "apparent" collapse of the wave function.
3. In particular, explanation of the collapse as a "mere" increase of information.
4. The "appearance" of "particles."
5. The "inconsistent" Copenhagen Interpretation - quantum "system," classical "apparatus."
6. The "insufficient" Ehrenfest Theorems.
Decoherence theorists admit that some problems remain to be addressed:
1. The "problem of outcomes." Without the collapse postulate, it is not clear how definite outcomes are to be explained.
As Tegmark and Wheeler put it:
The main motivation for introducing the notion of wave-function collapse had been to explain why experiments produced specific outcomes and not strange superpositions of is embarrassing that nobody has provided a testable deterministic equation specifying precisely when the mysterious collapse is supposed to occur.
Some of the controversial positions in decoherence theory, including the denial of collapses and particles, come straight from the work of Erwin Schrödinger, for example in his 1952 essays "Are There Quantum Jumps?" (Part I and Part II), where he denies the existence of "particles," claiming that everything can be understood as waves.
Other sources include: Hugh Everett III and his "relative state" or "many world" interpretations of quantum mechanics; Eugene Wigner's article on the problem of measurement; and John Bell's reprise of Schrödinger's arguments on quantum jumps.
Decoherence advocates therefore look to other attempts to formulate quantum mechanics. Also called "interpretations," these are more often reformulations, with different basic assumptions about the foundations of quantum mechanics. Most begin from the "universal" applicability of the unitary time evolution that results from the Schrödinger wave equation. They include:
• The DeBroglie-Bohm "pilot-wave" or "hidden variables" formulation.
• The Everett-DeWitt "relative-state" or "many worlds" formulation.
• The Ghirardi-Rimini-Weber "spontaneous collapse" formulation.
Note that these "interpretations" are often in serious conflict with one another. Where Erwin Schrödinger thinks that waves alone can explain everything (there are no particles in his theory), David Bohm thinks that particles not only exist but that every particle has a definite position that is a "hidden parameter" of his theory. H. Dieter Zeh, the founder of decoherence, sees
one of two possibilities: a modification of the Schrödinger equation that explicitly describes a collapse (also called "spontaneous localization") or an Everett type interpretation, in which all measurement outcomes are assumed to exist in one formal superposition, but to be perceived separately as a consequence of their dynamical autonomy resulting from decoherence.
It was John Bell who called Everett's many-worlds picture "extravagant,"
While this latter suggestion has been called "extravagant" (as it requires myriads of co-existing quasi-classical "worlds"), it is similar in principle to the conventional (though nontrivial) assumption, made tacitly in all classical descriptions of observation, that consciousness is localized in certain semi-stable and sufficiently complex subsystems (such as human brains or parts thereof) of a much larger external world. Occam's razor, often applied to the "other worlds", is a dangerous instrument: philosophers of the past used it to deny the existence of the interior of stars or of the back side of the moon, for example. So it appears worth mentioning at this point that environmental decoherence, derived by tracing out unobserved variables from a universal wave function, readily describes precisely the apparently observed "quantum jumps" or "collapse events."
The Information Interpretation of quantum mechanics also has explanations for the measurement problem, the arrow of time, and the emergence of adequately, i.e., statistically determined classical objects. However, I-Phi does it while accepting the standard assumptions of orthodox quantum physics. See below.
We briefly review the standard theory of quantum mechanics and compare it to the "decoherence program," with a focus on the details of the measurement process. We divide measurement into several distinct steps, in order to clarify the supposed "measurement problem" (mostly the lack of macroscopic state superpositions) and perhaps "solve" it.
The most famous example of probability-amplitude-wave interference is the two-slit experiment. Interference is between the probability amplitudes whose absolute value squared gives us the probability of finding the particle at various locations behind the screen with the two slits in it.
Finding the particle at a specific location is said to be a "measurement."
In standard quantum theory, a measurement is made when the quantum system is "projected" or "collapsed" or "reduced" into a single one of the system's allowed states. If the system was "prepared" in one of these "eigenstates," then the measurement will find it in that state with probability one (that is, with certainty).
However, if the system is prepared in an arbitrary state ψa, it can be represented as being in a linear combination of the system's basic energy states φn.
ψa = Σ cn | n >.
cn = < ψa | φn >.
It is said to be in "superposition" of those basic states. The probability Pn of its being found in state φn is
Pn = < ψa | φn >2 = cn2 .
Between measurements, the time evolution of a quantum system in such a superposition of states is described by a unitary transformation U (t, t0) that preserves the same superposition of states as long as the system does not interact with another system, such as a measuring apparatus. As long as the quantum system is completely isolated from any external influences, it evolves continuously and deterministically in an exactly predictable (causal) manner.
Whenever the quantum system does interact however, with another particle or an external field, its behavior ceases to be causal and it evolves discontinuously and indeterministically. This acausal behavior is uniquely quantum mechanical. Nothing like it is possible in classical mechanics. Most attempts to "reinterpret" or "reformulate" quantum mechanics are attempts to eliminate this discontinuous acausal behavior and replace it with a deterministic process.
We must clarify what we mean by "the quantum system" and "it evolves" in the previous two paragraphs. This brings us to the mysterious notion of "wave-particle duality." In the wave picture, the "quantum system" refers to the deterministic time evolution of the complex probability amplitude or quantum state vector ψa, according to the "equation of motion" for the probability amplitude wave ψa, which is the Schrödinger equation,
δψa/δt = H ψa.
The probability amplitude looks like a wave and the Schrödinger equation is a wave equation. But the wave is an abstract quantity whose absolute square is the probability of finding a quantum particle somewhere. It is distinctly not the particle, whose exact position is unknowable while the quantum system is evolving deterministically. It is the probability amplitude wave that interferes with itself. Particles, as such, never interfere (although they may collide).
Note that we never "see" the superposition of particles in distinct states. There is no microscopic superposition in the sense of the macroscopic superposition of live and dead cats (See Schrödinger's Cat).
When the particle interacts, with the measurement apparatus for example, we always find the whole particle. It suddenly appears. For example, an electron "jumps" from one orbit to another, absorbing or emitting a discrete amount of energy (a photon). When a photon or electron is fired at the two slits, its appearance at the photographic plate is sudden and discontinuous. The probability wave instantaneously becomes concentrated at the location of the particle.
There is now unit probability (certainty) that the particle is located where we find it to be. This is described as the "collapse" of the wave function. Where the probability amplitude might have evolved under the unitary transformation of the Schrödinger equation to have significant non-zero values in a very large volume of phase space, all that probability suddenly "collapses" (faster than the speed of light, which deeply bothered Albert Einstein) to the location of the particle.
Einstein said that some mysterious "spooky action-at-a-distance" must act to prevent the appearance of a second particle at a distant point where a finite probability of appearing had existed just an instant earlier.
Animation of a wave function collapsing - click to restart
Whereas the abstract probability amplitude moves continuously and deterministically throughout space, the concrete particle moves discontinuously and indeterministically to a particular point in space.
For this collapse to be a "measurement," the new information about which location (or state) the system has collapsed into must be recorded somewhere in order for it to be "observable" by a scientist. But the vast majority of quantum events - e.g., particle collisions that change the particular states of quantum particles before and after the collision - do not leave an indelible record of their new states anywhere (except implicitly in the particles themselves).
We can imagine that a quantum system initially in state ψa has interacted with another system and as a result is in a new state φn, without any macroscopic apparatus around to record this new state for a "conscious observer."
H. D. Zeh describes how quantum systems may be "measured" without the recording of information.
It is therefore a plausible experimental result that the interference disappears also when the passage [of an electron through a slit] is "measured" without registration of a definite result. The latter may be assumed to have become a "classical fact" as soon as the measurement has irreversibly "occurred". A quantum phenomenon may thus "become a phenomenon" without being observed. This is in contrast to Heisenberg's remark about a trajectory coming into being by its observation, or a wave function describing "human knowledge". Bohr later spoke of objective irreversible events occurring in the counter. However, what precisely is an irreversible quantum event? According to Bohr this event can not be dynamically analyzed.
Analysis within the quantum mechanical formalism demonstrates nonetheless that the essential condition for this "decoherence" is that complete information about the passage is carried away in some objective physical form. This means that the state of the environment is now quantum correlated (entangled) with the relevant property of the system (such as a passage through a specific slit). This need not happen in a controllable way (as in a measurement): the "information" may as well form uncontrollable "noise", or anything else that is part of reality. In contrast to statistical correlations, quantum correlations characterize real (though nonlocal) quantum states - not any lack of information. In particular, they may describe individual physical properties, such as the non-additive total angular momentum J2 of a composite system at any distance.
The Measurement Process
In order to clarify the measurement process, we separate it into several distinct stages, as follows:
• A particle collides with another microscopic particle or with a macroscopic object (which might be a measuring apparatus).
• In this scattering problem, we ignore the internal details of the collision and say that the incoming initial state ψa has changed asymptotically (discontinuously, and randomly = wave-function collapse) into the new outgoing final state φn.
• [Note that if we prepare a very large number of identical initial states ψa, the fraction of those ending up in the final state φn is just the probability < ψa | φn >2]
• The information that the system was in state ψa has been lost (its path information has been erased; it is now "noise," as Zeh describes it). New information exists (implicitly in the particle, if not stored anywhere else) that the particle is in state φn.
• If the collision is with a large enough (macroscopic) apparatus, it might be capable of recording the new system state information, by changing the quantum state of the apparatus into a "pointer state" correlated with the new system state.
"Pointers" could include the precipitated silver-bromide molecules of a photographic emulsion, the condensed vapor of a Wilson cloud chamber, or the cascaded discharge of a particle detector.
• But this new information will not be indelibly recorded unless the recording apparatus can transfer entropy away from the apparatus greater than the negative entropy equivalent of the new information (to satisfy the second law of thermodynamics). This is the second requirement in every two-step creation of new information in the universe.
• The new information could be useful (it is negative entropy) to an information processing system, for example, a biological cell like a brain neuron.
The collision of a sodium ion (Na+) with a sodium/potassium pump (an ion channel) in the cell wall could result in the sodium ion being transported outside the cell, resetting conditions for the next firing of the neuron's action potential, for example.
• The new information could be meaningful to an information processing agent who could not only observe it but understand it. Now neurons would fire in the mind of the conscious observer that John von Neumann and Eugene Wigner thought was necessary for the measurement process to occur at all.
Von Neumann (perhaps influenced by the mystical thoughts of Neils Bohr about mind and body as examples of his "complementarity.") saw three levels in a measurement;
1. the system to be observed, including light up to the retina of the observer.
2. the observer's retina, nerve tracts, and brain
3. the observer's abstract "ego."
• John Bell asked tongue-in-cheek whether no wave function could collapse until a scientist with a Ph.D. was there to observe it. He drew a famous diagram of what he called von Neumann's "shifty split."
Bell shows that one could place the arbitrary "cut" (Heisenberg called it the "Schnitt") at various levels without making any difference.
But an "objective" observer-independent measurement process ends when irreversible new information has been indelibly recorded (in the photographic plate of Bell's drawing).
Von Neumann's physical and mental levels are better discussed as the mind-body problem, not the measurement problem.
The Measurement Problem
So what exactly is the "measurement problem?"
For decoherence theorists, the unitary transformation of the Schrödinger equation cannot alter a superposition of microscopic states. Why then, when microscopic states are time evolved into macroscopic ones, don't macroscopic superpositions emerge? According to H. D. Zeh:
Because of the dynamical superposition principle, an initial superposition
Σ cn | n > does not lead to definite pointer positions (with their empirically observed frequencies). If decoherence is neglected, one obtains their entangled superposition Σ cn | n > | Φn >, that is, a state that is different from all potential measurement outcomes.
And according to Erich Joos, another founder of decoherence:
It remains unexplained why macro-objects come only in narrow wave packets, even though the superposition principle allows far more "nonclassical" states (while micro-objects are usually found in energy eigenstates). Measurement-like processes would necessarily produce nonclassical macroscopic states as a consequence of the unitary Schrödinger dynamics. An example is the infamous Schrödinger cat, steered into a superposition of "alive" and "dead".
The fact that we don't see superpositions of macroscopic objects is the "measurement problem," according to Zeh and Joos.
An additional problem is that decoherence is a completely unitary process (Schrödinger dynamics) which implies time reversibility. What then do decoherence theorists see as the origin of irreversibility? Can we time reverse the decoherence process and see the quantum-to-classical transition reverse itself and recover the original coherent quantum world?
To "relocalize" the superposition of the original system, we need only have complete control over the environmental interaction. This is of course not practical, just as Ludwig Boltzmann found in the case of Josef Loschmidt's reversibility objection.
Does irreversibility in decoherence have the same rationale - "not possible for all practical purposes" - as in classical statistical mechanics?
According to more conventional thinkers, the measurement problem is the failure of the standard quantum mechanical formalism (Schrödinger equation) to completely describe the nonunitary "collapse" process. Since the collapse is irreducibly indeterministic, the time of the collapse is completely unpredictable and unknowable. Indeterministic quantum jumps are one of the defining characteristics of quantum mechanics, both the "old" quantum theory, where Bohr wanted radiation to be emitted and absorbed discontinuously when his atom jumpped between staionary states, and the modern standard theory with the Born-Jordan-Heisenberg-Dirac "projection postulate."
To add new terms to the Schrödinger equation in order to control the time of collapse is to misunderstand the irreducible chance at the heart of quantum mechanics, as first seen clearly, in 1917, by Albert Einstein. When he derived his A and B coefficients for the emission and absorption of radiation, he found that an outgoing light particle must impart momentum hν/c to the atom or molecule, but the direction of the momentum can not be predicted! Neither can the theory predict the time when the light quantum will be emitted.
But the inability to predict both the time and direction of light particle emissions, said Einstein in 1917, is "a weakness in the theory..., that it leaves time and direction of elementary processes to chance (Zufall, ibid.)." It is only a weakness for Einstein, of course, because his God does not play dice. Decoherence theorists too appear to have what William James called an "antipathy to chance."
In the original "old" quantum mechanics, Neils Bohr made two assumptions. One was that atoms could only be found in what he called stationary energy states, later called eigenstates. The second was that the observed spectral lines were discontinuous sudden transitions of the atom between the states. The emission or absorption of quanta of light with energy equal to the energy difference between the states (or energy levels) with frequency ν was given by the formula
E2 - E1 = h ν,
where h is Planck's constant, derived from his radiation law that quantized the allowed values of energy.
In the now standard quantum theory, formulated by Werner Heisenberg, Max Born, Pascual Jordan, Erwin Schrödinger, Paul Dirac, and others, three foundational assumptions were made: the principle of superposition, the axiom of measurement, and the projection postulate. Since decoherence challenges some of these ideas, we review the standard definitions.
The Principle of Superposition
The fundamental equation of motion in quantum mechanics is Schrödinger's famous wave equation that describes the evolution in time of his wave function ψ,
i δψ/δt - Hψ.
For a single particle in idealized complete isolation, and for a Hamiltonian H that does not involve magnetic fields, the Schrödinger equation is a unitary transformation that is time-reversible (the principle of microscopic reversibility)
Max Born interpreted the square of the absolute value of Schrödinger's wave function as providing the probability of fi nding a quantum system in a certain state ψn.
The quantum (discrete) nature of physical systems results from there generally being a large number of solutions ψn (called eigenfunctions) of the Schrödinger equation in its time independent form, with energy eigenvalues En.
Hψn = Enψn,
The discrete energy eigenvalues En limit interactions (for example, with photons) to the energy di fferences En - Em, as assumed by Bohr. Eigenfunctions ψn are orthogonal to one another,
< ψn | ψm > = δnm,
where δnm is the Dirac delta-function, equal to 1 when n = m, and 0 otherwise. The sum of the diagonal terms in the matrix < ψn | ψm >, when n = m, must be normalized to 1 to be meaningful as Born rule probabilities.
Σ Pn = Σ < ψn | ψn >2 = 1.
The off-diagonal terms in the matrix, < ψn | ψm >, are interpretable as interference terms. When the matrix is used to calculate the expectation values of some quantum mechanical operator O, the off-diagonal terms < ψn | O | ψm > are interpretable as transition probabilities - the likelihood that the operator O will induce a transition from state ψn to ψm.
The Schrödinger equation is a linear equation. It has no quadratic or higher power terms, and this introduces a profound - and for many scientists and philosophers a disturbing - feature of quantum mechanics, one that is impossible in classical physics, namely the principle of superposition of quantum states. If ψa and ψb are both solutions of the equation, then an arbitrary linear combination of these, ψ = caψa + cbψb; with complex coefficients ca and cb, is also a solution.
Together with Born's probabilistic interpretation of the wave function, the principle of superposition accounts for the major mysteries of quantum theory, some of which we hope to resolve, or at least reduce, with an objective (observer-independent) explanation of information creation during quantum processes (which can often be interpreted as measurements).
The Axiom of Measurement
The axiom of measurement depends on the idea of "observables," physical quantities that can be measured in experiments. A physical observable is represented as a Hermitean operator A that is self-adjoint (equal to its complex conjugate, A* = A). The diagonal elements
< ψn | A | ψn > of the operator's matrix are interpreted as giving the expectation value for An (when we make a measurement). The off -diagonal n, m elements describe the uniquely quantum property of interference between wave functions and provide a measure of the probabilities for transitions between states n and m.
It is these intrinsic quantum probabilities that provide the ultimate source of indeterminism, and consequently of irreducible irreversibility, as we shall see. The axiom of measurement is then that a large number of measurements of the observable A, known to have eigenvalues An, will result in the number of measurements with value An being proportional to the probability of finding the system in eigenstate ψn with eigenvalue An.
The Projection Postulate
The third novel idea of quantum theory is often considered the most radical. It has certainly produced some of the most radical ideas ever to appear in physics, in attempts to deny it (as the decoherence program appears to do, as do also Everett relative-state interpretations, many worlds theories, and Bohm-de Broglie pilot waves). The projection postulate is actually very simple, and arguably intuitive as well. It says that when a measurement is made, the system of interest will be found in one of the possible eigenstates of the measured observable.
We have several possible alternatives for eigenvalues. Measurement simply makes one of these actual, and it does so, said Max Born, in proportion to the absolute square of the probability amplitude wave function ψn. In this way, ontological chance enters physics, and it is partly this fact of quantum randomness that bothered Albert Einstein ("God does not play dice") and Schrödinger (whose equation of motion is deterministic).
When Einstein derived the expressions for the probabilities of emission and absorption of photons in 1917, he lamented that the theory seemed to indicate that the direction of an emitted photon was a matter of pure chance (Zufall), and that the time of emission was also statistical and random, just as Rutherford had found for the time of decay of a radioactive nucleus. Einstein called it a "weakness in the theory."
What Decoherence Gets Right
Allowing the environment to interact with a quantum system, for example by the scattering of low-energy thermal photons or high-energy cosmic rays, or by collisions with air molecules, surely will suppress quantum interference in an otherwise isolated experiment. But this is because large numbers of uncorrelated (incoherent) quantum events will "average out" and mask the quantum phenomena. It does not mean that wave functions are not collapsing. They are, at every particle interaction.
Decoherence advocates describe the environmental interaction as "monitoring" of the system by continuous "measurements."
Decoherence theorists are correct that every collision between particles entangles their wave functions, at least for the short time before decoherence suppresses any coherent interference effects of that entanglement.
But in what sense is a collision a "measurement." At best, it is a "pre-measurement."
It changes the information present in the wave functions before the collision. But the new information may not be recorded anywhere (other than being implicit in the state of the system).
All interactions change the state of a system of interest, but not all leave the "pointer state" of some measuring apparatus with new information about the state of the system.
So environmental monitoring, in the form of continuous collisions by other particles, is changing the specific information content of both the system, the environment, and a measuring apparatus (if there is one). But if there is no recording of new information (negative entropy created locally), the system and the environment may be in thermodynamic equilibrium.
Equilibrium does not mean that decoherence monitoring of every particle is not continuing.
It is. There is no such thing as a "closed system." Environmental interaction is always present.
If a gas of particles is not already in equilibrium, they may be approaching thermal equilibrium. This happens when any non-equilibrium initial conditions (Zeh calls these a "conspiracy") are being "forgotten" by erasure of path information during collisions. Information about initial conditions is implicit in the paths of all the particles. This means that, in principle, the paths could be reversed to return to the initial, lower entropy, conditions (Loschmidt paradox).
Erasure of path information could be caused by quantum particle-particle scattering (our standard view) or by decoherence "monitoring." How are these two related?
The Two Steps Needed in a Measurement that Creates New Information
More than the assumed collapse of the wave function (von Neumann's Process 1, Pauli's measurement of the first kind) is needed. Indelibly recorded information, available for "observations" by a scientist, must also satisfy the second requirement for the creation of new information in the universe.
Everything created since the origin of the universe over ten billion years ago has involved just two fundamental physical processes that combine to form the core of all creative processes. These two steps occur whenever even a single bit of new information is created and survives in the universe.
• Step 1: A quantum process - the "collapse of the wave function."
The formation of even a single bit of information that did not previously exist requires the equivalent of a "measurement." This "measurement" does not involve a "measurer," an experimenter or observer. It happens when the probabilistic wave function that describes the possible outcomes of a measurement "collapses" and a eigenstate of a matter or energy particle is actually changed.
If the probability amplitude wave function did not collapse, unitary evolution would simply preserve the initial information.
• Step 2: A thermodynamic process - local reduction, but cosmic increase, in the entropy.
The second law of thermodynamics requires that the overall cosmic entropy always increases. When new information is created locally in step 1, some energy (with positive entropy greater than the negative entropy of the new information) must be transferred away from the location of the new bits or they will be destroyed, if local thermodynamical equilibrium is restored. This can only happen in a locality where flows of matter and energy with low entropy are passing through, keeping it far from equilibrium.
The two physical processes in the creative process, quantum physics and thermodynamics, are somewhat daunting subjects for philosophers, and even for many scientists, including decoherence advocates.
Quantum Level Interactions Do Not Create Lasting Information
The overwhelming number of collisions of microscopic particles like electrons, photons, atoms, molecules, etc, do not result in observable information about the collisions. The lack of observations and observers does not mean that there have been no "collapses" of wave functions. The idea that the time evolution of the deterministic Schrödinger equation continues forever in a unitary transformation that leaves the wave function of the whole universe undecided and in principle reversible at any time, is an absurd and unjustified extrapolation from the behavior of the ideal case of a single perfectly isolated particle.
The principle of microscopic reversibility applies only to such an isolated particle, something unrealizable in nature, as the decoherence advocates know with their addition of environmental "monitoring." Experimental physicists can isolate systems from the environment enough to "see" the quantum interference (but again, only in the statistical results of large numbers of identical experiments).
The Emergence of the Classical World
In the standard quantum view, the emergence of macroscopic objects with classical behavior arises statistically for two reasons involving large numbers:
1. The law of large numbers (from probability and statistics)
• When a large number of material particles is aggregated, properties emerge that are not seen in individual microscopic particles. These properties include ponderable mass, solidity, classical laws of motion, gravity orbits, etc.
• When a large number of quanta of energy (photons) are aggregated, properties emerge that are not seen in individual light quanta. These properties include continuous radiation fields with wavelike interference.
2. The law of large quantum numbers (Bohr Correspondence Principle).
Decoherence as "Interpreted" by Standard Quantum Mechanics
Can we explain the following in terms of standard quantum mechanics?
1. the decoherence of quantum interference effects by the environment
2. the measurement problem, viz., the absence of macroscopic superpositions of states
3. the emergence of "classical" adequately determined macroscopic objects
4. the logical compatibility and consistency of two dynamical laws - the unitary transformation and the "collapse" of the wave function
5. the entanglement of "distant" particles and the appearance of "nonlocal" effects such as those in the Einstein-Podolsky-Rosen experiment
Let's consider these point by point.
1. The standard explanation for the decoherence of quantum interference effects by the environment is that when a quantum system interacts with the very large number of quantum systems in a macroscopic object, the averaging over independent phases cancels out (decoheres) coherent interference effects.
2. In order to study interference effects, a quantum system is isolated from the environment as much as possible. Even then, note that microscopic interference is never "seen" directly by an observer. It is inferred from probabilistic theories that explain the statistical results of many identical experiments. Individual particles are never "seen" as superpositions of particles in different states. When a particle is seen, it is always the whole particle and nothing but the particle. The absence of macroscopic superpositions of states, such as the infamous linear superposition of live and dead Schrödinger Cats, is therefore no surprise.
3. The standard quantum-mechanical explanation for the emergence of "classical" adequately determined macroscopic objects is that they result from a combination of a) Bohr's correspondence principle in the case of large quantum numbers. together with b) the familiar law of large numbers in probability theory, and c) the averaging over the phases described in point 1. Heisenberg indeterminacy relations still apply, but the individual particles' indeterminacies average out, and the remaining macroscopic indeterminacy is practically unmeasurable.
4. Perhaps the two dynamical laws would be inconsistent if applied to the same thing at exactly the same time. But the "collapse" of the wave function (von Neumann's Process 1, Pauli's measurement of the first kind) and the unitary transformation that describes the deterministic evolution of the probability amplitude wave function (von Neumann's Process 2) are used in a temporal sequence.
first a wave of possibilities, then an actual particle.
The first process describes what happens when quantum systems interact, in a collision or a measurement, when they become indeterministically entangled. The second then describes their deterministic evolution (while isolated) along their mean free paths to the next collision or interaction. One dynamical law applies to the particle picture, the other to the wave picture.
5. The paradoxical appearance of nonlocal "influences" of one particle on an entangled distant particle, at velocities greater than light speed, are a consequence of a poor understanding of both the wave and particle aspects of quantum systems. The confusion usually begins with a statement such as "consider a particle A here and a distant particle B there." When entangled in a two-particle probability amplitude wave function, the two identical particles are "neither here nor there," just as the single particle in a two-slit experiment does not "go through" the slits.
It is the single-particle probability amplitude wave that must "go through" both slits if it is to interfere. For a two-particle probability amplitude wave that starts its deterministic time evolution when the two identical particles are produced, it is only the probability of finding the particles that evolves according to the unitary transformation of the Schrödinger wave equation. It says nothing about where the particles "are."
Now if and when a particle is measured somewhere, we can then label it particle A. Conservation of energy and momentum tell us immediately that the other identical particle is now symmetrically located on the other side of the central source of particles. If the particles are electrons (as in David Bohm's version of EPR), conservation of spin tells us that the now distant particle B must have its spin opposite to that of particle A is they were produced with a total spin of zero.
Nothing is sent from particle A to B. The deduced properties are the consequence of conservation laws that are true for much deeper reasons than the puzzles of nonlocal entanglement. The mysterious instantaneous values for the properties is exactly the same mystery that bothered Einstein about a single-particle wave function having values all over a photographic screen at one instant, then having values only at the position of the located particle in the next instant, apparently violating special relativity.
Animation of a two-particle wave function collapsing - click to restart
Compare the collapse of the two-particle probability amplitude above to the single-particle collapse here.
To summarize: Decoherence by interactions with environment can be explained perfectly by multiple "collapses" of the probability amplitude wave function during interactions with environment particles. Microscopic interference is never "seen" directly by an observer, therefore we do not expect ever to "see" macroscopic superpositions of live and dead cats. The "transition from quantum to classical" systems is the consequence of laws of large numbers. The quantum dynamical laws necessarily include two phases, one needed to describe the continuous deterministic motions of probability amplitude waves and the other the discontinuous indeterministic motions of physical particles. The mysteries of nonlocality and entanglement are no different from those of standard quantum mechanics as seen in the two-slit experiment. It is just that we now have two identical particles and their wave functions are nonseparable .
For Teachers
The Role of Decoherence in Quantum Mechanics, Stanford Encyclopedia of Philosophy
For Scholars
Chapter 3.7 - The Ergod Chapter 4.2 - The History of the Knowledge Problem
Part Three - Value Part Five - Problems
Normal | Teacher | Scholar |
bc4f4ec3dfad913b | Structure of Schrödinger’s Nucleon: Elastic Form-Factors and Radii
Journal of Applied Mathematics and Physics
Vol.03 No.10(2015), Article ID:60807,8 pages
Structure of Schrödinger’s Nucleon: Elastic Form-Factors and Radii
Gintautas P. Kamuntavičius
Vytautas Magnus University, Kaunas, Lithuania
Email: g.kamuntavicius@gmf.vdu.lt
Copyright © 2015 by author and Scientific Research Publishing Inc.
Received 23 September 2015; accepted 27 October 2015; published 30 October 2015
The Galilei invariant model of the nucleon as a system of three point particles, whose dynamics is governed by Schrödinger equation, after six Hamiltonian parameters fitting, predicts magnetic momenta, masses and charge radii of the proton and neutron with experimental precision. Now this model is applied in order to investigate nucleon charge, mass and magnetism distributions. The obtained electric and magnetic form factors at low values of momentum transfer are in satisfactory agreement with experimental information. The model predicts that neutron is a more compact system than proton.
Solutions of Wave Equations (Bound States), Potential Models, Proton and Neutron
1. Introduction
Different and significant changes of quantum systems, composing more complex structures (for example, atoms forming molecules or solid state) are obvious. The well-known experiments of atomic nuclei structure also indicate that a nucleon embedded in a nucleus is slightly modified in comparison with a free one [1] -[3] . Therefore, for the investigation of this effect, the simple model is necessary, which is compatible with the technique of atomic nuclei description and is able to predict the changes of nucleon structure once it appears in vicinity of other nucleons.
The Schrödinger’s model of nucleon [4] is introduced namely for the solution of this problem. The model considers proton and neutron as different systems of three point particles (PP) in correspondence with the Standard Model recommendations: proton as the system of two up (uPP) and one down (dPP) particle, while neutron ―as a system of one uPP and two dPP particles. These particles should not be identified with the quarks of the Standard Model because only their spins, charges (+2e/3 and −e/3) and baryon numbers (1/3) match the respective quarks quantum numbers. Both PP of our model are different; thus isospin quantum number is not necessary. Therefore, the color quantum number, which is used in the Standard Model to antisymmetrize the wave function, is also unnecessary. In our model antisymmetry is ensured with a smaller number of wave-func- tion’s degrees of freedom. Baryon number is necessary to prevent the possibility of system excitation when one or two PP escapes to continuous spectrum. The PPs, composing the nucleon, allow defining the magnetic momenta of structureless particles in Dirac’s way. The interactions of different pairs of PP (uu, ud and dd) contain the Coulomb and three-dimensional harmonic oscillator (spring) potentials, having four free parameters. Together with PP masses the model Hamiltonian has six free parameters. The conditions for this model are as follows. Firstly, it has to be Galilei invariant. Secondly, finite ranges of potential wells are applied in order to avoid the appearance of nonexistent bound excited states of the nucleon. Finally, the number of parameters of nucleon Hamiltonian has to be equal with the number of nucleon characteristics, applied for fitting. These are the best known characteristics of the proton and neutron―masses, magnetic momenta and charge distribution radii. The values of experimental results, given in Particle Data Group 2014 report [5] and corresponding model results are presented in Table 1. It is shown that six parameters of model Hamiltonian can be chosen so that the mentioned characteristics of nucleon could be predicted with experimental precision.
Thus, the eigenfunctions of introduced Hamiltonian application for nucleon structure investigation are the next interesting problem. This paper is devoted for electric and magnetic elastic form-factors and corresponding radii description.
2. The Galilei Invariant Form Factor Operator
The elastic form factor of nucleon is defined as density operator’s Fourier image:
Here denotes the proton, stands for the neutron, while
is density operator in the nucleon’s center-of-mass reference frame (where is center-of-mass radius vector). are k-th particle characteristics, given in Table 2.
For charge density they equal the PP charge, for magnetization density―the PP magnetic momentum operator, defined in [4] , while for mass distribution density―the mass of the PP’s. For PP density distribution they equal 1/3, i.e. the baryon number of PP. The values of and coincide due to indistinguishability of these PP that is why we will use only one of them in the following expressions. Applying this density definition, the form factor operator equals the sum
Table 1. Nucleon characteristics, applied for Hamiltonian parameters fitting (experiment; data are from Ref. [5] ), and results of present model (theory).
Table 2. Parameters and operators, present in density definition, Equation (2).
It is well-known that the form factors, defined experimentally, are independent of angles of the momentum transfer, therefore this expression, before estimating the mean value, needs to be averaged by spherical angles. All exponents have a common dependence on, so taking into account the integral of spherical harmonics scalar product, equal
one obtains that
Here is the spherical Bessel function. After this operation the form factor operator equals:
Written in Jacobi coordinates
it takes the form
Here is sum of PP masses. The mean value of this form factor operator at low q values provides important information about the system. When
it equals the nucleon charge, magnetic momentum, mass and unity correspondingly. The form factor presentation in a dimensionless and normalized form is
so that its value in zero equals one. The only exception is the electric form factor of the neutron, which equals zero. In order for to be dimensionless it is modified dividing by the elementary charge e. The next member of form factor expression in vicinity of equals:
is proportional to the square of corresponding radius operator, i.e.:
Inserting the charges of PP’s one obtains the expressions presented in [4] for the squared charge radius operators of the proton and neutron.
3. Expectation Values of Form Factor Operator
Elastic form-factors of nucleon as functions of momentum transfer are defined as ratio of mean values of form-factor operator, Equation (9) applying wave functions’ superposition defined in [4] :
Here both basic functions are bound angular and spin momenta functions
where the parentheses indicate the operation of momenta binding, is angular momentum of the first Jacobi coordinate, is spin momentum of the first particle, indicates the angular momentum of the second Jacobi coordinate, for which according to [6] the spin momenta of the second and third particles need to be set. The are total momenta of respective Jacobian subsystems. Their sum equals the nucleon momentum, i.e. its spin. The radial wave functions, dependent on different Jacobi variables, present in front of superposition (15) are given in Equation (19) of [4] :
where is the depth of dimensionless Hamiltonian well, which width equals. is eigenvalue (here is angular momentum quantum number, while equals the number of eigenfunction
nodes)., while denotes a degenerate hypergeometric function [7] [8] . In
the area where a potential equals zero, radial function equals spherical Hankel function of imaginary argument [8] . The conditions of equality of both parts of the function and their logarithmic derivatives at point
defines the constant L and eigenvalue while the constant N is defined by the normalization
Integration of the first member of right side of Equation (9) is straightforward. For calculation of the second and third integrals one needs spherical Bessel function expansion [8] :
The sum of two functions of this kind present in (9) equals
Having in mind the structure of nucleon wave function, only two first terms of expansion, corresponding give nonzero contribution to the form factor integral.
Finally, after some angular momentum algebra, the form factor can be presented as
Here, the radial wave functions as functions of, having the length dimension, are proportional to the present above functions in a following way:
where are potential wells parameters, given in [4] . These wave functions normalization condition looks as
Obviously, the momentum transfer q in all given expressions has dimension fm−1. The widely accepted dimension of this momentum Q is GeV/c. Therefore, the slight modification is necessary due to these momenta dependence:
Here, in square brackets the dimensions of corresponding quantities are written. The value of conversion factor is defined in Ref. [5] .
Operators, present in right hand side of Equation (13), having in mind the definitions (14), (11) and (10) are necessary for radii calculation. Together with precise calculation, the values of radii can be determined from the slopes of corresponding form factors in the limit of zero momentum transfer.
4. Results
The obtained charge, magnetic, mass and point particles radii are present in the “theory” column of Table 3.
The known experimental values of corresponding radii are given in the “experiment” column. The charge radii of nucleon are applied for fitting, hence their values are equal to the ones, recommended by the Particle Data Group 2014 report [5] . The straightforward evaluation of magnetic radii of nucleon is problematic, hence only few estimates of proton and only one―for neutron magnetic radius are known. The evaluations for mass and point particles radii are absent in literature.
Table 3. Values of different radii of the proton and neutron.
In Figure 1 and Figure 2, the electric and magnetic form factors for the proton and together with best fits of corresponding experimental results, given in [16] as ratio of two polynomials
with parameters and, are present.
The neutron electric form factor and double-polarization data, taken from [17] , are present in Figure 3.
The comparison of calculated and given by standard dipole approximation
with the neutron magnetic form factor is present in Figure 4.
The all four obtained form factors demonstrate good enough comparison with known experimental data at low values of momentum transfer, that characterises the nucleons, present in an atomic nucleus. Moreover, ratio of electric and magnetic form factors of neutron at obtained in [18] is 0.250 (58). Corresponding ratio of our form factors equals 0.257.
The mass and point particles form factors for proton, and neutron, are not present, due to the absence of experimental information and due to the trivial dependence on momentum transfer, looking correspondingly like the proton and neutron magnetic form factors with slightly different slopes at origin.
5. Conclusions
The most interesting result of radii calculation is that the neutron appears as a more compact system than the proton, although the results of the magnetism distribution radii, presented in [5] , show different relations. The obtained compactness of the neutron nicely fits with the well-known fact that the surface of the heavy nuclei is
Figure 1. The electric form factor of the proton (solid line). The fit from Ref. [16] is shown for comparison (dashed line).
Figure 2. The magnetic form factor of the proton (solid line). The fit from Ref. [16] is shown for comparison (dashed line).
Figure 3. The electric form factor of the neutron (solid line). The double-polarization data from Ref. [17] .
Figure 4. The magnetic form factor of the neutron (solid line). The standard dipole form factor, Equation (29) (dashed line).
more well-defined than the one of the proton [19] . As follows from our investigation, the proton’s density decreases smoothly and is much more spread out through the sphere than the one of the neutron. There is a surplus of the neutrons in heavy nuclei which usually distribute on the surface of the nucleus and can determine the mentioned effect. Moreover, compact distribution of neutron constituents gives larger probability of weak process, producing its decay.
Therefore, the obtained precision of nucleon description allows concluding that the calculations of other characteristics of the proton and neutron with obtained wave function may give some interesting and rather reliable results. It is well known that realistic potentials of nucleon-nucleon interaction, carefully fitted with the two- nucleon data, give smaller than experimental nuclear binding energies. It looks like that the introduced model is able to give a chance for this problem solution. As it is known from the solid state theory, when the distance between two potential wells decreases, the isolated levels of each well convert to the system of two levels, one of them is more bound than the other one. If Pauli principle allows the constituents of nucleon to occupy the best bound level, this may help us to improve the description of atomic nuclei, taking into account the changes of nucleon structure when merging into groups.
Cite this paper
Gintautas P.Kamuntavičius, (2015) Structure of Schrödinger’s Nucleon: Elastic Form-Factors and Radii. Journal of Applied Mathematics and Physics,03,1352-1360. doi: 10.4236/jamp.2015.310162
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a7ce2d380ba25f0f | 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.
Fig. 1 Friedrich Hund
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).
Fig. 3 George Gamow
Gamow was able to make an accurate approximation that allowed him to express the decay rate in terms of an exponential term
where Zα 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.
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[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. |
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On this blog, we often discuss the collapse of the wavefunction as the result of a measurement. This phenomenon is called the “measurement problem.” There are several reasons, for which the collapse of the wavefunction—part and parcel of the Copenhagen interpretation of quantum mechanics—is considered a problem. Firstly, it does not follow from the Schrödinger equation, the main equation of quantum mechanics that describes the evolution of the wavefunction in time, and is added ad hoc. Second, nobody knows how the collapse happens or how long the wave function takes to collapse. This is not even to consider that any notion that the collapse of the wavefunction is caused by human consciousness, as proposed by Von Neumann, leading to Cartesian dualism, is anathema to physicists. It is a problem, no matter how you look at it. What is the solution? The most radical, but the most accepted solution is the many-worlds interpretation of quantum mechanics.
Hugh Everett
Proposed by Hugh Everett in 1957 (H. Everett, Review of Modern Physics, July 1957) and developed by Bryce de Witt, (B. S. DeWitt and N. Graham, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Univ. Press, 1974) the many-worlds interpretation of quantum mechanics is, perhaps, the most outlandish yet the cleanest interpretation of the Schrödinger equation. This theory suggests that every transition between quantum states splits the universe into multiple copies or “branches,” in which all of the possible states are realized.
Bryce de Witt
This approach, as weird as it sounds—Bryce DeWitt called it “schizophrenia with vengeance”—is actually the most straightforward interpretation of the mathematical formalism of quantum mechanics because it does not rely on an ad hoc collapse of the wavefunction. Recall that the Schrödinger equation does not describe the evolution of a physical system per se, but the evolution in time of the wavefunction describing the state of the system.
Max Born
Max Born interpreted the wavefunction as the measure of the probability of finding the system in a particular state. More precisely, the squared amplitude of the wavefunction is the probability of finding the system in a certain region of the configurational space. Thus we cannot say anything certain about finding the system in any particular state. We can only speak of probabilities of finding the system in a particular state or place. However, when we measure the parameters of the system, such as its position or momentum, we always get a particular value of the parameter we measure. It is as if the cloud of probabilities has suddenly collapsed to a single point – the value we find in an experiment. Hence, the measurement problem. Instead, Everett suggested that no collapse takes place. Instead, all possible states are realized in different universes. Every time-irreversible event – be that a transition between quantum-mechanical states or the measurement – splits the world into as many branches as there are possible outcomes, which are all realized in respective branches of the universe.
A more recent variation on this theme is a parallel-universe interpretation. It differs from Everett’s original idea in two important aspects. Everett and DeWitt spoke of our universe branching out every time there was a transition between quantum states. So the world’s history looks like a huge tree, with the trunk in the past and an ever-increasing number of branches as time goes on. In the parallel-universe version, the multitude of universes exists all along, ab initio, and a wavefunction of a quantum-mechanical system is partitioned among these parallel universes. Another difference is that, unlike the many-worlds theory that completely prohibits any communication between different branches, parallel universes can merge under certain circumstances, such as during an interference experiment. For example, in a double-slit experiment, a wavefunction of a photon is partitioned between two universes: in one, the photon passes through one slit, and in another, it passes through the second slit in a completely deterministic manner. After that, due to interference, the two universes merge together producing a single tangible photon.
On this level, parallel universes remain an optional interpretation of quantum mechanics, which has its followers and its skeptics. On the level of quantum cosmology, however, we are almost compelled to adopt this interpretation. Indeed, in the quantum cosmology described by the Wheeler-DeWitt equation, the universal wavefunction Ψ(h, F, S) is defined on an ensemble of all possible space-like universes, and is interpreted as a probability amplitude to find a particular manifold S with a particular geometry h and non-gravitational fields F. The Anthropic principle is usually invoked to select that universe, which allows for the emergence of life and intelligent beings that are capable of asking the question: which particular universe we live in.
It is remarkable that the many-worlds interpretation or parallel universes idea boasts among its supporters such luminaries as Richard Feynman, Steven Hawking, Murray Gell-Mann, Steven Weinberg, and some of the other best theoretical physicists of the twentieth century.
The classical Jewish sources are replete with the notion of multiple worlds and parallel universes. Consider, for example, the universes of Tohu (Chaos) and Tikkun (Rectification) that coexist parallel to each other. Or the four worlds of ABYA: Atzilut (the world of Splendor), Briyah (the world of Creation), Yetzirah (the world of Formation) and Assiya (the world of Action), each of which is said to be subdivided into a myriad of parallel worlds. Needless to say, all these “universes” denote spiritual rather than physical worlds.
The most troubling aspect of the many-worlds approach is that it suggests that the observer also splits into multiple copies completely oblivious of each other – “schizophrenia with a vengeance!”
Let us look into this week’s Torah portion – Vayera. The second verse says:
The Zohar suggests that the three persons who came to visit Abraham in Mamre were no other than Abraham, Isaac, and Jacob. Here we have a “celestial copy” of Abraham visiting the “terrestrial copy” of Abraham, the two coexisting in parallel universes.
This idea is further stressed by the way we read the Torah scroll. According to the Mesorah, the Jewish Rabbinical tradition, the Torah scroll is read using cantillation marks (taamey ha-miqra or “trope”) that one can find in Tikun Sofrim or in most printed editions of the Chumash (Pentateuch). Later in this Torah portion, in the story of the binding of Isaac, the Akeida, an angel called to Abraham from heaven and said,
Abraham! Abraham! (Genesis 22:11).
There is a vertical line (a cantillation mark) between the first Abraham and the repetition of the name: “Abraham | Abraham.” This sign tells the reader to pause between the first Abraham and the second Abraham when reading the Torah scroll. The Kabbalah and Chassidic philosophy (see, for example, Hemshech Samech Vav by Rabbi Dovber of Lubavitch) explain that the pause is required to distinguish between the celestial Abraham and the terrestrial Abraham.
Abraham and three Angels, 1966 – Marc Chagall
In this Torah portion, Vayera, at least in the Zohar’s interpretation, we read about the terrestrial Abraham meeting his celestial counterpart. This situation is analogous to the parallel universe interpretation of quantum mechanics, which allows for the occasional merger of the parallel universes. Indeed, it is as if the spiritual universe – the abode of the celestial Abraham – merged for a moment with the physical universe – the abode of the terrestrial Abraham – to allow for their face-to-face encounter.
Biblical commentators struggle to explain the meaning of the language used by the Torah in the last week’s portion Lech Lecha, when G‑d tells Abram “lech lecha,” lit. go to yourself. Seemingly, it does not make sense in a literal translation. Some translators simply read out “to yourself” part, others translate it as “for your own good,” which is far from the literal meaning. Given our explanation above, perhaps it simply means a commandment for Abram to go to himself – his higher self, his celestial counterpart, with whom he ultimately meets in this Torah portion, Vayera.
Each Jew, a descendant of Abraham, has a celestial counterpart, the higher self – it is called the G‑dly soul – nefesh Elokit. When Abram was told by G‑d to leave his land and his father’s house and go to his higher self, Abram was not a Jew yet – this was before the covenant G‑d made with him, before he was given a new name – Abraham. He only merited to meet his higher self after his circumcision, after he became the first Jew. Children of Abraham, the Jewish people, have their higher self, their G‑dly soul, inside. As the Tanya says, every Jew possesses nefesh Elokit, which is helek Eloka memaal mamash – “a part of G‑d from above indeed.” Thus, our task is not going outside ourselves to seek our higher self, as our forefather Abraham had to do but to direct our attention inward, to return to our true G‑dly self. That is why the word teshuvah should not be translated as “repentance” but as “return,” which is what it literally means – return to our higher self. Our father Abraham paved the way for us.
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55e63587cfbb1615 | 2019 Physical Chemistry III
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Academic unit or major
Undergraduate major in Life Science and Technology
Sakurai Minoru Fujii Masaaki Ishiuchi Shun-Ichi Kitao Akio
Class Format
Media-enhanced courses
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Tue1-2(H121) Fri1-2(H121)
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Course description and aims
The course teaches the fundamentals of quantum theory and its applications to biological systems, including the electronic structures and spectroscopic properties of biological molecules. Quantum theory is important for understanding nature, and is essential for the study not only of life science, but also of other specialized sciences and engineering. Students learn the laws governing the motions of electrons in atoms and molecules together with the mathematical description of such motions, that is, the Schrödinger equation. They will be able to solve the equation for simple process (one- or two- dimensional translational, rotational and vibrational motions), and the electronic structures of diatomic molecules and the pi-electron systems of small conjugated double bond compounds. Together with quantum theory, this course provides brief reviews of classical mechanics, wave mechanics, electromagnetism and optics, which are helpful for understanding the origin of quantum theory. This course also provides a brief introduction to computer simulations that are currently indispensable for investigating biological molecules. By the end of this course,students will understand that quantum theory is essential to interpret and predict many spectroscopic data including ultraviolet/visible, fluorescence, vibration spectra.
Student learning outcomes
1. Understand the basic principles of quantum theory and its application to elementary processes
2. Understand the basic concept of molecular orbital theory and its application to small molecules
3. Understand the physical origins of various inter- and intra-molecular forces
4. Understand the electronic excited states, vibrational states and dynamic properties of biological molecules by means of spectroscopic experiments and computaer simulations.
5. Understand the basic principles of classical mechanics, wave mechanics, electromagnetism, and optics as a base of quantum mechanics.
quantum theory, Schrödinger equation, wavefunction, molecular orbital theory, intermolecular and interatomic interactions, molecular spectroscopy,
Competencies that will be developed
Class flow
At the beginning of each class, solutions to exercise problems that were assigned during the previous class are
Course schedule/Required learning
Course schedule Required learning
Class 1 Principles of quantum theory: Schrödinger equation, wavefunction, quantization, uncertainty principle Solve the Schrödinger equation for a particle that freely moves on the x-axis, and explain that the solution (wavefunction) satisfies the uncertainty principle.
Class 2 Application of quantum theory to simple processes such as translation, rotation and vibration motions Solve exercise problems 9・23, 9・23 and 9・27 on page 367 of textbook.
Class 3 The electronic structures of hydrogenic atoms: atomic orbitals and their energies Solve exercise problems 9・35, 9・36 and 9・38 on page 368 of textbook.
Class 4 The electronic stuctures of many electron atoms: the orbital approximation and the Pauli exclusion principle Find the electron configuration for each atom of H～Ca according to the Aufbau principle, and explain the relationship between the results and the periodic table.
Class 5 Valence bond theory: hybridized orbitals and diatomic molecules According to the concept of hybridization of atomic orbitals, explain the reason why the valence of carbon atom varies from 2 to 4.
Class 6 Molecular orbital theory: linear combination of atomic orbitals, homonuclear and heteronuclear diatomic molecules Solve exercise problems 10・23,10・24,10・29 and 10.30 on page 412 of textbook.
Class 7 Molecular orbital theory: polyatomic molecules and Hückel theory Solve exercise problems 10・32～10・35 on page 412 of textbook.
Class 8 Molecular orbital theory: d-Metal complexes, crystal fiels theory and computational biochemistry Explain the ligand-field theory.
Class 9 Intermolecular and interatomic interactions: electrostatic interaction, hydrogen bond and Lennard-Jones potential Solve exercise problems 11・27, 11・28 and 11・42 on pages 468～469 of textbook.
Class 10 Levels of structure: gases and liquids, the structures of biological macromolecules and membranes Solve exercise problems 11・42～11・44 and 11・50 on pages 469～470 of textbook.
Class 11 Computer simulation: molecular dynamics and Monte Carlo simulations, and quantitative structure-activity relationships Explain the difference between the molecular dynamics simulation and the Monte Carlo simulation.
Class 12 Biochemical spectroscopy: general features of spectroscopy Solve exercise problems 12・10, 11, 14, 15.
Class 13 Biochemical spectroscopy: principle of vibrational spectroscopy Solve exercise problems 12・22, 23.
Class 14 Biochemical spectroscopy: application of vibrational spectroscopy - IR and Raman spectroscopy Solve exercise problems 12・24~27.
Class 15 Biochemical spectroscopy: Electronic transition and Franck-Condon principle Explain Franck-Condon factor.
P. Atkins and J. D. Paula, Physical Chemistry for the Life Science, second edition、Oxford University Press.
Reference books, course materials, etc.
P. Atkins and J. D. Paula, Physical Chemistry, eight edition, Oxford University Press
I. Tinoco, K. Sauer, J. C. Wang, J. D. Puglisi, G. Harbison and D. Rovnyak, Physical Chemistry, Principles and Applications in Biological Sciences, fifth edition, PERSON.
D. A. McQuarrie and J. D. Simon, Physical Chemistry, A Molecular Approach, University Science Books.
Assessment criteria and methods
Learning achievement is evaluated by a final exam.
Related courses
• LST.A201 ： Physical Chemistry I
• LST.A206 ： Physical Chemistry II
• LST.A341 ： Biophysical Chemistry
LST.A201 ： Physical Chemistry I
LST.A206 ： Physical Chemistry II
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847de46e82c41f8c | Five Predictions On Watching Movies In 2022
confident businessman using smartphone on street The flow of feelings all through different types of movies. The causes corresponding to the enter sorts are assumed to be independent on each other. POSTSUBSCRIPT ≃ 40 brightest stars within the Northern hemisphere, as listed within the input catalog described above. POSTSUBSCRIPT ⟩ merely describes the leading term in an expansion of the angular distribution, as an illustration, by way of Chebyshev polynomials, see (2) in Supplementary Information. ⟩ in a closed-feedback-loop approach. From this experiment we conclude that the output of our automatic parsing strategy can function a substitute of manual annotations and permits to realize aggressive results. 2016), automated identification of character relationships Zhang et al. To mannequin the relationships between cyberlockers, we embed them into a series of graph buildings. Therefore, it does not preserve the temporal buildings of the original storyline, which is also an vital side in movie understanding. This procedure includes (1) deciphering some extent cloud as a noisy sampling of a topological area, (2) creating a world object by forming connections between proximate factors primarily based on a scale parameter, (3) determining the topological construction made by these connections, and (4) searching for buildings that persist across totally different scales.
We deploy numerous digital camera pictures by scripting the Camera sport object in Unity. Following strong-area multiple ionisation of the molecules, the generated charged fragments were projected by the VMI onto a mixed multichannel-plate (MCP) phosphor-display detector and skim out by a CCD camera. 1.75 µm was polarised perpendicularly to the detector aircraft to minimise the effects of ionisation selectivity, i. These pulses were linearly polarised parallel to the detector aircraft. This enables the simulation of the experiment by fixing the time-dependent Schrödinger equation for a rigid rotor coupled to a non-resonant ac electric subject representing the two laser pulses and a dc electric area representing the weak extraction subject within the VMI spectrometer. POSTSUPERSCRIPT such that the values for the focal diameter decided are composite entities consisting of the ratio of the laser selectivity and the actual focal diameter. The laser setup consisted of a commercial Ti:Sapphire laser system (KM labs) delivering pulses with 30303030 mJ pulse vitality, 35353535 fs (FWHM) pulse duration, and a central wavelength of 800 nm at a 1111 kHz repetition price. The optimisation parameters used had been the intensities and one common duration, iptv 2022 of Fourier-restricted Gaussian pulses, and the delay between the pulses within the case of two-pulse alignment.
1.Seventy five µm pulses so as to characterise the angular distribution of the molecules through Coulomb-explosion imaging. FLOATSUPERSCRIPT fragments, which resembled the orientation of the molecules in area on the instance of ionization, were recorded by a velocity map imaging (VMI) spectrometer Eppink and Parker (1997) for various time delays between the alignment pulse sequence and the probe pulse. Third, based mostly on the success of Long Short-Term Memory networks (LSTMs) (Hochreiter and Schmidhuber, 1997) for the image captioning drawback (Donahue et al., 2015; Karpathy and Fei-Fei, new iptv 2015; Kiros et al., 2015; Vinyals et al., 2015) we suggest our approachVisual-Labels. To reveal the effectiveness of our framework, we design an end-to-end memory network model that leverages our speaker naming model and achieves state-of-the-artwork results on the subtitles task of the MovieQA 2017 Challenge. Ng et al. (Yue-Hei Ng et al., 2015) thought of every body of a video as a phrase in a sentence and learnt an LSTM community to temporally embed the video. At intra-sentence degree – We perform this analysis at a sentence degree where every sentence is analyzed independently. 0.64, corresponding to the everlasting alignment degree.
Optimisation calculations were performed so as to predict the optimum pulse parameter for اشتراك iptv single and two-pulse discipline-free alignment. In the case of MPII-MD we have usually solely a single reference. P: customers have totally different interests at totally different, presumably shut-by deadlines. This motivates the seek for topological features, associated with the evolution of the frames of a hyperspectral film, within the corresponding points on the Grassmann manifold. This implies we don’t anticipate to take all frames of a whole movie in one step of studying, which is each prohibitively expensive (as a result of sheer quantity of data accommodates in a movie) and unnecessary (frames in a movie are extremely redundant). However, the current literature means that coaching on uncurated information yields considerably poorer representations compared to the curated options collected in supervised method, and the hole solely narrows when the quantity of knowledge considerably increases. For the purpose of a direct comparability with the experimental knowledge the 3D rotational wavepacket was reconstructed and, using a Monte-Carlo approach, projected onto a 2D-display using the radial distribution extracted from the experiment on the alignment revival at 120.78 ps. |
02c1b98f98583c93 | W2VLDA With the increase of online customer opinions in specialised websites and social networks, the necessity of automatic systems to help to organise and classify customer reviews by domain-specific aspect/categories and sentiment polarity is more important than ever. Supervised approaches to Aspect Based Sentiment Analysis obtain good results for the domain/language their are trained on, but having manually labelled data for training supervised systems for all domains and languages use to be very costly and time consuming. In this work we describe W2VLDA, an unsupervised system based on topic modelling, that combined with some other unsupervised methods and a minimal configuration, performs aspect/category classifiation, aspectterms/opinion-words separation and sentiment polarity classification for any given domain and language. We also evaluate the performance of the aspect and sentiment classification in the multilingual SemEval 2016 task 5 (ABSA) dataset. We show competitive results for several languages (English, Spanish, French and Dutch) and domains (hotels, restaurants, electronic-devices).
Waffle Chart / Square Pie Chart A little-known alternative to the round pie chart is the square pie or waffle chart. It consists of a square that is divided into 10×10 cells, making it possible to read values precisely down to a single percent. Depending on how the areas are laid out (as square as possible seems to be the best idea), it is very easy to compare parts to the whole.
Waikato Environment for Knowledge Analysis
Weka (Waikato Environment for Knowledge Analysis) is a popular suite of machine learning software written in Java, developed at the University of Waikato, New Zealand. Weka is free software available under the GNU General Public License.
Wait-if-Diff and Wait-if-Worse Agent (Cho and Esipova, 2016)
Incremental Decoding and Training Methods for Simultaneous Translation in Neural Machine Translation
wakefield wakefield is a Github based R package which is designed to quickly generate random data sets. The user passes n (number of rows) and predefined vectors to the r_data_frame function to produce a dplyr::tbl_df object.
Wake-Sleep Algorithm The wake-sleep algorithm is an unsupervised learning algorithm for a multilayer neural network (e.g. sigmoid belief net). Training is divided into two phases, ‘wake’ and ‘sleep’. In the ‘wake’ phase, neurons are driven by recognition connections (connections from what would normally be considered an input to what is normally considered an output), while generative connections (those from outputs to inputs) are modified to increase the probability that they would reconstruct the correct activity in the layer below (closer to the sensory input). In the ‘sleep’ phase the process is reversed: neurons are driven by generative connections, while recognition connections are modified to increase the probability that they would produce the correct activity in the layer above (further from sensory input).
Walk-Steered Convolution
Graph classification is a fundamental but challenging problem due to the non-Euclidean property of graph. In this work, we jointly leverage the powerful representation ability of random walk and the essential success of standard convolutional network work (CNN), to propose a random walk based convolutional network, called walk-steered convolution (WSC). Different from those existing graph CNNs with deterministic neighbor searching, we randomly sample multi-scale walk fields by using random walk, which is more flexible to the scalability of graph. To encode each-scale walk field consisting of several walk paths, specifically, we characterize the directions of walk field by multiple Gaussian models so as to better analogize the standard CNNs on images. Each Gaussian implicitly defines a directions and all of them properly encode the spatial layout of walks after the gradient projecting to the space of Gaussian parameters. Further, a graph coarsening layer using dynamical clustering is stacked upon the Gaussian encoding to capture high-level semantics of graph. Comprehensive evaluations on several public datasets well demonstrate the superiority of our proposed graph learning method over other state-of-the-arts for graph classification.
Walktrap Community Algorithm Tries to find densely connected subgraphs, also called communities in a graph via random walks. The idea is that short random walks tend to stay in the same community.
Wallaroo Wallaroo is a fast, elastic data processing engine that rapidly takes you from prototype to production by eliminating infrastructure complexity. Wallaroo is a fast and elastic data processing engine that rapidly takes you from prototype to production by making the infrastructure virtually disappear. We´ve designed it to handle demanding high-throughput, low-latency tasks where the accuracy of results is essential. Wallaroo takes care of mechanics of scaling, resilience, state management, and message delivery. We’ve designed Wallaroo to make it easy scale applications with no code changes, and allow programmers to focus on business logic.
Walsh Figure of Merit LowWAFOMNX
Ward Hierarchical Clustering “Ward’s Method”
Ward’s Hierarchical Clustering Method: Clustering Criterion and Agglomerative Algorithm
Ward’s Method In statistics, Ward’s method is a criterion applied in hierarchical cluster analysis. Ward’s minimum variance method is a special case of the objective function approach originally presented by Joe H. Ward, Jr. Ward suggested a general agglomerative hierarchical clustering procedure, where the criterion for choosing the pair of clusters to merge at each step is based on the optimal value of an objective function. This objective function could be ‘any function that reflects the investigator’s purpose.’ Many of the standard clustering procedures are contained in this very general class. To illustrate the procedure, Ward used the example where the objective function is the error sum of squares, and this example is known as Ward’s method or more precisely Ward’s minimum variance method.
Ward’s Method
WarpFlow WarpFlow is a fast, interactive data querying and processing system with a focus on petabyte-scale spatiotemporal datasets and Tesseract queries. With the rapid growth in smartphones and mobile navigation services, we now have an opportunity to radically improve urban mobility and reduce friction in how people and packages move globally every minute-mile, with data. WarpFlow speeds up three key metrics for data engineers working on such datasets — time-to-first-result, time-to-full-scale-result, and time-to-trained-model for machine learning.
WarpLDA Developing efficient and scalable algorithms for Latent Dirichlet Allocation (LDA) is of wide interest for many applications. Previous work has developed an $O(1)$ Metropolis-Hastings sampling method for each token. However, the performance is far from being optimal due to random accesses to the parameter matrices and frequent cache misses. In this paper, we propose WarpLDA, a novel $O(1)$ sampling algorithm for LDA. WarpLDA is a Metropolis-Hastings based algorithm which is designed to optimize the cache hit rate. Advantages of WarpLDA include 1) Efficiency and scalability: WarpLDA has good locality and carefully designed partition method, and can be scaled to hundreds of machines; 2) Simplicity: WarpLDA does not have any complicated modules such as alias tables, hybrid data structures, or parameter servers, making it easy to understand and implement; 3) Robustness: WarpLDA is consistently faster than other algorithms, under various settings from small-scale to massive-scale dataset and model. WarpLDA is 5-15x faster than state-of-the-art LDA samplers, implying less cost of time and money. With WarpLDA users can learn up to one million topics from hundreds of millions of documents in a few hours, at the speed of 2G tokens per second, or learn topics from small-scale datasets in seconds.
Wasserstein Auto-Encoder
We propose the Wasserstein Auto-Encoder (WAE)—a new algorithm for building a generative model of the data distribution. WAE minimizes a penalized form of the Wasserstein distance between the model distribution and the target distribution, which leads to a different regularizer than the one used by the Variational Auto-Encoder (VAE). This regularizer encourages the encoded training distribution to match the prior. We compare our algorithm with several other techniques and show that it is a generalization of adversarial auto-encoders (AAE). Our experiments show that WAE shares many of the properties of VAEs (stable training, encoder-decoder architecture, nice latent manifold structure) while generating samples of better quality, as measured by the FID score.
Wasserstein Barycenter Wasserstein barycenter is a single distribution that summarizes a collection of input measures while respecting their geometry.
Wasserstein CNN
Heterogeneous face recognition (HFR) aims to match facial images acquired from different sensing modalities with mission-critical applications in forensics, security and commercial sectors. However, HFR is a much more challenging problem than traditional face recognition because of large intra-class variations of heterogeneous face images and limited training samples of cross-modality face image pairs. This paper proposes a novel approach namely Wasserstein CNN (convolutional neural networks, or WCNN for short) to learn invariant features between near-infrared and visual face images (i.e. NIR-VIS face recognition). The low-level layers of WCNN are trained with widely available face images in visual spectrum. The high-level layer is divided into three parts, i.e., NIR layer, VIS layer and NIR-VIS shared layer. The first two layers aims to learn modality-specific features and NIR-VIS shared layer is designed to learn modality-invariant feature subspace. Wasserstein distance is introduced into NIR-VIS shared layer to measure the dissimilarity between heterogeneous feature distributions. So W-CNN learning aims to achieve the minimization of Wasserstein distance between NIR distribution and VIS distribution for invariant deep feature representation of heterogeneous face images. To avoid the over-fitting problem on small-scale heterogeneous face data, a correlation prior is introduced on the fully-connected layers of WCNN network to reduce parameter space. This prior is implemented by a low-rank constraint in an end-to-end network. The joint formulation leads to an alternating minimization for deep feature representation at training stage and an efficient computation for heterogeneous data at testing stage. Extensive experiments on three challenging NIR-VIS face recognition databases demonstrate the significant superiority of Wasserstein CNN over state-of-the-art methods.
Wasserstein Discriminant Analysis
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the ratio of two quantities: the dispersion of projected points coming from different classes, divided by the dispersion of projected points coming from the same class. To quantify dispersion, WDA uses regularized Wasserstein distances, rather than cross-variance measures which have been usually considered, notably in LDA. Thanks to the the underlying principles of optimal transport, WDA is able to capture both global (at distribution scale) and local (at samples scale) interactions between classes. Regularized Wasserstein distances can be computed using the Sinkhorn matrix scaling algorithm; We show that the optimization of WDA can be tackled using automatic differentiation of Sinkhorn iterations. Numerical experiments show promising results both in terms of prediction and visualization on toy examples and real life datasets such as MNIST and on deep features obtained from a subset of the Caltech dataset.
Wasserstein Distance “Wasserstein Metric”
Wasserstein GAN
Despite being impactful on a variety of problems and applications, the generative adversarial nets (GANs) are remarkably difficult to train. This issue is formally analyzed by \cite{arjovsky2017towards}, who also propose an alternative direction to avoid the caveats in the minmax two-player training of GANs. The corresponding algorithm, called Wasserstein GAN (WGAN), hinges on the 1-Lipschitz continuity of the discriminator. In this paper, we propose a novel approach to enforcing the Lipschitz continuity in the training procedure of WGANs. Our approach seamlessly connects WGAN with one of the recent semi-supervised learning methods. As a result, it gives rise to not only better photo-realistic samples than the previous methods but also state-of-the-art semi-supervised learning results. In particular, our approach gives rise to the inception score of more than 5.0 with only 1,000 CIFAR-10 images and is the first that exceeds the accuracy of 90% on the CIFAR-10 dataset using only 4,000 labeled images, to the best of our knowledge.
Wasserstein Identity Testing Problem Uniformity testing and the more general identity testing are well studied problems in distributional property testing. Most previous work focuses on testing under $L_1$-distance. However, when the support is very large or even continuous, testing under $L_1$-distance may require a huge (even infinite) number of samples. Motivated by such issues, we consider the identity testing in Wasserstein distance (a.k.a. transportation distance and earthmover distance) on a metric space (discrete or continuous). In this paper, we propose the Wasserstein identity testing problem (Identity Testing in Wasserstein distance). We obtain nearly optimal worst-case sample complexity for the problem. Moreover, for a large class of probability distributions satisfying the so-called ‘Doubling Condition’, we provide nearly instance-optimal sample complexity.
Wasserstein Introspective Neural Network
We present Wasserstein introspective neural networks (WINN) that are both a generator and a discriminator within a single model. WINN provides a significant improvement over the recent introspective neural networks (INN) method by enhancing INN’s generative modeling capability. WINN has three interesting properties: (1) A mathematical connection between the formulation of Wasserstein generative adversarial networks (WGAN) and the INN algorithm is made; (2) The explicit adoption of the WGAN term into INN results in a large enhancement to INN, achieving compelling results even with a single classifier on e.g., providing a 20 times reduction in model size over INN within texture modeling; (3) When applied to supervised classification, WINN also gives rise to greater robustness with an $88\%$ reduction of errors against adversarial examples — improved over the result of $39\%$ by an INN-family algorithm. In the experiments, we report encouraging results on unsupervised learning problems including texture, face, and object modeling, as well as a supervised classification task against adversarial attack.
Wasserstein Metric In mathematics, the Wasserstein (or Vasershtein) metric is a distance function defined between probability distributions on a given metric space M. Intuitively, if each distribution is viewed as a unit amount of ‘dirt’ piled on M, the metric is the minimum ‘cost’ of turning one pile into the other, which is assumed to be the amount of dirt that needs to be moved times the distance it has to be moved. Because of this analogy, the metric is known in computer science as the earth mover’s distance. The name ‘Wasserstein distance’ was coined by R. L. Dobrushin in 1970, after the Russian mathematician Leonid Vaseršteĭn who introduced the concept in 1969. Most English-language publications use the German spelling ‘Wasserstein’ (attributed to the name ‘Vasershtein’ being of German origin).
“Earth Mover’s Distance”
Wasserstein Distance
Wasserstein Transform We introduce the Wasserstein transform, a method for enhancing and denoising datasets defined on general metric spaces. The construction draws inspiration from Optimal Transportation ideas. We establish precise connections with the mean shift family of algorithms and establish the stability of both our method and mean shift under data perturbation.
Wasserstein Variational Gradient Descent Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of semi-discrete optimal transport. Instead of minimizing the KL divergence between the posterior and the variational approximation, we minimize a semi-discrete optimal transport divergence. The solution of the resulting optimal transport problem provides both a particle approximation and a set of optimal transportation densities that map each particle to a segment of the posterior distribution. We approximate these transportation densities by minimizing the KL divergence between a truncated distribution and the optimal transport solution. The resulting algorithm can be interpreted as a form of ensemble variational inference where each particle is associated with a local variational approximation.
Wasserstein Variational Inference This paper introduces Wasserstein variational inference, a new form of approximate Bayesian inference based on optimal transport theory. Wasserstein variational inference uses a new family of divergences that includes both f-divergences and the Wasserstein distance as special cases. The gradients of the Wasserstein variational loss are obtained by backpropagating through the Sinkhorn iterations. This technique results in a very stable likelihood-free training method that can be used with implicit distributions and probabilistic programs. Using the Wasserstein variational inference framework, we introduce several new forms of autoencoders and test their robustness and performance against existing variational autoencoding techniques.
Wasserstein-Wasserstein Auto-Encoder
To address the challenges in learning deep generative models (e.g.,the blurriness of variational auto-encoder and the instability of training generative adversarial networks, we propose a novel deep generative model, named Wasserstein-Wasserstein auto-encoders (WWAE). We formulate WWAE as minimization of the penalized optimal transport between the target distribution and the generated distribution. By noticing that both the prior $P_Z$ and the aggregated posterior $Q_Z$ of the latent code Z can be well captured by Gaussians, the proposed WWAE utilizes the closed-form of the squared Wasserstein-2 distance for two Gaussians in the optimization process. As a result, WWAE does not suffer from the sampling burden and it is computationally efficient by leveraging the reparameterization trick. Numerical results evaluated on multiple benchmark datasets including MNIST, fashion- MNIST and CelebA show that WWAE learns better latent structures than VAEs and generates samples of better visual quality and higher FID scores than VAEs and GANs.
Watanabe-Akaike Information Criteria
WAIC (the Watanabe-Akaike or widely applicable information criterion; Watanabe, 2010) can be viewed as an improvement on the deviance information criterion (DIC) for Bayesian models. DIC has gained popularity in recent years in part through its implementation in the graphical modeling package BUGS (Spiegelhalter, Best, et al., 2002; Spiegelhalter, Thomas, et al., 1994, 2003), but is known to have some problems, arising in part from it not being fully Bayesian in that it is based on a point estimate (van der Linde, 2005, Plummer, 2008). For example, DIC can produce negative estimates of the effective number of parameters in a model and it is not defined for singular models. WAIC is fully Bayesian and closely approximates Bayesian cross-validation. Unlike DIC, WAIC is invariant to parametrization and also works for singular models.
A Widely Applicable Bayesian Information Criterion
Watchdog AI
Artificial Intelligence (AI) technologies could be broadly categorised into Analytics and Autonomy. Analytics focuses on algorithms offering perception, comprehension, and projection of knowledge gleaned from sensorial data. Autonomy revolves around decision making, and influencing and shaping the environment through action production. A smart autonomous system (SAS) combines analytics and autonomy to understand, learn, decide and act autonomously. To be useful, SAS must be trusted and that requires testing. Lifelong learning of a SAS compounds the testing process. In the remote chance that it is possible to fully test and certify the system pre-release, which is theoretically an undecidable problem, it is near impossible to predict the future behaviours that these systems, alone or collectively, will exhibit. While it may be feasible to severely restrict such systems\textquoteright \ learning abilities to limit the potential unpredictability of their behaviours, an undesirable consequence may be severely limiting their utility. In this paper, we propose the architecture for a watchdog AI (WAI) agent dedicated to lifelong functional testing of SAS. We further propose system specifications including a level of abstraction whereby humans shepherd a swarm of WAI agents to oversee an ecosystem made of humans and SAS. The discussion extends to the challenges, pros, and cons of the proposed concept.
Waterfall Bandits A popular approach to selling online advertising is by a waterfall, where a publisher makes sequential price offers to ad networks for an inventory, and chooses the winner in that order. The publisher picks the order and prices to maximize her revenue. A traditional solution is to learn the demand model and then subsequently solve the optimization problem for the given demand model. This will incur a linear regret. We design an online learning algorithm for solving this problem, which interleaves learning and optimization, and prove that this algorithm has sublinear regret. We evaluate the algorithm on both synthetic and real-world data, and show that it quickly learns high quality pricing strategies. This is the first principled study of learning a waterfall design online by sequential experimentation.
Waterfall Chart A waterfall chart is a form of data visualization that helps in understanding the cumulative effect of sequentially introduced positive or negative values. The waterfall chart is also known as a flying bricks chart or Mario chart due to the apparent suspension of columns (bricks) in mid-air. Often in finance, it will be referred to as a bridge. Waterfall charts were popularized by the strategic consulting firm McKinsey & Company in its presentations to clients. The waterfall chart is normally used for understanding how an initial value is affected by a series of intermediate positive or negative values. Usually the initial and the final values are represented by whole columns, while the intermediate values are denoted by floating columns. The columns are color-coded for distinguishing between positive and negative values.
“Waterfall Chart”
Understanding Waterfall Plots
Waterfall plots – what and how?
Waterfall Plot A waterfall plot is a three-dimensional plot in which multiple curves of data, typically spectra, are displayed simultaneously. Typically the curves are staggered both across the screen and vertically, with ‘nearer’ curves masking the ones behind. The result is a series of ‘mountain’ shapes that appear to be side by side. The waterfall plot is often used to show how two-dimensional information changes over time or some other variable such as rpm. The term ‘waterfall plot’ is sometimes used interchangeably with ‘spectrogram’ or ‘Cumulative Spectral Decay’ (CSD) plot.
wav2letter++ This paper introduces wav2letter++, the fastest open-source deep learning speech recognition framework. wav2letter++ is written entirely in C++, and uses the ArrayFire tensor library for maximum efficiency. Here we explain the architecture and design of the wav2letter++ system and compare it to other major open-source speech recognition systems. In some cases wav2letter++ is more than 2x faster than other optimized frameworks for training end-to-end neural networks for speech recognition. We also show that wav2letter++’s training times scale linearly to 64 GPUs, the highest we tested, for models with 100 million parameters. High-performance frameworks enable fast iteration, which is often a crucial factor in successful research and model tuning on new datasets and tasks.
Introducing Wav2letter++
Wav2Pix Speech is a rich biometric signal that contains information about the identity, gender and emotional state of the speaker. In this work, we explore its potential to generate face images of a speaker by conditioning a Generative Adversarial Network (GAN) with raw speech input. We propose a deep neural network that is trained from scratch in an end-to-end fashion, generating a face directly from the raw speech waveform without any additional identity information (e.g reference image or one-hot encoding). Our model is trained in a self-supervised approach by exploiting the audio and visual signals naturally aligned in videos. With the purpose of training from video data, we present a novel dataset collected for this work, with high-quality videos of youtubers with notable expressiveness in both the speech and visual signals.
wav2vec We explore unsupervised pre-training for speech recognition by learning representations of raw audio. wav2vec is trained on large amounts of unlabeled audio data and the resulting representations are then used to improve acoustic model training. We pre-train a simple multi-layer convolutional neural network optimized via a noise contrastive binary classification task. Our experiments on WSJ reduce WER of a strong character-based log-mel filterbank baseline by up to 32% when only a few hours of transcribed data is available. Our approach achieves 2.78% WER on the nov92 test set. This outperforms Deep Speech 2, the best reported character-based system in the literature while using three orders of magnitude less labeled training data.
Wave Oriented Swarm Programming Paradigm
In this work, we present a programming paradigm allowing the control of swarms with a minimum communication bandwidth in a simple manner, yet allowing the emergence of diverse complex behaviors and autonomy of the swarm. Communication in the proposed paradigm is based on single bit ‘ping’-signals propagating as information-waves throughout the swarm. We show that even this minimum bandwidth communication between agents suffices for the design of a substantial set of behaviors in the domain of essential behaviors of a collective, including locomotion and self awareness of the swarm.
WaveGlow In this paper we propose WaveGlow: a flow-based network capable of generating high quality speech from mel-spectrograms. WaveGlow combines insights from Glow and WaveNet in order to provide fast, efficient and high-quality audio synthesis, without the need for auto-regression. WaveGlow is implemented using only a single network, trained using only a single cost function: maximizing the likelihood of the training data, which makes the training procedure simple and stable. Our PyTorch implementation produces audio samples at a rate of more than 500 kHz on an NVIDIA V100 GPU. Mean Opinion Scores show that it delivers audio quality as good as the best publicly available WaveNet implementation. All code will be made publicly available online.
Wavelet Convolutional Neural Network Spatial and spectral approaches are two major approaches for image processing tasks such as image classification and object recognition. Among many such algorithms, convolutional neural networks (CNNs) have recently achieved significant performance improvement in many challenging tasks. Since CNNs process images directly in the spatial domain, they are essentially spatial approaches. Given that spatial and spectral approaches are known to have different characteristics, it will be interesting to incorporate a spectral approach into CNNs. We propose a novel CNN architecture, wavelet CNNs, which combines a multiresolution analysis and CNNs into one model. Our insight is that a CNN can be viewed as a limited form of a multiresolution analysis. Based on this insight, we supplement missing parts of the multiresolution analysis via wavelet transform and integrate them as additional components in the entire architecture. Wavelet CNNs allow us to utilize spectral information which is mostly lost in conventional CNNs but useful in most image processing tasks. We evaluate the practical performance of wavelet CNNs on texture classification and image annotation. The experiments show that wavelet CNNs can achieve better accuracy in both tasks than existing models while having significantly fewer parameters than conventional CNNs.
WaveletFCNN Wind power, as an alternative to burning fossil fuels, is plentiful and renewable. Data-driven approaches are increasingly popular for inspecting the wind turbine failures. In this paper, we propose a novel classification-based anomaly detection system for icing detection of the wind turbine blades. We effectively combine the deep neural networks and wavelet transformation to identify such failures sequentially across the time. In the training phase, we present a wavelet based fully convolutional neural network (FCNN), namely WaveletFCNN, for the time series classification. We improve the original (FCNN) by augmenting features with the wavelet coefficients. WaveletFCNN outperforms the state-of-the-art FCNN for the univariate time series classification on the UCR time series archive benchmarks. In the detecting phase, we combine the sliding window and majority vote algorithms to provide the timely monitoring of the anomalies. The system has been successfully implemented on a real-world dataset from Goldwind Inc, where the classifier is trained on a multivariate time series dataset and the monitoring algorithm is implemented to capture the abnormal condition on signals from a wind farm.
Wavelet-like Auto-Encoder
Accelerating deep neural networks (DNNs) has been attracting increasing attention as it can benefit a wide range of applications, e.g., enabling mobile systems with limited computing resources to own powerful visual recognition ability. A practical strategy to this goal usually relies on a two-stage process: operating on the trained DNNs (e.g., approximating the convolutional filters with tensor decomposition) and fine-tuning the amended network, leading to difficulty in balancing the trade-off between acceleration and maintaining recognition performance. In this work, aiming at a general and comprehensive way for neural network acceleration, we develop a Wavelet-like Auto-Encoder (WAE) that decomposes the original input image into two low-resolution channels (sub-images) and incorporate the WAE into the classification neural networks for joint training. The two decomposed channels, in particular, are encoded to carry the low-frequency information (e.g., image profiles) and high-frequency (e.g., image details or noises), respectively, and enable reconstructing the original input image through the decoding process. Then, we feed the low-frequency channel into a standard classification network such as VGG or ResNet and employ a very lightweight network to fuse with the high-frequency channel to obtain the classification result. Compared to existing DNN acceleration solutions, our framework has the following advantages: i) it is tolerant to any existing convolutional neural networks for classification without amending their structures; ii) the WAE provides an interpretable way to preserve the main components of the input image for classification.
WaveletNet We present a logarithmic-scale efficient convolutional neural network architecture for edge devices, named WaveletNet. Our model is based on the well-known depthwise convolution, and on two new layers, which we introduce in this work: a wavelet convolution and a depthwise fast wavelet transform. By breaking the symmetry in channel dimensions and applying a fast algorithm, WaveletNet shrinks the complexity of convolutional blocks by an O(logD/D) factor, where D is the number of channels. Experiments on CIFAR-10 and ImageNet classification show superior and comparable performances of WaveletNet compared to state-of-the-art models such as MobileNetV2.
WaveNet Various sources have reported the WaveNet deep learning architecture being able to generate high-quality speech, but to our knowledge there haven’t been studies on the interpretation or visualization of trained WaveNets. This study investigates the possibility that WaveNet understands speech by unsupervisedly learning an acoustically meaningful latent representation of the speech signals in its receptive field; we also attempt to interpret the mechanism by which the feature extraction is performed. Suggested by singular value decomposition and linear regression analysis on the activations and known acoustic features (e.g. F0), the key findings are (1) activations in the higher layers are highly correlated with spectral features; (2) WaveNet explicitly performs pitch extraction despite being trained to directly predict the next audio sample and (3) for the said feature analysis to take place, the latent signal representation is converted back and forth between baseband and wideband components.
How WaveNet Works
Wavenilm Non-intrusive load monitoring (NILM) helps meet energy conservation goals by estimating individual appliance power usage from a single aggregate measurement. Deep neural networks have become increasingly popular in attempting to solve NILM problems; however, many of them are not causal which is important for real-time application. We present a causal 1-D convolutional neural network inspired by WaveNet for NILM on low-frequency data. We also study using various components of the complex power signal for NILM, and demonstrate that using all four components available in a popular NILM dataset (current, active power, reactive power, and apparent power) we achieve faster convergence and higher performance than state-of-the-art results for the same dataset.
W-Decorrelation Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method decorrelation procedure — W-decorrelation — for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the W-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic W-decorrelation procedure in two different adaptive data settings: the multi-armed bandits and autoregressive time series models.
Weakly Structured Information Processing and Exploration
WIPE is used for managing the graph traversal manipulation with BI-like data aggregation. WIPE stands for “Weakly-structured Information Processing and Exploration”. It is a data manipulation and query language built on top of the graph functionality in the SAP HANA Database. Like other domain specific languages provided by SAP HANA Database, WIPE is embedded in transactional context, which means that multiple WIPE statements can be executed concurrently, guaranteeing the atomicity, consistency, isolation and durability. With the help of this language, multiple graph operations such as inserting, updating or deleting a node and other query operations can be declared in one complex statement. It is the graph abstraction layer in the SAP HANA Database that provides interaction with the graph data stored in the database by exposing graph concepts directly to the application developer. The application developer can create or delete graphs, access the existing graphs, modify the vertices and edges of the graphs, or retrieve a set of vertices and edges based on their attributes. Besides retrieval and manipulation functions, a set of built-in graph operators are also provided by the SAP HANA Database. These operators, such as breadth-first or depth-first traversal algorithms, interact with the column store of the relational engine to execute efficiently and in a highly optimum manner.
Weakly-Supervised Neural Text Classification Deep neural networks are gaining increasing popularity for the classic text classification task, due to their strong expressive power and less requirement for feature engineering. Despite such attractiveness, neural text classification models suffer from the lack of training data in many real-world applications. Although many semi-supervised and weakly-supervised text classification models exist, they cannot be easily applied to deep neural models and meanwhile support limited supervision types. In this paper, we propose a weakly-supervised method that addresses the lack of training data in neural text classification. Our method consists of two modules: (1) a pseudo-document generator that leverages seed information to generate pseudo-labeled documents for model pre-training, and (2) a self-training module that bootstraps on real unlabeled data for model refinement. Our method has the flexibility to handle different types of weak supervision and can be easily integrated into existing deep neural models for text classification. We have performed extensive experiments on three real-world datasets from different domains. The results demonstrate that our proposed method achieves inspiring performance without requiring excessive training data and outperforms baseline methods significantly.
Weakly-supervised Temporal Activity Localization
Most activity localization methods in the literature suffer from the burden of frame-wise annotation requirement. Learning from weak labels may be a potential solution towards reducing such manual labeling effort. Recent years have witnessed a substantial influx of tagged videos on the Internet, which can serve as a rich source of weakly-supervised training data. Specifically, the correlations between videos with similar tags can be utilized to temporally localize the activities. Towards this goal, we present W-TALC, a Weakly-supervised Temporal Activity Localization and Classification framework using only video-level labels. The proposed network can be divided into two sub-networks, namely the Two-Stream based feature extractor network and a weakly-supervised module, which we learn by optimizing two complimentary loss functions. Qualitative and quantitative results on two challenging datasets – Thumos14 and ActivityNet1.2, demonstrate that the proposed method is able to detect activities at a fine granularity and achieve better performance than current state-of-the-art methods.
Weaver We introduce a new distributed graph store, called Weaver, which enables efficient, transactional graph analyses as well as strictly serializable read-write transactions on dynamic graphs. The key insight that enables Weaver to combine strict serializability with horizontal scalability and high performance is a novel request ordering mechanism called refinable timestamps. This technique couples coarse-grained vector timestamps with a fine-grained timeline oracle to pay the overhead of strong consistency only when needed.
Web Analytics Web analytics is the measurement, collection, analysis and reporting of web data for purposes of understanding and optimizing web usage. Web analytics is not just a tool for measuring web traffic but can be used as a tool for business and market research, and to assess and improve the effectiveness of a website. Web analytics applications can also help companies measure the results of traditional print or broadcast advertising campaigns. It helps one to estimate how traffic to a website changes after the launch of a new advertising campaign. Web analytics provides information about the number of visitors to a website and the number of page views. It helps gauge traffic and popularity trends which is useful for market research. There are two categories of web analytics; off-site and on-site web analytics. Off-site web analytics refers to web measurement and analysis regardless of whether you own or maintain a website. It includes the measurement of a website’s potential audience (opportunity), share of voice (visibility), and buzz (comments) that is happening on the Internet as a whole. On-site web analytics measure a visitor’s behavior once on your website. This includes its drivers and conversions; for example, the degree to which different landing pages are associated with online purchases. On-site web analytics measures the performance of your website in a commercial context. This data is typically compared against key performance indicators for performance, and used to improve a website or marketing campaign’s audience response. Google Analytics is the most widely used on-site web analytics service; although new tools are emerging that provide additional layers of information, including heat maps and session replay. Historically, web analytics has been used to refer to on-site visitor measurement. However, in recent years this meaning has become blurred, mainly because vendors are producing tools that span both categories.
Web of Data
Web Ontology Language
The Web Ontology Language (OWL) is a family of knowledge representation languages for authoring ontologies. Ontologies are a formal way to describe taxonomies and classification networks, essentially defining the structure of knowledge for various domains: the nouns representing classes of objects and the verbs representing relations between the objects. Ontologies resemble class hierarchies in object-oriented programming but there are several critical differences. Class hierarchies are meant to represent structures used in source code that evolve fairly slowly (typically monthly revisions) where as ontologies are meant to represent information on the Internet and are expected to be evolving almost constantly. Similarly, ontologies are typically far more flexible as they are meant to represent information on the Internet coming from all sorts of heterogeneous data sources. Class hierarchies on the other hand are meant to be fairly static and rely on far less diverse and more structured sources of data such as corporate databases. The OWL languages are characterized by formal semantics. They are built upon a W3C XML standard for objects called the Resource Description Framework (RDF). OWL and RDF have attracted significant academic, medical and commercial interest. In October 2007, a new W3C working group was started to extend OWL with several new features as proposed in the OWL 1.1 member submission. W3C announced the new version of OWL on 27 October 2009. This new version, called OWL 2, soon found its way into semantic editors such as Protégé and semantic reasoners such as Pellet, RacerPro, FaCT++ and HermiT. The OWL family contains many species, serializations, syntaxes and specifications with similar names. OWL and OWL2 are used to refer to the 2004 and 2009 specifications, respectively. Full species names will be used, including specification version (for example, OWL2 EL). When referring more generally, OWL Family will be used.
WebSeg In this paper, we improve semantic segmentation by automatically learning from Flickr images associated with a particular keyword, without relying on any explicit user annotations, thus substantially alleviating the dependence on accurate annotations when compared to previous weakly supervised methods. To solve such a challenging problem, we leverage several low-level cues (such as saliency, edges, etc.) to help generate a proxy ground truth. Due to the diversity of web-crawled images, we anticipate a large amount of ‘label noise’ in which other objects might be present. We design an online noise filtering scheme which is able to deal with this label noise, especially in cluttered images. We use this filtering strategy as an auxiliary module to help assist the segmentation network in learning cleaner proxy annotations. Extensive experiments on the popular PASCAL VOC 2012 semantic segmentation benchmark show surprising good results in both our WebSeg (mIoU = 57.0%) and weakly supervised (mIoU = 63.3%) settings.
WeCURE Missing data recovery is an important and yet challenging problem in imaging and data science. Successful models often adopt certain carefully chosen regularization. Recently, the low dimension manifold model (LDMM) was introduced by S.Osher et al. and shown effective in image inpainting. They observed that enforcing low dimensionality on image patch manifold serves as a good image regularizer. In this paper, we observe that having only the low dimension manifold regularization is not enough sometimes, and we need smoothness as well. For that, we introduce a new regularization by combining the low dimension manifold regularization with a higher order Curvature Regularization, and we call this new regularization CURE for short. The key step of solving CURE is to solve a biharmonic equation on a manifold. We further introduce a weighted version of CURE, called WeCURE, in a similar manner as the weighted nonlocal Laplacian (WNLL) method. Numerical experiments for image inpainting and semi-supervised learning show that the proposed CURE and WeCURE significantly outperform LDMM and WNLL respectively.
Weibull Distribution In probability theory and statistics, the Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. It is named after Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.
Weibull Hybrid Autoencoding Inference
To train an inference network jointly with a deep generative topic model, making it both scalable to big corpora and fast in out-of-sample prediction, we develop Weibull hybrid autoencoding inference (WHAI) for deep latent Dirichlet allocation, which infers posterior samples via a hybrid of stochastic-gradient MCMC and autoencoding variational Bayes. The generative network of WHAI has a hierarchy of gamma distributions, while the inference network of WHAI is a Weibull upward-downward variational autoencoder, which integrates a deterministic-upward deep neural network, and a stochastic-downward deep generative model based on a hierarchy of Weibull distributions. The Weibull distribution can be used to well approximate a gamma distribution with an analytic Kullback-Leibler divergence, and has a simple reparameterization via the uniform noise, which help efficiently compute the gradients of the evidence lower bound with respect to the parameters of the inference network. The effectiveness and efficiency of WHAI are illustrated with experiments on big corpora.
Weibull Time To Event Recurrent Neural Network
In this thesis we propose a new model for predicting time to events: the Weibull Time To Event RNN. This is a simple framework for time-series prediction of the time to the next event applicable when we have any or all of the problems of continuous or discrete time, right censoring, recurrent events, temporal patterns, time varying covariates or time series of varying lengths. All these problems are frequently encountered in customer churn, remaining useful life, failure, spike-train and event prediction. The proposed model estimates the distribution of time to the next event as having a discrete or continuous Weibull distribution with parameters being the output of a recurrent neural network. The model is trained using a special objective function (log-likelihood-loss for censored data) commonly used in survival analysis. The Weibull distribution is simple enough to avoid sparsity and can easily be regularized to avoid overfitting but is still expressive enough to encode concepts like increasing, stationary or decreasing risk and can converge to a point-estimate if allowed. The predicted Weibull-parameters can be used to predict expected value and quantiles of the time to the next event. It also leads to a natural 2d-embedding of future risk which can be used for monitoring and exploratory analysis. We describe the WTTE-RNN using a general framework for censored data which can easily be extended with other distributions and adapted for multivariate prediction. We show that the common Proportional Hazards model and the Weibull Accelerated Failure time model are special cases of the WTTE-RNN. The proposed model is evaluated on simulated data with varying degrees of censoring and temporal resolution. We compared it to binary fixed window forecast models and naive ways of handling censored data. The model outperforms naive methods and is found to have many advantages and comparable performance to binary fixed-window RNNs without the need to specify window size and the ability to train on more data. Application to the CMAPSS-dataset for PHM-run-to-failure of simulated Jet-Engines gives promising results.
Weight of Evidence
The Weight of Evidence or WoE value is a widely used measure of the ‘strength’ of a grouping for separating good and bad risk (default). It is computed from the basic odds ratio: (Distribution of Good Credit Outcomes) / (Distribution of Bad Credit Outcomes). Or the ratios of Distr Goods / Distr Bads for short, where Distr refers to the proportion of Goods or Bads in the respective group, relative to the column totals, i.e., expressed as relative proportions of the total number of Goods and Bads.
Why Use Weight of Evidence?
Weight Standardization
In this paper, we propose Weight Standardization (WS) to accelerate deep network training. WS is targeted at the micro-batch training setting where each GPU typically has only 1-2 images for training. The micro-batch training setting is hard because small batch sizes are not enough for training networks with Batch Normalization (BN), while other normalization methods that do not rely on batch knowledge still have difficulty matching the performances of BN in large-batch training. Our WS ends this problem because when used with Group Normalization and trained with 1 image/GPU, WS is able to match or outperform the performances of BN trained with large batch sizes with only 2 more lines of code. In micro-batch training, WS significantly outperforms other normalization methods. WS achieves these superior results by standardizing the weights in the convolutional layers, which we show is able to smooth the loss landscape by reducing the Lipschitz constants of the loss and the gradients. The effectiveness of WS is verified on many tasks, including image classification, object detection, instance segmentation, video recognition, semantic segmentation, and point cloud recognition. The code is available here: https://…/WeightStandardization.
Weighted Balanced Distribution Adaptation
“Balanced Distribution Adaptation”
Weighted Bootstrap Markov Chain Monte Carlo Many data sets, especially from surveys, are made available to users with weights. Where the derivation of such weights is known, this information can often be incorporated in the user’s substantive model (model of interest). When the derivation is unknown, the established procedure is to carry out a weighted analysis. However, with non-trivial proportions of missing data this is inefficient and may be biased when data are not missing at random. Bayesian approaches provide a natural approach for the imputation of missing data, but it is unclear how to handle the weights. We propose a weighted bootstrap Markov chain Monte Carlo algorithm for estimation and inference. A simulation study shows that it has good inferential properties. We illustrate its utility with an analysis of data from the Millennium Cohort Study.
Weighted Effect Coding Weighted effect coding refers to a specific coding matrix to include factor variables in generalised linear regression models. With weighted effect coding, the effect for each category represents the deviation of that category from the weighted mean (which corresponds to the sample mean). This technique has particularly attractive properties when analysing observational data, that commonly are unbalanced. The wec package is introduced, that provides functions to apply weighted effect coding to factor variables, and to interactions between (a.) a factor variable and a continuous variable and between (b.) two factor variables.
Weighted Entropy The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function.
Weighted Finite Automata
Approximating probabilistic models as weighted finite automata
Weighted Hausdorff Distance Recent advances in Convolutional Neural Networks (CNN) have achieved remarkable results in localizing objects in images. In these networks, the training procedure usually requires providing bounding boxes or the maximum number of expected objects. In this paper, we address the task of estimating object locations without annotated bounding boxes, which are typically hand-drawn and time consuming to label. We propose a loss function that can be used in any Fully Convolutional Network (FCN) to estimate object locations. This loss function is a modification of the Average Hausdorff Distance between two unordered sets of points. The proposed method does not require one to ‘guess’ the maximum number of objects in the image, and has no notion of bounding boxes, region proposals, or sliding windows. We evaluate our method with three datasets designed to locate people’s heads, pupil centers and plant centers. We report an average precision and recall of 94% for the three datasets, and an average location error of 6 pixels in 256×256 images.
Weighted Inverse Laplacian
Community detection was a hot topic on network analysis, where the main aim is to perform unsupervised learning or clustering in networks. Recently, semi-supervised learning has received increasing attention among researchers. In this paper, we propose a new algorithm, called weighted inverse Laplacian (WIL), for predicting labels in partially labeled networks. The idea comes from the first hitting time in random walk, and it also has nice explanations both in information propagation and the regularization framework. We propose a partially labeled degree-corrected block model (pDCBM) to describe the generation of partially labeled networks. We show that WIL ensures the misclassification rate is of order $O(\frac{1}{d})$ for the pDCBM with average degree $d=\Omega(\log n),$ and that it can handle situations with greater unbalanced than traditional Laplacian methods. WIL outperforms other state-of-the-art methods in most of our simulations and real datasets, especially in unbalanced networks and heterogeneous networks.
Weighted Label Smoothing Regularization
Conventional approaches used supervised learning to estimate off-line writer identifications. In this study, we improved the off-line writer identifications by semi-supervised feature learning pipeline, which trained the extra unlabeled data and the original labeled data simultaneously. In specific, we proposed a weighted label smoothing regularization (WLSR) method, which assigned the weighted uniform label distribution to the extra unlabeled data. We regularized the convolutional neural network (CNN) baseline, which allows learning more discriminative features to represent the properties of different writing styles. Based on experiments on ICDAR2013, CVL and IAM benchmark datasets, our results showed that semi-supervised feature learning improved the baseline measurement and achieved better performance compared with existing writer identifications approaches.
Weighted Majority Algorithm
In machine learning, Weighted Majority Algorithm (WMA) is a meta-learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes that we have no prior knowledge about the accuracy of the algorithms in the pool, but there are sufficient reasons to believe that one or more will perform well. There are many variations of the Weighted Majority Algorithm to handle different situations, like shifting targets, infinite pools, or randomized predictions. The core mechanism remain similar, with the final performances of the compound algorithm bounded by a function of the performance of the specialist (best performing algorithm) in the pool.
Weighted Mean Curvature In image processing tasks, spatial priors are essential for robust computations, regularization, algorithmic design and Bayesian inference. In this paper, we introduce weighted mean curvature (WMC) as a novel image prior and present an efficient computation scheme for its discretization in practical image processing applications. We first demonstrate the favorable properties of WMC, such as sampling invariance, scale invariance, and contrast invariance with Gaussian noise model; and we show the relation of WMC to area regularization. We further propose an efficient computation scheme for discretized WMC, which is demonstrated herein to process over 33.2 giga-pixels/second on GPU. This scheme yields itself to a convolutional neural network representation. Finally, WMC is evaluated on synthetic and real images, showing its superiority quantitatively to total-variation and mean curvature.
Weighted Multisource Tradaboost In this paper we propose an improved method for transfer learning that takes into account the balance between target and source data. This method builds on the state-of-the-art Multisource Tradaboost, but weighs the importance of each datapoint taking into account the amount of target and source data available. A comparative study is then presented exposing the performance of four transfer learning methods as well as the proposed Weighted Multisource Tradaboost. The experimental results show that the proposed method is able to outperform the base method as the number of target samples increase. These results are promising in the sense that source-target ratio weighing may be a path to improve current methods of transfer learning. However, against the asymptotic conjecture, all transfer learning methods tested in this work get outperformed by a no-transfer SVM for large number on target samples.
Weighted Network
In recent years, there has been increasing demand for automatic architecture search in deep learning. Numerous approaches have been proposed and led to state-of-the-art results in various applications, including image classification and language modeling. In this paper, we propose a novel way of architecture search by means of weighted networks (WeNet), which consist of a number of networks, with each assigned a weight. These weights are updated with back-propagation to reflect the importance of different networks. Such weighted networks bear similarity to mixture of experts. We conduct experiments on Penn Treebank and WikiText-2. We show that the proposed WeNet can find recurrent architectures which result in state-of-the-art performance.
Weighted Nonlinear Regression Nonlinear Least Squares
Weighted Object k-Means Weighted object version of k-means algorithm, robust against outlier data.
Weighted Ontology Approximation Heuristic
The present paper presents the Weighted Ontology Approximation Heuristic (WOAH), a novel zero-shot approach to ontology estimation for conversational agents development environments. This methodology extracts verbs and nouns separately from data by distilling the dependencies obtained and applying similarity and sparsity metrics to generate an ontology estimation configurable in terms of the level of generalization.
Weighted Ordered Weighted Aggregation
From a formal point of view, the WOWA operator is a particular case of Choquet integral (using a particular type of measure: a distorted probability).
Weighted Orthogonal Components Regression Analysis
In the multiple linear regression setting, we propose a general framework, termed weighted orthogonal components regression (WOCR), which encompasses many known methods as special cases, including ridge regression and principal components regression. WOCR makes use of the monotonicity inherent in orthogonal components to parameterize the weight function. The formulation allows for efficient determination of tuning parameters and hence is computationally advantageous. Moreover, WOCR offers insights for deriving new better variants. Specifically, we advocate weighting components based on their correlations with the response, which leads to enhanced predictive performance. Both simulated studies and real data examples are provided to assess and illustrate the advantages of the proposed methods.
Weighted Parallel SGD
Stochastic gradient descent (SGD) is a popular stochastic optimization method in machine learning. Traditional parallel SGD algorithms, e.g., SimuParallel SGD, often require all nodes to have the same performance or to consume equal quantities of data. However, these requirements are difficult to satisfy when the parallel SGD algorithms run in a heterogeneous computing environment; low-performance nodes will exert a negative influence on the final result. In this paper, we propose an algorithm called weighted parallel SGD (WP-SGD). WP-SGD combines weighted model parameters from different nodes in the system to produce the final output. WP-SGD makes use of the reduction in standard deviation to compensate for the loss from the inconsistency in performance of nodes in the cluster, which means that WP-SGD does not require that all nodes consume equal quantities of data. We also analyze the theoretical feasibility of running two other parallel SGD algorithms combined with WP-SGD in a heterogeneous environment. The experimental results show that WP-SGD significantly outperforms the traditional parallel SGD algorithms on distributed training systems with an unbalanced workload.
Weighted Quantile Sum
Weighted Random Survival Forest A weighted random survival forest is presented in the paper. It can be regarded as a modification of the random forest improving its performance. The main idea underlying the proposed model is to replace the standard procedure of averaging used for estimation of the random survival forest hazard function by weighted avaraging where the weights are assigned to every tree and can be veiwed as training paremeters which are computed in an optimal way by solving a standard quadratic optimization problem maximizing Harrell’s C-index. Numerical examples with real data illustrate the outperformance of the proposed model in comparison with the original random survival forest.
Weighted Score Table
Weighted Sigmoid Gate Unit
An activation function has crucial role in a deep neural network. A simple rectified linear unit (ReLU) are widely used for the activation function. In this paper, a weighted sigmoid gate unit (WiG) is proposed as the activation function. The proposed WiG consists of a multiplication of inputs and the weighted sigmoid gate. It is shown that the WiG includes the ReLU and same activation functions as a special case. Many activation functions have been proposed to overcome the performance of the ReLU. In the literature, the performance is mainly evaluated with an object recognition task. The proposed WiG is evaluated with the object recognition task and the image restoration task. Then, the expeirmental comparisons demonstrate the proposed WiG overcomes the existing activation functions including the ReLU.
Weighted Source-to-Distortion Ratio
Most deep learning-based models for speech enhancement have mainly focused on estimating the magnitude of spectrogram while reusing the phase from noisy speech for reconstruction. This is due to the difficulty of estimating the phase of clean speech. To improve speech enhancement performance, we tackle the phase estimation problem in three ways. First, we propose Deep Complex U-Net, an advanced U-Net structured model incorporating well-defined complex-valued building blocks to deal with complex-valued spectrograms. Second, we propose a polar coordinate-wise complex-valued masking method to reflect the distribution of complex ideal ratio masks. Third, we define a novel loss function, weighted source-to-distortion ratio (wSDR) loss, which is designed to directly correlate with a quantitative evaluation measure. Our model was evaluated on a mixture of the Voice Bank corpus and DEMAND database, which has been widely used by many deep learning models for speech enhancement. Ablation experiments were conducted on the mixed dataset showing that all three proposed approaches are empirically valid. Experimental results show that the proposed method achieves state-of-the-art performance in all metrics, outperforming previous approaches by a large margin.
Weighted Topological Overlaps
Weighted-SVD The Matrix Factorization models, sometimes called the latent factor models, are a family of methods in the recommender system research area to (1) generate the latent factors for the users and the items and (2) predict users’ ratings on items based on their latent factors. However, current Matrix Factorization models presume that all the latent factors are equally weighted, which may not always be a reasonable assumption in practice. In this paper, we propose a new model, called Weighted-SVD, to integrate the linear regression model with the SVD model such that each latent factor accompanies with a corresponding weight parameter. This mechanism allows the latent factors have different weights to influence the final ratings. The complexity of the Weighted-SVD model is slightly larger than the SVD model but much smaller than the SVD++ model. We compared the Weighted-SVD model with several latent factor models on five public datasets based on the Root-Mean-Squared-Errors (RMSEs). The results show that the Weighted-SVD model outperforms the baseline methods in all the experimental datasets under almost all settings.
Weight-Median Sketch We introduce a new sub-linear space data structure—the Weight-Median Sketch—that captures the most heavily weighted features in linear classifiers trained over data streams. This enables memory-limited execution of several statistical analyses over streams, including online feature selection, streaming data explanation, relative deltoid detection, and streaming estimation of pointwise mutual information. In contrast with related sketches that capture the most commonly occurring features (or items) in a data stream, the Weight-Median Sketch captures the features that are most discriminative of one stream (or class) compared to another. The Weight-Median sketch adopts the core data structure used in the Count-Sketch, but, instead of sketching counts, it captures sketched gradient updates to the model parameters. We provide a theoretical analysis of this approach that establishes recovery guarantees in the online learning setting, and demonstrate substantial empirical improvements in accuracy-memory trade-offs over alternatives, including count-based sketches and feature hashing.
“Waikato Environment for Knowledge Analysis”
Whale Optimization Algorithm
Whale Optimization Algorithm (WOA) is a recently proposed (2016) optimization algorithm mimicking the hunting mechanism of humpback whales in nature. It is worth mentioning here that bubble-net feeding is a unique behavior that can only be observed in humpback whales. In WOA the spiral bubble-net feeding maneuver is mathematically modeled in order to perform optimization.
A Systematic and Meta-analysis Survey of Whale Optimization Algorithm
What-If Tool What If… you could inspect a machine learning model, with no coding required? Building effective machine learning systems means asking a lot of questions. It’s not enough to train a model and walk away. Instead, good practitioners act as detectives, probing to understand their model better. But answering these kinds of questions isn’t easy. Probing ‘what if’ scenarios often means writing custom, one-off code to analyze a specific model. Not only is this process inefficient, it makes it hard for non-programmers to participate in the process of shaping and improving machine learning models. For us, making it easier for a broad set of people to examine, evaluate, and debug machine learning systems is a key concern. That’s why we built the What-If Tool. Built into the open-source TensorBoard web application – a standard part of the TensorFlow platform – the tool allows users to analyze an machine learning model without the need for writing any further code. Given pointers to a TensorFlow model and a dataset, the What-If Tool offers an interactive visual interface for exploring model results.
WHInter Learning sparse linear models with two-way interactions is desirable in many application domains such as genomics. l1-regularised linear models are popular to estimate sparse models, yet standard implementations fail to address specifically the quadratic explosion of candidate two-way interactions in high dimensions, and typically do not scale to genetic data with hundreds of thousands of features. Here we present WHInter, a working set algorithm to solve large l1-regularised problems with two-way interactions for binary design matrices. The novelty of WHInter stems from a new bound to efficiently identify working sets while avoiding to scan all features, and on fast computations inspired from solutions to the maximum inner product search problem. We apply WHInter to simulated and real genetic data and show that it is more scalable and two orders of magnitude faster than the state of the art.
White Noise In signal processing, white noise is a random signal with a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustic engineering, telecommunications, statistical forecasting, and many more. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal. A ‘white noise’ image. In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance; a single realization of white noise is a random shock. Depending on the context, one may also require that the samples be independent and have the same probability distribution (in other words i.i.d is a simplest representative of the white noise). In particular, if each sample has a normal distribution with zero mean, the signal is said to be Gaussian white noise. The samples of a white noise signal may be sequential in time, or arranged along one or more spatial dimensions. In digital image processing, the pixels of a white noise image are typically arranged in a rectangular grid, and are assumed to be independent random variables with uniform probability distribution over some interval. The concept can be defined also for signals spread over more complicated domains, such as a sphere or a torus. Some ‘white noise’ sound. An infinite-bandwidth white noise signal is a purely theoretical construction. The bandwidth of white noise is limited in practice by the mechanism of noise generation, by the transmission medium and by finite observation capabilities. Thus, a random signal is considered ‘white noise’ if it is observed to have a flat spectrum over the range of frequencies that is relevant to the context. For an audio signal, for example, the relevant range is the band of audible sound frequencies, between 20 to 20,000 Hz. Such a signal is heard as a hissing sound, resembling the /sh/ sound in ‘ash’. In music and acoustics, the term ‘white noise’ may be used for any signal that has a similar hissing sound. White noise draws its name from white light, although light that appears white generally does not have a flat spectral power density over the visible band. The term white noise is sometimes used in the context of phylogenetically based statistical methods to refer to a lack of phylogenetic pattern in comparative data. It is sometimes used in non technical contexts, in the metaphoric sense of ‘random talk without meaningful contents’.
White Noise Test
Whitening Transformation A whitening transformation is a decorrelation transformation that transforms a set of random variables having a known covariance matrix into a set of new random variables whose covariance is the identity matrix (meaning that they are uncorrelated and all have variance 1). The transformation is called “whitening” because it changes the input vector into a white noise vector. It differs from a general decorrelation transformation in that the latter only makes the covariances equal to zero, so that the correlation matrix may be any diagonal matrix. The inverse coloring transformation transforms a vector of uncorrelated variables (a white random vector) into a vector with a specified covariance matrix.
Whittemore This paper introduces Whittemore, a language for causal programming. Causal programming is based on the theory of structural causal models and consists of two primary operations: identification, which finds formulas that compute causal queries, and estimation, which applies formulas to transform probability distributions to other probability distribution. Causal programming provides abstractions to declare models, queries, and distributions with syntax similar to standard mathematical notation, and conducts rigorous causal inference, without requiring detailed knowledge of the underlying algorithms. Examples of causal inference with real data are provided, along with discussion of the implementation and possibilities for future extension.
Widely Applicable Bayesian Information Criterion
A statistical model or a learning machine is called regular if the map taking a parameter to a probability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant, the real log canonical threshold (RLCT), instead of half the number of parameters in BIC. Theoretical values of RLCTs in several statistical models are now being discovered based on algebraic geometrical methodology. However, it has been difficult to estimate the Bayes free energy using only training samples, because an RLCT depends on an unknown true distribution. In the present paper, we define a widely applicable Bayesian information criterion (WBIC) by the average log likelihood function over the posterior distribution with the inverse temperature 1/logn, where n is the number of training samples. We mathematically prove that WBIC has the same asymptotic expansion as the Bayes free energy, even if a statistical model is singular for or unrealizable by a statistical model. Since WBIC can be numerically calculated without any information about a true distribution, it is a generalized version of BIC onto singular statistical models.
“Watanabe-Akaike Information Criteria”
Widely Applicable Information Criterion
“Watanabe-Akaike Information Criteria”
Wiener Polarity Index The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u,v} in G such that the distance between u and v is equal to 3.
Wiener Process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, and physics. The Wiener process plays an important role both in pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm-Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a Gaussian white noise process, and so is useful as a model of noise in electronics engineering, instrument errors in filtering theory and unknown forces in control theory. The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker-Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman-Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black-Scholes option pricing model.
Wiener-Filter In signal processing, the Wiener Filter (Wiener-Kolmogorov Filter) is a filter used to produce an estimate of a desired or target random process by linear time-invariant filtering of an observed noisy process, assuming known stationary signal and noise spectra, and additive noise. The Wiener filter minimizes the mean square error between the estimated random process and the desired process.
WikiAtomicEdits We release a corpus of 43 million atomic edits across 8 languages. These edits are mined from Wikipedia edit history and consist of instances in which a human editor has inserted a single contiguous phrase into, or deleted a single contiguous phrase from, an existing sentence. We use the collected data to show that the language generated during editing differs from the language that we observe in standard corpora, and that models trained on edits encode different aspects of semantics and discourse than models trained on raw, unstructured text. We release the full corpus as a resource to aid ongoing research in semantics, discourse, and representation learning.
Wikibook-Bot A Wikipedia book (known as Wikibook) is a collection of Wikipedia articles on a particular theme that is organized as a book. We propose Wikibook-Bot, a machine-learning based technique for automatically generating high quality Wikibooks based on a concept provided by the user. In order to create the Wikibook we apply machine learning algorithms to the different steps of the proposed technique. Firs, we need to decide whether an article belongs to a specific Wikibook – a classification task. Then, we need to divide the chosen articles into chapters – a clustering task – and finally, we deal with the ordering task which includes two subtasks: order articles within each chapter and order the chapters themselves. We propose a set of structural, text-based and unique Wikipedia features, and we show that by using these features, a machine learning classifier can successfully address the above challenges. The predictive performance of the proposed method is evaluated by comparing the auto-generated books to existing 407 Wikibooks which were manually generated by humans. For all the tasks we were able to obtain high and statistically significant results when comparing the Wikibook-bot books to books that were manually generated by Wikipedia contributors
WikiConv We present a corpus that encompasses the complete history of conversations between contributors to Wikipedia, one of the largest online collaborative communities. By recording the intermediate states of conversations—including not only comments and replies, but also their modifications, deletions and restorations—this data offers an unprecedented view of online conversation. This level of detail supports new research questions pertaining to the process (and challenges) of large-scale online collaboration. We illustrate the corpus’ potential with two case studies that highlight new perspectives on earlier work. First, we explore how a person’s conversational behavior depends on how they relate to the discussion’s venue. Second, we show that community moderation of toxic behavior happens at a higher rate than previously estimated. Finally the reconstruction framework is designed to be language agnostic, and we show that it can extract high quality conversational data in both Chinese and English.
WikiLinkGraphs Wikipedia articles contain multiple links connecting a subject to other pages of the encyclopedia. In Wikipedia parlance, these links are called internal links or wikilinks. We present a complete dataset of the network of internal Wikipedia links for the $9$ largest language editions. The dataset contains yearly snapshots of the network and spans $17$ years, from the creation of Wikipedia in 2001 to March 1st, 2018. While previous work has mostly focused on the complete hyperlink graph which includes also links automatically generated by templates, we parsed each revision of each article to track links appearing in the main text. In this way we obtained a cleaner network, discarding more than half of the links and representing all and only the links intentionally added by editors. We describe in detail how the Wikipedia dumps have been processed and the challenges we have encountered, including the need to handle special pages such as redirects, i.e., alternative article titles. We present descriptive statistics of several snapshots of this network. Finally, we propose several research opportunities that can be explored using this new dataset.
Wikipedia WordNet Based QE Technique
Query expansion (QE) is a well known technique to enhance the effectiveness of information retrieval (IR). QE reformulates the initial query by adding similar terms that helps in retrieving more relevant results. Several approaches have been proposed with remarkable outcome, but they are not evenly favorable for all types of queries. One of the main reasons for this is the use of the same data source while expanding both the individual and the phrase query terms. As a result, the holistic relationship among the query terms is not well captured. To address this issue, we have selected separate data sources for individual and phrase terms. Specifically, we have used WordNet for expanding individual terms and Wikipedia for expanding phrase terms. We have also proposed novel schemes for weighting expanded terms: inlink score (for terms extracted from Wikipedia) and a tfidf based scheme (for terms extracted from WordNet). In the proposed Wikipedia WordNet based QE technique (WWQE), we weigh the expansion terms twice: first, they are scored by the weighting scheme individually, and then, the weighting scheme scores the selected expansion terms in relation to the entire query using correlation score. The experimental results show that the proposed approach successfully combines Wikipedia and WordNet as demonstrated through a better performance on standard evaluation metrics on FIRE dataset. The proposed WWQE approach is also suitable with other standard weighting models for improving the effectiveness of IR.
Wikipedia2Vec We present Wikipedia2Vec, an open source tool for learning embeddings of words and entities from Wikipedia. This tool enables users to easily obtain high-quality embeddings of words and entities from a Wikipedia dump with a single command. The learned embeddings can be used as features in downstream natural language processing (NLP) models. The tool can be installed via PyPI. The source code, documentation, and pretrained embeddings for 12 major languages can be obtained at http://wikipedia2vec.github.io.
WikiRank Keyphrase is an efficient representation of the main idea of documents. While background knowledge can provide valuable information about documents, they are rarely incorporated in keyphrase extraction methods. In this paper, we propose WikiRank, an unsupervised method for keyphrase extraction based on the background knowledge from Wikipedia. Firstly, we construct a semantic graph for the document. Then we transform the keyphrase extraction problem into an optimization problem on the graph. Finally, we get the optimal keyphrase set to be the output. Our method obtains improvements over other state-of-art models by more than 2% in F1-score.
Wikistat 2.0 Big data, data science, deep learning, artificial intelligence are the key words of intense hype related with a job market in full evolution, that impose to adapt the contents of our university professional trainings. Which artificial intelligence is mostly concerned by the job offers? Which methodologies and technologies should be favored in the training pprograms? Which objectives, tools and educational resources do we needed to put in place to meet these pressing needs? We answer these questions in describing the contents and operational ressources in the Data Science orientation of the speciality Applied Mathematics at INSA Toulouse. We focus on basic mathematics training (Optimization, Probability, Statistics), associated with the practical implementation of the most performing statistical learning algorithms, with the most appropriate technologies and on real examples. Considering the huge volatility of the technologies, it is imperative to train students in seft-training, this will be their technological watch tool when they will be in professional activity. This explains the structuring of the educational site https://…/wikistat into a set of tutorials. Finally, to motivate the thorough practice of these tutorials, a serious game is organized each year in the form of a prediction contest between students of Master degrees in Applied Mathematics for IA.
Wild Scale-Enhanced Bootstrap
Wildly-Unsupervised Domain Adaptation
Unsupervised domain adaptation (UDA) trains with clean labeled data in source domain and unlabeled data in target domain to classify target-domain data. However, in real-world scenarios, it is hard to acquire fully-clean labeled data in source domain due to the expensive labeling cost. This brings us a new but practical adaptation called wildly-unsupervised domain adaptation (WUDA), which aims to transfer knowledge from noisy labeled data in source domain to unlabeled data in target domain. To tackle the WUDA, we present a robust one-step approach called Butterfly, which trains four networks. Specifically, two networks are jointly trained on noisy labeled data in source domain and pseudo-labeled data in target domain (i.e., data in mixture domain). Meanwhile, the other two networks are trained on pseudo-labeled data in target domain. By using dual-checking principle, Butterfly can obtain high-quality target-specific representations. We conduct experiments to demonstrate that Butterfly significantly outperforms other baselines on simulated and real-world WUDA tasks in most cases.
Window-based Sentence Boundary Evaluation
Sentence Boundary Detection (SBD) has been a major research topic since Automatic Speech Recognition transcripts have been used for further Natural Language Processing tasks like Part of Speech Tagging, Question Answering or Automatic Summarization. But what about evaluation? Do standard evaluation metrics like precision, recall, F-score or classification error; and more important, evaluating an automatic system against a unique reference is enough to conclude how well a SBD system is performing given the final application of the transcript? In this paper we propose Window-based Sentence Boundary Evaluation (WiSeBE), a semi-supervised metric for evaluating Sentence Boundary Detection systems based on multi-reference (dis)agreement. We evaluate and compare the performance of different SBD systems over a set of Youtube transcripts using WiSeBE and standard metrics. This double evaluation gives an understanding of how WiSeBE is a more reliable metric for the SBD task.
Window-Bounded co-Occurrence This paper focuses on a traditional relation extraction task in the context of limited annotated data and a narrow knowledge domain. We explore this task with a clinical corpus consisting of 200 breast cancer follow-up treatment letters in which 16 distinct types of relations are annotated. We experiment with an approach to extracting typed relations called window-bounded co-occurrence (WBC), which uses an adjustable context window around entity mentions of a relevant type, and compare its performance with a more typical intra-sentential co-occurrence baseline. We further introduce a new bag-of-concepts (BoC) approach to feature engineering based on the state-of-the-art word embeddings and word synonyms. We demonstrate the competitiveness of BoC by comparing with methods of higher complexity, and explore its effectiveness on this small dataset.
Windowed Fourier Filtering Interferometric phase (InPhase) imaging is an important part of many present-day coherent imaging technologies. Often in such imaging techniques, the acquired images, known as interferograms, suffer from two major degradations: 1) phase wrapping caused by the fact that the sensing mechanism can only measure sinusoidal $2\pi$-periodic functions of the actual phase, and 2) noise introduced by the acquisition process or the system. This work focusses on InPhase denoising which is a fundamental restoration step to many posterior applications of InPhase, namely to phase unwrapping. The presence of sharp fringes that arises from phase wrapping makes InPhase denoising a hard-inverse problem. Motivated by the fact that the InPhase images are often locally sparse in Fourier domain, we propose a multi-resolution windowed Fourier filtering (WFF) analysis that fuses WFF estimates with different resolutions, thus overcoming the WFF fixed resolution limitation. The proposed fusion relies on an unbiased estimate of the mean square error derived using the Stein’s lemma adapted to complex-valued signals. This estimate, known as SURE, is minimized using an optimization framework to obtain the fusion weights. Strong experimental evidence, using synthetic and real (InSAR & MRI) data, that the developed algorithm, termed as SURE-fuse WFF, outperforms the best hand-tuned fixed resolution WFF as well as other state-of-the-art InPhase denoising algorithms, is provided.
Wire Data Wire data is the information that passes over computer and telecommunication networks defining communications between client and server devices. It is the result of decoding wire and transport protocols containing the bi-directional data payload. More precisely, wire data is the information that is communicated in each layer of the OSI model (Layer 1 not being included because those protocols are used to establish connections and do not communicate information).
Wisdom of Crowds
The wisdom of the crowd is the collective opinion of a group of individuals rather than that of a single expert. A large group’s aggregated answers to questions involving quantity estimation, general world knowledge, and spatial reasoning has generally been found to be as good as, and often better than, the answer given by any of the individuals within the group. An explanation for this phenomenon is that there is idiosyncratic noise associated with each individual judgment, and taking the average over a large number of responses will go some way toward canceling the effect of this noise.[1] This process, while not new to the Information Age, has been pushed into the mainstream spotlight by social information sites such as Wikipedia, Yahoo! Answers, Quora, and other web resources that rely on human opinion.[2] Trial by jury can be understood as wisdom of the crowd, especially when compared to the alternative, trial by a judge, the single expert. In politics, sometimes sortition is held as an example of what wisdom of the crowd would look like. Decision-making would happen by a diverse group instead of by a fairly homogenous political group or party. Research within cognitive science has sought to model the relationship between wisdom of the crowd effects and individual cognition.
WoCE: a framework for clustering ensemble by exploiting the wisdom of Crowds theory
Wishart Distribution In statistics, the Wishart distribution is a generalization to multiple dimensions of the chi-squared distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928. It is a family of probability distributions defined over symmetric, nonnegative-definite matrix-valued random variables (‘random matrices’). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector.
Wishart Matrix “Wishart Distribution”
Witness-Counting Problem Fast Witness Counting
W-Net Crowd management is of paramount importance when it comes to preventing stampedes and saving lives, especially in a country like China and India where the combined population is a third of the global population. Millions of people convene annually all around the nation to celebrate a myriad of events and crowd count estimation is the linchpin of the crowd management system that could prevent stampedes and save lives. We present a network for crowd counting which reports state of the art results on crowd counting benchmarks. Our contributions are, first, a U-Net inspired model which affords us to report state of the art results. Second, we propose an independent decoding Reinforcement branch which helps the network converge much earlier and also enables the network to estimate density maps with high Structural Similarity Index (SSIM). Third, we discuss the drawbacks of the contemporary architectures and empirically show that even though our architecture achieves state of the art results, the merit may be due to the encoder-decoder pipeline instead. Finally, we report the error analysis which shows that the contemporary line of work is at saturation and leaves certain prominent problems unsolved.
Wolfson Polarization Index affluenceIndex
Word Embedding Association Test
Universal Sentence Encoder
Word Embedding Attention Network
Most recent approaches use the sequence-to-sequence model for paraphrase generation. The existing sequence-to-sequence model tends to memorize the words and the patterns in the training dataset instead of learning the meaning of the words. Therefore, the generated sentences are often grammatically correct but semantically improper. In this work, we introduce a novel model based on the encoder-decoder framework, called Word Embedding Attention Network (WEAN). Our proposed model generates the words by querying distributed word representations (i.e. neural word embeddings), hoping to capturing the meaning of the according words. Following previous work, we evaluate our model on two paraphrase-oriented tasks, namely text simplification and short text abstractive summarization. Experimental results show that our model outperforms the sequence-to-sequence baseline by the BLEU score of 6.3 and 5.5 on two English text simplification datasets, and the ROUGE-2 F1 score of 5.7 on a Chinese summarization dataset. Moreover, our model achieves state-of-the-art performances on these three benchmark datasets.
Word Encoded Sequence Transducer
Most of the parameters in large vocabulary models are used in embedding layer to map categorical features to vectors and in softmax layer for classification weights. This is a bottle-neck in memory constraint on-device training applications like federated learning and on-device inference applications like automatic speech recognition (ASR). One way of compressing the embedding and softmax layers is to substitute larger units such as words with smaller sub-units such as characters. However, often the sub-unit models perform poorly compared to the larger unit models. We propose WEST, an algorithm for encoding categorical features and output classes with a sequence of random or domain dependent sub-units and demonstrate that this transduction can lead to significant compression without compromising performance. WEST bridges the gap between larger unit and sub-unit models and can be interpreted as a MaxEnt model over sub-unit features, which can be of independent interest.
Word ExtrAction for time SEries cLassification
Time series (TS) occur in many scientific and commercial applications, ranging from earth surveillance to industry automation to the smart grids. An important type of TS analysis is classification, which can, for instance, improve energy load forecasting in smart grids by detecting the types of electronic devices based on their energy consumption profiles recorded by automatic sensors. Such sensor-driven applications are very often characterized by (a) very long TS and (b) very large TS datasets needing classification. However, current methods to time series classification (TSC) cannot cope with such data volumes at acceptable accuracy; they are either scalable but offer only inferior classification quality, or they achieve state-of-the-art classification quality but cannot scale to large data volumes. In this paper, we present WEASEL (Word ExtrAction for time SEries cLassification), a novel TSC method which is both scalable and accurate. Like other state-of-the-art TSC methods, WEASEL transforms time series into feature vectors, using a sliding-window approach, which are then analyzed through a machine learning classifier. The novelty of WEASEL lies in its specific method for deriving features, resulting in a much smaller yet much more discriminative feature set. On the popular UCR benchmark of 85 TS datasets, WEASEL is more accurate than the best current non-ensemble algorithms at orders-of-magnitude lower classification and training times, and it is almost as accurate as ensemble classifiers, whose computational complexity makes them inapplicable even for mid-size datasets. The outstanding robustness of WEASEL is also confirmed by experiments on two real smart grid datasets, where it out-of-the-box achieves almost the same accuracy as highly tuned, domain-specific methods.
Word Sense Induction
In computational linguistics, word-sense induction (WSI) or discrimination is an open problem of natural language processing, which concerns the automatic identification of the senses of a word (i.e. meanings). Given that the output of word-sense induction is a set of senses for the target word (sense inventory), this task is strictly related to that of word-sense disambiguation (WSD), which relies on a predefined sense inventory and aims to solve the ambiguity of words in context.
Word Vectors Word vectors (also referred to as distributed representations) are an amazing alternative that sweep away most of the issues of dealing with NLP. They let us ignore the difficult-to-understand grammar & syntax of language while retaining the ability to ask and answer simple questions about a text.
Word2Bits Word vectors require significant amounts of memory and storage, posing issues to resource limited devices like mobile phones and GPUs. We show that high quality quantized word vectors using 1-2 bits per parameter can be learned by introducing a quantization function into Word2Vec. We furthermore show that training with the quantization function acts as a regularizer. We train word vectors on English Wikipedia (2017) and evaluate them on standard word similarity and analogy tasks and on question answering (SQuAD). Our quantized word vectors not only take 8-16x less space than full precision (32 bit) word vectors but also outperform them on word similarity tasks and question answering.
DL4J: Word2Vec
Wordswarm WordSwarm generates dynamic word clouds in which the word size changes as the animation moves forward through the corpus. The top words from the preprocessing are colored randomly or from an assigned pallet, sized according to their magnitude at the first date, and then displayed in a pseudo-random location on the screen. The animation progresses into the future by growing or shrinking each word according to its frequency in the corpus at the next date. Clash detection is achieved using a 2D physics engine, which also applies ‘gravitational force’ to each word, bringing the larger words closer to the center of the screen.
Work Stealing Load Balancing Algorithm A methodology for efficient load balancing of computational problems that can be easily decomposed into multiple tasks, but where it is hard to predict the computation cost of each task, and where new tasks are created dynamically during runtime. We present this methodology and its exploitation and feasibility in the context of graphics processors. Work-stealing allows an idle core to acquire tasks from a core that is overloaded, causing the total work to be distributed evenly among cores, while minimizing the communication costs, as tasks are only redistributed when required. This will often lead to higher throughput than using static partitioning.
Work Stealing with latency
Workflow Satisfiability Problem The Workflow Satisfiability Problem (WSP) Asks Whether There Exists an Assignment of Authorized Users to the Steps in a Workflow Specification That Satisfies the Constraints in the Specification. The Problem is NP-Hard in General, but Several Subclasses of the Problem are Known to be Fixed-Parameter Tractable (FPT) When Parameterized by the Number of Steps in the Specification.
Bounded and Approximate Strong Satisfiability in Workflows
Workforce Analytics Workforce analytics is a combination of software and methodology that applies statistical models to worker-related data, allowing enterprise leaders to optimize human resource management (HRM).
Workload-Aware Auto-Parallelization Framework
Deep neural networks (DNNs) have emerged as successful solutions for variety of artificial intelligence applications, but their very large and deep models impose high computational requirements during training. Multi-GPU parallelization is a popular option to accelerate demanding computations in DNN training, but most state-of-the-art multi-GPU deep learning frameworks not only require users to have an in-depth understanding of the implementation of the frameworks themselves, but also apply parallelization in a straight-forward way without optimizing GPU utilization. In this work, we propose a workload-aware auto-parallelization framework (WAP) for DNN training, where the work is automatically distributed to multiple GPUs based on the workload characteristics. We evaluate WAP using TensorFlow with popular DNN benchmarks (AlexNet and VGG-16), and show competitive training throughput compared with the state-of-the-art frameworks, and also demonstrate that WAP automatically optimizes GPU assignment based on the workload’s compute requirements, thereby improving energy efficiency.
WPU-Net Deep learning has driven great progress in natural and biological image processing. However, in materials science and engineering, there are often some flaws and indistinctions in material microscopic images induced from complex sample preparation, even due to the material itself, hindering the detection of target objects. In this work, we propose WPU-net that redesign the architecture and weighted loss of U-Net to force the network to integrate information from adjacent slices and pay more attention to the topology in this boundary detection task. Then, the WPU-net was applied into a typical material example, i.e., the grain boundary detection of polycrystalline material. Experiments demonstrate that the proposed method achieves promising performance compared to state-of-the-art methods. Besides, we propose a new method for object tracking between adjacent slices, which can effectively reconstruct the 3D structure of the whole material while maintaining relative accuracy.
Write Once, Deploy Anywhere
Write Once, Run Anywhere
Write once, run anywhere’ (WORA), or sometimes write once, run everywhere (WORE), is a slogan created by Sun Microsystems to illustrate the cross-platform benefits of the Java language. Ideally, this means Java can be developed on any device, compiled into a standard bytecode and be expected to run on any device equipped with a Java virtual machine (JVM). The installation of a JVM or Java interpreter on chips, devices or software packages has become an industry standard practice. This means a programmer can develop code on a PC and can expect it to run on Java enabled cell phones, as well as on routers and mainframes equipped with Java, without any adjustments. This is intended to save software developers the effort of writing a different version of their software for each platform or operating system they intend to deploy on. This idea originated as early as in the late 1970s, when the UCSD Pascal system was developed to produce and interpret p-code. UCSD Pascal (along with the Smalltalk virtual machine) was a key influence on the design of the Java virtual machine, as is cited by James Gosling. The catch is that since there are multiple JVM implementations, on top of a wide variety of different operating systems such as Windows, Linux, Solaris, NetWare, HP-UX, and Mac OS, there can be subtle differences in how a program may execute on each JVM/OS combination, which may require an application to be tested on various target platforms. This has given rise to a joke among Java developers, ‘Write Once, Debug Everywhere’. This architecture has sometimes been criticized as ‘Saying that Java is better because it works in all platforms is like saying that Anal Sex is better because it works with all genders.’. In comparison, the Squeak Smalltalk programming language and environment, boasts as being, ‘truly write once run anywhere’, because it ‘runs bit-identical images across its wide portability base’
WStream In the recent years, the scale of graph datasets has increased to such a degree that a single machine is not capable of efficiently processing large graphs. Thereby, efficient graph partitioning is necessary for those large graph applications. Traditional graph partitioning generally loads the whole graph data into the memory before performing partitioning; this is not only a time consuming task but it also creates memory bottlenecks. These issues of memory limitation and enormous time complexity can be resolved using stream-based graph partitioning. A streaming graph partitioning algorithm reads vertices once and assigns that vertex to a partition accordingly. This is also called an one-pass algorithm. This paper proposes an efficient window-based streaming graph partitioning algorithm called WStream. The WStream algorithm is an edge-cut partitioning algorithm, which distributes a vertex among the partitions. Our results suggest that the WStream algorithm is able to partition large graph data efficiently while keeping the load balanced across different partitions, and communication to a minimum. Evaluation results with real workloads also prove the effectiveness of our proposed algorithm, and it achieves a significant reduction in load imbalance and edge-cut with different ranges of dataset. |
49b1c37ad4937500 | 33. Schrödinger Equation.
The contention between Albert Einstein and Neils Bohr was carried forward and taken to a new level by their respective students – Erwin Schrödinger and Werner Heisenberg, who were hell-bent to prove that their theory about the atom was correct.
Let us begin talking about these very talented men, who went head-to-head to prove that their line of thought was inerrant.
Erwin Schrödinger
Erwin Schrödinger was an Austrian physicist, who is credited with developing ‘Schrödinger’s equation – a mathematical equation that can explain the behavior of all systems (macroscopic, sub-atomic, atomic, molecular) in the universe!
He was known to be an attractive, charming, suave, and promiscuous individual. He was a romantic at heart and had a very passionate personality.
In 1925, Schrödinger was getting burned out by his research work, so he decided to go out for Christmas vacation. He went to the Alpine resort in Switzerland with his former girlfriend. He returned two weeks later, with his equation of wave mechanics!! This equation was amazing as it could describe every element in the periodic table!
Erwin Schrödinger took de Broglie’s idea and took it further. He formulated his thesis by considering that the electron was itself a wave of energy. This wave vibrated so fast around the nucleus that it looked hazy- like a cloud of energy.
So, the exact position of an electron could not be determined, although the probability of finding the electron could be calculated.
What does the probability of finding an electron mean?
Probability is a mathematical concept that gives us a method of calculating the chance of a given outcome.
e.g. If we toss a coin, how likely is it that we get heads? Or how probable is the outcome of getting heads? Thus, mathematically we ask a question – what is the probability of getting heads? A coin has only two sides- heads and tails. So, the chances of getting heads after flipping the coin are half i.e 50%. Thus, the probability of getting heads is 50%. Thus, we cannot know what exactly we get after tossing the coin, but we can calculate the probability of an event occurring.
The higher the probability, the higher the chance of that event occurring. The lower the probability, the lower the chances of the event occurring.
Where would you find your mother at 3am every day? At this hour she must be definitely sleeping. So, the probability of finding her in the bedroom is very high. However, what if she got thirsty and went to the kitchen to drink some water? Then she could be in the kitchen as well but the probability of that happening is very low.
Similarly, Schrödinger stated that only the probability, of the location of an electron, can be calculated. The exact location of the electron was not definite. The distribution of these probabilities, around the nucleus, formed areas of space called orbitals. Thus, the probability of finding the electrons in the region of the orbitals is very high (as shown in the figure below).
An orbital is a wave function describing the state of a single electron in an atom. He proposed an equation for wave function, as follows –
H ⇒ Hamiltonian operator
ψ ⇒ Wave function
i ⇒ Imaginary number
ħ⇒ Reduced Planck’s Constant = (h/2π)
∂/∂t ⇒ Partial differentiation symbol.
It can also be represented as –
E ⇒ Binding energy i.e the energy that binds electrons to the nucleus.
Just as we use Newton’s equations to represent what happens to a ball when kicked, we use Schrodinger’s equation to understand the behavior of sub-atomic particles. (This is because Newtonian mechanics is not applicable at the sub-atomic level). Let us try to understand these terms one by one –
• An Operator is a symbol for a certain mathematical procedure, which transforms one function into another function. So, basically, it means that we are operating on an expression and changing it to another expression.
∴ Operator(function)= Another function.
• It is a mathematical procedure or an instruction to carry out certain operations.
e.g.- a square root sign ‘√’, is an operator. If we plug in a value in it, it will simply take the square root of that number and give us a new number e.g. √9 = 3. So the square root sign is an instruction given to a number. Similar instructions can be given to functions as well. An operator has to operate on a MATHEMATICAL FUNCTION.
• a symbol that represents a particular mathematical operation being carried out.
• The symbol for an operator is ∧.
• An operator has no meaning unless some quantity is put into it.
e.g. – 1)Square root ‘√ ‘ by itself has no meaning. When a value is put in it only then does it get its significance? √ 16=4.
2)d/dx is an operator.It transforms a function into its first derivative w.r.t ‘x’.
d/dx(xn) = n xn-1
• In Quantum Mechanics all operators are linear.
• In quantum mechanics, physical properties of classical mechanics like energy, time, linear momentum, angular momentum, etc are expressed by operators.
• The Hamiltonian operator is an operator corresponding with the total energy of the system i.e it is associated with the kinetic and potential energies at the sub-atomic level. When the Hamiltonian operator operates on the wave function Ψ(Si), we get Schrodinger’s equation.
• It is a mathematical model/function, which represents a wave equation. A wave equation describes the properties of the waves and the behavior of fields. Thus, Ψ is a mathematical description of the particles, which are behaving as waves, at the sub-atomic level.ψ represents a field of some quantity that exists at all points in space.
e.g. Field of temperature on the surface of the earth,ψtemp. The temperature varies in different parts of the world. A wave equation in ψtemp would describe temperatures in different parts of the earth.
• ψ has all the measurable information about the particle. We already know that at the sub-atomic level, particles behave as waves. Thus, Ψ will give us all the wave-like information the particle exhibits. Thus, Ψ is the description of the quantum mechanical state of a particle.
• We know that according to Schrödinger, we just get probabilities at the sub-atomic level. It is a wave equation in terms of the wave function which predicts analytically and precisely the probability of events or outcomes.
• The wave function describes the position and state of the electron and its square gives the probability density of electrons. It describes the quantum state of a particle or set of particles.
Ψ 2⇒Ψ, Ψ* ⇒Probability of finding an electron in unit volume.
Ψ = wave function which has no physical interpretation and could be real or imaginary.
Ψ*=the complex conjugate of Ψ.
Ψ 2 is called the probability density of finding a particle in a particular region.
When Ψ 2 is large → high probability of finding the particle in that region
When Ψ 2 is small → low probability of finding the particle in that region
• Thus, in Quantum Mechanics, the wave function describes the state of a system by the way of probabilities.
• When the operator operates onto the wave function, it extracts all the desired information from it. This information is called the EIGEN VALUE of the observable quantity.OPERATOR(WAVE FUNCTION) = EIGENVALUES.
The Nobel Prize in Physics 1933 was awarded jointly to Erwin Schrödinger and Paul Adrien Maurice Dirac “for the discovery of new productive forms of atomic theory”
Schrödinger’s equation can only be solved for very simple species like-
i) Particle in a box
ii)Harmonic oscillator
iii)Rigid rotator
iv) 1 electron systems.
Beyond these one has to use approximation methods to solve the equation. The methods commonly used are –
1)Perturbation method
2)Variation method.
Understanding the Schrödinger equation is a very complicated process. As students of chemistry, we lack the understanding of the mathematical modeling of particles. Thus, we are only expected to understand some basic ideas of this equation. It is impossible to comprehend this equation/ quantum mechanics in totality without an in-depth knowledge of mathematics. Thus, we just try to discuss the chemical aspects of this esoteric subject. We shall discuss this equation in greater detail when we start our discussions on Quantum Mechanics. For now, understanding these basic parameters would suffice. In the next post, we shall start talking about another genius who was Schrödinger’s rival.
Till then,
Be a perpetual student of life and keep learning…
Good day!
References and further reading –
5)MIT lecture 6,3.091 by Professor Sadoway.
Image source –
1)By Nobel foundation – http://nobelprize.org/nobel_prizes/physics/laureates/1933/schrodinger-bio.html, Public Domain, https://commons.wikimedia.org/w/index.php?curid=6209244
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cce6fe50e1f20f45 | Schrödinger's equation — in action
Marianne Freiberger Share this page
In the previous article we introduced Schrödinger's equation and its solution, the wave function, which contains all the information there is to know about a quantum system. Now it's time to see the equation in action, using a very simple physical system as an example. We'll also look at another weird phenomenon called quantum tunneling. (If you'd like to skip the maths you can go straight to the third article in this series which explores the interpretation of the wave function.)
The particle in a box
The particle in a box.
Suppose you have a particle bouncing back and forth between two walls in a box. Assume that the particle moves in one dimension only, along the $x$-axis, between vertical, impenetrable walls at $x=0$ and $x=L.$ There are no forces acting on the particle inside the box, so its potential energy is zero here: $V(x,t)=0$ for $0<x<L$. Infinitely large forces push the particle back when it hits a wall: the potential energy $V(x,t)$ is infinite for $x \leq 0$ and $x\geq L.$
Since in this example the potential energy does not depend on time we can use the time-independent one-dimensional Schrödinger equation within the box. Recalling that $V=0$ in the box we therefore have
\begin{equation} \frac{d^2 \psi }{dx^2} + \frac{8 \pi ^2 mE\psi }{h^2} = 0,\end{equation} (1)
where $m$ is the mass of the particle, $E$ is its total energy and $h=6.626068 \times 10^{-34} m^2kg/s$ is Planck’s constant.
A solution to equation (1) is a function $\psi (x)$ which when differentiated satisfies the equation. Any solution to equation (1) will take the form
\begin{equation} \psi (x) = A \cos {\left(\sqrt{\frac{8 \pi ^2 mE}{h^2}} x\right)} + B \sin {\left(\sqrt{\frac{8 \pi ^2 mE}{h^2} }x\right)}\end{equation} (2)
where $A$ and $B$ are constants.
Now $|\psi (x)|^2$ is related to the probability of finding the particle at position $x$ and time $t.$ We know that the particle can never be outside the box (in the region where $x<0$ or $x>L$) because it would need an infinite amount of energy to get there. This means that $\psi =0$ for $x<0$ and $x>L.$ And since $\psi $ is continuous at the boundary of the box, we deduce that $\psi $ is also equal to zero at $x=0$ and $x=L.$
The first condition, $\psi = 0$ at $x=0$, means that
\[ 0 = A \cos {0} + B \sin {0} = A, \]
so we can ignore the cosine term in (2) and our equation becomes
\begin{equation} \psi = B \sin {\left(\sqrt{\frac{8 \pi ^2 mE}{h^2} }x\right)}\end{equation} (3)
The second condition, $\psi =0$ at $x=L$, means that
\[ 0=B \sin {\left(\sqrt{\frac{8 \pi ^2 mE}{h^2}} L\right)}, \]
so either $B=0$ or the sine term is zero. The former would imply that $\psi $ is zero everywhere — this clearly can’t be the case as we know that the particle is somewhere in the box. So we deduce that
\[ \sin {\left(\sqrt{\frac{8 \pi ^2 mE}{h^2}} L\right)} =0. \]
Now $\sin {y}=0$ if and only if $y$ is a multiple of $\pi $, so we have
\[ \sqrt{\frac{8 \pi ^2 mE}{h^2}} L = 0, \pi , 2 \pi , 3\pi , ... \]
In other words
\[ \sqrt{\frac{8 \pi ^2 mE}{h^2}} L = n\pi \]
for $n$ a positive integer.
This tells us that the energy of the particle can only have discrete values
\begin{equation} E_ n = \frac{n^2 h^2}{8mL^2},\end{equation} (4)
for $n=0, 1,2,3,...$
The number $n$ corresponding to the energy level $E_ n$ is called the quantum number of $E_ n$.
The quantum number $n=0$ corresponds to zero energy — but it also gives a wave function $\psi _0$ which is zero everywhere in the box, which would mean the particle cannot be anywhere in the box. Thus, the quantum number $n=0$ is also ruled out, so the permissible energy levels are
for $n=1,2,3,...$
Wave-particle duality is a central concept in quantum mechanics.
The fact that the energy spectrum is discrete, ie that not all energies are permitted, and in particular that zero energy is not permitted, are results you don't get out of classical mechanics — in fact, they fly in the face of conventional wisdom, which holds that quantities such as energy should vary continuously: "nature does not make jumps" according to Gottfried Leibniz. Classical physics also tells us that the lowest energy state of a system (also called the ground state or the vacuum) should have zero energy. But these strange quantum results tally with the experimental observations of quantum systems, for example the discrete energy spectrum of the hydrogen atom.
The value of the constant B can be found by normalising the wave function. Recall that
\[ |\Psi (x)|^2 = |\psi (x)^2 e^{-(2 \pi i E/h)t}| = |\psi (x)^2||e^{-(2 \pi i E/h)t}|. \]
Now $e^{-(2 \pi i E/h)t}$ is a complex number, which can also be written as
\[ e^{-(2 \pi i E/h)t} = \cos {\left( -(2 \pi i E/h)t\right)} + i \sin {\left( -(2 \pi i E/h)t\right)}. \]
The modulus $|e^{-(2 \pi i E/h)t}|$ of this complex number is therefore
\[ \cos ^2{\left( -(2 \pi i E/h)t\right)} + \sin ^2{\left( -(2 \pi i E/h)t\right)}=1, \]
by the familiar trigonometric identity. Therefore we have
Now recall that $|\Psi (x)|^2$ is the probability density function for finding the particle at location $x$ at time $t.$ In other words, the probability that the particle is somewhere in the box is given by
\[ \int ^ L_0 |\Psi (x)|^2dx = \int ^ L_0 |\psi (x)|^2 dx. \]
Since we know for sure that the particle is somewhere in the box we have
\[ \int ^ L_0 |\Psi (x)|^2dx = \int ^ L_0 |\psi (x)|^2 dx = 1. \]
Substituting our expression for $\psi (x)$ from equation (3) we have
\[ \int ^ L_0{B^2 \sin ^2{\left(\sqrt{\frac{8 \pi ^2 mE}{h^2} }x\right)}dx}=1. \]
Using the fact that
\[ \int \sin ^2{(ax)} dx = -\frac{1}{2a}\sin {(ax)}\cos {(ax)}+\frac{x}{2}, \]
we can work out that
\[ B=\sqrt{\frac{2}{L.}} \]
\[ \]
Substituting the possible values $E_ n$ for $E$ from (4) we get infinitely many wave functions $\psi _ n(x)$ (one for each quantum number, ie permitted energy level):
for $0 < x < L$ and
\[ \psi _ n(x) = 0 \]
This gives us another surprising result: for any value of $n$ that is bigger than 1 we can find a value of $x$ that lies within the box for which $|\psi (x)|^2 = 0:$ if $x = \frac{kL}{n}$ for some integer $k < n$, then $0<x<L$ and $\sin {\frac{n \pi x}{L}= \sin {k \pi } = 0.}$
Since $|\psi (x)|^2=|\Psi (x)|^2$ is the probability density of finding the particle at the point $x$, this means that there are places in the box where the particle can never be found!
Thus we’ve seen how Schrödinger’s equation produces some very weird results that contradict our classical intuition. So why do we not see things like the discrete energy levels in macroscopic objects like billiard balls? The lowest permitted energy occurs for the quantum number $n=1$ and
\[ E_1 = \frac{h^2}{8mL^2}. \]
Planck’s constant $h$ is very small, $h=6.626068 \times 10^{-34} m^2kg/s.$ So for a large object, $m$ and $L$ will be comparatively huge. This means that $E_1$ will be so incredibly small that an object with energy $E_1$ becomes indistinguishable from one at rest. This is why in the macroscopic world a zero energy level does appear to be possible.
Now let’s compute the difference between two adjacent energy levels $E_ n$ and $E_{n+1}$ for a quantum number $n$:
\[ E_{n+1}-E_ n = \frac{(n+1)^2 h^2}{8mL^2} - \frac{n^2 h^2}{8mL^2} = \frac{h^2((n+1)^2-n^2)}{8mL^2} = \frac{h^2(2n+1)}{8mL^2}. \]
As the mass $m$ and the size $L$ of the box get large, this difference tends to zero. So for large objects the permitted energy levels are so close together that it’s impossible to distinguish them from the energy levels that are not permitted — the energy level appears to vary continuously.
Quantum tunneling
Quantum tunneling
Quantum tunneling: The vertical axis shows the potential energy of the particle, which is equal to V0 for x greater than 0 and less than L and zero elsewhere.
Now let’s change our set-up a little. Let’s still assume that our particle moves along the $x$-axis. But this time suppose that the potential energy $V$ is zero for $-\infty < x \leq 0$ and for $L \leq x < +\infty $ but non-zero between $0$ and $L$. So $V=V_0\neq 0$ for $0 < x < L.$ This is a potential barrier: classical physics tells us that if the particle is moving right along the negative $x$-axis towards 0, it can only penetrate the region between $0$ and $L$ if its energy $E$ is greater than $V_0.$
It’s a bit like a ball of mass $m$ rolling along and encountering a hill of height $H$: if it has enough energy to move to the top of the hill, then its potential energy there will be $V_0 = mgH,$ where $g$ is acceleration due to gravity. If its energy is less than $V_0$ the ball will never make it to the top. Only in our example the hill has a vertical slope because the potential energy jumps discontinuously between 0 and $V_0$ at $x=0$ and $x=L$, and it is flat on the top because the potential energy is constant for $0 < x < L.$
It turns out that in quantum mechanics the particle can make it to the top, and even the other side, of the "potential hill" even if its energy is less than $V_0$. We won’t go into the details here but solving Schrödinger’s equation (assuming $E<V_0$) with suitable boundary conditions gives a wave function that is non-zero on the entire $x$-axis. This means that a particle coming from the left actually has a non-zero probability of being found inside the barrier and even to the right of it. There is a small but non-zero probability that the particle will tunnel through the barrier to the other side, even though in classical terms it does not have enough energy to do so. As you would expect, this probability becomes smaller the thicker the barrier, that is, the larger the value of $L$.
Generally the term quantum tunneling refers to any situation in which a particle overcomes a potential barrier that it should not be able to overcome according to classical physics. Quantum tunneling does occur in nature, for example in when uranium decays to thorium in a form of radioactive decay known as alpha decay. Here the atomic nucleus emits an alpha particle (which consists of two protons and two neutrons and is structurally identical to a helium nucleus). According to classical physics, the process of emitting the particle should be impossible, as it requires more energy than the atom has available. It's through quantum tunneling that the atom accomplishes the feat.
The big question raised by all this mathematics is what Schrödinger's equation tells us about physical reality. How should we interpret its solution, the wave function? This is what we'll explore in the third article.
Read the final article Schrödinger's equation, what does it mean?
About the author
Marianne Freiberger is Editor of Plus. She would like to thank Jeremy Butterfield, a philosopher of physics, Nazim Bouatta, a Postdoctoral Fellow in Foundations of Physics, and Tony Short, a Royal Society Research Fellow in Foundations of Quantum Physics, all at the University of Cambridge, for their help in writing these articles.
This is a wonderful site and content. All students of math and physics should read this kind of material.
Many thanks! Keep up the good work!!
Question for Marianne:
Is it possible that h/2 is the fundamental quantum size? If we define a new Planck's constant (call it hs) such that hs = h/2, and plug that into the Schroedinger Equation, does that change the usefulness of the equation (or add any other complications to any aspect of physical laws or quantum mechanics)?
I have a reason for asking that, which we can discuss if your answer is no.
Thx for a very fine article. Very clever math! But I've been stumped by magicians who were as clever. The particle must know the location of L before he/she/it can leave x=0, or he/she/or it will not know when to pop back into reality. How does he/she/or it manage that? What was Schrodenger smoking? :>)
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12d9e0438668aef1 | . . . Thou shalt take of the first of all the fruit of the ground, which thou shalt bring in from thy land that the Lord thy G‑d giveth thee; and thou shalt put it in a basket and shalt go unto the place which the Lord thy G‑d shall choose to cause His name to dwell there. . . . And the priest shall take the basket out of thy hand, and set it down before the altar of the Lord thy G‑d.
Deuteronomy 26:2,4
And He garbed Himself with righteousness as a coat of mail, and a helmet of salvation upon His head . . .
Isaiah 59:17
The Torah portion Tavo begins with the commandment of the offering of the First Fruits (bikkurim). The simple meaning of this mitzvah is self-evident. We are commanded to express our gratitude to G‑d for bringing us to the promised land by bringing the First Fruits as a gift to the priests serving in the Holy Temple. However, the simplicity of this commandment betrays rich metaphors hidden therein. We have already discussed the metaphor of wave-particle duality symbolized by the fruits in the basket (see First Fruits and the Wave-Particle Duality of Nature).
Basket-Weave and the Lattice Field Theory
Now, let us focus on the symbolism of the wicker basket.
Figure 1. Fruit Basket. Shutterstock_1803756757
What do the tiny holes in the wicker basket mean? Every year, when I read this Torah portion, the wicker basket invariably invokes in my mind lattice geometry and its use in quantum field theory.
Figure 2. Lattice Pattern Basket Weave. Shutterstock_461309125
Indeed, the weave of a wicker basket is called “lattice.” However, it is not only the crossed wooden strips that make this weave into a lattice. Any regular arrangement of points or in space, such as atoms in a crystal, are called “lattice” in geometry.
Figure 3. A lattice in the Euclidean plane. By Jim.belk – Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=8313748
In the case of a basket, the regular arrangement of the holes between the interwoven strips represents a lattice pattern. As we shall see later, the holes play a very important role in this story.
It has been long understood in quantum theory that on the most fundamental level, space and time must be discrete, that is quantized. Indeed, a distance in space less than Planck length[1] is meaningless. So too, an interval of time less than Plank time[2] is meaningless. The question is how to quantize time, that is, how to make it discrete.[3] In quantum field theory, lattice geometry is one model to represent the quantization of space. This geometry is used in quantum field theory.[4]
If we explore the metaphor of the First Fruits offering further, things get even more interesting.
The First Fruits in Kabbalah
Rabbi Chaim Vital, in the name of the Arizal (Rabbi Isaac Luria Ashkenazi), comments on these verses:
The commandment of the First Fruits is the return of the lights of the Nukva to Chesed, which is embodied by the priest, in order that evil not be able to derive sustenance. This is the mystical meaning of “the priest will take the basket from your hand,” for the gematria (numerical value) of the word “the basket” (ha-tene) is the same as that of the name Adni.[5]
Ta’amei HaMitzvot and Sha’ar HaPesukim, parashat Tavo
Indeed, the numerical value of ha-tene is 65[6]—the same as that of Adni (or Ado-nai[7]).[8] Ado‑nai is a divine name representing the female principle, which the language of Kabbalah refers to as Nukva.[9] Nukva receives the abstract idea of the divine purpose[10] from her male counterpart, Zeir Anpin (Z”A), which embodies the six midot[11] from Chesed to Yesod. Here, as a shorthand, the Arizal identifies Z”A with the first and primary midahChesed. Z”A represents the abstract idea of the purpose of creation and the divine intention, as they exists in the mind of G‑d, whereas Nukva represents the drive to the actualization of this idea in the lower worlds.
However, there is always the danger that Nukva gets too involved with the lower worlds and loses her sense of purpose, the connection to the abstract idea. This, in kabbalistic terminology, leads to spillage of the light, which gives vitality to the forces of evil—those ideas that are opposed to the abstract idea of the divine. Thus, the Arizal explains, to prevent such spillage, Nukva must reconnect with her male counterpart—reconnect with the abstract idea and the initial intention. This is accomplished by Nukva reflecting back the excess light to Chesed. Thus far is Kabbalah’s insight into the mystical dimension of the commandment of bikkurim (the “First Fruits”). The most crucial point for us, in this commentary, is that the wicker basket symbolizes the sefirah of Malchut (Nukva).[12]
The First Fruits in Chasidic Philosophy of Chabad
Rabbi Moshe Wisnefsky connects this commentary of the Arizal with the Chasidic teaching of Rabbi Schneur Zalman of Liadi (the Alter Rebbe) in the Tanya.[13] The offering of the first fruits (bikkurim) is a form of tzedakah (charity), where farmers bring their first fruits to Jerusalem and give them to priests who would eat them. In Epistle 3 of Igeret HaKodesh, the Alter Rebbe explores the mystical dimension of charity—tzedakah.
The Alter Rebbe writes:
Therefore, by the act of charity and the performance of kindness [the fruits of which man enjoys in this world], there appear, metaphorically speaking, gaps in the supernal garment that encompasses the Body— [the kelim (vessels) of the ten sefirot ]—through which to irradiate and to diffuse light and abundance . . . .[14]
Tanya, Igeret HaKodesh, 3
There are several significant parallels in this narrative with the offering of the First Fruits as discussed by the Arizal. First, the act of charity parallels the offerings of the first fruits, as discussed above. The ten sefirot parallel the fruits in the basket. The kelim (vessels) of the ten sefirot parallel the basket that holds the fruits. Furthermore, the holes (or gaps) in the vessels of the sefirot parallel the holes in the wicker basket.[15]
Let us now recall that the Arizal explains that the basket is emblematic of the Nukva (the sefirah of Malchut). (Recall that “the basket,” that is hatene has the same numerical value, 65, as Ad-nai—a divine name associated with the sefirah of Malchut or Nukva.) Kabbalah teaches that space originates in Malchut. From this perspective, the wicker basket is a metaphor for space.[16]
Furthermore, the Alter Rebbe writes:
“And He garbed himself with tzedakah as a coat of mail, and a helmet of salvation upon His head.”[17] (On this verse) our sages, of blessed memory, commented: “Just as with chain mail each scale adds up to form a large mail, so it is with charity; each coin adds up to a large amount.”[18]
Tanya, Igeret HaKodesh, 3
Using the structural parallel between the wicker basket and space, let us apply the metaphor of chain-mail armor to physical space. A chain of mail is made of small interlocking rings. If our parallel is true, physical space would have to be made of tiny interconnected circles of sorts. And, in fact, this is exactly what loop quantum gravity says about the nature of space!
Figure 4. Chaim Mail Armor. Shutterstock_408551752
Loop Quantum Graivty
Loop quantum gravity was born out of the need to reconcile the two best theories of physics—general relativity and quantum theory. General relativity is very successful in describing cosmology and the universe on a large scale, but it fails when enormous mass becomes compressed in a tiny volume, as in a black hole or at the time of the Big Bang. Quantum theory, on the other hand, is extremely successful in describing subatomic particles but is incompatible with general relativity. The unification of these two great theories, called quantum gravity, has eluded physicists for almost a century –and not for the lack of effort.
The first significant step toward constructing the theory of quantum gravity was the formulation of the Wheeler–DeWitt equation, first derived in 1967, which is very similar to the static Schrödinger equation, except that the universal wave function accounts for a variety of possible geometries that the universe may have. The problem was that nobody knew how to solve this equation. The next breakthrough occurred in 1986, when Abhay Ashtekar reformulated general relativity in the language of what is called now Ashtekar variables. Ted Jacobson and Lee Smolin discovered that, when written in these Ashtekar variables, the Wheeler–DeWitt equation has solutions labeled by loops. Carlo Rovelli and Lee Smolin formulated quantum gravity theory in terms of these loop solutions.
Figure 5. Cover of a Book on Loop Quantum Gravity
In this theory, called loop quantum gravity (LOG), space is not continuous but rather is quantized, that is, discrete. Specifically, space appears to be woven from tiny interconnected loops. These loops are really small—on the order of magnitude of a Planck length—and are interlocked. When many of them are connected together, space appears to be like chain mail.
Let us now recall the Talmudic commentary on the verse from Isaiah (59:17), as cited by Rabbi Schneur Zalman: “Just as with chain mail each scale adds up to form a large mail, so it is with charity; each coin adds up to a large amount.”[19] Combining this Talmudic commentary with the kabbalistic commentaries of the Arizal and Rabbi Schneur Zalman, we have established earlier that this metaphor can also apply to space. That is, in its spiritual source in the sefirah of Malchut, space appears as chain mail, just as in loop quantum gravity—a leading candidate to stand in for the hypothetical theory of quantum gravity. Here is how one of the creators of loop quantum gravity, Carlo Rovelli, describes space in his book:
The theory describes these “atoms of space” in mathematical form and provides equations that determine their evolution. They are called “loops,” or rings, because they are linked to one another, forming a network of relations that weaves the texture of space, like the rings of a finely woven, immense chain mail.[20]
There we have it—a Chasidic Rebbe, Rabbi Schneur Zalman of Liadi,[21] and a leading contemporary theoretical physicist, both describing the structure of space in the same way—comparing it to chain mail.
Let us review the many steps that led us to this remarkable structural parallel between the mystical symbolism of the biblical commandment of bikurim (the First Fruits) and loop quantum gravity.
1. First, we note the symbolism of the basket weave (of the wicker basket in which the First Fruits are brought to the Temple) as a metaphor for lattice geometry used in quantum field theory.
2. We find in the writings of the Arizal (as recorded by his principal disciple, Rabbi Chayim Vital) the numerical value of hatenet (the basket) is the same as that of the divine name Ado-nai, which is associated with the sefirah of Malchut.
3. The Alter Rebbe (Rabbi Schneur Zalman of Liadi), in Igeret HaKodesh, connects the holes in the basket with the kabbalistic concept of the holes in curtain that screens the divine light—tzimtzum—diminishing its intensity. He uses a Talmudic teaching, based on the verse in Isaiah, to compare this pierced with the chain of mail.
4. Connecting this teaching of the Alter Rebbe with the teachings of the Arizal who identified the basket with the sefirah of Malchut, we note that, in Kabbalah and Chasidic philosophy, space originates in the sefirah of Malchut.
5. Connecting the dots allows us to conclude that the chain mail metaphor is applicable to the topology (structure) of space.
6. Finally, we observe that it is precisely the topology used in loop quantum gravity. In fact, the chain mail metaphor for the structure of space is widely used in loop quantum gravity—a leading contender for the hypothetical theory of quantum gravity.
This structural parallel between the kabbalistic/Chasidic view of space and the conception of space in quantum loop gravity is remarkable, to say the least.
[1] Planck length is the smallest conceivable unit of length, P is ~ 1.616 ´ 10-35 m.
[2] Planck time is the smallest conceivable unit of time, tP is ~ 5.39 ´ 10-44 s.
[3] The notion of discreteness of space and time is very near and dear to me. When I first studied quantum theory as a teenager, I immediately realized that space and time cannot be continuous in quantum field theory as they are in quantum mechanics or relativity. Space and time must be discrete. I started thinking how to represent quantized space-time mathematically and immediately ran into a problem: quantum theory, as well as all of theoretical physics, employs calculus, which presupposes continuity and smoothness of space and time. However, if there cannot be an interval of time shorter than Planck time, tP = 5.39×10−44 s, then how do you define a derivative over time? Similarly, when we consider fields, we calculate partial derivatives of field potential, where the partial derivatives are similarly defined as limits. If the smallest unit of length is Planck length, how do you define partial derivatives over space coordinates? However, if we postulate that the space is quantized, the length can never be less than that Planck length, P. I was stuck. Existing mathematics seemed not to allow the application of calculus in quantized space-time. By the time I turned thirteen, I developed what I thought was new mathematics, where calculus was built on small but finite elements replacing limits. I shared my discovery with my math teacher, and she thought I really managed to build a new mathematical theory. She advised me to travel to Moscow and present my theory at Moscow University. I did just that. I showed up at the theoretical physics seminar of Prof. Dmitri D. Ivanenko of the physics department of Moscow University, where I described my theory. I was politely told that this theory was already well known and was called the theory of finite differences. I was offered a cup of tea as a consolation. As I should have expected, I had reinvented the wheel. But my naïve intuition that space and time must be quantized was correct. As I later learned, this intuition was shared by many physicists, including Lee Smolin and Carlo Rovelli, who finally implemented this idea in their concept of loop quantum gravity.
[4] One example of the use of lattice geometry in quantum field theory is the lattice gauge theory—a group of gauge theories, such as quantum electrodynamics, chromodynamics, and the standard model, defined on quantized space-time.
[5] Ta’amei HaMitzvot and Sha’ar HaPesukim, parashat Tavo. (English translation by Rabbi Moshe Wisnefsky, Apples from the Orchard: Mystical Insights on the Weekly Torah Portion [Malibu, CA: Thirty Seven Books, 2008], p. 989.
[6] Hatene: heh-tet-nun-alef = 5 + 9 + 50 + 1 = 65.
[7] It is a prevalent Jewish custom to change the spelling of a divine name to avoid its possible desecration, if the paper on which it is printed gets discarded. The translator of the writings of Rabbi Chaim Vital chose to chose to change the spelling to Adni, whereas the convention adopted in this blog is to insert a hyphen.
[8] Ado-nai: alef-dalet-nun-yud = 1 + 4 + 50 + 10 = 65.
[9] Nukva is an Aramaic (the language of the Talmud and Zohar) equivalent of the Hebrew nekevah, meaning “female.” The root of this word is nekev (“hole”). Thus, the literal translation of the words nukva and nekevah is “full of holes.” The reason for this strange etymology is explained in the Talmud, which points out that female body has one more opening than a male body. It is not coincidental then that the wicker basket is connected through the gematria to the female principle. First, the basket represents a keli (“vessel”). So too in Kabbalah, the female is always a vessel; physically, a woman’s body is a vessel for the male seed. Conceptually, female refers to the receiving principle (vessel), whereas male is always the giving principle. Second, just as a female body is “full of holes,” so too is the wicker basket full of holes.
[10] This abstract idea of the divine purpose inspires Nukva to implement and actualize the divine purpose in the lower realm. Alternatively, the books of Kabbalah speak of Nukva receiving the divine light from Z”A (for which “seed” is sometimes used as a physical metaphor, meaning that Z”A “impregnates” Nukva with the seminal idea of the divine purpose.)
[11] The six lower sefirot are Chesed, Gevurah, Tiferet, Netzach, Hod, and Yesod.
[12] The sefirah of Malchut is a vessel that receives from the male Z”A.
[13] Rabbi Moshe Wisnefsky, Apples from the Orchard: Mystical Insights on the Weekly Torah Portion (Malibu, CA: Thirty Seven Books, 2008), p. 990.
[14] Rabbi Schneur Zalman of Liadi, Tanya, Igeret HaKodesh, Epistle 3. (See online at https://www.chabad.org/library/tanya/tanya_cdo/aid/1029262/jewish/Epistle-3.htm.)
[15] On a mystical level, the holes in the kelim (“vessels”) of the ten sefirot echo the holes in the Nukva, which is symbolized by the wicker basket in the commentary of the Arizal (see footnote 4 above).
[16] As the basket is a vessel that hold the fruit, space is seen in Kabbalah as a vessel that holds the stars and the planets. (CITE)
[17] Isaiah 59:17.
[18] Babylonian Talmud, tr. Bava Batra 9b.
[19] Babylonian Talmud, tr. Bava Batra 9b.
[20] Carlo Rovelli, Seven Brief Lessons on Physics (Riverhead Books, 2014), pp. 42–44.
[21] This metaphor appears in Igeret HaKodesh published posthumously as a part of the Tanya in 1814. Igeret HaKodesh is a collection of letters in which Rabbi Schneur Zalman explains to his disciples the mystical meaning of the Torah commandments. These letters were written presumably sometime between 1788 and 1812. Let us recall, however, that originally the metaphor of chain mail was used by the prophet Isaiah and later discussed in the Talmud, albeit in a different context.
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2cb48f7ab79521c1 | Approximability of Optimization Problems through Adiabatic Quantum Computation
By William Cruz-Santos, Guillermo Morales-Luna
, 113 pages © 2014
ISBN 1627055568
Availability: Out of stock
Price: US $40.00 Add to Cart
The adiabatic quantum computation (AQC) is based on the adiabatic theorem to approximate solutions of the Schrödinger equation. The design of an AQC algorithm involves the construction of a Hamiltonian that describes the behavior of the quantum system. This Hamiltonian is expressed as a linear interpolation of an initial Hamiltonian whose ground state is easy to compute, and a final Hamiltonian whose ground state corresponds to the solution of a given combinatorial optimization problem. The adiabatic theorem asserts that if the time evolution of a quantum system described by a Hamiltonian is large enough, then the system remains close to its ground state. An AQC algorithm uses the adiabatic theorem to approximate the ground state of the final Hamiltonian that corresponds to the solution of the given optimization problem. In this book, we investigate the computational simulation of AQC algorithms applied to the MAX-SAT problem. A symbolic analysis of the AQC solution is given in order to understand the involved computational complexity of AQC algorithms. This approach can be extended to other combinatorial optimization problems and can be used for the classical simulation of an AQC algorithm where a Hamiltonian problem is constructed. This construction requires the computation of a sparse matrix of dimension 2ⁿ × 2ⁿ, by means of tensor products, where n is the dimension of the quantum system. Also, a general scheme to design AQC algorithms is proposed, based on a natural correspondence between optimization Boolean variables and quantum bits. Combinatorial graph problems are in correspondence with pseudo-Boolean maps that are reduced in polynomial time to quadratic maps. Finally, the relation among NP-hard problems is investigated, as well as its logical representability, and is applied to the design of AQC algorithms. It is shown that every monadic second-order logic (MSOL) expression has associated pseudo-Boolean maps that can be obtained by expanding the given expression, and also can be reduced to quadratic forms. Table of Contents: Preface / Acknowledgments / Introduction / Approximability of NP-hard Problems / Adiabatic Quantum Computing / Efficient Hamiltonian Construction / AQC for Pseudo-Boolean Optimization / A General Strategy to Solve NP-Hard Problems / Conclusions / Bibliography / Authors' Biographies
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40354765fd228d36 | Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\pi],\mu_h)$ known? Here $d\mu_h(x)=Z^{-1}e^{h\cos x}dx$ and $Z>0$ is chosen such that $\mu_h$ is a probability measure. I suspect it involves Bessel functions.
N.B. The Sturm-Liouville problem is unitarily equivalent to the Schrödinger operator $$ Hf(x)=-f''(x)+\frac{h}{4}(h\sin^2(x)-2\cos(x))f(x) $$ on $L^2([-\pi,\pi])$.
Edit: By the hint of Sascha, we transform $H$ into a Schrödinger operator with Whittaker-Hill potential. Set $y=x/2$, then $$ Hf(y)=-\frac14 f''(y)+\left(\frac{h^2}{8}-\frac{h^2}{8}\cos(4y)-\frac{h}{2}\cos(2y)\right)f(y). $$ Hence, $4H-h^2/2$ is a Schrödinger operator with Whittaker-Hill potential with parameters $\alpha=h/2$, $s=1$ (in the convention of the paper by Hemery and Veselov).
Is anything known about the spectral gap of this operator (as noted by Hemery and Veselov, the ground state eigenvalue is (not very surprisingly) $-h^2/2$)?
• $\begingroup$ The operator $L$ is not self-adjoint, right? Could you thus specify what exactly you mean by "eigenvalue decomposition"? $\endgroup$ – Jochen Glueck Feb 27 '19 at 19:07
• $\begingroup$ My bad. I missed some crucial information. I edited the question accordingly. $\endgroup$ – julian Feb 27 '19 at 19:16
• $\begingroup$ I'd be surprised if you could find the eigenvalues explicitly here, but I might be wrong of course. $\endgroup$ – Christian Remling Feb 27 '19 at 19:42
• $\begingroup$ So what do precise information on the spectral gap do you need? $\endgroup$ – Sascha Feb 28 '19 at 14:51
• $\begingroup$ I need the second lowest eigenvalue. $\endgroup$ – julian Feb 28 '19 at 14:52
I think this is one of the quasi exactly solvable potentials for the Schrödinger equation, see this paper.
To say a bit more: For particular choices of $h$ certain eigenfunctions are explicit but not all of them.
• $\begingroup$ Thank you for this hint. It works out in fact (see the edited question). $\endgroup$ – julian Feb 28 '19 at 14:06
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4853001f1476798b | Saturday, November 14, 2020
Understanding Quantum Mechanics #8: The Tunnel Effect
[This is a transcript of the video embedded below. Parts of the text will not make sense without the graphics in the video.]
Have you heard that quantum mechanics is impossible to understand? You know what, that’s what I was told, too, when I was a student. But twenty years later, I think the reason so many people believe one cannot understand quantum mechanics is because they are constantly being told they can’t understand it. But if you spend some time with quantum mechanics, it’s not remotely as strange and weird as they say. The strangeness only comes in when you try to interpret what it all means. And there’s no better way to illustrate this than the tunnel effect, which is what we will talk about today.
Before we can talk about tunneling, I want to quickly remind you of some general properties of wave-functions, because otherwise nothing I say will make sense. The key feature of quantum mechanics is that we cannot predict the outcome of a measurement. We can only predict the probability of getting a particular outcome. For this, we describe the system we are observing – for example a particle – by a wave-function, usually denoted by the Greek letter Psi. The wave-function takes on complex values, and probabilities can be calculated from it by taking the absolute square.
But how to calculate probabilities is only part of what it takes to do quantum mechanics. We also need to know how the wave-function changes in time. And we calculate this with the Schrödinger equation. To use the Schrödinger equation, you need to know what kind of particle you want to describe, and what the particle interacts with. This information goes into this thing labeled H here, which physicists call the “Hamiltonian”.
To give you an idea for how this works, let us look at the simplest possible case, that’s a massive particle, without spin, that moves in one dimension, without any interaction. In this case, the Hamiltonian merely has a kinetic part which is just the second derivative in the direction the particle travels, divided by twice the mass of the particle. I have called the direction x and the mass m. If you had a particle without quantum behavior – a “classical” particle, as physicists say – that didn’t interact with anything, it would simply move at constant velocity. What happens for a quantum particle? Suppose that initially you know the position of the particle fairly well, so the probability distribution is peaked. I have plotted here an example. Now if you solve the Schrödinger equation for this initial distribution, what happens is the following.
The peak of the probability distribution is moving at constant velocity, that’s the same as for the classical particle. But the width of the distribution is increasing. It’s smearing out. Why is that?
That’s the uncertainty principle. You initially knew the position of the particle quite well. But because of the uncertainty principle, this means you did not know its momentum very well. So there are parts of this wave-function that have a somewhat larger momentum than the average, and therefore a larger velocity, and they run ahead. And then there are some which have a somewhat lower momentum, and a smaller velocity, and they lag behind. So the distribution runs apart. This behavior is called “dispersion”.
Now, the tunnel effect describes what happens if a quantum particle hits an obstacle. Again, let us first look at what happens with a non-quantum particle. Suppose you shoot a ball in the direction of a wall, at a fixed angle. If the kinetic energy, or the initial velocity, is large enough, it will make it to the other side. But if the kinetic energy is too small, the ball will bounce off and come back. And there is a threshold energy that separates the two possibilities.
What happens if you do the same with a quantum particle? This problem is commonly described by using a “potential wall.” I have to warn you that a potential wall is in general not actually a wall, in the sense that it is not made of bricks or something. It is instead just generally a barrier for which a classical particle would have to have an energy above a certain threshold.
So it’s kind of like in the example I just showed with the classical particle crossing over an actual wall, but that’s really just an analogy that I have used for the purpose of visualization.
Mathematically, a potential wall is just a step function that’s zero everywhere except in a finite interval. You then add this potential wall as a function to the Hamiltonian of the Schrödinger equation. Now that we have the equation in place, let us look at what the quantum particle does when it hits the wall. For this, I have numerically integrated the Schrödinger equation I just showed you.
The following animations are slow-motion compared to the earlier one, which is why you cannot see that the wave-function smears out. It still does, it’s just so little that you have to look very closely to see it. It did this because it makes it easier to see what else is happening. Again, what I have plotted here is the probability distribution for the position of the particle.
We will first look at the case when the energy of the quantum particle is much higher than the potential wall. As you can see, not much happens. The quantum particle goes through the barrier. It just gets a few ripples.
Next we look at the case where the energy barrier of the potential wall is much, much higher than the energy of the particle. As you can see, it bounces off and comes back. This is very similar to the classical case.
The most interesting case is when the energy of the particle is smaller than the potential wall but the potential wall is not extremely much higher. In this case, a classical particle would just bounce back. In the quantum case, what happens is this. As you can see, part of the wave-function makes it through to the other side, even though it’s energetically forbidden. And there is a remaining part that bounces back. Let me show you this again.
Now remember that the wave-function tells you what the probability is for something to happen. So what this means is that if you shoot a particle at a wall, then quantum effects allow the particle to sometimes make it to the other side, when this should actually be impossible. The particle “tunnels” through the wall. That’s the tunnel effect.
I hope that these little animations have convinced you that if you actually do the calculation, then tunneling is half as weird as they say it is. It just means that a quantum particle can do some things that a classical particle can’t do. But, wait, I forgot to tell you something...
Here you see the solutions to the Schrödinger equation with and without the potential wall, but for otherwise identical particles with identical energy and momentum. Let us stop this here. If you compare the position of the two peaks, the one that tunneled and the one that never saw a wall, then the peak of the tunneled part of the wave-function has traveled a larger distance in the same time.
If the particle was travelling at or very close by the speed of light, then the peak of the tunneled part of the wave-function seems to have moved faster than the speed of light. Oops.
What is happening? Well, this is where the probabilistic interpretation of quantum mechanics comes to haunt you. If you look at where the faster-than light particles came from in the initial wave-function, then you find that they were the ones which had a head-start at the beginning. Because, remember, the particles did not all start from exactly the same place. They had an uncertainty in the distribution.
Then again, if the wave-function really describes single particles, as most physicists today believe it does, then this explanation makes no sense. Because then only looking at parts of the wave-function is just not an allowed way to define the particle’s time of travel. So then, how do you define the time it takes a particle to travel through a wall? And can the particle really travel faster than the speed of light? That’s a question which physicists still argue about today.
This video was sponsored by Brilliant which is a website that offers interactive courses on a large variety of topics in science and mathematics. I hope this video has given you an idea how quantum mechanics works. But if you really want to understand the tunnel effect, then you have to actively engage with the subject. Brilliant is a great starting point to do exactly this. To get more background on this video’s content, I recommend you look at their courses on quantum objects, differential equations, and probabilities.
To support this channel and learn more about Brilliant, go to and sign up for free. The first 200 subscribers using this link will get 20 percent off their annual premium subscription.
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1. It amazes me that this phenomenon was used to create practical device: a scanning tunneling microscope.
2. Dr. Hossenfelder: On the last point, the peak traveling faster than light: I don't understand why there is a controversy; the two waveforms are each still a distribution, I presume with infinite overlap, so who cares where the peaks are? Those are no more definitive of position than the rest of the distribution.
I presume upon measurement in either distribution, the particle will NOT have traveled faster than light, within the bounds of its original positional uncertainty.
So all that has changed with a wall is the positional probability; not the actual position.
The controversy seems (to me) like a misinterpretation of statistical probability. Can you explain why physicists see a controversy?
1. Very interesting. I think they are wrong about one thing, though:
The article says: "Why, though, couldn’t you blast tons of particles at the ultra-thick barrier in the hopes that one will make it through superluminally? Wouldn’t just one particle be enough to convey your message and break physics? Steinberg, who agrees with the statistical view of the situation, argues that a single tunneled particle can’t convey information. A signal requires detail and structure, and any attempt to send a detailed signal will always be faster sent through the air than through an unreliable barrier."
Passing even one bit is a message, yes or no. As the article said earlier, this experiment can be done with multiple kinds of atoms.
Thus in the scenario of "blasting tons of particles", I could theoretically associate different atomic elements with different answers, blast a ton of a particular kind of atom at the barrier, some which traverse it superluminally, and the receiver knows the intended message based upon the kind of atom received.
That message could reasonably consist of 3 or 4 bits of information.
2. While the Quanta article is excellent and a lot of fun to read, I did have one quibble with its presentation of the issue.
For any classically-initiated test of the tunneling velocity of a particle, the front edge of its wave packet will necessarily be well-defined, as opposed to fading out indefinitely. This must be the case because it must begin as a particle that is classically visible to the tester. Thus no matter how one interprets the internal state of the particle -- its phase for example, or some other version of an internal clock -- the detectable front edge of its wave packet will always and only propagate as a quite ordinary Schrodinger wave that cannot travel faster than the speed of light, even for photon wave packets.
This finite velocity of a wave packet with a well-defined leading edge means there is never a possibility that any component of the wave packet can reach a remote location in a way that affects classical causality, regardless of internal phase states that may perplexingly suggest otherwise.
After all, just because I can turn off my car clock as I enter the Baltimore Harbor Tunnel and turn it back on as I exit the other end does not mean that I traveled through the tunnel at infinite speed. Similarly, just because the phase of a particle froze as it tunneled through a barrier does not mean that it traveled through that barrier at higher than c velocity.
3. Terry Bollinger: Not a physicist here; but I thought the wave packet of a particle with indefinite position, as shown in the video, was a distribution with infinite tails.
How is a rubidium atom "classically visible", and even if it were, isn't that measuring its position and therefore invalidating the experiment?
“That message could reasonably consist of 3 or 4 bits of information.”
Dr. A. M. Castaldo, that’s a very interesting thought experiment. I was racking my brain, or what’s left of it, being an older person (and brain cells decrease with age), to figure out a loophole in your argument, that would prevent information being transferred superluminally via multiple species of atoms passing through the barrier to provide distinguishable information bits. There might be something quite subtle in the quantum mechanical toolbox that would frustrate efforts to superluminally signal via that strategy. With my knowledge base I’m poorly equipped to come up with a solution. But I love a challenge, and I’m going to keep thinking about it. But, likely, much brainier, and more knowledgeable, people here will figure it out long before I do, if I ever do.
5. Hi Dr A.M., nice to hear from you and I hope you are well!
An answer in two parts:
No. To be precise, such wave functions cannot even exist in the real universe. That’s because they would require infinite time and space to form infinite tails via Schrödinger’s equation, and the real universe is finite in both size and time
More importantly for small-scale quantum experiments, and regardless of one’s philosophical stance regarding “wave collapse”, any localization of a wave function that results in an irreversible historical record — actual bits of information, a “detection” in space — unavoidably and completely erases all traces (and thus leading and trailing edges) of that wave function outside of the xyz box in which it was relocalized (“found”). That is what wave collapse is: the complete removal — not just diminution, but erasure — of probability amplitudes outside of the xyz detection box.
The trailing edges then must reform “from scratch”, so to speak, and can only do so via Schrödinger’s equation. In the context of a wave function that has been collapsed into a well-defined initial xyz region, Schrödinger’s equation behaves like an entirely conventional wave function, one that in the case of photons is isomorphic to the electromagnetic wave equation. Since this equation is bound by the speed of light, there is never a leading edge beyond the light cone leading out form the initial xyz box location. The spread of this quantum wave thus is classical in every way except for the final Born interpretation, which is the real spanner in the monkey works (and yes, I couldn’t resist mixing US and UK metaphors there… :)
The idea of a pure sinusoidal particle state is quite nonsensical in this context, for the simple reason that all actual, experimentally meaningful wave functions created naturally or in the lab necessarily have associated with them some classical-origin boundary boxes in xyz. Thus all wave functions are more correctly described as wave packets — wave pulses with sharp edges beyond which their amplitudes are not just very small, but exactly zero. The unreal case of a pure sinusoidal state — meaning a quantum entity with a completely undefined location in the universe — is a state that real wave packets can only approach, and even then only if provided with infinite time and space.
I am genuinely a bit baffled at why so many quantum textbooks are sloppy about this point, since there’s nothing radical about it. The sections about pure states almost always correctly point out that such states can only be approached, but then they start slipping into treating Dirac deltas is if they are real states in the real world, resulting in sloppy equations that assert quantum entities to be in places where they cannot be experimentally.
It’s usually not that bit of a problem, but it can become a big one when folks start talking about things like group velocities in waves, which that can create illusion of faster-than-c propagation. I say “illusion” because faster-than-c group velocities in waves are no more a violation of c than the dot you could get by sweeping a very powerful laser sideways across the surface of the moon at a sufficiently high angular velocity. Such a dot would appear to earth telescopes to move across the moon at faster-than-c velocity, just as some group velocities can appear to create wave that move faster than c. However, in both cases no independent signal is carried from one edge to the other, only the illusion of motion.
When you bound the group velocities by a sharp wave edge, even the illusion disappears as the group velocity waves strike edge of reality and disappear.
6. >… How is a rubidium atom "classically visible", and even if it were, isn't that measuring its position and therefore invalidating the experiment?
It’s classically visible because it’s inside the apparatus. It’s hard to play with rubidium atoms that aren’t there! The game in experiments like this is that you let the location of the rubidium atom wave function get large enough to accommodate some slop in the measured speed of light.
That is all any of this ever is, which is why I’m not personally much impressed by any of this work, even though I fully acknowledge there are some interesting math issues. For example, and as nicely Sabine described, the peak of the tunneled particle looks like it moved faster than light. That sounds impressive until you realize that all that is really going on is that some of the wave components closer to its speed-of-light-bound leading-edge are getting a bigger share of expectation amplitude than they normally would receive.
If that’s breaking the speed of light, then so giving a few dozen folks in a marathon a ride in a car so they can finish closer to but not in front of the actual winner. And since that winner, coincidentally name W. Schrödinger, is never permitted any such boost, the Schrödinger edge-boundary (and classical causality) remain nicely intact, despite the subsequent clump of tunnel-cheaters.
7. Terry Bollinger: Wow, thanks, that is enlightening.
If all that happened is the barrier introduced some skew in a non-infinite distribution and the leading edges are still the same, then I still see no reason for FTL angst. I'd expect the barrier to change the distribution somehow.
I suppose there is some mystery, which might be resolved just by the math, as to why the leading edge of the distribution seems to get a greater than expected through-rate than the rest of the distribution (which would be, to me, the plausible cause of the skewing).
Since the expectations have a not-flat shape, it would be interesting to see what the shape is; e.g. actual probability of tunneling vs. probability of position.
Would you think that a non-zero probability of tunneling requires the leading edge of the wave packet to be on the other side of the barrier? (as opposed to within the barrier).
8. Hi Dr A.M.,
Thanks, I appreciate the feedback! For some reason my own logic always sounds a lot clearer inside my head than it probably does to folks outside said head. (Anyone else ever feel that way?… :)
Good question, one with a nicely straightforward answer: yes.
That is, some non-zero amplitude (a leading edge) must exist on the far side of the barrier for tunneling through the barrier to occur, always. That's because at the Feynman QED level, the leading edge of the wave is really nothing more than the collection of all possible paths by which the particle might get there, subject to the limits of physics and the potentials affecting its motion. Stated that way, it becomes almost a tautology: The wave is the sum of all real paths by which a real particle could get there, not some separate entity. It only looks like a wave because quantum histories interfere and reinforce each other, so that even a single particle looks just like an entire front of electromagnetic waves.
Tangent 1 of 2: The beauty and predictive power of the Feynman path integral approach is also why I don’t buy into the de Broglie / Bohm pilot wave concept, even though I fully agree that it can be a very powerful and useful analytical tool when used in the right way. My concern is simple: If the best and most precise way to define the ephemeral “pilot wave” is simply to map out every possible path of the real particle, while using phase tracking and mutual interference to keep track of which paths are most energetically favorable… well, then why in the heck do you need a separate, much more magical wave that “just happens” to look like the collection of all those possible particle paths?
Tangent 2 of 2: The presence of massive redundancy turns out to be a powerful and quite generic analytical heuristic for uncovering logical inconsistencies in theories. For example, massive redundancy also provides a powerful argument against the late-1970s presumption (that’s all it ever was) that gravity “must” me a quantum force, just like the electromagnetic, strong, and weak forces. There’s a bit of a problem with that idea, however, and the problem is very much one of redundancy. Ask yourself this: If gravitons in any way resemble the bosons of the other three forces, then just like those bosons, then they must travel and interact over an xyz-type space, right? So far so good. But now ask yourself the follow-up question: If you then curve the space over which such gravitons are traveling, what do you get in terms of the impact of that curvature on those same particles? Gravity again… the Einstein topological version!
My point is simply that the very concept of boson-based gravity amounts to a sort of theoretical cotton candy, a hypothetical way of constructing a force that kind of looks like gravity in the sense that it pulls together any two objects with “mass charge”. However, on closer examination this mechanism remains flatly unrelated to the actual, topological gravity that Einstein figured out. Smart fellow, that Einstein.
9. I should think you have one or the other, gravitons or curvature; and the relationship between them is analogous to how the equations of fluid dynamics (e.g. Navier-Stokes and others) accurately describe the aggregate behavior of what are ultimately many trillions of discrete atoms; but (as we've seen even with QD) does not capture the behavior of individual particles.
Meaning, Einstein's curvature formulation might be just a very accurate approximation of how trillions of gravitons behave, but might not accurately capture the behavior of just a few gravitons.
3. By statistics you can imagine classically that if the particle is measured as tunneled, it (its info) was actually travelled at front in the wave. Logically you can also imagine that the tunneling particle is actually from the barrier matter structure via energetic interactions.
Still, the human imagination does not produce new physics but only helps avoid obstacles for it. The same other way: no equally varying interpretation is physics but a new prediction really is if it's more accurate than earlier.
4. As a young nuclear engineer, I accepted radioactive behavior without explanation. I genuinely appreciate having a better understanding - however slight - of radioactive decay as a manifestation of tunneling.
5. This is an excellent video, Sabine! I liked in particular your strategy of starting with how particles smear out, so that folks can see how the idea of a single particle with a well-defined location breaks down in quantum mechanics.
One thought that occurred to me as you showed a ball bouncing near the top of the wall is that there is also an even closer analogy possible: A very slow-moving neutron heading toward the top of a quite literal wall of dense metal. Viewed as a point, the neutron would either bounce off of (or more likely, be absorbed by) the wall, or it would sail over it, exactly like a classical wall. But because the neutron wave function smears out in exactly the fashion you described at the start of your video, there will instead be a part of the neutron wave function that, if it is very close to the top of the metal wall, will diffract over the wall, allowing some neutrons to cross over the wall even though the same neutrons as points would not be able to do so.
6. Tunneling in part is manifestated by the occurrence of a wave function in a region that is forbidden classically. Classically if the energy of the particle is smaller than the potential the particle bounces off. The generator of the unitary motion for a particle in a potential barrier V is ε = √[2m/ħ(E – V)]. For V > E this is complex valued and the complex valued e^{-iEt/ħ} becomes real valued. It is also exponential decaying. One result is that we can write εt/ħ with ε = √[2m/ħ(E – V)] as ε = i√|2m/ħ(E – V)| → iε and associate the i = √-1 with time so it = τ. Then for this we have ετ/ħ = ε/kT for some pseudo-temperature T = ħτ/k.
I have thought this pseudo-temperature might be interpreted as a temperature of nonlocal hidden variables, maybe local if we cast realism aside, that in a statistical sense define this pseudo-temperature. This is also the imaginary Euclidean time above. This would the correspond to the time of tunneling.
Quantum tunneling is a big cornerstone of a lot of technology, in particular quantum electronics.
7. One of the cornerstones of Quantum Mechanics the Quantum Tunneling that appears to be well understood, being controllable with many applications in solid state physics e.g. semiconductors, it is actually non-understood at all. The wave function and the uncertainty principle serve as the veil of Quantum weirdness that ignore the cause behind the effect pointing to Quantum Mechanics incompleteness.
Below are listed the two inconsistencies being saved by the probabilistic observational justification:
a) A quantum particle overcomes a potential barrier having less energy would normally lead to violation of energy conservation if the particle was classical
b) A quantum particle overcomes a potential barrier in the name of the Wave function (not an intrinsic property of the particle) and the uncertainty principle corresponding the first to the probabilistic (pure mathematics unrelated to physics) outcome and the later a justification that has nothing to do with the observation but to the determination (uncertainty) of the position (Δx) and momentum (Δp) or the energy (ΔE) and time (Δt).
Obviously, due to (a) and (b) no one understands why the particle was actually able to tunnel through the barrier, because the Schrodinger equation has nothing to say about it except to calculate the probability of seeing something on the other side (beyond the barrier).
The question remains: If one tells you the probability to experience/measure an outcome that under classical conditions defies the energy conservation, it cannot be justified just by using the definition of "quantum particle" (see Schrodinger equation).
The missing part in the puzzle is according to my opinion, a shielding effect that leads to the reduction of quantum particle effective charge. All those mentioned in the video clip make really sense when the cause e.g. shielding effect is integrated in the Schrodinger equation, otherwise the Quantum Tunneling tends to be classified as an illusion than a genuine effect.
1. It is interesting to consider that any type of wave can tunnel even sound waves. Both Yang and Robertson have shown that acoustic waves can tunnel They found that inside the stop band the acoustic group delay was relatively insensitive to the length of the structure, a verification of the Hartman effect. Furthermore the group velocity increased with length and was greater than the speed of sound, a phenomenon they refer to as "breaking the sound barrier".
2. John,
The potential of the barrier is an average one. You do not actually know what was the potential at the exact time and place the particle was there. It might be the case it was low enough so that no violation of energy conservation is required.
8. The Hartman effect is the tunnelling effect through a barrier where the tunnelling time tends to a constant for thick enough barriers. This was first described by Thomas E. Hartman in 1962. This leads to the conclusion that the wave does not travel through the barrier but just appears at the other side of the barrier in the same constant time no matter how long the barrier happens to be.
But the probability of transmission through such a thick barrier becomes vanishingly small, since the probability density inside the barrier is an exponentially decreasing function of barrier length.
For waves traveling at the speed of light, this effect implies that for a thick barrier, the wave may go superluminal.
1. I am familiar with this. My ancient memory on this was jogged a while back with news of experimental data on this. I will say that I do not think this determines the speed information crosses a barrier. The Hartman effect is the time for phase velocity to cross or the phase time.
I suggested above this time is related to a pseudo-temperature determined by statistics on hidden variables. Either this HV is nonlocal, or if local one abandons realism as with Wigner's friend and related developments of late. This would then have no sensitivity to the scale of the potential barrier.
9. In a previous universe, a photon could have tunnels through a seemingly insurmountable barrier. Since this event has a non-zero probability, given enough time, such an event will happen.
With its passage through the barrier, this tunneling event would have greatly amplified the energy of the photon to the point of big bang causation and subsequent superluminal inflation. This process is referred to as Quantum Creation.
10. The particle can be anywhere and everywhere within distributions both before tunneling and after. The distribution peak after tunneling appears ahead of the peak without tunneling so lets behave ourselves and bound the distribution after tunneling by x=tc. Steinberg says the probability of tunneling is extraordinarily low but he didn't say to bound the distribution by 'c'. I suppose this model looses fidelity when the barrier is the strong force and we examine nuclei collisions on the Sun?
11. There must be a typo error in the Quanta magazine article that Sabine referenced. It states: “The researchers reported that the rubidium atoms spent, on average, 0.61 milliseconds inside the barrier, in line with Larmor clock times theoretically predicted in the 1980s. That’s less time than the atoms would have taken to travel through free space.” In .61 milliseconds a light beam, in free space, traverses 183 kilometers. So, that’s a rather big laboratory for this experiment. It must have been pico-seconds, which would correspond to .183 millimeters for the width of the laser beam. Or, maybe, it was a really wide laser beam at 183 millimeters, or about 7.2 inches. At least those would fit into a terrestrial laboratory.
1. The atoms don't move with the speed of light. The next sentence explains it.
2. Sabine, thanks, I didn't make that connection.
12. lagunastreets: Statistically speaking, if you bound an infinite tail of a distribution at some point X, the volume of probabilities in the excluded region (beyond X) must be somehow incorporated back into the acceptable, realizable portion of the distribution.
Which may not be much, admittedly, but would still suggest the Schrödinger equation is not accurately capturing the probabilities; but consistently falling short.
Not being a physicist, I would have guessed such a discrepancy between theoretical and realized probabilities would have been noticed by now.
1. Mathematically, Schrödinger's equation is akin to a diffusion equation, and inaccurate in the context discussed here. A relativistic equation (Klein-Gordon or Dirac) should be used. The solutions could then actually have sharp edges instead of long tail precursors.
Incidentally, the tunnel effect in the case of photons is adequately described by classical optics ("evanescent waves"). The distinction between "classical" and "quantum" effects is not Nature's, but of how we choose to describe them.
13. As someone with an understanding of physics limited by not having the math involved, I still found this to be a very clarifying description of tunneling. Thanks, Dr. Hossenfelder!
14. Referring to the last part of your video, I noticed how the 2007 conjectures of Günter Nimtz - the superluminal tunneling of information and the violation of Special Relativity - can be debunked with a single stroke. kudos!
15. The tunnel effect can also be understood by the particle model of Louis de Broglie; which means in a more classical way which is helpful for imagination and avoids much of the weirdness of QM.
A potential wall in this context is realized by a field which repels the charges of the particle trying to pass. These field forces build the potential wall which the particle has to pass. This is the field of the particles which are the cause of the wall. The oscillations which are present in all particles cause the internal charges to build an external oscillating field. If there are several particles, which is the normal case, then the field is built by a random superposition of the individual fields. So the repelling force is in a permanent change. If this constellation is momentarily such that the resulting field is smaller than the average, then the particle in view can pass even if its energy is too small for passing at the average potential. The probability that the superposition of the single fields cause a considerably reduced one is statistically a rare case. In such rare case the arriving particle may pass at a comparatively small energy.
This probability is described by the Schrödinger equation.
And I think that the apparent case of a superluminal velocity can be explained by the difficulty to determine the position of a particle, if only the surrounding wave is detectable.
16. I have to wonder what happens to the portion of the wavefuction that was associated with the tunneled particle, but is reflected at the barrier and doesn’t continue with the particle? This, I imagine, impinges on the old debate whether the wavefunction is something real, or is just knowledge of the observer. My understanding is that the Copenhagen interpretation would fit in the latter category. Hidden variable theories like the de Broglie-Bohm model by imputing reality to the wavefunction (as an ensemble of pilot waves associated with the particle, I think?), would (presumably) treat this reflected portion of the wavefunction as objectively real and able to exist on its own.
But, I’m guessing that this depiction of the reflected portion of the wavefunction, originally associated with the particle, in both the Quanta magazine article, Sabine’s excellent video, and textbooks in general, is just for illustration and not intended to represent reality. In the de Broglie-Bohm pilot wave model perhaps the reflected part of the wavefunction (as an ensemble of pilot waves), in a real world situation like a radioactive nucleus, just continues its existence inside the potential barrier of the nucleus and either mixes with other pilot waves or dissipates eventually. Don’t have time this AM to research this in more detail, with a pending appointment, but will check it out later today.
17. Commonly tunnel effects are noticeable through diminishing hustle and bustle.
18. To Engineers, i.e. people who know PDEs, the Fourier theory, and a bit about numerical solutions. I write this for you. (Sorry, laymen! I just don't know how to explain it *briefly* enough---and I have RSI.):
See if this helps. (Sabine, could you please to chime in if I am describing it wrong somewhere.)
Explanatory Comments:
A neat applet quite similar to what Sabine has shown, is by Prof. Dr. Daniel V. Schroeder. (Google on his name + "applet" + "BarrierScattering.html".)
I now assume you've played a bit with this applet.
This applet has thankfully been written in HTML5, and so I could easily take a peek at the code (right click, View Page Source). Very brief comments follow. (I assume that the scheme for Sabine's simulation goes similarly.):
1. There is a finite box with infinitely large potential walls on its two extreme ends.
2. Assume no PE (i.e. V =0) i.e. no middle barrier. In Schroeder's applet, you can't make V zero, but you can make it ("Barrier energy") very small; smallest possible is 0.001.
3. The simulation puts a Gaussian wave-packet in this box, as the initial condition. Its initial "width" is small.
(Technically, the Gaussian *always* fills all the space of any domain---finite or infinite. The "width" for the Gaussian is therefore defined differently. (Recall/refer to the normal distribution.))
4. The peak of the initial Gaussian is put into the left half of the box. Though the Gaussian's tail is not zero anywhere in the box, it's almost zero in the right hand-side half.
5. With time, due to the Schrodinger evolution, the Gaussian wave-packet spreads. (It "diffuses", though this is not quite the exact word.)
Schroeder implements the time-marching through the leap-frog, not FFT, but yes, you can always think about the time evolution using the Fourier theoretical terms.
6. The interaction of the Gaussian wave-packet with the boundaries (and the potential barrier, if any), together with the Schrodinger evolution, makes the peak go to the right.
This rightward motion of the peak is not a primitive; it is a result of a process. The motion does not exactly occur at a constant speed, but I guess it might be OK to describe it thus in a pop-sci video.
7. Next, you increase V ("Barrier energy") to some significant value, and repeat.
8. With V = 0, there is only one peak. With V = nonzero value suitable for tunnelling, there are two peaks. Yes, it's a bit complicated, but basically it's directly a result of the Fourier theory (and the Schroedinger equation).
9. In both cases, the peak is a result of superpositions of sinusoidals of *all* frequencies (f = upto half the number of cells used in the simulation; Shannon's theorem). Remember, Gaussian in theory carries a continuum of pure frequencies.
10. The motion of the peak (e.g. its speed) thus is a result of superpositions of plane waves (sinusoidals).
11. Different frequencies "react" differently to the potential wall of the specified width and height.
Think of it this way: each sinusoidal has to maintain continuity. To appreciate this point, check out the PhET simulation on quantum tunnelling. (QM *theory* requires only C0 continuity, but the IC is Gaussian, so C1 also would get satisfied.)
12. Now, feel free to interpret what(ever) it all means.
PS: Guess it would be a good idea to turn this reply into a blog post at my blog too!
1. Hi Ajit,
Thanks for pointing out, I hadn't seen this app! I have really just dumbly forward integrated the Schr eq. Anything more sophisticated seemed to me an overkill given that I'd only need a few seconds of it. I did not put infinite walls on the side, and the Gaussian you see in my sim actually isn't exactly a Gaussian because it goes to zero at a point you can't see (which produces some artifacts). Oh, and the color in my animation doesn't show the phase, I found that too be too confusing and also superfluous somehow.
2. Hi Sabine,
1. Sure! Also see the PhET simulation. You can try plane-waves in it.
2. For pop-sci videos, the forward Euler is quite great! I vaguely recall that Chad Orzel used it for showing how precession (with nutation) arises, and Rhett Allain for demo'ing 3-body simulation, in their blog posts.
(BTW, I always check out *others'* code first, before implementing anything. After all, I am, ahem, a *professional* programmer.)
3. What approximate Gaussian with smoothly approached zero-ends did you use? Any systematic means to construct such functions for numerical work? In any case, looks like it gives great results for this case. Would be a handy tool to have. (For my work, I was actually thinking of just "pulling down" the whole Gaussian so that its support becomes finite!)
4. Phases *are* confusing. I don't "get" much anything using them. I have to see the separate Re and Im plots, really speaking. But for pop-sci, prob. density is best, I think. No confusions!
5. I caught a mistake I made in my comment above:
QM theory requires only square-normalization, not even the C0 continuity. E.g., in theory, you can have a wave-function whose Re and Im parts *separately* go like ideal square waves (with the Re and Im squares not even matching). Neither Re nor Im part would be, technically, even C0 continuous.
3. Sabine,
Sorry to bother you again, but I've another question:
If you didn't put infinite walls on the sides, then how come the peak ends up travelling in one direction (to the right)?
...On second viewing of the video, looks like your wavefunction isn't symmetrical around the peak---it has a sharper drop on the right hand side. That can explain it.
(May be, if you don't mind, could you please share the code? But of course, a hint would be good enough too.)
4. My "code" isn't a code really, and it's in no condition to share. I'm not sure I understand your question, I just used the initial condition for a wave-packet with a non-vanishing momentum, that's why it travels to the right. I don't know why you think it's not symmetric. It should be. Maybe that's from the shading? The shading may not be symmetric. (It's actually just a blurred gradient that I didn't do myself, so god knows what that is.) The issue with the boundary condition is that the program I used expected the boundary condition to be constant with t. So I set it to zero at some initial time, but then if the wave-packet gets close to the boundary, that produces nonsense. I didn't have the patience to deal with that, so I just put the boundary very far away from the center of the Gaussian, so it's to excellent precision zero and all is fine. Hope that clarifies it.
5. Sabine,
1. OK, I got it! I mean the travel to the right. (Stupid me! Somehow, had got stuck in the special case of k_0 = 0 in my imagination, once I *began* with that case).
2. Well, the profile in the video does seem to become asymmetric as it moves. Could be a numerical artifact, but guess it's best to leave the matter at that, because the overall picture *is* quite clear now. Thanks!
6. Hi Ajit,
I am still not sure what asymmetry you mean. Could you let me know what time in the video you are referring to so I can look into this?
7. Hi Sabine,
Check out at 02.49 in the video. The peak is under your left palm. I took a screenshot and verified. While the absolute number of pixels would depend on the size of the window of the video when the screenshot was taken, it is clear that the x-axis extents of the left- and right halves of the profile, as measured at the bottom of the profile, are roughly in the 0.44 : 0.56 proportion. Not 0.5 : 0.5.
8. Oops. The left half is longer (~0.56), and the right half is shorter (~0.44). --Ajit
9. Hi Ajit,
Ok, thanks, I see now what you mean. I thought you were talking about the tunneling part. Yes, I suspect that this comes from the boundary condition. You see, the boundary condition will try to push the value of the Gaussian down to zero somewhere off to the right, though actually it isn't zero. I guess I should have put the boundary farther away, but that'd have brought up the computation time. Sorry about that & thanks for pointing out.
19. Dr. Hossenfelder:
Is there some mass limit (or complexity limit) to tunneling? For example, can a water molecule tunnel through a physical barrier? Or is it impossible to make a physical barrier that thin?
(I have long thought pancake syrup tunnels; I have no other explanation for how it can get everywhere.)
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Scattering of He Atoms from He Surfaces
E. Krotscheck and R. Zillich Institut für Theoretische Physik, Johannes Kepler Universität, A 4040 Linz, Austria
We develop a first principles, microscopic theory of impurity atom scattering from inhomogeneous quantum liquids such as adsorbed films, slabs, or clusters of He. The theory is built upon a quantitative, microscopic description of the ground state of both the host liquid as well as the impurity atom. Dynamic effects are treated by allowing all ground–state correlation functions to be time–dependent.
Our description includes both the elastic and inelastic coupling of impurity motion to the excitations of the host liquid. As a specific example, we study the scattering of He atoms from adsorbed He films. We examine the dependence of “quantum reflection” on the substrate, and the consequences of impurity bound states, resonances, and background excitations for scattering properties.
A thorough analysis of the theoretical approach and the physical circumstances point towards the essential role played by inelastic processes which determine almost exclusively the reflection probabilities. The coupling to impurity resonances within the film leads to a visible dependence of the reflection coefficient on the direction of the impinging particle.
I Introduction
Dynamic scattering processes of helium atoms from low temperature liquid He films and the bulk fluid in the vicinity of a free surface continue to be a subject of considerable interest. Experimental information is available mostly for He scattering processes, connected with quantum reflection and quantum evaporation [1, 2, 3, 4], as well as the surface reflectivity [5, 6, 7, 8, 1]. Due to experimental difficulties, there are only few data for He scattering[9], but there is also interest (experimental [10, 11, 12, 13], and theoretical [14, 15, 16] ) in the dynamics of atomic Hydrogen atoms on He surfaces for which our theory also applies.
This paper follows up on a line of work studying the properties and the dynamic features of quantum liquid films from a manifestly microscopic point of view. Most relevant for the present work are papers designing the theory for the background host liquid[17], its excitations[18, 19], and the dynamics of atomic impurities[20]. In that work, we have used the method of correlated variational wave functions which has in many situation proven to be a computationally efficient, precise, and robust method for the purpose of studying strongly interacting quantum liquids. Even the simplest approximation of the theory has in the past given quite satisfactory results on the nature of the impurity states[21], their effective mass[22] and the impurity-impurity interaction[23] in inhomogeneous geometries. The reason for the qualitative success of the theory is that it contains a consistent treatment of both the short- and the long-range structure of the system. This implies that both the low- and the high-lying excitations are treated accurately.
The present paper complements a similar study of the scattering of He atoms from He slabs[24]; the problem at hand is somewhat simpler since there is no need to fully symmetrize the wave function of the background system and the impinging particle. Another major physical difference to the scattering of He particles is that in the latter case one might observe[25, 26, 27] the coupling to the Bose-Einstein condensate, whereas in the present case one can couple both to phonon-like and to single particle excitations. Nevertheless we will see that many similarities exits between the two problems: The scattering process is dominated by inelastic channels, mostly the coupling to ripplonic excitations.
Generally, the impinging particle can, in the presence of other particles like the film of He under consideration here, scatter into three types of channels:
1. Elastic reflection: The incoming particle, characterized by the wave vector , is elastically reflected with a probability . It creates virtual excitations of the background, but transfers no energy.
2. Inelastic scattering: with a probability the particle loses some energy to an excitation of the film, and retains enough energy to leave the attractive potential of the film and the substrate. The film excitation can be either a collective wave (ripplon, phonon), or a single He that is elevated above the chemical potential and leaves the film. The creation of several excitations is in principle also included in our theoretical description, but it is ignored in the linearized treatment of the equations of motion.
3. Adsorption: as in the previous case, the film is excited, but the particle is adsorbed to the film. The corresponding sticking coefficient is the probability for this process.
These three types of processes are depicted in Fig. 1. Because of the hermiticity of the many–body Hamiltonian for He atoms and the He impurity, we have
This work focusses on the calculation of elastic scattering because the impinging particle couples, in particular at low energies, predominantly to the low–lying, bound excitations of the background film and the impurity atom. We shall argue below that, basically for phase-space reasons, inelastic processes are expected to be less important than either elastic, or total absorption processes.
Since most of the theoretical tools of the present study have been derived in Ref. 20, we outline in Sec. II only briefly the theoretical methods and the basic equations to be solved. The scattering problem will be be formulated in terms of a non-local, energy dependent “optical potential” which depends explicitly on the coupling of the impinging particle to background and impurity excitations.
The results of our calculations are discussed in section IV. To cover a variety of physical situations, we will present results for several of the systems that were studied extensively in our previous calculations: These will range from strongly bound films on a model graphite substrate that is covered with two layers of solid helium, to a very weakly bound model, described by a rather thick, metastable film on a Cesium substrate. We first discuss the possible excitations of the background systems, and then present results for the surface reflectivity as a function of impact energy and angle for some of those systems. At very low energies, we will encounter the effect of “quantum reflection” [28, 29, 30, 16, 31, 14, 15]; with increasing impact energies we also can analyze the influence of surface excitations (ripplons) and the Andreev state, phonon/roton creation, and under certain circumstance the coupling to an “Andreev resonance” of the impurity particle close to the substrate.
Ii Microscopic Theory
The theoretical description of He films and impurity properties starts with a description of the ground state of the background system. Next, a single impurity is added, and finally this impurity is allowed to move. The technical derivation and in particular the important verification of our theoretical tools have been presented in a series of previous papers[32, 17, 20], we will therefore discuss the theoretical background only briefly.
ii.1 The Background Liquid
In the first step, one calculates the properties of the background helium film. The only phenomenological input to the theory is the microscopic Hamiltonian
where is the He-He interaction, and is the external “substrate” potential. The many–body wave function is modeled by the Jastrow-Feenberg ansatz
An essential part of the method is the optimization of the many-body correlations by solving the Euler equations
where is the energy expectation value of the -particle Hamiltonian (2) with respect to the wave function (3),
The energy is evaluated using the hyper-netted chain (HNC) hierarchy of integral equations[33]; “elementary diagrams” and triplet correlations have been treated as described in Ref. 17.
The HNC equations also provide relationships between the correlation functions and the corresponding -body densities. One of the quantities of primary interest is the pair distribution function and the associated ( real-space ) static structure function
The static structure function and the effective one-body Hamiltonian
define the Feynman excitation spectrum through the generalized eigenvalue problem
which is readily identified with the inhomogeneous generalization[34] of the well-known Feynman dispersion relation[49] . The states , their associated energies , and the adjoint states
are useful quantities for the impurity problem and for the representation of the dynamic structure function of the background film.
ii.2 The Static Impurity Atom
The Hamiltonian of the particle system consisting of He atoms and one impurity is
We adopt the convention that coordinate refers to the impurity particle and coordinates , with to the background particles. Note that the substrate potentials and , as well as the interactions and , can be different functions for different particle species.
The generalization of the wave function (3) for an inhomogeneous -particle Bose system with a single impurity atom is
The energy necessary for (or gained by) adding one impurity atom into a system of background atoms is the impurity chemical potential
Here, is to be understood as the energy expectation value of the Hamiltonian (10) with respect to the wave function (11). The further steps parallel those of the derivation of the background structure.
The impurity density is calculated by minimizing the chemical potential (12) with respect to . This leads to an effective Hartree equation
where is an effective, self-consistent one-body potential for the single impurity. The lowest eigenvalue of Eq. (13) is the impurity chemical potential , and the corresponding eigenfunction the density of the impurity ground state, .
In the systems studied below, translational invariance in the plane is assumed, and the states are characterized by two quantum numbers, and , associated with the motion perpendicular () and parallel to the symmetry plane (). When unambiguous, as in the states and , we shall use the single label (e.g. ) to collectively represent both quantum numbers. In particular, the states depend only trivially on the parallel coordinate,
The unit volume is chosen as the size of the normalization volume. The corresponding energies are
where are the eigenvalues of Eq. (13) for .
ii.3 Impurity dynamics
It is tempting to identify the higher-lying eigenstates of the “Hartree-equation” (13) with the excited states of the impurity. This is legitimate only in a static approximation for the impurity features. However, such a simplification misses two important features:
• If the momentum is a good quantum number, low-lying excited states can be discussed in terms of an effective mass. In our geometry, a “hydrodynamic effective mass” is associated with the motion of an impurity particle parallel to the surface; it is caused by the coupling of the impurity motion to the excitations of the background liquid. The local Hartree–equation (13) misses this effect.
• The effective Hartree-potential is real, i.e. all “excitations” defined by the local equation (13) have an infinite lifetime. A more realistic theory should describe resonances and allow for their decay by the coupling to the low-lying background excitations of the host film.
Hence, a static equation of the type (13) is appropriate for the impurity ground state only. The natural generalization of the variational approach to a dynamic situation is to allow for time–dependent correlation functions . We write the time dependent variational wave function in the form
Consistent with the general strategy of variational methods, we include the time dependence in the one-particle and two-particle impurity-background correlations, i.e. we write
The time independent part remains the same as defined in Eq. (11). The time–dependent correlations are determined by searching for a stationary state of the action integral
where is the Hamiltonian (10) of the impurity-background system.
The derivation of a set of useful equations of motion for the impurity have been given in Ref. 20. The final result is readily (and expectedly) identified with a Green’s function expression, where the three-body vertex function describes an impurity atom scattering off a phonon, and is given in terms of quantities calculated in the ground-state theory. The motion of the impurity particle is determined by an effective Schrödinger equation of the form
where is the effective one–body potential of Eq. (13), and is the impurity self–energy. Within the chosen level of the theory, is describes three–body processes,
where is the three–body vertex function that describes the coupling between an incoming He particle to an outgoing He in the state as well as an outgoing phonon in state . The detailed form of these matrix elements follows from the microscopic theory that has been described in length in Ref. 20, it is not illuminating for the further considerations.
The structure of Eqs. (19) and (20) is of the expected form of an energy-dependent Hartree-equation with a self-energy correction involving the energy loss or gain of the impurity particle by coupling to the excitations of the background system. It is the simplest form that contains the desired physical effects.
The energy denominator in Eq. (20) contains the Feynman excitation energies defined in Eq. (8) and the Hartree impurity energies of Eq. (13). These energies are too high, and we expect therefore that three–body effects are somewhat underestimated. A lowering of the spectra in the energy denominator by an impurity effective mass or by a more quantitative phonon/roton spectrum should have the effect of enhancing the importance of multi-particle scattering processes. Hence, it is expected that the binding energy of the surface resonance is still somewhat too high compared with experiments. On the other hand, it is not expected that a more quantitative spectrum in the self–energy should change the effective mass of the Andreev state considerably because the hydrodynamic backflow causing this effective mass is mostly caused by the coupling to ripplons, which are well described within the Feynman approximation.
Iii The physical models
We consider liquid He adsorbed to a plane attractive substrate which is translationally invariant in the plane, i.e. . The systems under consideration are characterized by the substrate potential and the surface coverage
where is the density profile of the He host system. This density profile is, along with the energetics, structure functions, and excitations of the film, obtained through the optimization of the ground-state (3) as outlined above; the procedure has been described in detail in Ref. 17.
iii.1 Ground state
We have in this work studied the scattering properties of He atoms for a number of selected substrate potentials and surface coverages; we have selected four cases for the purpose of a detailed discussion. The substrate potentials and the corresponding density profiles are shown in Figs. 2 and 3. The surface coverages are for each substrate potential; additionally we have considered the case for a Cs substrates as well as Mg for a case that is somewhat more attractive than the screened graphite, but also has a long range.
Alkali metal substrate potentials are simple potentials characterized by their range and their well depth . They have the form
The range parameters of these potentials have been calculated by Zaremba and Kohn[35], the short–range term is phenomenological and fitted to reproduce the binding energies of a single atom on these substrates. Slightly more complicated is our model of a graphite substrate covered with two solid layers of He. Most important for low–energy scattering properties is the coefficient of the long–range attraction, the values of for our substrates of graphite, Cs, Na, and Mg are , , and K , respectively[35, 36]. The graphite– potential is relatively short-range but deep and produces a very visible layering structure of the background film; thus one obtains a rather “stiff” system[17].
Fig. 2 provides a comparison of these four different potentials. It is seen that the alkali metal potentials are longer ranged, the magnesium substrate has the deepest potential well. At the opposite end of the potential strength is the Cs substrates. This substrate has received much attention in recent years because of the experimental finding that it is non-wetting[37, 38, 39]. Note that the Cs-adsorbed films are metastable; they were examined with two purposes in mind. One is to generate a situation that is reasonably close the infinite half–space limit. Therefore, we have studied this case also for the larger surface coverage . The second reason is that the nature of the low–lying excitations[40] as well as that of the impurity states[41] is somewhat different than those for the graphite model as will be seen below.
The third case, a Na substrate, is an intermediate case which is of some interest for the nature of the He bound states, whereas the Mg substrate is both deeper and longer ranged than the screened graphite.
iii.2 Background Excitations
Our earlier work[18, 19, 42] has discussed extensively the excitations of quantum liquid films adsorbed to various substrates. These studies have been concerned with the interpretation of neutron scattering experiments[43, 44, 45], they have therefore focussed on excitations propagating parallel to the film. Typically, four types of modes were found:
1. Surface excitations: At long wavelengths and on strong substrates, these are substrate potential driven modes with a linear dispersion relation
where is the speed of third sound. At shorter wavelengths and in the case of an infinite half–space, the surface–mode is driven by the surface tension and has a dispersion relation
where is the surface tension, and the density of the bulk liquid. In practice, the dispersion relation is linear only in a rather small momentum regime, and the ripplon dispersion relation (24) is a quite good approximation[42] for the surface—mode dispersion relation up to wavelengths of about . The theoretically predicted surface energy obtained from Eq. (24) by a fit to the dispersion relation is which compares favourably with the most recent experimental value[46, 47] of .
2. Bulk Rotons: Films with a thickness of two or more liquid layers show already a quite clear phonon/roton spectrum. The spectrum starts at finite energy in the long—wavelength limit and contributes, in this momentum regime, very little to the strength. It takes over most of the strength in the regime of the roton minimum.
3. Layer Rotons: Films with a strongly layered structure also show excitations (identified as sound–like through their longitudinal current pattern) that propagate essentially within one atomic layer. These excitations have a two–dimensional roton with an energy below the bulk roton, and have been identified with a “shoulder” in the neutron scattering spectrum below the ordinary roton minimum.
4. Interfacial Ripplons: On very weak substrates, like cesium, one can also have an “interface ripplon[40, 42]”. Its appearance can be understood easily from the following consideration: Consider first a film with two free surfaces. Obviously, this film would exhibit two ripplon modes, one at each surface[48]. Now, a weak substrate is moved against one of the two surfaces. The character of the “ripplon” at this surface will not change abruptly; rather the circular motion of the particles will be somewhat inhibited, and the energy of the mode will rise. This is precisely what is seen in the energetics and the current pattern of this second mode on Cs. Stronger substrate potentials suppress this interface mode; to distinguish between an “interfacial ripplon” and a “layer phonon” one must look at the current pattern of the excitation[42].
The above list of excitations is restricted to modes that can be characterized legitimately by a wave vector parallel to the surface. To calculate the response to particles impinging normally on the surface, one must also look at the types of excitations perpendicular to the surface. These cannot be rigorously classified by a wave number, but one should basically expect standing waves or resonances at discrete frequencies, approaching the excitations of a bulk system as the film becomes thicker. No ripplonic excitations or layer–modes should be visible in this case.
The character of excitations is intelligently discussed by examining the dynamic structure function . A general procedure has been developed in Refs. 18, 19 to use time–dependent correlations for a quantitative calculation of the dynamic structure function. The simplest version of the theory is analogous to the Feynman approximation [49, 34]; the dynamic structure function in that approximation can be calculated directly from the solutions of Eq. (8)
where the are adjoint states (9) of the solutions of Eq. (8) for energy . The Feynman approximation has its well known deficiencies, and methods for its improvements have been derived which provide quantitative agreement with experiments.
Previous work has concentrated on the theoretical interpretation of neutron–scattering experiments, it was therefore concerned with momenta parallel to the liquid surface. In the present situation we must allow for both parallel and perpendicular momentum transfer. We show in Figs. 4 and 5 the dynamic structure function for parallel and perpendicular momentum transfer. Fig. 4 shows the picture familiar from previous work[18, 19, 42]: a low-lying excitation which can be identified with a ripplon by its dispersion relation and its particle motion, and a high density of states on the roton regime; note that the second lowest dispersion branch corresponds to the interfacial ripplon mentioned above. Note also that the modes below the continuum energy are discrete; they have been broadened by a Lorentzian of the same strength to make them visible.
The situation is quite different for perpendicular scattering. Again, the discrete excitations below the the evaporation energy have been broadened. We see a dominant ridge basically along the dispersion relation of a Feynman phonon, and a high density of states in the regime of the roton. The ridge shows a number of “echoes” at shorter wavelengths; this is due to the finite–size of the film. But there are — expectedly — no excitations corresponding to the (interfacial) ripplons.
iii.3 Impurity Excitations
Calculations of low-lying, bound states including the dynamic self–energy have been discussed extensively in Ref. 20, we list here the most important ones demonstrating both the theoretical consistency as well as the quantitative reliability and highlight their relevance for scattering processes:
• When applied to the bulk liquid, the ground state theory produces the correct chemical potentials of He and hydrogenic impurities[50].
• In an inhomogeneous geometry, the static theory reproduces the binding energy of the Andreev state[51]. The theory also predicts, even in its most primitive version[32], the existence of a surface resonance.
• The dynamic theory predicts a hydrodynamic effective mass of the Andreev state of , to be compared to the value on 1.38 given by Higley at al. [52] somewhat larger than the value of , reported by Valles et al. [53] at the lower end of the value given by Edwards and Saam [54]. In other words, our theoretical prediction is within the spread of experimental values.
• The energy of the first excited surface state is lowered from about -2.2 K to -2.8 K, improving the agreement with the experimental value[51] of approximately -3.2 K notably.
Similar to the obvious existence of interfacial ripplons, one also expects, on weak substrates, the appearance of an interfacial Andreev state. The binding energy of this state was found in Ref. 20 to be approximately -4.3 K. which is somewhat higher than the experimental value[41] of -4.8 K. We attribute the difference to uncertainties in the substrate potential and the certainly oversimplified assumption of a perfectly flat surface. This state — being confined to a smaller area than the surface state — has always an energy that is higher than the Andreev state. Although it can in principle decay into a surface bound state, it has negligible overlap and hence its lifetime is practically infinite. With increasing potential strength, the energy of the substrate bound state increases; the state disappears completely on substrates somewhat more attractive than Na. Then, the “interfacial Andreev state” turns into a resonance to which a scattering particle can couple. Similar “resonances” can be found on Mg substrates even in the second layer; we shall return to this point further below.
The two surface–bound states (and, if applicable, the interfacial Andreev state) can be described in the energy regime we are interested in reasonably well by an . Above the solvation energy of a He atom, a sequence of impurity states can exist that are spread out throughout the film; the detailed energetics of these states depends on the thickness of the film and the corrugation of the background liquid.
Iv Scattering states
The background and impurity excitations discussed in the previous section specify the possible energy loss channels for a scattering particle; we can now turn to the analysis of our results.
The previous work has concentrated on the properties of bound impurity atoms, their effective masses, and the lifetime of resonances. Scattering processes are treated within the same theory, imposing asymptotic plane–wave boundary conditions on the solution of the effective Schrödinger equation (19):
One of the key quantities of the theory is the elastic reflection coefficient because it is directly influenced by the coupling of the motion of the impinging particle to the excitations of the quantum liquid. The absolute value of the reflection coefficient can differ from unity only if the self–energy is non-hermitian. This happens when the energy denominator in the self–energy (20) has zeroes; note that the quantum numbers and include both the motion of the particles parallel to the surface as well as the discrete or continuous degrees of freedom in the -direction.
Superficially, we appear to be describing a single–particle quantum mechanical scattering problem. In fact, a number of notions can be carried over from single particle models can be carried over, and simple phenomenological descriptions can be constructed at the level of a one–body theory. But the actual situation is far richer: Since the scattering film is composed of helium atoms, this is a generically non-local problem when viewed at the one-body level. Moreover, the film is dynamic: the incoming particle may produce excited states of the background. This may result in the capture of the particle and/or the emission of particles in states other than the elastic channel.
iv.1 Quantum reflection
Generally, the amplitude of the wave function of an impinging particle of low energy is suppressed inside an attractive potential by the mismatch of the wave lengths inside and outside the potential if its range is small compared to the wavelength of the particle. As a consequence, the particles are almost totally reflected even if there is dissipation inside the potential (caused by the imaginary part of the self energy operator (20) in our case)
and, consequently, and . The effect is called universal quantum reflection[55, 56].
Quantum reflection can be described phenomenologically in an effective single particle picture with a complex optical potential. The many-body aspect of the problem is to determine the physical origin, the magnitude, and the shape as well as possible non–locality of that “optical potential”. The energy range where quantum reflection is visible in a many-body system like one of those considered here depends sensitively on the energy–loss mechanisms and calls for a quantitative calculation. Even in the limit of zero incident energy, the self energy (20) is non-hermitian, and thus allows in principle for sticking. Furthermore, this energy range is strongly affected by the long range features of the substrate potentials[29, 30, 15, 14].
Specifically, in the 3-9 substrate potential models (22), the sticking coefficient depends on the strength of the potential: For a local potential with the asymptotic form as , one can show[30] that the amplitude of the wave function inside the potential depends linearly on the normal momentum of the incoming particle. Increasing makes the potential appear smoother for particles with long wave length, thus increasing the penetration depth and the probability to reach the film. Indeed, a calculation of the sticking coefficient from the non-Hermitian effective Schrödinger equation (19) gives, already in the distorted wave Born approximation (DWBA), .
Inelastic scattering is, at low incident energies, only possible by coupling to ripplons. An analysis of the imaginary part of reveals that the contribution of the inelastic channels is proportional to which gives in the DWBA . In other words, inelastic processes are negligible in the low–energy regime.
Although it is not the main thrust of our paper, we have examined the low–energy reflection probabilities. Fig. 6 shows three examples for the dependence of the sticking probability on the the incident energy for normal incidence. While on graphite adsorbed films, quantum sticking is readily observable in the sense that the sticking coefficient starts to drop monotonically for wavelengths longer than 0.1, corresponding to energies less than 0.1 K, the linear dependence of on begins only at energies that are two to three orders of magnitude less for Cs adsorbed films (and similarly Mg and Na).
Once the origin and properties of the optical potential for low–energy scattering are understood from a microscopic point of view, one may a posteriori construct simple, analytic models that provide, within the range suggested by the estimated accuracy of the microscopic picture, some flexibility to examine the dependence of on features of the optical potential. A simple model consists of a local potential that approaches the substrate potential in the asymptotic region and that is approximated by a square well with a depth estimated from the binding energy of the Andreev state and a width of 15 . The energy dissipation term can be included through a localized imaginary part of the typical magnitude of our self–energy. Such a model reproduces qualitatively the large values of in the mK energy regime. Of course, the model fails to explain the dependence on , see Fig. 11. For completeness, we should also add that retardation should be taken into account for quantitative results below 1-10 mK[15].
iv.2 Ripplon coupling
“Quantum reflection” as a generic phenomenon needs only some damping mechanism; we now turn to the task of many–body theory to identify and examine the physics that leads to damping. The basic physics is contained in the self–energy (20) used in our calculation; it includes the energy loss of an incoming particle with energy to a background excitation , leaving the particle in the state . Within this model, damping is expected to be somewhat underestimated because the possibility to emit two or more phonons has been neglected.
Unless there is negligible overlap of the wave functions, the most efficient energy loss mechanism is the coupling to the lowest–lying excitation. These lowest lying excitations of the helium film are the surface waves (ripplons), hence one expects that the energy loss of the He particle is dominated by the emission of a ripplon. This serves as a qualitative argument. However, the reality is more complicated for He scattering because several states are accessible. The condition that an excitation contributes to the imaginary part of the self–energy is that the energy denominator of the self–energy (20) vanishes, i.e. , and there are several open channels even for vanishing incident energy. First, the particle can, although less efficiently, also couple to higher film excitations and can be promoted into either the second Andreev state or into a bound state in the bulk liquid. The reflection coefficients also depends visibly on the real part of the self–energy, and no quantitative statement can be made without proper treatment of both. The argument holds even at normal incidence, and infinitesimal asymptotic energy of the impinging particle.
We show in Figs. 7-10 a few typical examples of the reflection probability for scattering from He films adsorbed on graphite, Na, and Cs substrates. In contrast to experiments on atomic scattering of He from free He surfaces[7], there is evidently a strong dependence on the parallel wave vector which needs to be explained in terms the possible decay channels discussed above. Since it is unlikely that a specific feature is due to a delicate cooperation between film and impurity degrees of freedom, it is legitimate to discuss film– and single particle excitations independently.
The fact that ripplon coupling is the dominant energy loss mechanism can be verified in various ways. The simplest one is the inspection of the self–energy (20): The imaginary part of the self–energy is, with a few exceptions to be discussed below, localized in the surface region where the ripplon lives. The consequence is that, at energies below the roton, the wave functions of the impinging particle decays basically within the surface region. The effect can be seen in the wave functions and even better in the probability currents which basically decay within the surface region. A “resonance” in Figs. 12 and 13 will be discussed momentarily.
From looking at Figs. 7-10, it appears that quantum reflection is seen only for the graphite substrate. As explained above, this is simply a consequence of the fact that the reflection becomes visible only at much lower energies on the alkali metal substrates. To demonstrate this, we have magnified in Fig. 11 the low–energy region for the Cs substrate; consistent with Fig. 6, it is seen that the reflectivity starts to rise at impact energies of less than .001 K.
iv.3 Single particle resonances
While the generic many–body aspect of all scattering and in particular damping mechanisms must be kept in mind, one–body pictures can occasionally — as above for quantum reflection — provide useful paradigms in cases where the process under consideration can be described in terms of the degrees of freedom of a single particle. Such an effect is the coupling to single–particle resonances. A convenient and physically illustrative definition of a resonance at an energy is a large probability in the region of interaction. The resulting large dissipation will render small.
The peak of the wave function close to the substrate at and shown in Fig. 12 is a very pronounced example of such a resonance. It displays exactly the phenomenon discussed above that the interfacial Andreev state turns into a resonance as the substrate strength is increased. The energy of this resonance is significantly reduced by the coupling to virtual phonons: The resonance has an energy of approximately 6 K in the static approximation (13). Including the dynamic self–energy corrections through the (real part of) , the resonance energy drops to approximately 1.3 K. The energy where the wave function has a strong peak in the vicinity of the substrate coincides with that of the dip in the reflection coefficient. Fig. 7 shows this for the special case of zero parallel momentum, but the agreement between the peak of the wave function and the minimum of the reflection probability persists at all parallel momenta. Also seen clearly in Fig. 12 is the change in the phase of the wave as the resonance is crossed as a function of energy.
The elliptic ridge of the reflection coefficient as a function of (, ) can be explained by the coupling of the interfacial Andreev resonance discussed above to the virtual excitations of the film. This has the consequence that the resonance acquires an effective mass[20] . At zero parallel momentum, the position of the dip in the reflection coefficient agrees with the location of the resonance seen in Fig. 12. The shape of the ridge can be explained by assuming that all of the energy of the impinging particle is deposited in that resonance. Energy conservation and momentum conservation parallel to the substrate then leads to the relationship
where is the energy of the resonance. Following the peak of the wave function in the resonance in the plane leads, within the accuracy that can be expected from such a relatively crude argument, to the same conclusion. Basically – and expectedly — the same resonances occur at other surface coverages; the precise location of the dip in the reflection varies due to the multitude of other open scattering channels. A similar resonance occurs in the more strongly attractive Mg substrates, the corresponding wave functions are shown in Fig. 13. In this case, one finds a second resonance in the second layer which is, however, less pronounced. A list of energies and effective masses is given, together with the values for the Andreev state and the results of Ref. 20 of the bound states, in Table 1. The effective masses were obtained by fitting the curve defined by Eq. (28) to reproduce the location of the peak of the wave function within the visible region. As pointed out above, the weaker substrate Cs has a bound state to which the scattering particle cannot couple, whereas the Na substrate is a marginal case.
In all cases considered here we have found a significant dependence of the reflection coefficient on the parallel momentum, cf. Figs. 7-10. Such a feature can not be explained within a local, complex single–particle model, is not also not seen in experiments on He scattering off He films/surfaces[7]. The feature is most pronounced on graphite and Mg substrates, cf. Fig. 7.
Also for the other substrates (see Figs. 8, 9 and 10), depends on the parallel component of the momentum. One sees similar, but broader ridges in the reflection coefficient, but no sharp peaks in the wave function, cf. Fig. 14. The effect can also be explained by the features of the impurity states within the film. But this time, the impurity states are not localized but are extended states that will, with increasing film thickness, develop to He states dissolved in the He liquid. Consistent with this picture, the energy of the resonances in Na and Cs adsorbed films decreases with increasing surface coverage , until they become bound states, cf. 16. This slipping below the threshold is best seen in the phase shift at , which jumps whenever this occurs (in case of Cs: at and ).
iv.4 Roton coupling
The presence of roton excitations affects the scattering properties at two levels. First, at all energies, the coupling to virtual rotons is a significant contribution to the real part of the self–energy; omitting these contributions by, for example, restricting the state sums in the self–energy (20) to energies below the roton minimum, leads to reflection coefficients that are, even at energies well below the roton minimum, about a factor of 2 smaller than when excitations in the roton regime are included. This is to some extent plausible since the roton is a reflection of the short–range structure of the system which is dominated by the core repulsion, and such effects should make the film look “stiffer”.
At higher energies, the coupling to roton excitations also opens a new damping mechanism. As seen in Fig. 5, “roton–like” excitations appear also for film excitations perpendicular to the surface, and the impinging particle can couple to these excitations. In our calculations, the effect of roton coupling should become visible at an energy of about 15 K, this is because we have used a Feynman-spectrum in the energy denominator of Eq. (20). In previous work[19], we have scaled the energy denominators in the self-energy by an amount such that the roton is placed at roughly the right energy. We have refrained from this phenomenological modification since this procedure would also scale the ripplon away from its correct value which is already obtained in the Feynman approximation.
Above 15 K, the film loses its elastic reflectivity for He atoms completely. There is, of course, still the possibility of some inelastic scattering, but we consider this scenario unlikely from our experience with the propagation of He impurities in bulk He[50]. Hence, we expect that He atoms will be completely absorbed by He films when the impact energy is above the energy of the bulk roton. The effect is also seen quite clearly in the wave function of the scattering particle which does not penetrate into the film at all at energies above that of the roton.
V Summary
We have set in this paper the basic scenario for calculations of atom scattering processes from inhomogeneous He. This work parallels similar research on scattering of He atoms[24] and also provides the groundwork for applications on the presently active area of atom scattering from He clusters.
Technically, the calculations presented here are somewhat simpler than those for He atom scattering[24] (since the Hartree impurity spectra appearing in the energy denominators in 20 are decoupled in parallel and perpendicular motion in contrast to the ripplon/phonon/roton spectra) which enabled us to do a systematic study of the dependence of the reflection coefficient on the parallel momentum.
When applicable, our general conclusions are very similar to those that we have drawn for He atom scattering: Most of the physics happens due to ripplon coupling, the wave function is substantially damped in the surface region. “Quantum reflection” does not come to bear until energies as low as on a graphite substrate, and or less on alkali metals. A second damping mechanism happening at higher energies is the coupling to rotons, this effect dampens the impurity motion completely; an equivalent effect is expected, and found, in bulk He.
A new aspect specific to He scattering is the coupling to single–particle resonances within the film; such an effect will not be seen for He scattering. We have demonstrated that the properties of the remaining reflected particles are directly influenced by the features of the impurity states within these films and that scattering experiments can directly measure the energy and the “effective mass” of these resonances. Fully acknowledging the experimental difficulty of the task, we hope that these findings will inspire further measurements on He scattering off He surfaces and films.
Unfortunately it is difficult to make direct comparisons with experiments available today[7]. One reason is that we are with our calculations apparently still too far from the bulk limit that a comparison is meaningful. This is most clearly seen in the oscillatory dependence of the reflection coefficient on the energy of the incoming particle even in a case if a relatively thick film without localized resonances within the film (Fig. 14). There is also still pronounced non-monotonic dependence of the reflection coefficient on the surface coverage, cf. Fig. 16.
Further applications of our work are twofold: One is the application to scattering of hydrogen isotopes off He surface. While experimental efforts in this area have been significantly stronger[11, 12, 13], the situation is less rich: The H impurity is only very weakly bound and can lose its energy only to the ripplon, in other words the imaginary part of the self–energy (20) comes from one state only. Moreover, the H atom must overcome a potential barrier of about 10 K to penetrate into the bulk liquid which makes the coupling to any interior degrees of freedom negligible.
Similarly interesting is the possibility of scattering experiments of both He and He atoms off He droplets. These experiments, to be carried out in the energy regime of a few tenth to a few degrees, would also couple to both surface and volume modes and should also more clearly display the coupling to excitations inside the droplets. In the spherical geometry, inelastic processes of the kind described here cannot occur at low energy because the continuous quantum number is replaced by the discrete angular momentum. Hence, all low-lying modes are discrete and the energy denominator of Eq. (20) will normally be non–zero, in other words the self-energy is hermitean. Second, that these systems are not contaminated by substrate effects and should therefore allow for a cleaner interpretation of the results. We have learned that such scattering experiments have meanwhile been performed[57] and show indeed the expected coupling of He particles to roton-like excitations inside the droplets.
Such experiments and calculations would also provide an ideal scenario to test ideas and procedures established in nuclear physics in a much better controlled and — in terms of the underlying Hamiltonian — better understood physics.
Calculations in this direction are in progress and will be published elsewhere.
Vi Acknowledgements
This work was supported, in part, by the Austrian Science Fund under project P11098-PHY. We thank S. E. Campbell, R. B. Hallock, J. Klier, and M. Saarela for valuable discussions.
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Substrate Energy - \dec-5.4 1.3 Cs \dec-4.3 1.7 C \dec1.30.3 1.70.3 Mg \dec4.30.5 1.60.2
Table 1: Resonance energies and effective masses of the interfacial Andreev state on various substrates. The first line gives, for reference, the data of the Andreev state at the free surface, and the second the interfacial Andreev state on a Cs substrate (From Ref. 20). The last three lines give the results obtained here from scattering properties. Energies are given in K, the row labeled with “C” refers to the graphite plus two solid layers of He model used in this work.
The three classes of scattering channels are illustrated.
The incoming particle can be (a) scattered elastically
(top figure),
(b) inelastically, (middle figure), or (c) adsorbed to the
film (bottom figure).
Figure 1: The three classes of scattering channels are illustrated. The incoming particle can be (a) scattered elastically (top figure), (b) inelastically, (middle figure), or (c) adsorbed to the film (bottom figure).
The figure shows the three substrate potentials for the films
under consideration here: Graphite
Figure 2: The figure shows the three substrate potentials for the films under consideration here: Graphite plus two solid helium layers (solid line), Mg (long dashed line), Na (short dashed line) and Cs (dotted line).
The figure shows the four density profiles of the background
liquid (solid lines) and the impurity location (long dashed lines)
for which most of the present calculations were done. Graphite
substrate results are marked with
Figure 3: The figure shows the four density profiles of the background liquid (solid lines) and the impurity location (long dashed lines) for which most of the present calculations were done. Graphite substrate results are marked with -symbols, Na results with stars, and Cs results with crosses. Also shown is the interfacial Andreev state on a Cs substrate (short-dashed line marked with crosses). Coverages are for Cs, Na, and graphite, and for Cs. Profiles on Mg have been left out for clarity.
Dynamic structure function
Figure 4: Dynamic structure function in Feynman approximation for a film with coverage of on a Cs substrate and parallel momentum transfer. The solid curve shows the continuum boundary and the dashed line the bulk Feynman spectrum.
Same as Fig.
Figure 5: Same as Fig. 4 for momentum transfer perpendicular to the film. The horizontal solid line shows the continuum boundary and the dashed line the bulk Feynman spectrum.
Sticking on a graphite, Na, and Cs–adsorbed
film of
Figure 6: Sticking on a graphite, Na, and Cs–adsorbed film of . The square boxes in the upper left of the plot are the data of Ref. 7. Note that these data were taken at an impact angle of 60.
Dependence of reflection coefficient on wave vector magnitude
and angle from a graphite film of density
Figure 7: Dependence of reflection coefficient on wave vector magnitude and angle from a graphite film of density .
Same as
Figure 8: Same as 7 for a Cs film of density .
Same as
Figure 9: Same as 7 for a Na film of density .
Same as
Figure 10: Same as 7 for a Cs film of density .
Figure 11: Fig. 8 is magnified into the regime of low to demonstrate that finally approaches unity.
The figure shows the wave function
Figure 12: The figure shows the wave function of a He as a function of distance and perpendicular wave number for normal incidence. The left face shows, for reference, the density profile of the film and the back face the reflection coefficient . The substrate is graphite plus two solid helium layers, the surface coverage is .
Same as
Figure 13: Same as 12 for a film of on a Mg substrate.
Same as
Figure 14: Same as 12 for a film of on a Cs substrate.
Reflection coefficient
Figure 15: Reflection coefficient for normal incidence on a film with on a Cs-adsorbed film. The solid line is the result when all relevant intermediate states are kept in the state sums (20) whereas the dashed line is the result when the sum over intermediate states is truncated below the roton minimum. A new scattering channel opens at 3.5 K.
Reflection coefficient
Figure 16: Reflection coefficient for normal incidence on a a sequence films with surface coverages between and on a Cs-adsorbed film.
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b41ccb4f142adf60 | Quantum tunnelling
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Quantum tunnelling was developed from the study of radioactivity,[3] which was discovered in 1896 by Henri Becquerel.[4] Radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903.[4] Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlrausch. The idea of the half-life and the impossibility of predicting decay was created from their work.[3]
In 1901, Robert Francis Earhart, while investigating the conduction of gases between closely spaced electrodes using the Michelson interferometer to measure the spacing, discovered an unexpected conduction regime. J. J. Thomson commented the finding warranted further investigation. In 1911 and then 1914, then-graduate student Franz Rother, employing Earhart's method for controlling and measuring the electrode separation but with a sensitive platform galvanometer, directly measured steady field emission currents. In 1926, Rother, using a still newer platform galvanometer of sensitivity 26 pA, measured the field emission currents in a "hard" vacuum between closely spaced electrodes.[5]
Friedrich Hund was the first to take notice of tunnelling in 1927 when he was calculating the ground state of the double-well potential.[4] Its first application was a mathematical explanation for alpha decay, which was done in 1928 by George Gamow and independently by Ronald Gurney and Edward Condon.[6][7][8][9] The two researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunnelling.
After attending a seminar by Gamow, Max Born recognised the generality of tunnelling. He realised that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems.[3] Shortly thereafter, both groups considered the case of particles tunnelling into the nucleus. The study of semiconductors and the development of transistors and diodes led to the acceptance of electron tunnelling in solids by 1957. The work of Leo Esaki, Ivar Giaever and Brian Josephson predicted the tunnelling of superconducting Cooper pairs, for which they received the Nobel Prize in Physics in 1973.[3] In 2016, the quantum tunneling of water was discovered.[10]
Introduction to the concept[edit]
Animation showing the tunnel effect and its application to an STM
Quantum tunnelling through a barrier. The energy of the tunnelled particle is the same but the probability amplitude is decreased.
Quantum tunnelling through a barrier. At the origin (x=0), there is a very high, but narrow potential barrier. A significant tunnelling effect can be seen.
Quantum tunnelling falls under the domain of quantum mechanics: the study of what happens at the quantum scale. This process cannot be directly perceived, but much of its understanding is shaped by the microscopic world, which classical mechanics cannot adequately explain. To understand the phenomenon, particles attempting to travel between potential barriers can be compared to a ball trying to roll over a hill; quantum mechanics and classical mechanics differ in their treatment of this scenario. Classical mechanics predicts that particles that do not have enough energy to classically surmount a barrier and will not be able to reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down. Or, lacking the energy to penetrate a wall, it would bounce back (reflection) or in the extreme case, bury itself inside the wall (absorption). In quantum mechanics, these particles can, with a very small probability, tunnel to the other side, thus crossing the barrier. Here, the "ball" could, in a sense, borrow energy from its surroundings to tunnel through the wall or "roll over the hill", paying it back by making the reflected electrons more energetic than they otherwise would have been.[11]
The reason for this difference comes from the treatment of matter in quantum mechanics as having properties of waves and particles. One interpretation of this duality involves the Heisenberg uncertainty principle, which defines a limit on how precisely the position and the momentum of a particle can be known at the same time.[4] This implies that there are no solutions with a probability of exactly zero (or one), though a solution may approach infinity if, for example, the calculation for its position was taken as a probability of 1, the other, i.e. its speed, would have to be infinity. Hence, the probability of a given particle's existence on the opposite side of an intervening barrier is non-zero, and such particles will appear on the 'other' (a semantically difficult word in this instance) side with a relative frequency proportional to this probability.
An electron wavepacket directed at a potential barrier. Note the dim spot on the right that represents tunnelling electrons.
Quantum tunnelling in the phase space formulation of quantum mechanics. Wigner function for tunnelling through the potential barrier in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian .
The tunnelling problem[edit]
The wave function of a particle summarises everything that can be known about a physical system.[12] Therefore, problems in quantum mechanics center around the analysis of the wave function for a system. Using mathematical formulations of quantum mechanics, such as the Schrödinger equation, the wave function can be solved. This is directly related to the probability density of the particle's position, which describes the probability that the particle is at any given place. In the limit of large barriers, the probability of tunnelling decreases for taller and wider barriers.
For simple tunnelling-barrier models, such as the rectangular barrier, an analytic solution exists. Problems in real life often do not have one, so "semiclassical" or "quasiclassical" methods have been developed to give approximate solutions to these problems, like the WKB approximation. Probabilities may be derived with arbitrary precision, constrained by computational resources, via Feynman's path integral method; such precision is seldom required in engineering practice.[citation needed]
Related phenomena[edit]
There are several phenomena that have the same behaviour as quantum tunnelling, and thus can be accurately described by tunnelling. Examples include the tunnelling of a classical wave-particle association,[13] evanescent wave coupling (the application of Maxwell's wave-equation to light) and the application of the non-dispersive wave-equation from acoustics applied to "waves on strings". Evanescent wave coupling, until recently, was only called "tunnelling" in quantum mechanics; now it is used in other contexts.
These effects are modelled similarly to the rectangular potential barrier. In these cases, there is one transmission medium through which the wave propagates that is the same or nearly the same throughout, and a second medium through which the wave travels differently. This can be described as a thin region of medium B between two regions of medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to these other effects provided that the wave equation has travelling wave solutions in medium A but real exponential solutions in medium B.
In optics, medium A is a vacuum while medium B is glass. In acoustics, medium A may be a liquid or gas and medium B a solid. For both cases, medium A is a region of space where the particle's total energy is greater than its potential energy and medium B is the potential barrier. These have an incoming wave and resultant waves in both directions. There can be more mediums and barriers, and the barriers need not be discrete; approximations are useful in this case.
Tunnelling occurs with barriers of thickness around 1-3 nm and smaller,[14] but is the cause of some important macroscopic physical phenomena. For instance, tunnelling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague high-speed and mobile technology; it is considered the lower limit on how small computer chips can be made.[15]
Nuclear fusion in stars[edit]
Main article: Nuclear fusion
Quantum tunnelling is essential for nuclear fusion in stars. Temperature and pressure in the core of stars are insufficient for nuclei to overcome the Coulomb barrier in order to achieve a thermonuclear fusion. However, there is some probability to penetrate the barrier due to quantum tunnelling. Though the probability is very low, the extreme number of nuclei in a star generates a steady fusion reaction over millions or even billions of years - a precondition for the evolution of life in insolation habitable zones.[16]
Radioactive decay[edit]
Main article: Radioactive decay
Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunnelling into the nucleus is electron capture). This was the first application of quantum tunnelling and led to the first approximations. Radioactive decay is also a relevant issue for astrobiology as this consequence of quantum tunnelling is creating a constant source of energy over a large period of time for environments outside the circumstellar habitable zone where insolation would not be possible (subsurface oceans) or effective.[16]
Astrochemistry in interstellar clouds[edit]
By including quantum tunnelling the astrochemical syntheses of various molecules in interstellar clouds can be explained such as the synthesis of molecular hydrogen, water (ice) and the prebiotic important Formaldehyde.[16]
Quantum Biology[edit]
Quantum tunnelling is among the central non trivial quantum effects in Quantum biology. Here it is important both as electron tunnelling and proton tunnelling. Electron tunnelling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis while proton tunnelling is a key factor in spontaneous mutation of DNA.[16]
Spontaneous mutation of DNA occurs when normal DNA replication takes place after a particularly significant proton has defied the odds in quantum tunnelling in what is called "proton tunnelling"[17] (quantum biology). A hydrogen bond joins normal base pairs of DNA. There exists a double well potential along a hydrogen bond separated by a potential energy barrier. It is believed that the double well potential is asymmetric with one well deeper than the other so the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower of the two potential wells. The movement of the proton from its regular position is called a tautomeric transition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised causing a mutation.[18] Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix (quantum bio). Other instances of quantum tunnelling-induced mutations in biology are believed to be a cause of ageing and cancer.[19]
Cold emission[edit]
Main article: Semiconductor devices
Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field.[20] These materials are important for flash memory, vacuum tubes, as well as some electron microscopes.
Tunnel junction[edit]
Main article: Tunnel junction
A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires quantum tunnelling.[21] Josephson junctions take advantage of quantum tunnelling and the superconductivity of some semiconductors to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields,[20] as well as the multijunction solar cell.
A working mechanism of a resonant tunnelling diode device, based on the phenomenon of quantum tunnelling through the potential barriers.
Tunnel diode[edit]
Main article: Tunnel diode
Diodes are electrical semiconductor devices that allow electric current flow in one direction more than the other. The device depends on a depletion layer between N-type and P-type semiconductors to serve its purpose; when these are very heavily doped the depletion layer can be thin enough for tunnelling. Then, when a small forward bias is applied the current due to tunnelling is significant. This has a maximum at the point where the voltage bias is such that the energy level of the p and n conduction bands are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically.[22]
Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage is increased. This peculiar property is used in some applications, like high speed devices where the characteristic tunnelling probability changes as rapidly as the bias voltage.[22]
The resonant tunnelling diode makes use of quantum tunnelling in a very different manner to achieve a similar result. This diode has a resonant voltage for which there is a lot of current that favors a particular voltage, achieved by placing two very thin layers with a high energy conductance band very near each other. This creates a quantum potential well that have a discrete lowest energy level. When this energy level is higher than that of the electrons, no tunnelling will occur, and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage is increased further tunnelling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.[23]
Tunnel field-effect transistors[edit]
A European research project has demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunnelling rather than by thermal injection, reducing gate voltage from ~1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they will significantly improve the performance per power of integrated circuits.[24]
Quantum conductivity[edit]
While the Drude model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunnelling to explain the nature of the electron's collisions.[20] When a free electron wave packet encounters a long array of uniformly spaced barriers the reflected part of the wave packet interferes uniformly with the transmitted one between all barriers so that there are cases of 100% transmission. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons will tunnel through the metal as free electrons, leading to an extremely high conductance, and that impurities in the metal will disrupt it significantly.[20]
Scanning tunnelling microscope[edit]
The scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, allows imaging of individual atoms on the surface of a metal.[20] It operates by taking advantage of the relationship between quantum tunnelling with distance. When the tip of the STM's needle is brought very close to a conduction surface that has a voltage bias, by measuring the current of electrons that are tunnelling between the needle and the surface, the distance between the needle and the surface can be measured. By using piezoelectric rods that change in size when voltage is applied over them the height of the tip can be adjusted to keep the tunnelling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor.[20] STMs are accurate to 0.001 nm, or about 1% of atomic diameter.[23]
Faster than light[edit]
It is possible for spin zero particles to travel faster than the speed of light when tunnelling.[3] This apparently violates the principle of causality, since there will be a frame of reference in which it arrives before it has left. However, careful analysis of the transmission of the wave packet shows that there is actually no violation of relativity theory. In 1998, Francis E. Low reviewed briefly the phenomenon of zero time tunnelling.[25] More recently experimental tunnelling time data of phonons, photons, and electrons have been published by Günter Nimtz.[26]
Mathematical discussions of quantum tunnelling[edit]
The following subsections discuss the mathematical formulations of quantum tunnelling.
The Schrödinger equation[edit]
The time-independent Schrödinger equation for one particle in one dimension can be written as
where is the reduced Planck's constant, m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ is the Schrödinger wave function, V is the potential energy of the particle (measured relative to any convenient reference level), E is the energy of the particle that is associated with motion in the x-axis (measured relative to V), and M(x) is a quantity defined by V(x) – E which has no accepted name in physics.
The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x) is positive or negative. When M(x) is constant and negative, then the Schrödinger equation can be written in the form
The solutions of this equation represent traveling waves, with phase-constant +k or -k. Alternatively, if M(x) is constant and positive, then the Schrödinger equation can be written in the form
The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M(x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M(x) determines the nature of the medium, with negative M(x) corresponding to medium A as described above and positive M(x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence creating a potential barrier.
The mathematics of dealing with the situation where M(x) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A discussion of the semi-classical approximate method, as found in physics textbooks, is given in the next section. A full and complicated mathematical treatment appears in the 1965 monograph by Fröman and Fröman noted below. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.
The WKB approximation[edit]
Main article: WKB approximation
The wave function is expressed as the exponential of a function:
, where
is then separated into real and imaginary parts:
, where A(x) and B(x) are real-valued functions.
Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:
To solve this equation using the semiclassical approximation, each function must be expanded as a power series in . From the equations, the power series must start with at least an order of to satisfy the real part of the equation; for a good classical limit starting with the highest power of Planck's constant possible is preferable, which leads to
with the following constraints on the lowest order terms,
At this point two extreme cases can be considered.
Case 1 If the amplitude varies slowly as compared to the phase and
which corresponds to classical motion. Resolving the next order of expansion yields
Case 2
If the phase varies slowly as compared to the amplitude, and
which corresponds to tunnelling. Resolving the next order of the expansion yields
In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points . Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.
To start, choose a classical turning point, and expand in a power series about :
Keeping only the first order term ensures linearity:
Using this approximation, the equation near becomes a differential equation:
This can be solved using Airy functions as solutions.
Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the 2 coefficients on one side of a classical turning point, the 2 coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.
Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between and are
With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunnelling through a single potential barrier is
where are the 2 classical turning points for the potential barrier.
For a rectangular barrier, this expression is simplified to:
See also[edit]
1. ^ Serway; Vuille (2008). College Physics. 2 (Eighth ed.). Belmont: Brooks/Cole. ISBN 978-0-495-55475-2.
2. ^ Taylor, J. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. p. 234. ISBN 0-13-805715-X.
3. ^ a b c d e f Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. pp. 4, 462. ISBN 9812564888.
4. ^ a b c d Nimtz; Haibel (2008). Zero Time Space. Wiley-VCH. p. 1.
5. ^ Thomas Cuff. "The STM (Scanning Tunneling Microscope) [The forgotten contribution of Robert Francis Earhart to the discovery of quantum tunneling.]". ResearchGate.
6. ^ Gurney, R. W.; Condon, E. U. (1928). "Quantum Mechanics and Radioactive Disintegration". Nature. 122 (3073): 439. Bibcode:1928Natur.122..439G. doi:10.1038/122439a0.
7. ^ Gurney, R. W.; Condon, E. U. (1929). "Quantum Mechanics and Radioactive Disintegration". Phys. Rev. 33 (2): 127–140. Bibcode:1929PhRv...33..127G. doi:10.1103/PhysRev.33.127.
8. ^ Bethe, Hans (27 October 1966). "Hans Bethe - Session I". Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA (Interview). Interview with Charles Weiner; Jagdish Mehra. Cornell University. Retrieved 1 May 2016.
9. ^ Friedlander, Gerhart; Kennedy, Joseph E.; Miller, Julian Malcolm (1964). Nuclear and Radiochemistry (2nd ed.). New York: John Wiley & Sons. pp. 225–7. ISBN 978-0-471-86255-0.
10. ^ Kolesnikov et al. (22 April 2016). "Quantum Tunneling of Water in Beryl: A New State of the Water Molecule". Physical Review Letters. doi:10.1103/PhysRevLett.116.167802. Retrieved 23 April 2016.
11. ^ Davies, P. C. W. (2005). "Quantum tunneling time" (PDF). American Journal of Physics. 73: 23. arXiv:quant-ph/0403010Freely accessible. Bibcode:2005AmJPh..73...23D. doi:10.1119/1.1810153.
12. ^ Bjorken and Drell, "Relativistic Quantum Mechanics", page 2. Mcgraw-Hill College, 1965.
13. ^ Eddi, A.; Fort, E.; Moisy, F.; Couder, Y. (16 June 2009). "Unpredictable Tunneling of a Classical Wave-Particle Association" (PDF). Physical Review Letters. 102 (24). Bibcode:2009PhRvL.102x0401E. doi:10.1103/PhysRevLett.102.240401. Retrieved 1 May 2016.
14. ^ Lerner; Trigg (1991). Encyclopedia of Physics (2nd ed.). New York: VCH. p. 1308. ISBN 0-89573-752-3.
15. ^ "Applications of tunneling" Archived 23 July 2011 at the Wayback Machine.. Simon Connell 2006.
17. ^ Matta, Cherif F. (2014). Quantum Biochemistry: Electronic Structure and Biological Activity. Weinheim: Wiley-VCH. ISBN 978-3-527-62922-0.
18. ^ Majumdar, Rabi (2011). Quantum Mechanics: In Physics and Chemistry with Applications to Bioloty. Newi: PHI Learning. ISBN 9788120343047.
19. ^ Cooper, WG (June 1993). "Roles of Evolution, Quantum Mechanics and Point Mutations in Origins of Cancer". Cancer Biochemistry Biophysics. 13 (3): 147–70. PMID 8111728.
20. ^ a b c d e f Taylor, J. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. p. 479. ISBN 0-13-805715-X.
21. ^ Lerner; Trigg (1991). Encyclopedia of Physics (2nd ed.). New York: VCH. pp. 1308–1309. ISBN 0-89573-752-3.
22. ^ a b Krane, Kenneth (1983). Modern Physics. New York: John Wiley and Sons. p. 423. ISBN 0-471-07963-4.
23. ^ a b Knight, R. D. (2004). Physics for Scientists and Engineers: With Modern Physics. Pearson Education. p. 1311. ISBN 0-321-22369-1.
24. ^ Ionescu, Adrian M.; Riel, Heike (2011). "Tunnel field-effect transistors as energy-efficient electronic switches". Nature. 479 (7373): 329–337. Bibcode:2011Natur.479..329I. doi:10.1038/nature10679. PMID 22094693.
25. ^ Low, F. E. (1998). "Comments on apparent superluminal propagation". Ann. Phys. Leipzig. 7 (7–8): 660–661. Bibcode:1998AnP...510..660L. doi:10.1002/(SICI)1521-3889(199812)7:7/8<660::AID-ANDP660>3.0.CO;2-0.
26. ^ Nimtz, G. (2011). "Tunneling Confronts Special Relativity". Found. Phys. 41 (7): 1193–1199. arXiv:1003.3944Freely accessible. Bibcode:2011FoPh...41.1193N. doi:10.1007/s10701-011-9539-2.
Further reading[edit]
External links[edit] |
3e7010c6481390f3 | Take the 2-minute tour ×
I'm reading the Wikipedia page for the Dirac equation:
The continuity equation is as before. Everything is compatible with relativity now, but we see immediately that the expression for the density is no longer positive definite - the initial values of both ψ and ∂tψ may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.
I am not sure how one gets a new $\rho$ and $J^\mu$. How does one do to derive these two? And can anyone show me why the expression for density not positive definite?
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any comment...? – Paul Reubens Oct 7 '12 at 6:41
please see below, hope that helps – Hal Swyers Oct 7 '12 at 17:17
1 Answer 1
up vote 1 down vote accepted
This particular writing of the problem in the article I have always thought was sloppy as well. The most confusing part of the discussion is the statement "The continuity equation is as before". At first one writes the continuity equation as:
$$\nabla \cdot J + \dfrac{\partial\rho}{\partial t} = 0$$
Although the del operator can be defined to be infinite dimensional, it is frequently reserved for three dimensions and so the construction of the sentence does not provide a clear interpretation. If you look up conserved current you find the 4-vector version of the continuity equation:
$$\partial_\mu j^\mu = 0$$
What is important about the derivation in the wikipedia article is the conversion of the non time dependent density to a time dependent density, or rather:
$$\rho = \phi^*\phi$$
$$\rho = \dfrac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*)$$
the intent is clear, the want to make the time component have the same form as the space components. The equation of the current is now:
$$J^\mu = \dfrac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*)$$
which now contains the time component. So the continuity equation that should be used is:
$$\partial_\mu J^\mu = 0$$
where the capitalization of $J$ appears to be arbitrary choice in the derivation.
One can verify that this is the intent by referring to the article on probability current.
From the above I can see that the sudden insertion of the statement that one can arbitrarily pick $$\psi$$ and $$\dfrac{\partial \psi}{\partial t}$$ isn't well explained. This part the article was a source of confusion for me as well until one realized that the author was trying to get to a discussion about the Klein Gordon equation
A quick search of web for "probability current and klein gordan equation" finds good links, including a good one from the physics department at UC Davis. If you follow the discussion in the paper you can see it confirms that the argument is really trying to get to a discussion about the Klein Gordon equation and make the connection to probability density.
Now, if one does another quick search for "negative solutions to the klein gordan equation" one can find a nice paper from the physics department of the Ohio University. There we get some good discussion around equation 3.13 in the paper which reiterates that, when we redefined the density we introduced some additional variability. So the equation:
$$\rho = \dfrac{i\hbar}{2mc^2}(\psi^*\partial_t\psi - \psi\partial_t\psi^*)$$
(where in the orginal, c was set at 1) really is at the root of the problem (confirming the intent in the original article). However, it probably still doesn't satisfy the question,
"can anyone show me why the expression for density not positive definite?",
but if one goes on a little shopping spree you can find the book Quantum Field Theory Demystified by David McMahon (and there are some free downloads out there, but I won't link to them out of respect for the author), and if you go to pg 116 you will find the discussion:
Remembering the free particle solution $$\varphi(\vec{x},t) = e^{-ip\cdot x} = e^{-i(Et- px)}$$ the time derivatives are $$\dfrac{\partial\varphi}{\partial t} = -iEe^{-i(Et- px)}$$ $$\dfrac{\partial\varphi^*}{\partial t} = iEe^{i(Et- px)}$$ We have $$\varphi^*\dfrac{\partial\varphi}{\partial t} = e^{i(Et- px)}[-iEe^{-i(Et- px)}] = -iE$$ $$\varphi\dfrac{\partial\varphi^*}{\partial t} = e^{-i(Et- px)}[iEe^{i(Et- px)}] = iE$$ So the probability density is $$\rho = i(\varphi^*\dfrac{\partial\varphi}{\partial t} - \varphi\dfrac{\partial\varphi^*}{\partial t}) = i(-iE-iE) = 2E$$ Looks good so far-except for those pesky negative energy solutions. Remember that $$E = \pm\sqrt{p^2+m^2}$$ In the case of the negative energy solution $$\rho = 2E =-2\sqrt{p^2+m^2}<0$$ which is a negative probability density, something which simply does not make sense.
Hopefully that helps, the notion of a negative probability does not make sense because we define probability on the interval [0,1], so by definition negative probabilities have no meaning. This point is sometimes lost on people when they try to make sense of things, but logically any discussion of negative probabilities is non-sense. This is why QFT ended up reinterpreting the Klein Gordan equation and re purposing it for an equation that governs creation and annihilation operators.
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cb312479b70b81a8 | Evanescent field
From Wikipedia, the free encyclopedia
(Redirected from Evanescent wave)
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In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field which does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents). Even when there in fact is an electromagnetic wave produced (e.g., by a transmitting antenna) one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths (such as the far field of a transmitting antenna).
The hallmark of an evanescent field is that there is no net energy flow in that region. Since the net flow of electromagnetic energy is given by the average Poynting vector, that means that the Poynting vector in these regions, as averaged over a complete oscillation cycle, is zero.[note 1]
Schematic representation of a surface wave propagating along a metal-dielectric interface. The fields away from the surface die off exponentially (right hand graph) and those fields are thus described as evanescent in the z direction
Usage of the term[edit]
In many cases one cannot simply say that a field is or is not evanescent. For instance, in the above illustration energy is indeed transmitted in the horizontal direction. The field strength drops off exponentially away from the surface, leaving it concentrated in a region very close to the interface, for which reason this is referred to as a surface wave. However there is no propagation of energy away from (or toward) the surface (in the z direction), so that one could properly describe the field as being "evanescent in the z direction." This is one illustration of the inexactness of the term. In most cases where they exist, evanescent fields are simply thought of and referred to as electric or magnetic fields, without the evanescent property (zero average Poynting vector in one or all directions) ever being pointed out. The term is especially applied to differentiate a field or solution from cases where one normally expects a propagating wave
Everyday electronic devices and electrical appliances are surrounded by large fields which have this property. Their operation involves alternating voltages (producing an electric field between them) and alternating currents (producing a magnetic field around them). The term "evanescent" is never heard in this ordinary context. Rather, there may be concern with inadvertent production of a propagating electromagnetic wave and thus discussion of reducing radiation losses (since the propagating wave steals power from the circuitry) or interference. On the other hand "evanescent field" is used in various contexts where there is a propagating (even if confined) electromagnetic wave involved, to describe accompanying electromagnetic components which do not have that property. Or in some cases where there would normally be an electromagnetic wave (such as light refracted at the interface between glass and air) the term is invoked to describe the field when that wave is suppressed (such as with light in glass incident on an air interface beyond the critical angle).
Although all electromagnetic fields are classically governed according to Maxwell's equations, different technologies or problems have certain types of expected solutions, and when the primary solutions involve wave propagation the term "evanescent" is frequently applied to field components or solutions which do not share that property. For instance, the propagation constant of a hollow metal waveguide is a strong function of frequency (a so-called dispersion relation). Below a certain frequency (the cut-off frequency) the propagation constant becomes an imaginary number. A solution to the wave equation having an imaginary wavenumber does not propagate as a wave but falls off exponentially, so the field excited at that lower frequency is considered evanescent. It can also be simply said that propagation is "disallowed" for that frequency. The formal solution to the wave equation can describe modes having an identical form, but the change of the propagation constant from real to imaginary as the frequency drops below the cut-off frequency totally changes the physical nature of the result. One can describe the solution then as a "cut-off mode" or an "evanescent mode";[1] while a different author will just state that no such mode exists. Since the evanescent field corresponding to the mode was computed as a solution to the wave equation, it is often discussed as being an "evanescent wave" even though its properties (such as not carrying energy) are inconsistent with the definition of wave.
Although this article concentrates on electromagnetics, the term evanescent is used similarly in fields such as acoustics and quantum mechanics where the wave equation arises from the physics involved. In these cases, solutions to the wave equation resulting in imaginary propagation constants are likewise termed "evanescent" and have the essential property that no net energy is transmitted even though there is a non-zero field.
Evanescent wave applications[edit]
In optics and acoustics, evanescent waves are formed when waves traveling in a medium undergo total internal reflection at its boundary because they strike it at an angle greater than the so-called critical angle.[2][3] The physical explanation for the existence of the evanescent wave is that the electric and magnetic fields (or pressure gradients, in the case of acoustical waves) cannot be discontinuous at a boundary, as would be the case if there was no evanescent wave field. In quantum mechanics, the physical explanation is exactly analogous—the Schrödinger wave-function representing particle motion normal to the boundary cannot be discontinuous at the boundary.
Electromagnetic evanescent waves have been used to exert optical radiation pressure on small particles to trap them for experimentation, or to cool them to very low temperatures, and to illuminate very small objects such as biological cells or single protein and DNA molecules for microscopy (as in the total internal reflection fluorescence microscope). The evanescent wave from an optical fiber can be used in a gas sensor, and evanescent waves figure in the infrared spectroscopy technique known as attenuated total reflectance.
In electrical engineering, evanescent waves are found in the near-field region within one third of a wavelength of any radio antenna. During normal operation, an antenna emits electromagnetic fields into the surrounding nearfield region, and a portion of the field energy is reabsorbed, while the remainder is radiated as EM waves.
Recently, a graphene-based Bragg grating (one-dimensional photonic crystal) has been fabricated and demonstrated its competence for excitation of surface electromagnetic waves in the periodic structure using a prism coupling technique.[4]
In quantum mechanics, the evanescent-wave solutions of the Schrödinger equation give rise to the phenomenon of wave-mechanical tunneling.
In microscopy, systems that capture the information contained in evanescent waves can be used to create super-resolution images. Matter radiates both propagating and evanescent electromagnetic waves. Conventional optical systems capture only the information in the propagating waves and hence are subject to the diffraction limit. Systems that capture the information contained in evanescent waves, such as the superlens and near field scanning optical microscopy, can overcome the diffraction limit; however these systems are then limited by the system's ability to accurately capture the evanescent waves.[5] The limitation on their resolution is given by
where is the maximum wave vector that can be resolved, is the distance between the object and the sensor, and is a measure of the quality of the sensor.
More generally, practical applications of evanescent waves can be classified in the following way:
1. Those in which the energy associated with the wave is used to excite some other phenomenon within the region of space where the original traveling wave becomes evanescent (for example, as in the total internal reflection fluorescence microscope)
2. Those in which the evanescent wave couples two media in which traveling waves are allowed, and hence permits the transfer of energy or a particle between the media (depending on the wave equation in use), even though no traveling-wave solutions are allowed in the region of space between the two media. An example of this is so-called wave-mechanical tunnelling, and is known generally as evanescent wave coupling.
Total internal reflection of light[edit]
Top to bottom: representation of a refracted incident wave and an evanescent wave at an interface.
For example, consider total internal reflection in two dimensions, with the interface between the media lying on the x axis, the normal along y, and the polarization along z. One might naively expect that for angles leading to total internal reflection, the solution would consist of an incident wave and a reflected wave, with no transmitted wave at all, but there is no such solution that obeys Maxwell's equations. Maxwell's equations in a dielectric medium impose a boundary condition of continuity for the components of the fields E||, H||, Dy, and By. For the polarization considered in this example, the conditions on E|| and By are satisfied if the reflected wave has the same amplitude as the incident one, because these components of the incident and reflected waves superimpose destructively. Their Hx components, however, superimpose constructively, so there can be no solution without a non-vanishing transmitted wave. The transmitted wave cannot, however, be a sinusoidal wave, since it would then transport energy away from the boundary, but since the incident and reflected waves have equal energy, this would violate conservation of energy. We therefore conclude that the transmitted wave must be a non-vanishing solution to Maxwell's equations that is not a traveling wave, and the only such solutions in a dielectric are those that decay exponentially: evanescent waves.
Mathematically, evanescent waves can be characterized by a wave vector where one or more of the vector's components has an imaginary value. Because the vector has imaginary components, it may have a magnitude that is less than its real components. If the angle of incidence exceeds the critical angle, then the wave vector of the transmitted wave has the form
which represents an evanescent wave because the y component is imaginary. (Here α and β are real and i represents the imaginary unit.)
For example, if the polarization is perpendicular to the plane of incidence, then the electric field of any of the waves (incident, reflected, or transmitted) can be expressed as
where is the unit vector in the z direction.
Substituting the evanescent form of the wave vector k (as given above), we find for the transmitted wave:
where α is the attenuation constant and β is the propagation constant.
Evanescent-wave coupling[edit]
Plot of 1/e-penetration depth of the evanescent wave against angle of incidence in units of wavelength for different refraction indices.
Especially in optics, evanescent-wave coupling refers to the coupling between two waves due to physical overlap of what would otherwise be described as the evanescent fields corresponding to the propagating waves.[6]
One classical example is frustrated total internal reflection in which the evanescent field very close (see graph) to the surface of a dense medium at which a wave normally undergoes total internal reflection overlaps another dense medium in the vicinity. This disrupts the totality of the reflection, diverting some power into the second medium.
Coupling between two optical waveguides may be effected by placing the fiber cores close together so that the evanescent field generated by one element excites a wave in the other fiber. This is used to produce fiber optic splitters and in fiber tapping. At radio (especially microwave) frequencies, such a device is called a directional coupler
Evanescent-wave coupling is synonymous with near field interaction in electromagnetic field theory. Depending on the nature of the source element, the evanescent field involved is either predominantly electric (capacitive) or magnetic (inductive), unlike (propagating) waves in the far field where these components are connected (identical phase, in the ratio of the impedance of free space). The evanescent wave coupling takes place in the non-radiative field near each medium and as such is always associated with matter; i.e., with the induced currents and charges within a partially reflecting surface. Other commonplace examples are the coupling between the primary and secondary coils of a transformer, or between the two plates of a capacitor. In quantum mechanics the wave function interaction may be discussed in terms of particles and described as quantum tunneling.
See also[edit]
1. ^ or expressing the fields E and H as phasors, the complex Poynting vector has a zero real part
1. ^ IEEE Standard Dictionary of Electrical and Electronics Terms (IEEE STD 100-1992 ed.). New York, NY: The Institute of Electrical and Electronics Engineers, Inc. 1992. p. 458. ISBN 1-55937-2400.
2. ^ Tineke Thio (2006). "A Bright Future for Subwavelength Light Sources". American Scientist. American Scientist. 94 (1): 40–47. doi:10.1511/2006.1.40.
3. ^ Marston, Philip L.; Matula, T.J. (May 2002). "Scattering of acoustic evanescent waves...". Journal of the Acoustical Society of America. 111 (5): 2378. Bibcode:2002ASAJ..111.2378M. doi:10.1121/1.4778056.
4. ^ Sreekanth, Kandammathe Valiyaveedu; Zeng, Shuwen; Shang, Jingzhi; Yong, Ken-Tye; Yu, Ting (2012). "Excitation of surface electromagnetic waves in a graphene-based Bragg grating". Scientific Reports. 2. Bibcode:2012NatSR...2E.737S. doi:10.1038/srep00737. PMC 3471096free to read. PMID 23071901.
5. ^ Neice, A., "Methods and Limitations of Subwavelength Imaging", Advances in Imaging and Electron Physics, Vol. 163, July 2010
6. ^ Zeng, Shuwen; Yu, Xia; Law, Wing-Cheung; Zhang, Yating; Hu, Rui; Dinh, Xuan-Quyen; Ho, Ho-Pui; Yong, Ken-Tye (2013). "Size dependence of Au NP-enhanced surface plasmon resonance based on differential phase measurement". Sensors and Actuators B: Chemical. 176: 1128. doi:10.1016/j.snb.2012.09.073.
7. ^ Fan, Zhiyuan; Zhan, Li; Hu, Xiao; Xia, Yuxing (2008). "Critical process of extraordinary optical transmission through periodic subwavelength hole array: Hole-assisted evanescent-field coupling". Optics Communications. 281 (21): 5467. Bibcode:2008OptCo.281.5467F. doi:10.1016/j.optcom.2008.07.077.
8. ^ Karalis, Aristeidis; J.D. Joannopoulos; Marin Soljačić (February 2007). "Efficient wireless non-radiative mid-range energy transfer". Annals of Physics. 323: 34. arXiv:physics/0611063v2free to read. Bibcode:2008AnPhy.323...34K. doi:10.1016/j.aop.2007.04.017.
9. ^ "'Evanescent coupling' could power gadgets wirelessly", Celeste Biever, NewScientist.com, 15 November 2006
10. ^ Wireless energy could power consumer, industrial electronicsMIT press release
11. ^ Axelrod, D. (1 April 1981). "Cell-substrate contacts illuminated by total internal reflection fluorescence". The Journal of Cell Biology. 89 (1): 141–145. doi:10.1083/jcb.89.1.141. PMC 2111781free to read. PMID 7014571.
External links[edit] |
83e12a7410e0d866 | March 15, 2012
More Designer Electrons- Artificial Molecular Graphene used to Mimic Higgs Field and Relativity
Researchers arranged carbon monoxide molecules to form the same hexagonal pattern found in graphene, except that they could adjust molecular spacing slightly. They placed individual molecules of carbon monoxide onto a copper sheet. The material's electrons behave remarkably like relativistic particles, with a "speed of light" that they can adjust. Additionally, the researchers could change the spacing between molecules in a way that the masses of the quasiparticles changed, or cause them to behave as though they are interacting with electric and magnetic fields—without actually applying those fields to the material.
Manoharan has indicated that his team will be working on using the new material as a test bed for future exploitation as well as creating new nanoscale materials with new properties.
This is a follow up to the design electron article from yesterday
Manoharan lab covers their own work here
The work could lead to new materials and devices.
Graphical summary of this work. Artificial “molecular” graphene is fabricated via atom manipulation, and then imaged and locally probed via scanning tunneling microscopy (STM). Guided by theory, we fabricate successively more exotic variants of graphene. From left to right: pristine graphene exhibiting emergent massless Dirac fermions; graphene with a Kekulé distortion dresses the Dirac fermions with a scalar gauge field that creates mass; graphene with a triaxial strain distortion embeds a vector gauge field which condenses a time-reversal-invariant relativistic quantum Hall phase. In the theory panel, images are color representations of the strength of the carbon-carbon bonds (corresponding to tight-binding hopping parameters t), and the curves shown are calculated electronic density of states (DOS) from tight-binding (TB) theory. In the experiment panel, images are STM topographs acquired after molecular assembly, and the curves shown are normalized conductance spectra obtained from the associated nanomaterial.
In this work we combine a central tenet of condensed matter physics—how electronic band structure emerges from a periodic potential in a crystal—with the most advanced imaging and atomic manipulation techniques afforded by the scanning tunnelling microscope. We synthesize a completely artificial form of graphene (“molecular graphene”) in which Dirac fermions can be materialized, probed, and tailored in ways unprecedented in any other known materials. We do this by using single molecules, bound to a two-dimensional surface, to craft designer potentials that transmute normal electrons into exotic charge carriers. With honeycomb symmetry, electrons behave as massless relativistic particles as they do in natural graphene. With altered symmetry and texturing, these Dirac particles can be given a tunable mass, or even be married with a fictitious electric or magnetic field (a so-called gauge field) such that the carriers believe they are in real fields and condense into the corresponding ground state. We show an array of new phenomena emerging from: patterning Dirac carrier densities with atomic precision, without need for conventional gates (corresponding to locally uniform electric fields which adjust chemical potential); spatially texturing the electron bonds such that the Dirac point is split by an energy gap (corresponding to a nonuniform scalar gauge field); straining the bonds in such a way that a quantum Hall effect emerges even without breaking time-reversal symmetry (corresponding to a vector gauge field). Along the way, we make use of several theoretical predictions for real graphene which have never been realized in experiment
Nature - Designer Dirac fermions and topological phases in molecular graphene
Phantom Fields
A version of molecular graphene in which the electrons respond as if they're experiencing a very high magnetic field (red areas) when none is actually present. Scientists from Stanford and SLAC National Accelerator Laboratory calculated the positions where carbon atoms in graphene should be to make its electrons believe they were being exposed to a magnetic field of 60 Tesla, more than 30 percent higher than the strongest continuous magnetic field ever achieved on Earth. (A 1 Tesla magnetic field is about 20,000 times stronger than the Earth's.) The researchers then used a scanning tunneling microscope to place carbon monoxide molecules (black circles) at precisely those positions. The electrons responded by behaving exactly as expected — as if they were exposed to a real field, but no magnetic field was turned on in the laboratory. Image credit: Hari Manoharan / Stanford University.
Schrödinger Meets Dirac
Visualization depicting the transformation of an electron moving under the influence of the non-relativistic Schrödinger equation (upper planar quantum waves) into an electron moving under the prescription of the relativistic Dirac equation (lower honeycomb quantum waves). The light blue line shows a quasiclassical path of one such electron as it enters the molecular graphene lattice made of carbon monoxide molecules (black/red atoms) positioned individually by an STM tip (comprised of iridium atoms, dark blue). The path shows that the electron becomes trapped in synthetic chemical bonds that bind it to a honeycomb lattice and allow it to quantum mechanically tunnel between neighboring honeycomb sites, just like graphene. The underlying electron density in a honeycomb pattern (lower part of image, yellow-orange) is the quantum superposition formed from all such electron paths as they transmute into a new tunable species of massless Dirac fermions. Image credit: Hari Manoharan / Stanford University.
Designer Electrons
This graphic shows the effect that a specific pattern of carbon monoxide molecules (black/red) has on free-flowing electrons (orange/yellow) atop a copper surface. Ordinarily the electrons behave as simple plane waves (background). But the electrons are repelled by the carbon monoxide molecules, placed here in a hexagonal pattern. This forces the electrons into a honeycomb shape (foreground) mimicking the electronic structure of graphene, a pure form of carbon that has been widely heralded for its potential in future electronics. The molecules are precisely positioned with the tip of a scanning tunneling microscope (dark blue). Image credit: Hari Manoharan / Stanford University.
Molecular Graphene PNP Junction Device
Stretching or shrinking the bond lengths in molecular graphene corresponds to changing the concentrations of Dirac electrons present. This image shows three regions of alternating lattice spacing sandwiched together. The two regions on the ends contain Dirac "hole" particles (p-type regions), while the region in the center contains Dirac "electron" particles (n-type region). A p-n-p structure like this is of interest in graphene transistor applications. Image credit: Hari Manoharan / Stanford University.
The observation of massless Dirac fermions in monolayer graphene has generated a new area of science and technology seeking to harness charge carriers that behave relativistically within solid-state materials. Both massless and massive Dirac fermions have been studied and proposed in a growing class of Dirac materials that includes bilayer graphene, surface states of topological insulators and iron-based high-temperature superconductors. Because the accessibility of this physics is predicated on the synthesis of new materials, the quest for Dirac quasi-particles has expanded to artificial systems such as lattices comprising ultracold atoms. Here we report the emergence of Dirac fermions in a fully tunable condensed-matter system—molecular graphene—assembled by atomic manipulation of carbon monoxide molecules over a conventional two-dimensional electron system at a copper surface5. Using low-temperature scanning tunnelling microscopy and spectroscopy, we embed the symmetries underlying the two-dimensional Dirac equation into electron lattices, and then visualize and shape the resulting ground states. These experiments show the existence within the system of linearly dispersing, massless quasi-particles accompanied by a density of states characteristic of graphene. We then tune the quantum tunnelling between lattice sites locally to adjust the phase accrual of propagating electrons. Spatial texturing of lattice distortions produces atomically sharp p–n and p–n–p junction devices with two-dimensional control of Dirac fermion density and the power to endow Dirac particles with mass. Moreover, we apply scalar and vector potentials locally and globally to engender topologically distinct ground states and, ultimately, embedded gauge fields wherein Dirac electrons react to ‘pseudo’ electric and magnetic fields present in their reference frame but absent from the laboratory frame. We demonstrate that Landau levels created by these gauge fields can be taken to the relativistic magnetic quantum limit, which has so far been inaccessible in natural graphene. Molecular graphene provides a versatile means of synthesizing exotic topological electronic phases in condensed matter using tailored nanostructures.
14 pages of supplemental material
Molecular graphene assembly
Molecular graphene assembly. A movie shows the nanoscale assembly sequence of an electronic honeycomb lattice by manipulating individual CO molecules on the Cu(111) two-dimensional electron surface state with the STM tip. The video comprises 52 topographs (30 × 30 nm2, bias voltage V = 10 mV, tunnel current I = 1 nA) acquired during the construction phase and between manipulation steps.
Tunable Pseudomagnetic Field
Molecular Manipulation
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783e949cb03eca26 | Legendre polynomials
From Wikipedia, the free encyclopedia
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For Legendre's Diophantine equation, see Legendre's equation.
Associated Legendre polynomials are the most general solution to Legendre's Equation and Legendre polynomials are solutions that are azimuthally symmetric.
In mathematics, Legendre functions are solutions to Legendre's differential equation:
{d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.
The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial).
These solutions for n = 0, 1, 2, ... (with the normalization Pn(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:
P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right].
That these polynomials satisfy the Legendre differential equation (1) follows by differentiating n + 1 times both sides of the identity
(x^2-1)\frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n
and employing the general Leibniz rule for repeated differentiation.[1] The Pn can also be defined as the coefficients in a Taylor series expansion:[2]
\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n.
In physics, this ordinary generating function is the basis for multipole expansions.
Recursive definition[edit]
Expanding the Taylor series in Equation (2) for the first two terms gives
P_0(x) = 1,\quad P_1(x) = x
for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (2) is differentiated with respect to t on both sides and rearranged to obtain
\frac{x-t}{\sqrt{1-2xt+t^2}} = (1-2xt+t^2) \sum_{n=1}^\infty n P_n(x) t^{n-1}.
Replacing the quotient of the square root with its definition in (2), and equating the coefficients of powers of t in the resulting expansion gives Bonnet’s recursion formula
(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x).\,
This relation, along with the first two polynomials P0 and P1, allows the Legendre Polynomials to be generated recursively.
Explicit representations include
\begin{align}P_n(x)&= \frac 1 {2^n} \sum_{k=0}^n {n\choose k}^2 (x-1)^{n-k}(x+1)^k \\ &=\sum_{k=0}^n {n\choose k} {-n-1\choose k} \left( \frac{1-x}{2} \right)^k \\&= 2^n\cdot \sum_{k=0}^n x^k {n \choose k}{\frac{n+k-1}2\choose n},\end{align}
where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient.
The first few Legendre polynomials are:
n P_n(x)\,
0 1\,
1 x\,
2 \begin{matrix}\frac12\end{matrix} (3x^2-1) \,
3 \begin{matrix}\frac12\end{matrix} (5x^3-3x) \,
4 \begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,
5 \begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,
6 \begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,
7 \begin{matrix}\frac1{16}\end{matrix} (429x^7-693x^5+315x^3-35x)\,
8 \begin{matrix}\frac1{128}\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\,
9 \begin{matrix}\frac1{128}\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\,
10 \begin{matrix}\frac1{256}\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\,
The graphs of these polynomials (up to n = 5) are shown below:
An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval −1 ≤ x ≤ 1:
\int_{-1}^{1} P_m(x) P_n(x)\,dx = {2 \over {2n + 1}} \delta_{mn}
(where δmn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise). In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram–Schmidt process on the polynomials {1, xx2, ...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a Sturm–Liouville problem, where the Legendre polynomials are eigenfunctions of a Hermitian differential operator:
{d \over dx} \left[ (1-x^2) {d \over dx} P(x) \right] = -\lambda P(x),
where the eigenvalue λ corresponds to n(n + 1).
Applications of Legendre polynomials in physics[edit]
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre[3] as the coefficients in the expansion of the Newtonian potential
\frac{1}{\left| \mathbf{x}-\mathbf{x}^\prime \right|} = \frac{1}{\sqrt{r^2+r^{\prime 2}-2rr'\cos\gamma}} = \sum_{\ell=0}^{\infty} \frac{r^{\prime \ell}}{r^{\ell+1}} P_{\ell}(\cos \gamma)
where r and r' are the lengths of the vectors \mathbf{x} and \mathbf{x}^\prime respectively and \gamma is the angle between those two vectors. The series converges when r>r'. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.
Legendre polynomials occur in the solution of Laplace's equation of the static potential, \nabla^2 \Phi(\mathbf{x})=0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where \widehat{\mathbf{z}} is the axis of symmetry and \theta is the angle between the position of the observer and the \widehat{\mathbf{z}} axis (the zenith angle), the solution for the potential will be
\Phi(r,\theta)=\sum_{\ell=0}^{\infty} \left[ A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right] P_\ell(\cos\theta).
A_\ell and B_\ell are to be determined according to the boundary condition of each problem.[4]
They also appear when solving Schrödinger equation in three dimensions for a central force.
Legendre polynomials in multipole expansions[edit]
Figure 2
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
\frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x)
which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.
As an example, the electric potential \Phi(r, \theta) (in spherical coordinates) due to a point charge located on the z-axis at z=a (Figure 2) varies like
\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^{2} + a^{2} - 2ar \cos\theta}}.
If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials
\Phi(r, \theta) \propto
\frac{1}{r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k}
P_{k}(\cos \theta)
where we have defined η = a/r < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.
Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion.
Legendre polynomials in trigonometry[edit]
The trigonometric functions \cos n\theta, also denoted as the Chebyshev polynomials T_n(\cos\theta)\equiv\cos n\theta, can also be multipole expanded by the Legendre polynomials P_n(\cos\theta). The first several orders are as follows:
T_2(\cos\theta)=\cos 2\theta=\frac{1}{3}(4P_2(\cos\theta)-P_0(\cos\theta))
T_3(\cos\theta)=\cos 3\theta=\frac{1}{5}(8P_3(\cos\theta)-3P_1(\cos\theta))
T_4(\cos\theta)=\cos 4\theta=\frac{1}{105}(192P_4(\cos\theta)-80P_2(\cos\theta)-7P_0(\cos\theta))
T_5(\cos\theta)=\cos 5\theta=\frac{1}{63}(128P_5(\cos\theta)-56P_3(\cos\theta)-9P_1(\cos\theta))
T_6(\cos\theta)=\cos 6\theta=\frac{1}{1155}(2560P_6(\cos\theta)-1152P_4(\cos\theta)-220P_2(\cos\theta)-33P_0(\cos\theta))
Additional properties of Legendre polynomials[edit]
Legendre polynomials are symmetric or antisymmetric, that is
P_n(-x) = (-1)^n P_n(x). \,[2]
Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not unity) by being scaled so that
P_n(1) = 1. \,
The derivative at the end point is given by
P_n'(1) = \frac{n(n+1)}{2}. \,
As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula
{x^2-1 \over n} {d \over dx} P_n(x) = xP_n(x) - P_{n-1}(x).
Useful for the integration of Legendre polynomials is
(2n+1) P_n(x) = {d \over dx} \left[ P_{n+1}(x) - P_{n-1}(x) \right].
From the above one can see also that
{d \over dx} P_{n+1}(x) = (2n+1) P_n(x) + (2(n-2)+1) P_{n-2}(x) + (2(n-4)+1) P_{n-4}(x) + \ldots
or equivalently
{d \over dx} P_{n+1}(x) = {2 P_n(x) \over \| P_n(x) \|^2} + {2 P_{n-2}(x) \over \| P_{n-2}(x) \|^2}+\ldots
where \| P_n(x) \| is the norm over the interval −1 ≤ x ≤ 1
\| P_n(x) \| = \sqrt{\int _{- 1}^{1}(P_n(x))^2 \,dx} = \sqrt{\frac{2}{2 n + 1}}.
From Bonnet’s recursion formula one obtains by induction the explicit representation
P_n(x) = \sum_{k=0}^n (-1)^k \begin{pmatrix} n \\ k \end{pmatrix}^2 \left( \frac{1+x}{2} \right)^{n-k} \left( \frac{1-x}{2} \right)^k.
The Askey–Gasper inequality for Legendre polynomials reads
\sum_{j=0}^n P_j(x)\ge 0\qquad (x\ge -1).
A sum of Legendre polynomials is related to the Dirac delta function for -1\leq y\leq 1 and -1\leq x\leq1
\delta(y-x) = \frac12\sum_{\ell=0}^{\infty} (2\ell + 1) P_\ell(y)P_\ell(x)\,.
The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using
P_{\ell}({r}\cdot {r'})=\frac{4\pi}{2\ell + 1}\sum_{m=-\ell}^{\ell} Y_{\ell m}(\theta,\phi)Y_{\ell m}^*(\theta',\phi')\,.
where the unit vectors r and r' have spherical coordinates (\theta,\phi) and (\theta',\phi'), respectively.
Asymptotically for \ell\rightarrow \infty for arguments less than unity
P_{\ell}(\cos \theta)
J_0(\ell\theta) + \mathcal{O}(\ell^{-1})
\frac{2}{\sqrt{2\pi \ell \sin \theta}}\cos\left[\left(\ell + \frac{1}{2}\right)\theta - \frac{\pi}{4}\right]
and for arguments greater than unity
P_{\ell}\left(\frac{1}{\sqrt{1-e^2}}\right) =
I_0(\ell e) + \mathcal{O}(\ell^{-1})
\frac{1}{\sqrt{2\pi \ell e}} \frac{(1+e)^{(\ell+1)/2}}{(1-e)^{\ell/2}}
+ \mathcal{O}(\ell^{-1})\,,
where J_0 and I_0 are Bessel functions.
Shifted Legendre polynomials[edit]
The shifted Legendre polynomials are defined as \tilde{P_n}(x) = P_n(2x-1). Here the "shifting" function x\mapsto 2x-1 (in fact, it is an affine transformation) is chosen such that it bijectively maps the interval [0, 1] to the interval [−1, 1], implying that the polynomials \tilde{P_n}(x) are orthogonal on [0, 1]:
\int_{0}^{1} \tilde{P_m}(x) \tilde{P_n}(x)\,dx = {1 \over {2n + 1}} \delta_{mn}.
An explicit expression for the shifted Legendre polynomials is given by
The analogue of Rodrigues' formula for the shifted Legendre polynomials is
\tilde{P_n}(x) = \frac{1}{n!} {d^n \over dx^n } \left[ (x^2 -x)^n \right].\,
The first few shifted Legendre polynomials are:
n \tilde{P_n}(x)
0 1
1 2x-1
2 6x^2-6x+1
3 20x^3-30x^2+12x-1
4 70x^4-140x^3+90x^2-20x+1
Legendre functions of the Second Kind (Q_n)[edit]
As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series. These are the Legendre functions of the second kind, denoted by Q_n(x).
The differential equation
{d \over dx} \left[ (1-x^2) {d \over dx} f(x) \right] + n(n+1)f(x) = 0
has the general solution
where A and B are constants.
Legendre functions of fractional degree[edit]
Main article: Legendre function
Legendre functions of fractional degree exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional degree Legendre function Pn agrees with the associated Legendre polynomial P0
See also[edit]
1. ^ Courant & Hilbert 1953, II, §8
2. ^ a b George B. Arfken, Hans J. Weber (2005), Mathematical Methods for Physicists, Elsevier Academic Press, p. 743, ISBN 0-12-059876-0
3. ^ M. Le Gendre, "Recherches sur l'attraction des sphéroïdes homogènes," Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées, Tome X, pp. 411–435 (Paris, 1785). [Note: Legendre submitted his findings to the Academy in 1782, but they were published in 1785.] Available on-line (in French) at: http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf .
4. ^ Jackson, J.D. Classical Electrodynamics, 3rd edition, Wiley & Sons, 1999. page 103
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cc614bcb4f46ea75 | Take the 2-minute tour ×
The Schrödinger Equation provides a Probability Density map of the atom. In light of that, are either of the following possible:
1. The orbital/electron cloud converges to a 2d surface without heat (absolute zero)?
2. heat is responsible for the probability density variation from the above smooth surface?
I have taken two calculus based physics, and Modern Physics with the Schrödinger equation, Heisenberg Uncertainty Principle, Etc.
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1 Answer 1
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1.) No. All the calculations one does in elementary quantum mechanics courses are at zero temperature. If they were at a finite temperature, you could never reliably say what quantum mechanical state your system is in; it would always be in an ensemble of different states. Since the ground-state wavefunction and ground-state density is not a 2d surface, you don't get one at $T = 0$.
2.) No. At zero temperature, the probability density of your electron is given by the ground state wavefunction: $$\varrho(x) = \psi_0^*(x) \psi_0(x)$$ At finite temperature, your system is best described by an ensemble of states. Basically, you get $$\varrho(x) = \sum_i p_i \psi_i^*(x) \psi_i(x)$$ where $p_i$ is the ensemble-probability for your system to be in state $\psi_i(x)$. For a canonical ensemble, for example, you have $p_i \sim e^{-E_i/kT}$ if your $\psi_i(x)$ are the energy-eigenstates with eigenenergies $E_i$.
The same is true for any other expectation value: $$\langle \hat A \rangle = \sum_i p_i \langle \psi_i | \hat A | \psi_i \rangle$$ Note the two different expectation value here: One is $\langle \psi_i | \hat A | \psi_i \rangle$, the quantum mechanical expectation value of $\hat A$ when the system is in state $| \psi_i \rangle$. The sum over these, together with the $p_i$, then gives the thermodynamic expectation value
This framework is used everywhere in physics and has been proven to be mind-bogglingly exact.
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+1. This is a very good statement of the state of affairs, according to standard quantum theory. For completeness, it's probably worth adding that this theory is incredibly well-tested experimentally. For instance, when people do atomic physics experiments, they do them at a very wide range of temperatures. The wavefunctions corresponding to the various atomic energy levels do not vary as functions of temperature. (I mention this only because it seems possible that the questioner is asking whether standard theory might be wrong, as opposed to asking what standard theory says. It isn't.) – Ted Bunn Apr 18 '11 at 22:11
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A drawing of a Lithium atom. In the middle is the nucleus, which in this case has four neutrons (blue) and three protons (red). Orbiting it are its three electrons.
Lithium atom model
Showing nucleus with four neutrons (blue),
three protons (red) and,
orbited by three electrons (black).
Smallest recognised division of a chemical element
Mass: 1.66 x 10(−27) to 4.52 x 10(−25) kg
Electric charge: zero
An atom is the basic unit that makes up all matter. There are many different types of atoms, each with its own name, mass and size. These different types of atoms are called chemical elements. The chemical elements are organized on the periodic table. Examples of elements are hydrogen and gold. Atoms are very small, but the exact size changes depending on the element. Atoms range from 0.1 to 0.5 nanometers in width.[1] One nanometer is around 100,000 times smaller than the width of a human hair.[2] This makes atoms impossible to see without special tools. Equations must be used to see the way they work and how they interact with other atoms.
Atoms come together to make molecules or particles: for example, two hydrogen atoms and one oxygen atom combine to make a water molecule, a form of a chemical reaction.
Atoms themselves are made up of three kinds of smaller particles, called protons (which are positively charged), neutrons (which have no charge) and electrons (which are negatively charged). The protons and neutrons are in the middle of the atom. They are called the nucleus. They are surrounded by a cloud of electrons which are attracted to the nucleus' positive charge. This attraction is called electromagnetic force.
Protons and neutrons are made up of even smaller particles called quarks. Electrons are elementary or fundamental particles; they cannot be split into smaller parts.
The number of protons, neutrons and electrons an atom has determines what element it is. Hydrogen, for example, has one proton, no neutrons and one electron; the element sulfur has 16 protons, 16 neutrons and 16 electrons.
Atoms move faster when in gas form (as they are free to move) than liquid and solid matter. In solid materials, the atoms are tightly next to each other so they vibrate, but are not able to move (there is no room) as atoms in liquids do.
History[change | change source]
The word "atom" comes from the Greek (ἀτόμος), indivisible, from (ἀ)-, not, and τόμος, a cut. The first historical mention of the word atom came from works by the Greek philosopher Democritus, around 400 BC.[3] Atomic theory stayed as a mostly philosophical subject, with not much actual scientific investigation or study, until the development of chemistry in the 1650s.
In 1777 French chemist Antoine Lavoisier defined the term element for the first time. He said that an element was any basic substance that could not be broken down into other substances by the methods of chemistry. Any substance that could be broken down was a compound.[4]
In 1803, English philosopher John Dalton suggested that elements were tiny, solid spheres made of atoms. Dalton believed that all atoms of the same element have the same mass. He said that compounds are formed when atoms of more than one element combine. According to Dalton, in a compound, atoms of different elements always combine the same way.
In 1827, British scientist Robert Brown looked at pollen grains in water and used Dalton's atomic theory to describe patterns in the way they moved. This was called Brownian Motion. In 1905 Albert Einstein used mathematics to prove that the seemingly random movements were down to the reactions of atoms, and by doing so he conclusively proved the existence of the atom.[5] In 1869 scientist Dmitri Mendeleev published the first version of the periodic table. The periodic table groups atoms by their atomic number (how many protons they have. This is usually the same as the number of electrons). Elements in the same column, or period, usually have similar properties. For example helium, neon, argon, krypton and xenon are all in the same column and have very similar properties. All these elements are gases that have no colour and no smell. Together they are known as the noble gases.[4]
The physicist J.J. Thomson was the first man to discover electrons. This happened while he was working with cathode rays in 1897. He realized they had a negative charge, unlike protons (positive) and neutrons (no charge). Thomson created the plum pudding model, which stated that an atom was like plum pudding: the dried fruit (electrons) were stuck in a mass of pudding (protons). In 1909, a scientist named Ernest Rutherford used the Geiger–Marsden experiment to prove that most of an atom is in a very small space called the atomic nucleus. Rutherford took a photo plate and surrounded it with gold foil, and then shot alpha particles at it. Many of the particles went through the gold foil, which proved that atoms are mostly empty space. Electrons are so small they make up only 1% of an atom's mass.[6]
Ernest Rutherford
In 1913, Niels Bohr introduced the Bohr model. This model showed that electrons orbit the nucleus in fixed circular orbits. This was more accurate than the Rutherford model. However, it was still not completely right. Improvements to the Bohr model have been made since it was first introduced.
In 1925, chemist Frederick Soddy found that some elements in the periodic table had more than one kind of atom.[7] For example any atom with 2 protons should be a helium atom. Usually, a helium nucleus also contains two neutrons. However, some helium atoms have only one neutron. This means they are still helium, as the element is defined by the number of protons, but they are not normal helium either. Soddy called an atom like this, with a different number of neutrons, an isotope. To get the name of the isotope we look at how many protons and neutrons it has in its nucleus and add this to the name of the element. So a helium atom with two protons and one neutron is called helium-3, and a carbon atom with six protons and six neutrons is called carbon-12. However, when he developed his theory Soddy could not be certain neutrons actually existed. To prove they were real, physicist James Chadwick and a team of others created the mass spectrometer.[8] The mass spectrometer actually measures the mass and weight of individual atoms. By doing this Chadwick proved that to account for all the weight of the atom, neutrons must exist.
In 1937, German chemist Otto Hahn became the first person to create nuclear fission in a laboratory. He discovered this by chance when he was shooting neutrons at a uranium atom, hoping to create a new isotope.[9] However, he noticed that instead of a new isotope the uranium simply changed into a barium atom. This was the world's first recorded nuclear fission reaction. This discovery eventually led to the creation of the atomic bomb.
Further into the 20th century physicists went deeper into the mysteries of the atom. Using particle accelerators they discovered that protons and neutrons were actually made of other particles, called quarks.
The most accurate model so far comes from the Schrödinger equation. Schrödinger realized that the electrons exist in a cloud around the nucleus, called the electron cloud. In the electron cloud, it is impossible to know exactly where electrons are. The Schrödinger equation is used to find out where an electron is likely to be. This area is called the electron's orbital.
Structure and parts[change | change source]
Parts[change | change source]
A helium atom, with the nucleus shown in red (and enlarged), embeded in a cloud of electrons. If this drawing were to scale, the gray area around the center would be about 5m in diameter.
The complex atom is made up of three main particles; the proton, the neutron and the electron. The isotope of Hydrogen Hydrogen-1 has no neutrons, and a positive hydrogen ion has no electrons. These are the only known exceptions, all other atoms have at least one proton, neutron and electron each.
Electrons are by far the smallest of the three, their mass and size is too small to be measured using current technology.[10] They have a negative charge. Protons and neutrons are of similar size to each other[10] Protons are positively charged and neutrons have no charge. Most atoms have a neutral charge; because the number of protons (positive) and electrons (negative) are the same, the charges balance out to zero. However in ions (different number of electrons) this is not always the case and they can have a positive or a negative charge. Protons and Neutrons are made out of quarks, of two types; up quarks and down quarks. A proton is made of two up quarks and one down quark and a neutron is made of two down quarks and one up quark.
Nucleus[change | change source]
The nucleus is in the middle of an atom. It is made up of protons and neutrons. Usually in nature, two things with the same charge repel or shoot away from each other. So for a long time it was a mystery to scientists how the positively charged protons in the nucleus stayed together. They solved this by finding a particle called a Gluon. Its name comes from the word glue as Gluons act like atomic glue, sticking the protons together using the strong nuclear force. It is this force which also holds the quarks together that make up the protons and neutrons.
A diagram showing the main difficulty in nuclear fusion, the fact that protons, which have positive charges, repel each other when forced together.
The number of neutrons in relation to protons defines whether the nucleus is stable or goes through radioactive decay. When there are too many neutrons or protons, the atom tries to make the numbers the same by getting rid of the extra particles. It does this by emitting radiation in the form of alpha, beta or gamma decay.[11] Nuclei can change through other means too. Nuclear fission is when the Nucleus splits into two smaller nuclei, releasing a lot of stored energy. This release energy is what makes nuclear fission useful for making bombs and electricity, in the form of nuclear power. The other way nuclei can change is through nuclear fusion, when two nuclei join together, or fuse, to make a heavier nucleus. This process requires extreme amounts of energy in order to overcome the electrostatic repulsion between the protons, as they have the same charge. Such high energies are most common in stars like our Sun, which fuses hydrogen for fuel.
Electrons[change | change source]
Electrons orbit or go around the nucleus. They are called the atom's electron cloud. They are attracted towards the nucleus because of the electromagnetic force. Electrons have a negative charge and the nucleus always has a positive charge, so they attract each other. Around the nucleus some electrons are further out than others. These are called electron shells. In most atoms the first shell has two electrons, and all after that have eight. Exceptions are rare, but they do happen and are difficult to predict.[12] The further away the electron is from the nucleus, the weaker the pull of the nucleus on it. This is why bigger atoms, with more electrons, react more easily with other atoms. The electromagnetism of the nucleus is not enough to hold onto their electrons and they lose them to the strong attraction of smaller atoms. [13]
Radioactive decay[change | change source]
Some elements, and many isotopes, have what is called an unstable nucleus. This means the nucleus is either too big to hold itself together[14] or has too many protons, electrons or neutrons. When this happens the nucleus has to get rid of the excess mass or particles. It does this through radiation. An atom that does this can be called radioactive. Unstable atoms continue to be radioactive until they lose enough mass/particles that they become stable. All atoms above atomic number 82 (82 protons) are radioactive.[14]
There are three main types of radioactive decay; alpha, beta and gamma.[15]
• Alpha decay is when the atom shoots out a particle having two protons and two neutrons. This is essentially a helium nucleus. The result is an element with atomic number two less than before. So for example if a beryllium atom (atomic number 4) went through alpha decay it would become helium (atomic number 2). Alpha decay happens when an atom is too big and needs to get rid of some mass.
• Beta decay is when a neutron turns into a proton or a proton turns into a neutron. In the first case the atom shoots out an electron, in the second case it is a positron (like an electron but with a positive charge). The end result is an element with one higher or one lower atomic number than before. Beta decay happens when an atom has either too many protons, or too many neutrons.
• Gamma decay is when an atom shoots out a gamma ray, or wave. It happens when there is a change in the energy of the nucleus. This is usually after a nucleus has already gone through alpha or beta decay. There is no change in the mass, or atomic number or the atom, only in the stored energy inside the nucleus.
Every radioactive element or isotope has something called a half life. This is how long it takes half of any sample of atoms of that type to decay until they become a different stable isotope or element.[16] Large atoms, or isotopes with a big difference between the number of protons and neutrons will therefore have a long half life.
References[change | change source]
1. "Size of an Atom".
2. "Diameter of a Human Hair".
3. "History of Atomic Theory".
4. 4.0 4.1 "A Brief History of the Atom".
5. "Brownian motion - a history".
6. "Ernest Rutherford on Nuclear spin and Alpha Particle interaction.".
7. "Frederick Soddy, the Nobel Prize in chemistry: 1921".
8. "James Chadwick: The Nobel Prize in Physics 1935, a lecture on the Neutron and its properties".
9. "Otto Hahn, Liese Meitner and Fritz Strassman".
10. 10.0 10.1 "Particle Physics - Structure of a Matter".
11. "How does radioactive decay work?".
12. "Chemtutor on atomic structure".
13. "Chemical reactivity".
14. 14.0 14.1 "Radioactivity".
15. "S-Cool: Types of radiation".
16. "What is half-life?".
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67f575f4ecfac96f | Take the 2-minute tour ×
This is not a homework question, just a question I have developed to get a better conceptual understanding of the results of the Schrödinger equation.
If I had a 3D spherical container or radius R, containing 2 particles of opposite charge, say a proton and an electron, what does the solution to the resulting Schrödinger equation look like?
How does the solution compare to the solution of the Schrödinger equation for a simple hydrogen atom? What happens as R approaches infinity?
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up vote 4 down vote accepted
The hamiltonian of this system is quite simply the sum of hamiltonian of hydrogen atom and wall potentials for two particles: $$ H = \frac{1}{2 m_1} p_1^2 + \frac{1}{2 m_2} p_2^2 - \frac{e^2}{|\mathbf{r}_1-\mathbf{r}_2|} + V^\text{box}_1(r_1)+V^\text{box}_2(r_2), $$ where $V^\text{box}$ are the confining box potentials. For impenetrable box we can set $V^\text{box}_1(r)=V^\text{box}_2(r)=\infty \cdot \theta(r - R)$, with $\theta$ the Heaviside function.
Additionally, if there is considerable difference in masses $m_1$ and $m_2$ (like for masses of proton and electron) the problem could be essentially reduced to a motion of a single electron while the proton sits at the center of the cavity, essentially (after separating the angular variables) giving the following one dimensional Schrödinger equation: \begin{equation} \left[ -\frac{d^{2}}{dr^{2}}+\frac{l(l+1)}{r^{2}}-\frac{A}{r}\right] \psi (r)=E\psi (r),~\psi (0)=\psi (R)=0 \end{equation} This problem could be easily analyzed using variety of methods. Additional degeneracy of hydrogen atom associated with conserved Lenz vector usually disappears in this problem, however for some specific values of $R$ this degeneracy reappears.
There is quite a lot of literature on this problem. The first results go back to 1937:
Michels, A., J. De Boer, and A. Bijl. "Remarks concerning molecural interaction and their influence on the polarisability." Physica 4.10 (1937): 981-994.
For the overview of results let us look at one of the recent papers:
Ciftci, H., Hall, R. L., & Saad, N. "Study of a confined hydrogen‐like atom by the asymptotic iteration method." International Journal of Quantum Chemistry 109.5 (2009): 931-937. Arxiv:0807.4135.
From it we learn:
The concept of a confined quantum system goes back to the early work of Michels et al [1] who studied the properties of an atomic system under very high pressures. They suggested to replace the interaction of the atoms with surrounding atoms by a uniform pressure on a sphere within which the atom is considered to be en closed. This led them to consider the problem of hydrogen with modified external boundary conditions [2]. Since then, the confined hydrogen atom attracted widespread attention [2]-[33].
Many researchers have carried out accurate calculations of eigenvalues of the confined hydrogen atom using various techniques. Some of these are variational methods [18]-[27], finite element methods [28], and algebraic methods [29].
The authors then present analysis of the problem, including exact solutions for some specific values of $R$.
Another approach (originally by Wigner) to the problem of confined hydrogen atom is to start with free particle(s) in a box and use the Coulomb potential as a perturbation, obtaining the expansion in terms of $e^2$. This method is explained in:
Aguilera-Navarro, V. C., W. M. Kloet, and A. H. Zimerman. "Application of the Rayleigh-Schrödinger perturbation theory to hydrogen atom". Instituto de Fisica Teorica, Sao Paulo, Brazil, 1971. online version
This method is useful for small values of $R$ however the limit $R \to \infty$ presents porblems:
By numerical computation we found that in the perturbation series for the energy the sign of each term (with the exception of the unperturbed energy) is always negative, making it in our opinion improbable that the series is convergent for $R \to \infty$
(in Wigner's paper [1] the possibility is discussed that, although each term in the perturbation series from the third order on in $e^2$ is more and more divergent for $R\to \infty$, the whole series could converge to the actual value as $R\to \infty$).
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Comparison of homogeneous dust model with ESA/NAVCAM Rosetta images.
The shape of the universe
The following post is contributed by Peter Kramer.
hyperbolic dodecahedron
Shown are two faces of a hyberbolic dodecahedron.
The red line from the family of shortest lines (geodesics) connects both faces. Adapted from CRM Proceedings and Lecture Notes (2004), vol 34, p. 113, by Peter Kramer.
The new Planck data on the cosmic microwave background (CMB) has come in. For cosmic topology, the data sets contain interesting information related to the size and shape of the universe. The curvature of the three-dimensional space leads to a classification into hyperbolic, flat, or spherical cases. Sometimes in popular literature, the three cases are said to imply an inifinite (hyperbolic, flat) or finite (spherical) size of the universe. This statement is not correct. Topology supports a much wider zoo of possible universes. For instance, there are finite hyperbolic spaces, as depicted in the figure (taken from Group actions on compact hyperbolic manifolds and closed geodesics, arxiv version). The figure also shows the resulting geodesics, which is the path of light through such a hyperbolic finite sized universe. The start and end-points must be identified and lead to smooth connection.
Recent observational data seem to suggest a spherical space. Still, it does not resolve the issue of the size of the universe.
Instead of a fully filled three-sphere, already smaller parts of the sphere can be closed topologically and thus lead to a smaller sized universe. A systematic exploration of such smaller but still spherical universes is given in my recent article
Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis.
In physics, it is important to give specific predictions for observations of the topology, for instance by predicting the ratio of the different angular modes of the cosmic microwave background. It is shown that this is indeed the case and for instance in a cubic (still spherical!) universe, the ratio of 4th and 6th multipole order squared are tied together in the proportion 7 : 4, see Table 5. On p. 35 of ( the Planck collaboration article) the authors call for models yielding such predictions as possible explanations for the observed anisotropy and the ratio of high and low multipole moments.
When two electrons collide. Visualizing the Pauli blockade.
The upper panel shows two (non-interacting) electrons approaching with small relative momenta, the lower panel with larger relative momenta.
The upper panel shows two electrons with small relative momenta colliding, in the lower panel with larger relative momenta.
From time to time I get asked about the implications of the Pauli exclusion principle for quantum mechanical wave-packet simulations.
I start with the simplest antisymmetric case: a two particle state given by the Slater determinant of two Gaussian wave packets with perpendicular directions of the momentum:
φa(x,y)=e-[(x-o)2+(y-o)2]/(2a2)-ikx+iky and φb(x,y)=e-[(x+o)2+(y-o)2]/(2a2)+ikx+iky
This yields the two-electron wave function
The probability to find one of the two electrons at a specific point in space is given by integrating the absolute value squared wave function over one coordinate set.
The resulting single particle density (snapshots at specific values of the displacement o) is shown in the animation for two different values of the momentum k (we assume that both electrons are in the same spin state).
For small values of k the two electrons get close in phase space (that is in momentum and position). The animation shows how the density deviates from a simple addition of the probabilities of two independent electrons.
If the two electrons differ already by a large relative momentum, the distance in phase space is large even if they get close in position space. Then, the resulting single particle density looks similar to the sum of two independent probabilities.
The probability to find the two electrons simultaneously at the same place is zero in both cases, but this is not directly visible by looking at the single particle density (which reflects the probability to find any of the electrons at a specific position).
For further reading, see this article [arxiv version].
The impact of scientific publications – some personal observations
I will resume posting about algorithm development for computational physics. To put these efforts in a more general context, I start with some observation about the current publication ranking model and explore alternatives and supplements in the next posts.
Solvey congress 19...
Solvey congress 1970, many well-known nuclear physicists are present, including Werner Heisenberg.
Working in academic institutions involves being part of hiring committees as well as being assessed by colleagues to measure the impact of my own and other’s scientific contributions.
In the internet age it has become common practice to look at various performance indices, such as the h-index, number of “first author” and “senior author” articles. Often it is the responsibility of the applicant to submit this data in electronic spreadsheet format suitable for an easy ranking of all candidates. The indices are only one consideration for the final decision, albeit in my experience an important one due to their perceived unbiased and statistical nature. Funding of whole university departments and the careers of young scientists are tied to the performance indices.
I did reflect about the usefulness of impact factors while I collected them for various reports, here are some personal observations:
1. Looking at the (very likely rather incomplete) citation count of my father I find it interesting that for instance a 49 year old contribution by P Kramer/M Moshinsky on group-theoretical methods for few-body systems gains most citations per year after almost 5 decades. This time-scale is well beyond any short-term hiring or funding decisions based on performance indices. From colleagues I hear about similar cases.
2. A high h-index can be a sign of a narrow research field, since the h-index is best built up by sticking to the same specialized topic for a long time and this encourages serialised publications. I find it interesting that on the other hand important contributions have been made by people working outside the field to which they contributed. The discovery of three-dimensional quasicrystals discussed here provides a good example. The canonical condensed matter theory did not envision this paradigmatic change, rather the study of group theoretical methods in nuclear physics provided the seeds.
3. The full-text search provided by the search engines offers fascinating options to scan through previously forgotten chapters and books, but it also bypasses the systematic classification schemes previously developed and curated by colleagues in mathematics and theoretical physics. It is interesting to note that for instance the AMS short reviews are not done anonymously and most often are of excellent quality. The non-curated search on the other hand leads to a down-ranking of books and review articles, which contain a broader and deeper exposition of a scientific topic. Libraries with real books grouped by topics are deserted these days, and online services and expert reviews did in general not gain a larger audience or expert community to write reports. One exception might be the public discussion of possible scientific misconduct and retracted publications.
4. Another side effect: searching the internet for specific topics diminishes the opportunity to accidentally stumble upon an interesting article lacking these keywords, for instance by scanning through a paper volume of a journal while searching for a specific article. I recall that many faculty members went every monday to the library and looked at all the incoming journals to stay up-to-date about the general developments in physics and chemistry. Today we get email alerts about citation counts or specific subfields, but no alert contains a suggestion what other article might pick our intellectual curiosity – and looking at the rather stupid shopping recommendations generated by online-warehouses I don’t expect this to happen anytime soon.
5. On a positive note: since all text sources are treated equally, no “high-impact journals” are preferred. In my experience as a referee for journals of all sorts of impact numbers, the interesting contributions are not necessarily published or submitted to highly ranked journals.
To sum up, the assessment of manuscripts, contribution of colleagues, and of my own articles requires humans to read them and to process them carefully – all of this takes a lot of time and consideration. It can take decades before publications become alive and well cited. Citation counts of the last 10 years can be poor indicators for the long-term importance of a contribution. Counting statistics provides some gratification by showing immediate interest and are the (less personal) substitute for the old-fashioned postcards requesting reprints. People working in theoretical physics are often closely related by collaboration distance, which provides yet another (much more fun!) factor. You can check your Erdos number (mine is 4) or Einstein number (3, thanks to working with Marcos Moshinsky) at the AMS website.
How to improve the current situation and maintain a well curated and relevant library of scientific contributions – in particular involving numerical results and methods? One possibility is to make a larger portion of the materials surrounding a publication available. In computational physics it is of interest to test and recalculate published results shown in journals. The nanohub.org platform is in my view a best practice case for providing supplemental information on demand and to ensure a long-term availability and usefulness of scientific results by keeping the computational tools running and updated. It is for me a pleasure and excellent experience to work with the team around nanohub to maintain our open quantum dynamics tool. Another way is to provide and test background materials in research blogs. I will try out different approaches with the next posts.
Better than Slater-determinants: center-of-mass free basis sets for few-electron quantum dots
Error analysis of eigenenergies of the standard configuration interaction (CI) method (right black lines). The left colored lines are obtained by explicitly handling all spurious states.
Error analysis of eigenenergies of the standard configuration interaction (CI) method (right black lines). The left colored lines are obtained by explicitly handling all spurious states. The arrows point out the increasing error of the CI approach with increasing center-of-mass admixing.
Solving the interacting many-body Schrödinger equation is a hard problem. Even restricting the spatial domain to a two-dimensions plane does not lead to analytic solutions, the trouble-maker are the mutual particle-particle interactions. In the following we consider electrons in a quasi two-dimensional electron gas (2DEG), which are further confined either by a magnetic field or a harmonic oscillator external confinement potential. For two electrons, this problem is solvable for specific values of the Coulomb interaction due to a hidden symmetry in the Hamiltonian, see the review by A. Turbiner and our application to the two interacting electrons in a magnetic field.
For three and more electrons (to my knowledge) no analytical solutions are known. One standard computational approach is the configuration interaction (CI) method to diagonalize the Hamiltonian in a variational trial space of Slater-determinantal states. Each Slater determinant consists of products of single-particle orbitals. Due to computer resource constraints, only a certain number of Slater determinants can be included in the basis set. One possibility is to include only trial states up to certain excitation level of the non-interacting problem.
The usage of Slater-determinants as CI basis-set introduce severe distortions in the eigenenergy spectrum due to the intrusion of spurious states, as we will discuss next. Spurious states have been extensively analyzed in the few-body problems arising in nuclear physics but have rarely been mentioned in solid-state physics, where they do arise in quantum-dot systems. The basic defect of the Slater-determinantal CI method is that it brings along center-of-mass excitations. During the diagonalization, the center-of-mass excitations occur along with the Coulomb-interaction and lead to an inflated basis size and also with a loss of precision for the eigenenergies of the excited states. Increasing the basis set does not uniformly reduce the error across the spectrum, since the enlarged CI basis set brings along states of high center-of-mass excitations. The cut-off energy then restricts the remaining basis size for the relative part.
The cleaner and leaner way is to separate the center-of-mass excitations from the relative-coordinate excitations, since the Coulomb interaction only acts along the relative coordinates. In fact, the center-of-mass part can be split off and solved analytically in many cases. The construction of the relative-coordinate basis states requires group-theoretical methods and is carried out for four electrons here Interacting electrons in a magnetic field in a center-of-mass free basis (arxiv:1410.4768). For three electrons, the importance of a spurious state free basis set was emphasized by R Laughlin and is a design principles behind the Laughlin wave function.
Slow or fast transfer: bottleneck states in light-harvesting complexes
Exciton dynamics in LHCII.
High-performance OpenCL code for modeling energy transfer in spinach
With increasing computational power of massively-parallel computers, a more accurate modeling of the energy-transfer dynamics in larger and more complex photosynthetic systems (=light-harvesting complexes) becomes feasible – provided we choose the right algorithms and tools.
OpenCL cross platform performance for tracking energy-transfer in the light-harvesting complex II found in spinach.
OpenCL cross platform performance for tracking energy-transfer in the light-harvesting complex II found in spinach, see Fig. 1 in the article . Shorter values show higher perfomance. The program code was originally written for massively-parallel GPUs, but performs also well on the AMD opteron setup. The Intel MIC OpenCL variant does not reach the peak performance (a different data-layout seems to be required to benefit from autovectorization).
The diverse character of hardware found in high-performance computers (hpc) seemingly requires to rewrite program code from scratch depending if we are targeting multi-core CPU systems, integrated many-core platforms (Xeon PHI/MIC), or graphics processing units (GPUs).
To avoid the defragmentation of our open quantum-system dynamics workhorse (see the previous GPU-HEOM posts) across the various hpc-platforms, we have transferred the GPU-HEOM CUDA code to the Open Compute Language (OpenCL). The resulting QMaster tool is described in our just published article Scalable high-performance algorithm for the simulation of exciton-dynamics. Application to the light harvesting complex II in the presence of resonant vibrational modes (collaboration of Christoph Kreisbeck, Tobias Kramer, Alan Aspuru-Guzik). This post details the computational challenges and lessons learnt, the application to the light-harvesting complex II found in spinach will be the topic of the next post.
In my experience, it is not uncommon to develop a nice GPU application for instance with CUDA, which later on is scaled up to handle bigger problem sizes. With increasing problem size also the memory demands increase and even the 12 GB provided by the Kepler K40 are finally exhausted. Upon reaching this point, two options are possible: (a) to distribute the memory across different GPU devices or (b) to switch to architectures which provide more device-memory. Option (a) requires substantial changes to existing program code to manage the distributed memory access, while option (b) in combination with OpenCL requires (in the best case) only to adapt the kernel-launch configuration to the different platforms.
The OpenCL device fission extension allows to investigate the scaling of the QMaster code with the number of CPU cores. We observe a linear scaling up to 48 cores.
The OpenCL device fission extension allows us to investigate the scaling of the QMaster code with the number of CPU cores. We observe a linear scaling up to 48 cores.
QMaster implements an extension of the hierarchical equation of motion (HEOM) method originally proposed by Tanimura and Kubo, which involves many (small) matrix-matrix multiplications. For GPU applications, the usage of local memory and the optimal thread-grids for fast matrix-matrix multiplications have been described before and are used in QMaster (and the publicly available GPU-HEOM tool on nanohub.org). While for GPUs the best performance is achieved using shared/local memory and assign one thread to each matrix element, the multi-core CPU OpenCL variant performs better with fewer threads, but getting more work per thread done. Therefore we use for the CPU machines a thread-grid which computes one complete matrix product per thread (this is somewhat similar to following the “naive” approach given in NVIDIA’s OpenCL programming guide, chapter 2.5). This strategy did not work very well for the Xeon PHI/MIC OpenCL case, which requires additional data structure changes, as we learnt from discussions with the distributed algorithms and hpc experts in the group of Prof. Reinefeld at the Zuse-Institute in Berlin.
The good performance and scaling across the 64 CPU AMD opteron workstation positively surprised us and lays the groundwork to investigate the validity of approximations to the energy-transfer equations in the spinach light-harvesting system, the topic for the next post.
Flashback to the 80ies: filling space with the first quasicrystals
This post provides a historical and conceptional perspective for the theoretical discovery of non-periodic 3d space-fillings by Peter Kramer, later experimentally found and now called quasicrystals. See also these previous blog entries for more quasicrystal references and more background material here.
The following post is written by Peter Kramer.
When sorting out old texts and figures from 1981 of mine published in Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells, Acta Cryst. (1982). A38, 257-264), I came across the figure of a regular pentagon of edge length L, which I denoted as p(L). In the left figure its red-colored edges are star-extending up to their intersections. Straight connection of these intersection points creates a larger blue pentagon. Its edges are scaled up by τ2, with τ the golden section number, so the larger pentagon we call p(τ2 L). This blue pentagon is composed of the old red one plus ten isosceles triangles with golden proportion of their edge length. Five of them have edges t1(L): (L, τ L, τ L), five have edges t2(L): (τ L,τ L, τ2 L). We find from Fig 1 that these golden triangles may be composed face-to-face into their τ-extended copies as t1(τ L) = t1(L) + t2(L) and t2(τ L) = t1(L) + 2 t2(L).
Moreover we realize from the figure that also the pentagon p(τ2 L) can be composed from golden triangles as p(τ2 L) = t1(τ L) + 3 t2(τ L) = 4 t1(L) + 7 t2(L).
This suggests that the golden triangles t1,t2 can serve as elementary cells of a triangle tiling to cover any range of the plane and provide the building blocks of a quasicrystal. Indeed we did prove this long range property of the triangle tiling (see Planar patterns with fivefold symmetry as sections of periodic structures in 4-space).
An icosahedral tiling from star extension of the dodecahedron.
The star extension of the dodecahedron.
Star extension of the dodecahedron d(L) to the icosahedron i(τ2L) and further to d(τ3L) and i(τ5L) shown in Fig 3 of the 1982 paper. The vertices of these polyhedra are marked by filled circles; extensions of edges are shown except for d(L).
In the same paper, I generalized the star extension from the 2D pentagon to the 3D dodecahedron d(L) of edge length L in 3D (see next figure) by the following prescription:
• star extend the edges of this dodecahedron to their intersections
• connect these intersections to form an icosahedron
The next star extension produces a larger dodecahedron d(τ3L), with edges scaled by τ3. In the composition of the larger dodecahedron I found four elementary polyhedral shapes shown below. Even more amusing I also resurrected the paper models I constructed in 1981 to actually demonstrate the complete space filling!
These four polyhedra compose their copies by scaling with τ3. As for the 2D case arbitrary regions of 3D can be covered by the four tiles.
Elementary cells The paper models I built in 1981 are still around and complete enough to fill the 3D space.
The four elementary cells shown in the 1982 paper, Fig. 4. The four shapes are named dodecahedron (d) skene (s), aetos (a) and tristomos (t). The paper models from 1981 are still around in 2014 and complete enough to fill the 3D space without gaps. You can spot all shapes (d,s,a,t) in various scalings and they all systematically and gapless fill the large dodecahedron shell on the back of the table.
The only feature missing for quasicrystals is aperiodic long-range order which eventually leads to sharp diffraction patterns of 5 or 10 fold point-symmetries forbidden for the old-style crystals. In my construction shown here I strictly preserved central icosahedral symmetry. Non-periodicity then followed because full icosahedral symmetry and periodicity in 3D are incompatible.
In 1983 we found a powerful alternative construction of icosahedral tilings, independent of the assumption of central symmetry: the projection method from 6D hyperspace (On periodic and non-periodic space fillings of Em obtained by projection) This projection establishes the quasiperiodicity of the tilings, analyzed in line with the work Zur Theorie der fast periodischen Funktionen (i-iii) of Harald Bohr from 1925 , as a variant of aperiodicity (more background material here).
Tutorial #1: simulate 2d spectra of light-harvesting complexes with GPU-HEOM @ nanoHub
The computation and prediction of two-dimensional (2d) echo spectra of photosynthetic complexes is a daunting task and requires enormous computational resources – if done without drastic simplifications. However, such computations are absolutely required to test and validate our understanding of energy transfer in photosyntheses. You can find some background material in the recently published lecture notes on Modelling excitonic-energy transfer in light-harvesting complexes (arxiv version) of the Latin American School of Physics Marcos Moshinsky.
The ability to compute 2d spectra of photosynthetic complexes without resorting to strong approximations is to my knowledge an exclusive privilege of the Hierarchical Equations of Motion (HEOM) method due to its superior performance on massively-parallel graphics processing units (GPUs). You can find some background material on the GPU performance in the two conference talks Christoph Kreisbeck and I presented at the GTC 2014 conference (recored talk, slides) and the first nanoHub users meeting.
GPU-HEOM 2d spectra computed at nanohub
GPU-HEOM 2d spectra computed at nanohubComputed 2d spectra for the FMO complex for 0 picosecond delay time (upper panel) and 1 ps (lower panel). The GPU-HEOM computation takes about 40 min on the nanohub.org platform and includes all six Liouville pathways and averages over 4 spatial orientations.
1. login on nanoHub.org (it’s free!)
2. switch to the gpuheompop tool
3. click the Launch Tool button (java required)
4. for this tutorial we use the example input for “FMO coherence, 1 peak spectral density“.
You can select this preset from the Example selector.
5. we stick with the provided Exciton System parameters and only change the temperature to 77 K to compare the results with our published data.
6. in the Spectral Density tab, leave all parameters at the the suggested values
7. to compute 2d spectra, switch to the Calculation mode tab
8. for compute: choose “two-dimensional spectra“. This brings up input-masks for setting the directions of all dipole vectors, we stick with the provided values. However, we select Rotational averaging: “four shot rotational average” and activate all six Liouville pathways by setting ground st[ate] bleach reph[asing , stim[ulated] emission reph[asing], and excited st[ate] abs[orption] to yes, as well as their non-rephasing counterparts (attention! this might require to resize the input-mask by pulling at the lower right corner)
9. That’s all! Hit the Simulate button and your job will be executed on the carter GPU cluster at Purdue university. The simulation takes about 40 minutes of GPU time, which is orders of magnitude faster than any other published method with the same accuracy. You can close and reopen your session in between.
10. Voila: your first FMO spectra appears.
11. Now its time to change parameters. What happens at higher temperature?
12. If you like the results or use them in your work for comparison, we (and the folks at nanoHub who generously develop and provide the nanoHub platform and GPU computation time) appreciate a citation. To make this step easy, a DOI number and reference information is listed at the bottom of the About tab of the tool-page.
With GPU-HEOM we and now you (!) can not only calculate the 2d echo spectra of the Fenna-Matthews-Olson (FMO) complex, but also reveal the strong link between the continuum part of the vibrational spectral density and the prevalence of long-lasting electronic coherences as written in my previous posts
GPU and cloud computing conferences in 2014
Two conferences are currently open for registration related to GPU and cloud computing. I will be attending and presenting at both, please email me if you want to get in touch at the meetings.
Oscillations in two-dimensional spectroscopy
Transition from electronic coherence to a vibrational mode.
Transition from electronic coherence to a vibrational mode made visible by Short Time Fourier Transform (see text).
Over the last years, a debate is going on whether the observation of long lasting oscillatory signals in two-dimensional spectra are reflecting vibrational of electronic coherences and how the functioning of the molecule is affected. Christoph Kreisbeck and I have performed a detailed theoretical analysis of oscillations in the Fenna-Matthews-Olson (FMO) complex and in a model three-site system. As explained in a previous post, the prerequisites for long-lasting electronic coherences are two features of the continuous part of the vibronic mode density are: (i) a small slope towards zero frequency, and (ii) a coupling to the excitonic eigenenergy (ΔE) differences for relaxation. Both requirements are met by the mode density of the FMO complex and the computationally demanding calculation of two-dimensional spectra of the FMO complex indeed predicts long-lasting cross-peak oscillations with a period matching h/ΔE at room temperature (see our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes or arXiv version). The persistence of oscillations is stemming from a robust mechanism and does not require adding any additional vibrational modes at energies ΔE (the general background mode density is enough to support the relaxation toward a thermal state). But what happens if in addition to the background vibronic mode density additional vibronic modes are placed within the vicinity of the frequencies related electronic coherences? This fine-tuning model is sometimes discussed in the literature as an alternative mechanism for long-lasting oscillations of vibronic nature. Again, the answer requires to actually compute two-dimensional spectra and to carefully analyze the possible chain of laser-molecule interactions. Due to the special way two-dimensional spectra are measured, the observed signal is a superposition of at least three pathways, which have different sensitivity for distinguishing electronic and vibronic coherences. Being a theoretical physicists now pays off since we have calculated and analyzed the three pathways separately (see our recent publication Disentangling Electronic and Vibronic Coherences in Two-Dimensional Echo Spectra or arXiv version). One of the pathways leads to an enhancement of vibronic signals, while the combination of the remaining two diminishes electronic coherences otherwise clearly visible within each of them. Our conclusion is that estimates of decoherence times from two-dimensional spectroscopy might actually underestimate the persistence of electronic coherences, which are helping the transport through the FMO network. The fine tuning and addition of specific vibrational modes leaves it marks at certain spots of the two-dimensional spectra, but does not destroy the electronic coherence, which is still there as a Short Time Fourier Transform of the signal reveals.
Computational physics on GPUs: writing portable code
GPU-HEOM code comparison for various hardware.
Runtime in seconds for our GPU-HEOM code on various hardware and software platforms.
I am preparing my presentation for the simGPU meeting next week in Freudenstadt, Germany, and performed some benchmarks.
In the previous post I described how to get an OpenCL program running on a smartphone with GPU. By now Christoph Kreisbeck and I are getting ready to release our first smartphone GPU app for exciton dynamics in photosynthetic complexes, more about that in a future entry.
Getting the same OpenCL kernel running on laptop GPUs, workstation GPUs and CPUs, and smartphones/tablets is a bit tricky, due to different initialisation procedures and the differences in the optimal block sizes for the thread grid. In addition on a smartphone the local memory is even smaller than on a desktop GPU and double-precision floating point support is missing. The situation reminds me a bit of the “earlier days” of GPU programming in 2008.
Besides being a proof of concept, I see writing portable code as a sort of insurance with respect to further changes of hardware (however always with the goal to stick with the massively parallel programming paradigm). I am also amazed how fast smartphones are gaining computational power through GPUs!
Same comparison for smaller memory consumption. Note the drop in OpenCL performance for the NVIDIA K20c GPU.
Here some considerations and observations:
1. Standard CUDA code can be ported to OpenCL within a reasonable time-frame. I found the following resources helpful:
AMDs porting remarks
Matt Scarpinos OpenCL blog
2. The comparison of OpenCL vs CUDA performance for the same algorithm can reveal some surprises on NVIDIA GPUs. While on our C2050 GPU OpenCL works a bit faster for the same problem compared to the CUDA version, on a K20c system for certain problem sizes the OpenCL program can take several times longer than the CUDA code (no changes in the basic algorithm or workgroup sizes).
3. The comparison with a CPU version running on 8 cores of the Intel Xeon machine is possible and shows clearly that the GPU code is always faster, but requires a certain minimal systems size to show its full performance.
4. I am looking forward to running the same code on the Intel Xeon Phi systems now available with OpenCL drivers, see also this blog.
[Update June 22, 2013: I updated the graphs to show the 8-core results using Intels latest OpenCL SDK. This brings the CPU runtimes down by a factor of 2! Meanwhile I am eagerly awaiting the possibility to run the same code on the Xeon Phis…]
Computational physics on the smartphone GPU
/home/tkramer/android-ndk-r8d/build/tools/make-standalone-toolchain.sh \
adb pull /system/lib/libOpenCL.so
rm plasma_disk_gpu
-I. \
-Llib \
-lOpenCL \
-o plasma_disk_gpu plasma_disk.cpp
scp -P 2222 integrate_eom_kernel.cl root@192.168.0.NNN:
scp -P 2222 plasma_disk_gpu root@192.168.0.NNN:
9. ssh into your phone and run the GPU program:
ssh -p 2222 root@192.168.0.NNN
./plasma_disk_gpu 64 16
cd /data/data/jackpal.androidterm
mkdir gpu
chmod 777 gpu
adb push integrate_eom_kernel.cl /data/data/jackpal.androidterm/gpu/
adb push plasma_disk_gpu /data/data/jackpal.androidterm/gpu/
adb shell
cd /data/data/jackpal.androidterm/gpu/
./plasma_disk_gpu 64 16
Computational physics & GPU programming: exciton lab for light-harvesting complexes (GPU-HEOM) goes live on nanohub.org
User interface of the GPU-HEOM tool for light-harvesting complexes at nanohub.org.
Christoph Kreisbeck and I are happy to announce the public availability of the Exciton Dynamics Lab for Light-
Harvesting Complexes (GPU-HEOM) hosted on nanohub.org. You need to register a user account (its free), and then you are ready to use GPU-HEOM for the Frenkel exciton model of light harvesting complexes. In release 1.0 we support
• calculating population dynamics
• tracking coherences between two eigenstates
• obtaining absorption spectra
• two-dimensional echo spectra (including excited state absorption)
• … and all this for general vibronic spectral densities parametrized by shifted Lorentzians.
I will post some more entries here describing how to use the tool for understanding how the spectral density affects the lifetime of electronic coherences (see also this blog entry).
In the supporting document section you find details of the implemented method and the assumptions underlying the tool. We are appreciating your feedback for further improving the tool.
We are grateful for the support of Prof. Gerhard Klimeck, Purdue University, director of the Network for Computational Nanotechnology to bring GPU computing to nanohub (I believe our tool is the first GPU enabled one at nanohub).
If you want to refer to the tool you can cite it as:
Christoph Kreisbeck; Tobias Kramer (2013), “Exciton Dynamics Lab for Light-Harvesting Complexes (GPU-HEOM),” https://nanohub.org/resources/gpuheompop. (DOI:10.4231/D3RB6W248).
and you find further references in the supporting documentation.
I very much encourage my colleagues developing computer programs for theoretical physics and chemistry to make them available on platforms such as nanohub.org. In my view, it greatly facilitates the comparison of different approaches and is the spirit of advancing science by sharing knowledge and providing reproducible data sets.
Good or bad vibrations for the Fenna-Matthews-Olson complex?
Electronic and vibronic coherences in the FMO complex using GPU HEOM by Kreisbeck and Kramer
Time-evolution of the coherence for the FMO complex (eigenstates 1 and 5 ) calculated with GPU-HEOM by Kreisbeck and Kramer, J. Phys. Chem Lett. 3, 2828 (2012).
Due to its known structure and relative simplicity, the Fenna-Matthews-Olson complex of green sulfur bacteria provides an interesting test-case for our understanding of excitonic energy transfer in a light-harvesting complex.
The experimental pump-probe spectra (discussed in my previous post catching and tracking light: following the excitations in the Fenna-Matthews-Olson complex) show long-lasting oscillatory components and this finding has been a puzzle for theoretician and led to a refinement of the well-established models. These models show a reasonable agreement with the data and the rate equations explain the relaxation and transfer of excitonic energy to the reaction center.
However, the rate equations are based on estimates for the relaxation and dephasing rates. As Christoph Kreisbeck and I discuss in our article Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes (arxiv version), an exact calculation with GPU-HEOM following the best data for the Hamiltonian allows one to determine where the simple approach is insufficient and to identify a key-factor supporting electronic coherence:
Wendling spectral density for FMO complex
Important features in the spectral density of the FMO complex related to the persistence of cross-peak oscillations in 2d spectra.
It’s the vibronic spectral density – redrawn (in a different unit convention, multiplied by ω2) from the article by M. Wendling from the group of Prof. Rienk van Grondelle. We did undertake a major effort to proceed in our calculations as close to the measured shape of the spectral density as the GPU-HEOM method allows one. By comparison of results for different forms of the spectral density, we identify how the different parts of the spectral density lead to distinct signatures in the oscillatory coherences. This is illustrated in the figure on the rhs. To get long lasting oscillations and finally to relax, three ingredients are important
1. a small slope towards zero frequency, which suppresses the pure dephasing.
2. a high plateau in the region where the exciton energy differences are well coupled. This leads to relaxation.
3. the peaked structures induce a “very-long-lasting” oscillatory component, which is shown in the first figure. In our analysis we find that this is a persistent, but rather small (<0.01) modulation.
2d spectra are smart objects
FMO spectrum calculated with GPU-HEOM
FMO spectrum calculated with GPU-HEOM for a 3 peak approximation of the measured spectral density, including disorder averaging but no excited state absorption.
The calculation of 2d echo spectra requires considerable computational resources. Since theoretically calculated 2d spectra are needed to check how well theory and experiment coincide, I conclude with showing a typical spectrum we obtain (including static disorder, but no excited state absorption for this example). One interesting finding is that 2d spectra are able to differentiate between the different spectral densities. For example for a a single-peak Drude-Lorentz spectral density (sometimes chosen for computational convenience), the wrong peaks oscillate and the life-time of cross-peak oscillations is short (and becomes even shorter with longer vibronic memory). But this is for the experts only, see the supporting information of our article.
Are vibrations good or bad? Probably both… The pragmatic answer is that the FMO complex lives in an interesting parameter regime. The exact calculations within the Frenkel exciton model do confirm the well-known dissipative energy transfer picture. But on the other hand the specific spectral density of the FMO complex supports long-lived coherences (at least if the light source is a laser beam), which require considerable theoretical and experimental efforts to be described and measured. Whether the seen coherence has any biological relevance is an entirely different topic… maybe the green-sulfur bacteria are just enjoying a glimpse into Schrödinger’s world of probabilistic uncertainty.
Computational physics & GPU programming: interacting many-body simulation with OpenCL
Trajectories in a two-dimensional interacting plasma simulation, reproducing the density and pair-distribution function of a Laughlin state relevant for the quantum Hall effect. Figure taken from Interacting electrons in a magnetic field: mapping quantum mechanics to a classical ersatz-system.
In the second example of my series on GPU programming for scientists, I discuss a short OpenCL program, which you can compile and run on the CPU and the GPUs of various vendors. This gives me the opportunity to perform some cross-platform benchmarks for a classical plasma simulation. You can expect dramatic (several 100 fold) speed-ups on GPUs for this type of system. This is one of the reasons why molecular dynamics code can gain quite a lot by incorporating the massively parallel-programming paradigm in the algorithmic foundations.
The Open Computing Language (OpenCL) is relatively similar to its CUDA pendant, in practice the setup of an OpenCL kernel requires some housekeeping work, which might make the code look a bit more involved. I have based my interacting electrons calculation of transport in the Hall effect on an OpenCL code. Another examples is An OpenCL implementation for the solution of the time-dependent Schrödinger equation on GPUs and CPUs (arxiv version) by C. Ó Broin and L.A.A. Nikolopoulos.
Now to the coding of a two-dimensional plasma simulation, which is inspired by Laughlin’s mapping of a many-body wave function to an interacting classical ersatz dynamics (for some context see my short review Interacting electrons in a magnetic field: mapping quantum mechanics to a classical ersatz-system on the arxiv).
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Computational physics & GPU programming: Solving the time-dependent Schrödinger equation
I start my series on the physics of GPU programming by a relatively simple example, which makes use of a mix of library calls and well-documented GPU kernels. The run-time of the split-step algorithm described here is about 280 seconds for the CPU version (Intel(R) Xeon(R) CPU E5420 @ 2.50GHz), vs. 10 seconds for the GPU version (NVIDIA(R) Tesla C1060 GPU), resulting in 28 fold speed-up! On a C2070 the run time is less than 5 seconds, yielding an 80 fold speedup.
autocorrelation function in a uniform force field
Autocorrelation function C(t) of a Gaussian wavepacket in a uniform force field. I compare the GPU and CPU results using the wavepacket code.
The description of coherent electron transport in quasi two-dimensional electron gases requires to solve the Schrödinger equation in the presence of a potential landscape. As discussed in my post Time to find eigenvalues without diagonalization, our approach using wavepackets allows one to obtain the scattering matrix over a wide range of energies from a single wavepacket run without the need to diagonalize a matrix. In the following I discuss the basic example of propagating a wavepacket and obtaining the autocorrelation function, which in turn determines the spectrum. I programmed the GPU code in 2008 as a first test to evaluate the potential of GPGPU programming for my research. At that time double-precision floating support was lacking and the fast Fourier transform (FFT) implementations were little developed. Starting with CUDA 3.0, the program runs fine in double precision and my group used the algorithm for calculating electron flow through nanodevices. The CPU version was used for our articles in Physica Scripta Wave packet approach to transport in mesoscopic systems and the Physical Review B Phase shifts and phase π-jumps in four-terminal waveguide Aharonov-Bohm interferometers among others.
Here, I consider a very simple example, the propagation of a Gaussian wavepacket in a uniform potential V(x,y)=-Fx, for which the autocorrelation function of the initial state
⟨x,y|ψ(t=0)⟩=1/(a√π)exp(-(x2+y2)/(2 a2))
is known in analytic form:
⟨ψ(t=0)|ψ(t)⟩=2a2m/(2a2m+iℏt)exp(-a2F2t2/(4ℏ2)-iF2t3/(24ℏ m)).
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The physics of GPU programming
GPU cluster
Me pointing at the GPU Resonance cluster at SEAS Harvard with 32x448=14336 processing cores. Just imagine how tightly integrated this setup is compared to 3584 quad-core computers. Picture courtesy of Academic Computing, SEAS Harvard.
From discussions I learn that while many physicists have heard of Graphics Processing Units as fast computers, resistance to use them is widespread. One of the reasons is that physics has been relying on computers for a long time and tons of old, well trusted codes are lying around which are not easily ported to the GPU. Interestingly, the adoption of GPUs happens much faster in biology, medical imaging, and engineering.
I view GPU computing as a great opportunity to investigate new physics and my feeling is that todays methods optimized for serial processors may need to be replaced by a different set of standard methods which scale better with massively parallel processors. In 2008 I dived into GPU programming for a couple of reasons:
1. As a “model-builder” the GPU allows me to reconsider previous limitations and simplifications of models and use the GPU power to solve the extended models.
2. The turn-around time is incredibly fast. Compared to queues in conventional clusters where I wait for days or weeks, I get back results with 10000 CPU hours compute time the very same day. This in turn further facilitates the model-building process.
3. Some people complain about the strict synchronization requirements when running GPU codes. In my view this is an advantage, since essentially no messaging overhead exists.
4. If you want to develop high-performance algorithm, it is not good enough to convert library calls to GPU library calls. You might get speed-ups of about 2-4. However, if you invest the time and develop your own know-how you can expect much higher speed-ups of around 100 times or more, as seen in the applications I discussed in this blog before.
This summer I will lecture about GPU programming at several places and thus I plan to write a series of GPU related posts. I do have a complementary background in mathematical physics and special functions, which I find very useful in relation with GPU programming since new physical models require a stringent mathematical foundation and numerical studies.
Catching and tracking light: following the excitations in the Fenna-Matthews-Olson complex
Peak oscillations in the FMO complex calculated using GPU-HEOM
The animation shows how peaks in the 2d echo-spectra are oscillation and changing for various delay times. For a full explanation, see Modelling of Oscillations in Two-Dimensional Echo-Spectra of the Fenna-Matthews-Olson Complex by B.Hein, C. Kreisbeck, T. Kramer, M. Rodríguez, New J. of Phys., 14, 023018 (2012), open access.
Efficient and fast transport of electric current is a basic requirement for the functioning of nanodevices and biological systems. A neat example is the energy-transport of a light-induced excitation in the Fenna-Matthews-Olson complex of green sulfur bacteria. This process has been elucidated by pump-probe spectroscopy. The resulting spectra contain an enormous amount of information about the couplings of the different pigments and the pathways taken by the excitation. The basic guide to a 2d echo-spectrum is as follows:
You can find peaks of high intensity along the diagonal line which are roughly representing a more common absorption spectrum. If you delay the pump and probe pulses by several picoseconds, you will find a new set of peaks at a horizontal axis which indicates that energy of the excitation gets redistributed and the system relaxes and transfers part of the energy to vibrational motion. This process is nicely visible in the spectra recorded by Brixner et al.
A lot of excitement and activity on photosynthetic complexes was triggered by experiments of Engel et al showing that besides the relaxation process also periodic oscillations are visible in the oscillations for more than a picosecond.
What is causing the oscillations in the peak amplitudes of 2d echo-spectra in the Fenna-Matthews Olson complex?
A purely classical transport picture should not show such oscillations and the excitation instead hops around the complex without interference. Could the observed oscillations point to a different transport mechanism, possibly related to the quantum-mechanical simultaneous superposition of several transport paths?
The initial answer from the theoretical side was no, since within simplified models the thermalization occurs fast and without oscillations. It turned out that the simple calculations are a bit too simplistic to describe the system accurately and exact solutions are required. But exact solutions (even for simple models) are difficult to obtain. Known exact methods such as DMRG work only reliable at very low temperatures (-273 C), which are not directly applicable to biological systems. Other schemes use the famous path integrals but are too slow to calculate the pump-probe signals.
Our contribution to the field is to provide an exact computation of the 2d echo-spectra at the relevant temperatures and to see the difference to the simpler models in order to quantify how much coherence is preserved. From the method-development the computational challenge is to speed-up the calculations several hundred times in order to get results within days of computational run-time. We did achieve this by developing a method which we call GPU-hierarchical equations of motion (GPU-HEOM). The hierarchical equations of motions are a nice scheme to propagate a density matrix under consideration of non-Markovian effects and strong couplings to the environment. The HEOM scheme was developed by Kubo, Tanimura, and Ishizaki (Prof. Tanimura has posted some material on HEOM here).
However, the original computational method suffers from the same problems as path-integral calculations and is rather slow (though the HEOM method can be made faster and applied to electronic systems by using smart filtering as done by Prof. YiJing Yan). The GPU part in GPU-HEOM stands for Graphics Processing Units. Using our GPU adoption of the hierarchical equations (see details in Kreisbeck et al[JCTC, 7, 2166 (2011)] ) allowed us to cut down computational times dramatically and made it possible to perform a systematic study of the oscillations and the influence of temperature and disorder in our recent article Hein et al [New J. of Phys., 14, 023018 (2012), open access] .
The Nobel Prize 2011 in Chemistry: press releases, false balance, and lack of research in scientific writing
To get this clear from the beginning: with this posting I am not questioning the great achievement of Prof. Dan Shechtman, who discovered what is now known as quasicrystal in the lab. Shechtman clearly deserves the prize for such an important experiment demonstrating that five-fold symmetry exists in real materials.
My concern is the poor quality of research and reporting on the subject of quasicrystals starting with the press release of the Swedish Academy of Science and lessons to be learned about trusting these press releases and the reporting in scientific magazines. To provide some background: with the announcement of the Nobel prize a press release is put online by the Swedish academy which not only announces the prize winner, but also contains two PDFs with background information: one for the “popular press” and another one with for people with a more “scientific background”. Even more dangerously, the Swedish Academy has started a multimedia endeavor of pushing its views around the world in youtube channels and numerous multimedia interviews with its own members (what about asking an external expert for an interview?).
Before the internet age journalists got the names of the prize winners, but did not have immediately access to a “ready to print” explanation of the subject at hand. I remember that local journalists would call at the universities and ask a professor who is familiar with the topic for advice or get at least the phone number of somebody familiar with it. Not any more. This year showed that the background information prepared in advance by the committee is taken over by the media outlets basically unchanged. So far it looks as business as usual. But what if the story as told by the press release is not correct? Does anybody still have time and resources for some basic fact checking, for example by calling people familiar with the topic, or by consulting the archives of their newspaper/magazine to dig out what was written when the discovery was made many years ago? Should we rely on the professor who writes the press releases and trust that this person adheres to scientific and ethic standards of writing?
For me, the unfiltered and unchecked usage of press releases by the media and even by scientific magazines shows a decay in the quality of scientific reporting. It also generates a uniformity and self-referencing universe, which enters as “sources” in online encyclopedias and in the end becomes a “self-generated” truth. However it is not that difficult to break this circle, for example by
1. digging out review articles on the topic and looking up encyclopedias for the topic of quasicrystals, see for example: Pentagonal and Icosahedral Order in Rapidly Cooled Metals by David R. Nelson and Bertrand I. Halperin, Science 19 July 1985:233-238, where the authors write: “Independent of these experimental developments, mathematicians and some physicists had been exploring the consequences of the discovery by Penrose in 1974 of some remarkable, aperiodic, two-dimensional tilings with fivefold symmetry (7). Several authors suggested that these unusual tesselations of space might have some relevance to real materials (8, 9). MacKay (8) optically Fourier-transformed a two-dimensional Penrose pattern and found a tenfold symmetric diffraction pattern not unlike that shown for Al-Mn in Fig. 2. Three-dimensional generalizations of the Penrose patterns, based on the icosahedron, have been proposed (8-10). The generalization that appears to be most closely related to the experiments on Al-Mn was discovered by Kramer and Neri (11) and, independently, by Levine and Steinhardt (12).
2. identifying from step 1 experts and asking for their opinion
3. checking the newspaper and magazine archives. Maybe there exists already a well researched article?
4. correcting mistakes. After all mistakes do happen. Also in “press releases” by the Nobel committee, but there is always the option to send out a correction or to amend the published materials. See for example the letter in Science by David R. Nelson
Icosahedral Crystals in Perspective, Science 13 July 1990:111 again on the history of quasicrystals:
“[…] The threedimensional generalization of the Penrose tiling most closely related to the experiments was discovered by Peter Kramer and R. Neri (3) independently of Steinhardt and Levine (4). The paper by Kramer and Neri was submitted for publication almost a year before the paper of Shechtman et al. These are not obscure references: […]
Since I am working in theoretical physics I find it important to point out that in contrast to the story invented by the Nobel committee actually the theoretical structure of quasicrystals was published and available in the relevant journal of crystallography at the time the experimental paper got published. This sequence of events is well documented as shown above and in other review articles and books.
I am just amazed how the press release of the Nobel committee creates an alternate universe with a false history of theoretical and experimental publication records. It does give false credits for the first theoretical work on three-dimensional quasicrystals and at least in my view does not adhere to scientific and ethic standards of scientific writing.
Prof. Sven Lidin, who is the author of the two press releases of the Swedish Academy has been contacted as early as October 7 about his inaccurate and unbalanced account of the history of quasicrystals. In my view, a huge responsibility rests on the originator of the “story” which was put in the wild by Prof. Lidin, and I believe he and the committee members are aware of their power since they use actively all available electronic media channels to push their complete “press package” out. Until today no corrections or updates have been distributed. Rather you can watch on youtube the (false) story getting repeated over and over again. In my view this example shows science reporting in its worst incarnation and undermines the credibility and integrity of science.
Quasicrystals: anticipating the unexpected
The following guest entry is contributed by Peter Kramer
Dan Shechtman received the Nobel prize in Chemistry 2011 for the experimental discovery of quasicrystals. Congratulations! The press release stresses the unexpected nature of the discovery and the struggles of Dan Shechtman to convince the fellow experimentalists. To this end I want to contribute a personal perspective:
From the viewpoint of theoretical physics the existence of icosahedral quasicrystals as later discovered by Shechtman was not quite so unexpected. Beginning in 1981 with Acta Cryst A 38 (1982), pp. 257-264 and continued with Roberto Neri in Acta Cryst A 40 (1984), pp. 580-587 we worked out and published the building plan for icosahedral quasicrystals. Looking back, it is a strange and lucky coincidence that unknown to me during the same time Dan Shechtman and coworkers discovered icosahedral quasicrystals in their seminal experiments and brought the theoretical concept of three-dimensional non-periodic space-fillings to live.
More about the fascinating history of quasicrystals can be found in a short review: gateways towards quasicrystals and on my homepage.
Time to find eigenvalues without diagonalization
Solving the stationary Schrödinger (H-E)Ψ=0 equation can in principle be reduced to solving a matrix equation. This eigenvalue problem requires to calculate matrix elements of the Hamiltonian with respect to a set of basis functions and to diagonalize the resulting matrix. In practice this time consuming diagonalization step is replaced by a recursive method, which yields the eigenfunctions for a specific eigenvalue.
A very different approach is followed by wavepacket methods. It is possible to propagate a wavepacket without determining the eigenfunctions beforehand. For a given Hamiltonian, we solve the time-dependent Schrödinger equation (i ∂t-H) Ψ=0 for an almost arbitrary initial state Ψ(t=0) (initial value problem).
The reformulation of the determination of eigenstates as an initial value problem has a couple of computational advantages:
• results can be obtained for the whole range of energies represented by the wavepacket, whereas a recursive scheme yields only one eigenenergy
• the wavepacket motion yields direct insight into the pathways and allows us to develop an intuitive understanding of the transport choreography of a quantum system
• solving the time-dependent Schrödinger equation can be efficiently implemented using Graphics Processing Units (GPU), resulting in a large (> 20 fold) speedup compared to CPU code
Aharnov-Bohm Ring conductance oscillations
The Zebra stripe pattern along the horizontal axis shows Aharonov-Bohm oscillations in the conductance of a half-circular nanodevice due to the changing magnetic flux. The vertical axis denotes the Fermi energy, which can be tuned experimentally. For details see our paper in Physical Review B.
The determination of transmissions requires now to calculate the Fourier transform of correlation functions <Ψ(t=0)|Ψ(t)>. This method has been pioneered by Prof. Eric J. Heller, Harvard University, and I have written an introductory article for the Latin American School of Physics 2010 (arxiv version).
Recently, Christoph Kreisbeck has done a detailed calculations on the gate-voltage dependency of the conductance in Aharonov-Bohm nanodevices, taking full adventage of the simultaneous probing of a range of Fermi energies with one single wavepacket. A very clean experimental realization of the device was achieved by Sven Buchholz, Prof. Saskia Fischer, and Prof. Ulrich Kunze (RU Bochum), based on a semiconductor material grown by Dr. Dirk Reuter and Prof. Anreas Wieck (RU Bochum). The details, including a comparison of experimental and theoretical results shown in the left figure, are published in Physical Review B (arxiv version).
Cosmic topology from the Möbius strip
Fig 1. The Möbius twist.
The following article is contributed by Peter Kramer.
Einstein’s fundamental theory of Newton’s gravitation relates the interaction of masses to the curvature of space. Modern cosmology from the big bang to black holes results from Einstein’s field equations for this relation. These differential equations by themselves do not yet settle the large-scale structure and connection of the cosmos. Theoretical physicists in recent years tried to infer information on the large-scale cosmology from Cosmic microwave background radiation (CMBR), observed by satellite observations. In the frame of large-scale cosmology, the usual objects of astronomy from solar systems to galaxy clusters are smoothed out, and conditions imprinted in the early stage of the universe dominate.
Fig 2: The planar Möbius crystal cm
In mathematical language one speaks of cosmic topology. Topology is often considered to be esoteric. Here we present topology from the familiar experience with the twisted Möbius strip. This strip on one hand can be seen as a rectangular crystallographic lattice cell whose copies tile the plane, see Fig. 2. The Möbius strip is represented as a rectangular cell, located between the two vertical arrows, of a planar crystal. A horizontal dashed line through the center indicates a glide-reflection line. A glide reflection is a translation along the dashed line by the horizontal length of the cell, followed by a reflection in this line. The crystallographic symbol for this planar crystal is cm. In three-dimensional space the planar Möbius crystal (top panel of Fig. 1) is twisted (middle panel of Fig. 1). The twist is a translation along the dashed line, combined with a rotation by 180 degrees around that line. A final bending (bottom panel of Fig. 1) of the dashed line and a smooth gluing of the arrowed edges yields the familiar Möbius strip.
Fig 3: Cubic twist N3.
Given this Möbius template in two dimension, we pass to manifolds of dimension three. We present in Fig. 3 a new cubic manifold named N3. Three cubes are twisted from an initial one. A twist here is a translation along one of the three perpendicular directions, combined with a right-hand rotation by 90 degrees around this direction. To follow the rotations, note the color on the faces. The three neighbor cubes can be glued to the initial one. If the cubes are replaced by their spherical counterparts on the three-sphere, the three new cubes can pairwise be glued with one another, with face gluings indicated by heavy lines. The complete tiling of the three-sphere comprises 8 cubes and is called the 8-cell. The gluings shown here generate the so-called fundamental group of a single spherical cube on the three-sphere with symbol N3. This spherical cube is a candidate for the cosmic topology inferred from the cosmic microwave background radiation. A second cubic twist with a different gluing and fundamental group is shown in Fig. 4. Here, the three twists combine translations along the three directions with different rotations.
The key idea in cosmic topology is to pass from a topological manifold to its eigen- or normal modes. For the Möbius strip, these eigenmodes are seen best in the planar crystal representation of Fig. 2. The eigenmodes can be taken as sine or cosine waves of wave length \lambda which repeat their values from edge to edge of the cell. It is clear that the horizontal wavelength \lambda of these modes has as upper bound the length L of the rectangle. The full Euclidean plane allows for infinite wavelength, and so the eigenmodes of the Möbius strip obey a selection rule that characterizes the topology. Moreover the eigenmodes of the Möbius strip must respect its twisted connection.
Fig 4: Cubic twist N2.
Similarly, the eigenmodes of the spherical cubes in Fig. 3 must repeat themselves when going from cube to neighbor cube. It is intuitively clear that the cubic eigenmodes must have a wavelength smaller than the edge length of the cubes. The wave length of the eigenmodes of the full three-sphere are bounded by the equator length of the three-sphere. Seen on a single cube, the different twists and gluings of the manifolds N2 and N3 shown in Figs. 3 and 4 form different boundary value problems for the cubic eigenmodes.
Besides of these spherical cubic manifolds, there are several other competing polyhedral topologies with multiple connection or homotopy. Among them are the famous Platonic polyhedra. Each of them gives rise to a Platonic tesselation of the three-sphere. Everitt has analyzed all their possible gluings in his article Three-manifolds from platonic solids in Topology and its applications, vol 138 (2004), pp. 253-263. In my contribution Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules, Class. Quant. Grav., vol. 27 (2010), 095013 (freely available on the arxiv) I display them and present a full analysis of their corresponding eigenmodes and selection rules.
Since terrestrial observations measure the incoming radiation in terms of its spherical multipoles as functions of their incident direction, the eigenmodes must be transformed to a multipole expansion as done in my work. New and finer data on the CMB radiation are expected from the Planck spacecraft launched in 2009. These data, in conjunction with the theoretical models, will promote our understanding of cosmic space and possible twists in its topology.
Hot spot: the quantum Hall effect in graphene
Hall potential in a graphene device due to interactions and equipotential boundary conditions at the contacts.
An interesting and unfinished chapter of condensed mater theory concerns the quantum Hall effect. Especially the integer quantum Hall effect (IQHE) is actually not very well understood. The fancy cousin of the IQHE is the fractional quantum Hall effect (FQHE). The FQHE is easier to handle since there is agreement about the Hamiltonian which is to be solved (although the solutions are difficult to obtain): the quantum version of the very Hamiltonian used for the classical Hall effect, namely the one for interacting electrons in a magnetic field. The Hamiltonian is still lacking the specification of the boundary conditions, which can completely alter the results for open and current carrying systems (as in the classical Hall effect) compared to interacting electrons in a box.
Surprisingly no agreement about the Hamiltonian underlying the IQHE exists. It was once hoped that it is possible to completely neglect interactions and still to obtain a theoretical model describing the experiments. But if we throw out the interactions, we throw out the Hall effect itself. Thus we have to come up with the correct self-consistent solution of a mean field potential which incorporates the interactions and the Hall effect.
Is it possible to understand the integer quantum Hall effect without including interactions – and if yes, how does the effectively non-interacting Hamiltonian look like?
Starting from a microscopic theory we have constructed the self-consistent solution of the Hall potential in our previous post for the classical Hall effect. Two indispensable factors caused the emergence of the Hall potential:
1. repulsive electronic interactions and
2. equipotential boundary conditions at the contacts.
The Hall potential which emerges from our simulations has been directly imaged in GaAs Hall-devices under conditions of a quantized conductance by electro-optical methods and by scanning probe microscopy using a single electron transistor. Imaging requires relatively high currents in order to resolve the Hall potential clearly.
In graphene the dielectric constant is 12 times smaller than in GaAs and thus the Coulomb repulsion between electrons are stronger (which should help to generate the Hall potential). The observation of the FQHE in two-terminal devices has led the authors of the FQHE measurments to conjecture that hot-spots are also present in graphene devices [Du, Skachko, Duerr, Luican Andrei Nature 462, 192-195 (2009)].
These observations are extremely important, since the widely used theoretical model of edge-state transport of effectively non-interacting electrons is not readily compatible with these findings. In the edge-state model conductance quantization relies on the counter-propagation of two currents along the device borders, whereas the shown potential supports only a unidirectional current from source to drain diagonally across the device.
Moreover the construction of asymptotic scattering states is not possible, since no transverse lead-eigenbasis exists at the contacts. Electrons moving strictly along one side of the device from one contact to the other one would locally increase the electron density within the contact and violate the metallic boundary condition (see our recent paper on the Self-consistent calculation of electric potentials in Hall devices [Phys. Rev. B, 81, 205306 (2010)]).
Are there models which support a unidirectional current and at the same time support a quantized conductance in units of the conductivity quantum?
We put forward the injection model of the quantum Hall effect, where we take the Hall potential as being the self-consistent mean-field solution of the interacting and current carrying device. On this potential we construct the local density of states (LDOS) next to the injection hot spot and calculate the resulting current flow. In our model, the conductivity of the sample is completely determined by the injection processes at the source contact where the high electric field of the hot spots leads to a fast transport of electrons into the device. The LDOS is broadened due to the presence of the electric Hall field during the injection and not due to disorder. Our model is described in detail in our paper Theory of the quantum Hall effect in finite graphene devices [Phys. Rev. B, 81, 081410(R) (2010), free arxiv version] and the LDOS in a conventional semiconductor in electric and magnetic fields is given in a previous paper on electron propagation in crossed magnetic and electric fields. The tricky part is to prove the correct quantization, since the absence of any translational symmetries in the Hall potential obliterates the use of “Gedankenexperimente” relying on periodic boundary conditions or fancy loop topologies.
In order to propel the theoretical models forward, we need more experimental images of the Hall potential in a device, especially in the vicinity of the contacts. Experiments with graphene devices, where the Hall potential sits close to the surface, could help to establish the potential distribution and to settle the question which Hamiltonian is applicable for the quantum Hall effects. Is there anybody out to take up this challenge?
Trilobites revived: fragile Rydberg molecules, Coulomb Green’s function, Lambert’s theorem
The trilobite state
The trilobite Rydberg molecule can be modeled by the Coulomb Green’s function, which represents the quantized version of Lambert’s orbit determination problem.
The recent experimental realization observation of giant Rydberg molecules by Bendkowsky, Butscher, Nipper, Shaffer, Löw, Pfau [theoretically studied by Greene and coworkers, see for example Phys. Rev. Lett. 85, 2458 (2000)] shows Coulombic forces at work at large atomic distances to form a fragile molecule. The simplest approach to Rydberg molecules employs the Fermi contact potential (also called zero range potential), where the Coulomb Green’s function plays a central role. The quantum mechanical expression for the Coulomb Green’s function was derived in position space by Hostler and in momentum space by Schwinger. The quantum mechanical expression does not provide immediate insights into the peculiar nodal structure shown on the left side and thus it is time again to look for a semiclassical interpretation, which requires to translate an astronomical theorem into the Schrödinger world, one of my favorite topics.
Johann Heinrich Lambert was a true “Universalgelehrter”, exchanging letters with Kant about philosophy, devising a new color pyramid, proving that π is an irrational number, and doing physics. His career did not proceed without difficulties since he had to educate himself after working hours in his father’s tailor shop. After a long journey Lambert ended up in Berlin at the academy (and Euler choose to “escape” to St. Petersburg).
Lambert followed Kepler’s footsteps and tackled one of the most challenging problems of the time: the determination of celestial orbits from observations. In 1761 Lambert did solve the problem of orbit determination from two positions measurements. Lambert’s Theorem is a cornerstone of astronavigation (see for example the determination of Sputnik’s orbit using radar range measurements and Lambert’s theorem). Orbit determination from angular information alone (without known distances) is another problem and requires more observations.
Lambert poses the following question [Insigniores orbitae cometarum proprietates (Augsburg, 1761), p. 120, Lemma XXV, Problema XL]: Data longitudine axis maioris & situ foci F nec non situ punctorum N, M, construere ellipsin [Given the length of the semi-major axis, the location of one focal point, the points N,M, construct the two possible elliptical orbits connecting both points.]
Lambert's construction of two ellipses.
Lambert’s construction to find all possible trajectories from N to M and to map them to a ficticious 1D motion from n to m.
Lambert finds the two elliptic orbits [Fig. XXI] with an ingenious construction: he maps the rather complicated two-dimensional problem to the fictitious motion along a degenerate linear ellipse. Some physicists may know how to relate the three-dimensional Kepler problem to a four-dimensional oscillator via the Kustaanheimo–Stiefel transformation [see for example The harmonic oscillator in modern physics by Moshinsky and Smirnov]. But Lambert’s quite different procedure has its advantages for constructing the semiclassical Coulomb Green’s function, as we will see in a moment.
Shown are two ellipses with the same lengths of the semimajor axes 1/2 A1B1=1/2 A2 B2 and a common focus located at F. The centers of the two ellipses are denoted by C1 and C2. Lambert’s lemma allows to relate the motion from N to M on both ellipses to a common collinear motion on the degenerate linear ellipse Fb, where the points n and m are chosen such that the time of flight (TOF) along nm equals the TOF
along the elliptical arc NM on the first ellipse. On the second ellipse the TOF along the arc NB2M equals the TOF along nbm. The points n and m are found by marking the point G halfway between N and M. Then the major axis Fb=A1 B1=A2 B2 of the linear ellipse is drawn starting at F and running through G. On this line the point g is placed at the distance Fg=1/2(FN+FM). Finally n and m are given by the intersection points of a circle around g with radius GN=GM. This construction shows that the sum of the lengths of the shaded triangle α±=FN + FM ± NM is equal to α±=fn+ fm ± nm. The travel time depends only on the distances entering α±, and all calculations of the travel times etc. are given by one-dimensional integrations along the ficticious linear ellipse.
Lambert did find all the four possible trajectories from N to M which have the same energy (=semimajor axis a), regardless of their eccentricity (=angular momentum). The elimination of the angular momentum from Kepler’s equation is a tremendous achievement and the expression for the action is converted from Kepler’s form
• [Kepler] W(r,r‘;E)=√μ a Kc [ξ + ε sin(ξ) – ξ’ – ε sin(ξ’)], with eccentricity ε, eccentric anomaly ξ to
• [Lambert] W(r,r‘;E)=√μ a Kc[γ + sin(γ) – δ – sin(δ)], with
sin2(γ/2)=(r+r’+ |r‘-r|)/(4a) and sin2(δ/2)=(r+r’- |r‘-r|)/(4a).
The derivation is also discussed in detail in our paper [Kanellopoulos, Kleber, Kramer: Use of Lambert’s Theorem for the n-Dimensional Coulomb Problem Phys. Rev. A, 80, 012101 (2009), free arxiv version here]. The Coulomb problem of the hydrogen atom is equivalent to the gravitational Kepler problem, since both are subject to a 1/r potential. Some readers might have seen the equation for the action in Gutzwiller’s nice book Chaos in classical and quantum mechanics, eq. (1.14). It is worthwhile to point out that the series solution given by Lambert (and Gutzwiller) for the time of flight can be summed up easily and is denoted today by an inverse sine function (for hyperbolic motion a hyperbolic sine, a function later introduced by Riccati and Lambert). Again, the key-point is the introduction of the linear ficticious ellipse by Lambert which avoids integrating along elliptical arcs.
The surprising conclusion: the nodal pattern of the hydrogen atom can be viewed as resulting from a double-slit interference along two principal ellipses. The interference determines the eigenenergies and the eigenstates. Even the notorious difficult-to-calculate van Vleck-Pauli-Morette (VVPM) determinant can be expressed in short closed form with the help of Lambert’s theorem and our result works even in higher dimensions. The analytic form of the action and the VVPM determinant becomes essential for our continuation of the classical action into the forbidden region, which corresponds to a tunneling process, see the last part of our paper.
Lambert is definitely a very fascinating person. Wouldn’t it be nice to discuss with him about philosophy, life, and science?
Determining the affinities of electrons OR: seeing semiclassics in action
Electron trajectories for photodetachment in an electric field.
Negatively charged ions are an interesting species, having managed to bind one more electron than charge neutrality grants them [for a recent review see T. Andersen: Atomic negative ions: structure, dynamics and collisions, Physics Reports 394 p. 157-313 (2004)]. The precise determination of the usually small binding energy is best done by shining a laser beam of known wave length on the ions and detect at which laser frequency the electron gets detached from the atomic core.
For some ions (oxygen, sulfur, or hydrogen fluoride and many more) the most precise values given at NIST are obtained by Christophe Blondel and collaborators with an ingenious apparatus based on an idea by Demkov, Kondratovich, and Ostrovskii in Pis’ma Zh. Eksp. Teor. Fiz. 34, 425 (1981) [JETP Lett. 34, 403 (1981)]: the photodetachment microscope. Here, in addition to the laser energy, the energy of the released electron is measured via a virtual double-slit experiment. The ions are placed in an electric field, which makes the electronic wave running against the field direction turn back and interfere with the wave train emitted in the field direction. The electric-field induced double-slit leads to the build up of a circular interference pattern of millimeter size (!) on the detector shown in the left figure (the animation was kindly provided by C. Blondel, W. Chaibi, C. Delsart, C. Drag, F. Goldfarb & S. Kröger, see their orginal paper The electron affinities of O, Si, and S revisited with the photodetachment microscope, Eur. Phys. J. D 33 (2005) 335-342).
Observed time-dependent emergence of the interference pattern in an electric field. Video shown with kind permission of C. Blondel et al. (see text for full credit)
I view this experiment as one of the best illustrations of how quantum and classical mechanics are related via the classical actions along trajectories. The two possible parabolic trajectories underlying the quantum mechanical interference pattern were described by Galileo Galilei in his Discourses & Mathematical Demonstrations Concerning Two New Sciences Pertaining to Mechanics & Local Motions in proposition 8: Le ampiezze de i tiri cacciati con l’istesso impeto, e per angoli egualmente mancanti, o eccedenti l’angolo semiretto, sono eguali. Ironically the “old-fashioned” parabolic motion was removed from the latest Gymnasium curriculum in Baden-Württemberg to make space for modern quantum physics.
At the low energies of the electrons, their paths are easily deflected by the magnetic field of the Earth and thus require either excellent shielding of the field or an active compensation, which was achieved recently by
Chaibi, Peláez, Blondel, Drag, and Delsart in Eur. Phys. J. D 58, 29-37 (2010). The new paper demonstrates nicely the focusing effect of the combined electric an magnetic fields, which Christian Bracher, John Delos, Manfred Kleber, and I have analyzed in detail and where one encounters some of the seven elementary catastrophies since the magnetic field allows one to select the number of interfering paths.
We have predicted similar fringes for the case of matter waves in the gravitational field around us originating from trapped Bose-Einstein condensates (BEC), but we are not aware of an experimental observation of similar clarity as in the case of the photodetachment microscope.
Mathematically, the very same Green’s function describes both phenomena, photodetachment and atomlasers. For me this universality demonstrates nicely how mathematical physics allows us to understand phenomena within a language suitable for so many applications.
Interactions: from galaxies to the nanoscale
Microscopic model of a Hall bar
(a) Device model
(b) phenomenological potential
(c) GPU result
For a while we have explored the usage of General Purpose Graphics Processing Units (GPGPU) for electronic transport calculations in nanodevices, where we want to include all electron-electron and electron-donor interactions. The GPU allows us to drastically (250 fold !!!) boost the performance of N-body codes and we manage to propagate 10,000 particles over several million time-steps within days. While GPU methods are now rather popular within the astrophysics crowd, we haven’t seen many GPU applications for electronic transport in a nanodevice. Besides the change from astronomical units to atomic ones, gravitational forces are always attractive, whereas electrons are affected by electron-donor charges (attractive) and electron-electron repulsion. Furthermore we have a magnetic field present, leading to deflections. Last, the space where electrons can spread out is limited by the device borders. In total the force on the kth electron is given by \vec{F}_{k}=-\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1}}^{N_{\rm donor}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1\\l\ne k}}^{N_{\rm elec}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+e \dot{\vec{r}}_k\times\vec{B}
Our recent paper in Physical Review B (also freely available on the arxiv) gives the first microscopic description of the classical Hall effect, where interactions are everything: without interactions no Hall field and no drift transport. The role and importance of the interactions is surprisingly sparsely mentioned in the literature, probably due to a lack of computational means to move beyond phenomenological models. A notable exception is the very first paper on the Hall effect by Edwin Hall, where he writes “the phenomena observed indicate that two currents, parallel and in the same direction, tend to repel each other”. Note that this repulsion works throughout the device and therefore electrons do not pile up at the upper edge, but rather a complete redistribution of the electronic density takes place, yielding the potential shown in the figure.
Another important part of our simulation of the classical Hall effect are the electron sources and sinks, the contacts at the left and right ends of the device. We have developed a feed-in and removal model of the contacts, which keeps the contact on the same (externally enforced) potential during the course of the simulation.
Mind-boggling is the fact that the very same “classical Hall potential” has also been observed in conjunction with a plateau of the integer quantum Hall effect (IQHE) [Knott et al 1995 Semicond. Sci. Technol. 10 117 (1995)]. Despite these observations, many theoretical models of the integer quantum Hall effect do not consider the interactions between the electrons. In our classical model, the Hall potential for non-interacting electrons differs dramatically from the solution shown above and transport proceeds then (and only then) along the lower and upper edges. However the edge current solution is not compatible with the contact potential model described above where an external reservoir enforces equipotentials within each contact. |
1cfcd1820d3f15c4 | Sofja Kovalevskaja Award 2006 - Award Winners
Jens Bredenbeck
Molecule dynamics - mainspring of chemistry and biology
Elementary vital functions, chemical reactions, the behaviour of substances in our environment - the driving force behind all of these phenomena is the movement of molecules. Molecules interact and change their structure, sometimes slowly, sometimes at incredible speed. Jens Bredenbeck is developing new measuring techniques which can keep up with the molecules' pace. Multidimensional infrared spectroscopy is the name given to the method which measures molecular motion with ultrashort infrared laser pulses. This molecular motion detector should help us to understand what important processes on the molecular level look like in real time, such as how biomolecules fold themselves into the right structure and how they fulfil their vitally important tasks.
Host Institute: Frankfurt a.M. University, Institute of Biophysics
Host: Prof. Dr. Josef Wachtveitl
• Jens BredenbeckDr. Jens Bredenbeck,
born in Germany in 1975, studied chemistry at Darmstadt Technical University, Göttingen University and Zürich University, Switzerland, where he completed his doctorate at the Institute of Physical Chemistry in 2005. He is currently continuing his research at the FOM Institute for Atomic and Molecular Physics in Amsterdam, Netherlands.
Jure Demsar
Solid State Physics
New impulse for developing usable superconductors
Greenhouse gases, climate change and rising prices - the consequences of our use of energy are onerous. The idea that it might be possible to conduct electrical current without loss, to transform it and use it in engines - in a completely new way - sounds rather like a fairytale. Precisely this, however, i.e. the superconductivity of certain materials, has long since become reality. But only in the lab. Problems with the materials as well as the low temperatures required make it difficult to transform the new superconductors into usable electricity conductors. Thus, Jure Demsar is investigating novel so-called strongly-correlated high temperature superconductors. With the aid of ultrafast laser pulses he is observing in real time how electrons and other excitations behave and interact in this highly correlated superconducting state, and drawing inferences for optimising the material. The dream of loss-free conductors and other new electronic applications could move a step closer thanks to superconductivity research.
Host Institute: Konstanz University, Department of Physics, Modern Optics and Photonics
Host: Prof. Dr. Thomas Dekorsy
• Jure DemsarDr. Jure Demsar,
born in Slovenia in 1970, studied physics at Ljubljana University, where he took his doctorate in 2000. He continued his research in Ljubljana at the Jo¿ef Stefan Institute, in the Complex Matter Department, before receiving a two-year fellowship to research at the Los Alamos National Laboratory in the United States. Since then, Demsar has been working at the Jo¿ef Stefan Institute where he attained his professorial qualification (Habilitation) in 2005.
Felix Engel
Cell biology
Hearts that heal themselves
The human heart is a unique organ in the true sense of the word: adult heart cells are unable to divide. If they die off as a result of a heart attack, for instance, the tissue cannot rebuild itself. Felix Engel is searching for a way of encouraging adult heart cells to divide - a capacity inherent in youthful cells, but one which they lose shortly after birth. As Felix Engel and his colleagues discovered, responsibility for this lies with a protein. If it is blocked, the cell regains its capacity to divide. What has worked in experiments on animals is now supposed to be used to treat humans successfully and thus be developed as an alternative to the controversial treatment with stem cells.
Host Institute: Max Planck Institute for Heart and Lung Research, Bad Nauheim
Host: Prof. Dr. Thomas Braun
• Felix EngelDr. Felix Engel,
born in Germany in 1971, previously worked at the Children's Hospital/Harvard Medical School in Boston. Engel studied biotechnology at Berlin Technical University and completed his doctorate there in 2001 after working on his thesis externally at the Max Delbrück Centre for Molecular Medicine in Berlin.
Natalia Filatkina
Historical Philology
Vom Kerbholz zur Datenbank
What did the German expression, einen blauen Mantel umhängen, mean in the Middle Ages? What is a tally stick, what has it got to do with committing a criminal offence and how does it come about that this term is still used in the same context in modern German? These are the questions being answered by Natalia Filatkina who is investigating the history of such formulaic figures of speech in German. These so-called phraseologisms are, after all, a salient feature of all languages and essential for understanding them. What are the social, historical and cultural phenomena underlying these ancient phraseologisms? What conclusions can be drawn for modern language? So far, there have only been fragmentary investigations in this field. In her pioneering work, which is combining historical philology with the international technologies of markup languages, Natalia Filatkina is preparing an electronic body of texts from the 8th to the 17th centuries and interpreting them according to modern linguistic criteria. In this way, a data base is being created that will bring a part of cultural history nearer not only to an interdisciplinary circle of experts but also to a broad non-academic public and will generate new knowledge for the present day.
Host Institute: Trier University, Department of German, Older German Philology
Host: Prof. Dr. Claudine Moulin
• Natalia FilatkinaDr. Natalia Filatkina,
born in the Russian Federation in 1975, studied at the Moscow State Linguistic University, the Humboldt University Berlin on a DAAD scholarship, the University of Luxembourg, and Bamberg University where she took her doctorate in 2003. Her dissertation on the Luxembourg language was awarded the Prix d'encouragement for young researchers by the University of Luxembourg. She is working in the field of Older German Philology in the Department of German at Trier University.
Olga Holtz
Numerical Analysis
The way out of the data jungle
Whether you are looking at the handling and flying qualities of the new Airbus, developing a new drug to combat Aids or designing the ideal underground timetable for a city with more than a million inhabitants - at some time or other you will have to do some complicated computations. The amount of data computers have to cope with is extremely large, we are talking in terms of millions of equations and unknowns, and they only have a finite number of digits for representing a number. In order to solve this problem using reliable and fast algorithms you need to know as much about computers as mathematics. Olga Holtz is working at the interface of pure and applied mathematics. She is searching for methods which are both fast and reliable - which in this field of applied mathematics is usually a contradiction in terms. Her project, developing a method of matrix multiplication, should provide the solution to a multitude of computational calculations in science and engineering.
Host Institute: Berlin Technical University, Institute of Mathematics
Host: Prof. Dr. Volker Mehrmann
• Olga HoltzDr. Olga Holtz,
born in the Russian Federation in 1973, studied applied mathematics in her own country at the Chelyabinsk State Technical University and at the University of Wisconsin Madison in the United States, where she received a doctorate in mathematics in 2000 and subsequently continued her research in the Department of Computer Sciences. She was a Humboldt Research Fellow at Berlin Technical University before being appointed to the University of California, Berkeley, where she has been working ever since.
Reinhard Kienberger
Electron and Quantum Optics
Using x-ray flashes to visualise inconceivable speed
If you want to observe and understand how chemical bonds evolve, how electrons move in semi-conductors or how light is turned into chemical energy through photosynthesis, you have to be pretty fast, because in these chemical, atomic or biological processes we are dealing with fractions of a second, so-called attoseconds, which last no longer than a trillionth of a second. Reinhard Kienberger has significantly contributed to developing observation methods which use ultra fast, intensive x-ray flashes on the attosecond scale to visualise and, in future maybe even to be able to control what has so far been unobservable. Novel lasers based on ultraviolet light or x-rays as well as improved radiation therapies in medicine are just a few of the possible future applications ensuing from the young discipline of attosecond research.
Host Institute: Max Planck Institute of Quantum Optics, Laboratory for Attosecond and High-Field Physics, Garching, near Munich
Host: Prof. Dr. Ferenc Krausz
• Reinhard Kienberger Dr. Reinhard Kienberger,
studied at Vienna Technical University, Austria, and completed his doctorate there with a dissertation on quantum mechanics in 2002. He subsequently became a fellow of the Austrian Academy of Sciences, researching at Stanford University's Stanford Linear Accelerator Center, Menlo Park in California. He is currently working at the Max Planck Institute of Quantum Optics in Garching.
Marga Cornelia Lensen
Macromolecular Chemistry
Turning to nature: made-to-measure hydrogels for medical systems
If the first thing you associate with a happy baby is a dry nappy, it probably does not occur to you that both the parents and the baby actually have the blessings of biomaterial research to thank for this satisfactory state of affairs. The reason for this is that nappies and other hygiene products for absorbing moisture contain the magic anti-moisture ingredients known as hydrogels. These are three-dimensional polymer networks which can store many times their own weight in water and release it again. Humans have copied this principle from nature where hydrogels proliferate, in plants for instance. But hydrogels have much greater potential than this, for example in bioresearch or medicine. They might release doses of drugs in the body or act as sensors. They might also be used as artificial muscles or to bond natural tissue with artificial implants. This would require gels with properties made-to-measure through utilising nanotechnology. To lay the foundations for this, Marga Cornelia Lensen is investigating ways of changing the structure of the gels and how they interact with cells. Consequently, one of the things she is going to do is to use novel nanoimprint technology, which, so far, has largely been tested on hard material, to structure hydrogels and insert them as carriers for experiments on living cells.
Host Institute: RWTH Aachen, German Wool Research Institute
Host: Prof. Dr. Martin Möller
• Marga Cornelia LensenDr. Marga Cornelia Lensen,
born in the Netherlands in 1977, studied chemistry at Wageningen University and at Radboud University Nijmegen, where she took her doctorate in 2005. As a Humboldt Research Fellow she has been working at her host institute at RWTH Aachen, where she will continue her research as a Kovalevskaja Award Winner, since October 2005.
Martin Lövden
Developmental Psychology
Tracking down the secret of life-long learning
In our aging societies in Europe, the idea of life-long learning has gained a special relevance. But although the learning ability of young brains is considerable and has been well researched, there are not many studies on the reasons for the deterioration of learning ability in old age and how to deal with it. Martin Lövden is investigating the neurochemical, neuroanatomical and neurofunctional conditions for successful learning in old age and the consequences for everyday life. To this end, he uses neuroimaging methods, such as functional resonance imaging and resonance spectroscopy, by which he can observe the brains of old and young test subjects during memory training in order to track down the neurological secret of successful learning and its limitations in old age.
Host Institute: Max Planck Institute for Human Development, Research Area Lifespan Psychology, Berlin
Host: Prof. Dr. Ulman Lindenberger
• Martin LövdenDr. Martin Lövden,
born in Sweden in 1972, studied psychology at Salzburg University in Austria and at the universities of Lund and Stockholm in Sweden as well as neuroscience at the Karolinska Institute in Stockholm. He was awarded his doctorate at Stockholm University in 2002. He continued his research at the Saarland University in Saarbrücken and is currently working at the Max Planck Institute for Human Development in Berlin.
Thomas Misgeld
Nerve fibres: the brain's fast wire
In the nervous system information is transported in the form of electrical impulses. To this end, every nerve cell has an appendage, the function of which is similar to that of a telephone cable - the nerve fibres, also called axons. Axons run through the brain and the spinal cord to the switch points at the nerve roots and have a certain capacity for learning. They are able to adapt to new requirements. Not a lot is known about how this adaptation functions and how axons protect themselves against damage. So, Thomas Misgeld is investigating the axons of living mice using high resolution microscopy. He wants to discover how nerve fibres are nourished, adapted and maintain their efficiency in a healthy organism. This basic information could lead to the development of new therapies for diseases like multiple sclerosis or for spinal cord injuries.
Host Institute: Munich Technical University, Institute of Neurosciences
Host: Prof. Dr. Arthur Konnerth
• Thomas MisgeldDr. Thomas Misgeld,
born in Germany in 1971, studied medicine at Munich Technical University where he completed his doctorate in 1999. He continued his research in the department of clinical neuroimmunology at the Max Planck Institute of Neurobiology in Martinsried and at Washington University in St. Louis. His most recent position was at Harvard University in Cambridge. In 2005, he was granted the first ever Wyeth Multiple Sclerosis Junior Research Award and the Robert Feulgen Prize by the Society for Histochemistry.
Benjamin Schlein
Mathematical Physics
Seeking evidence in the quantum world
In the first half of the 20th century, when physicists observed that new properties were revealed by light interacting with material, classical physics reached its limits. It was the birth of quantum mechanics, the principles of which are part of common knowledge in physics nowadays, such as the fact that material particles exhibit waves, just like light. This is a principle used in modern electron microscopes. One of the main pillars of quantum mechanics is the Schrödinger equation which, to this day, has been very successful in predicting experiments. But when it comes to examining macroscopic systems - i.e. systems composed of multitudes of the tiniest particles - the amount of data is so enormous that even the most modern computers are not powerful enough to solve the Schrödinger equation. Benjamin Schlein is trying to develop mathematical methods which will make it possible to derive simpler equations to describe the dynamics of macroscopic systems. He wants to create a solid mathematical basis on which to assess and develop further applications in quantum mechanics.
Host Institute: Munich University, Institute of Mathematics
Host: Prof. Dr. Laszlo Erdös
• Benjamin SchleinDr. Benjamin Schlein,
born in Switzerland in 1975, studied theoretical physics at the Swiss Federal Institute of Technology (ETH) in Zürich and completed his doctorate there with a dissertation on mathematical physics in 2002. He subsequently continued his research in the United States, at the universities of New York, Stanford, Harvard and California in Davis.
Taolei Sun
Medical Biochemistry
Novel biocompatible materials for medical systems
"Surfaces are a creation of the devil", the famous physicist and Nobel Prize Winner, Wolfgang Pauli, once remarked when he realised how much more complex the surfaces of materials were than their massive substance. Many technical, indeed everyday applications depend on the properties of material surfaces and their interactions, which is especially important in the biomedical fields. Just consider the surfaces of artificial joints and other implants, or artificial access to the human bloodstream in intensive medicine or cancer treatment. All of them have to get along really well with the surfaces of human tissue or human cells. Taolei Sun is working on biocompatible, artificial implants and medical devices, combining modern nanotechnology with chemical surface modification. His aim is to use nanostructured polymeric surfaces with special wettability as a platform for the emergence of a new generation of biocompatible materials.
Host Institute: Universität Münster, Physikalisches Institut
Host: Prof. Dr. Harald Fuchs
• Taolei SunDr. Taolei Sun,
born in China in 1974, studied at Wuhan University and at the Technical Institute of Physics and Chemistry in the Chinese Academy of Sciences in Beijing, where he took his doctorate in 2002, and then continued his research. He subsequently worked at the National Center for Nanosciences and Technology of China in Beijing before becoming a Humboldt Research Fellow in the Institute of Physics at Münster University where he will now carry out research as a Kovalevskaja Award Winner.
Kristina Güroff
Kerstin Schweichhart
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f64198315b20efa3 | Inorganic Chemistry/Chemical Bonding/Molecular orbital theory
In chemistry, molecular orbital theory (MO theory) is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.[1] In this theory, each molecule has a set of molecular orbitals, in which it is assumed that the molecular orbital wave function ψf may be written as a simple weighted sum of the n constituent atomic orbitals χi, according to the following equation:[2]
\psi_j = \sum_{i=1}^{n} c_{ij} \chi_i
The cij coefficients may be determined numerically by substitution of this equation into the Schrödinger equation and application of the variational principle. This method is called the linear combination of atomic orbitals approximation and is used in computational chemistry.
Molecular orbital theory was developed, in the years after valence bond theory (1927) had been established, primarily through the efforts of Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones.[3] MO theory was originally called the Hund-Mulliken theory.[4] The word orbital was introduced by Mulliken in 1932.[4] By 1933, the molecular orbital theory had become accepted as a valid and useful theory.[5] According to German physicist and physical chemist Erich Hückel, the first quantitative use of molecular orbital theory was the 1929 paper of Lennard-Jones.[6] The first accurate calculation of a molecular orbital wavefunction was that made by Charles Coulson in 1938 on the hydrogen molecule.[7] By 1950, molecular orbitals were completely defined as eigenfunctions (wave functions) of the self-consistent field Hamiltonian and it was at this point that molecular orbital theory became fully rigorous and consistent.[8]
Molecular orbital (MO) theory uses a linear combination of atomic orbitals to form molecular orbitals which cover the whole molecule. These are often divided into bonding orbitals, anti-bonding orbitals, and non-bonding orbitals. A molecular orbital is merely a Schrödinger orbital which includes several, but often only two nuclei. If this orbital is of type in which the electron(s) in the orbital have a higher probability of being between nuclei than elsewhere, the orbital will be a bonding orbital, and will tend to hold the nuclei together. If the electrons tend to be present in a molecular orbital in which they spend more time elsewhere than between the nuclei, the orbital will function as an anti-bonding orbital and will actually weaken the bond. Electrons in non-bonding orbitals tend to be in deep orbitals (nearly atomic orbitals) associated almost entirely with one nucleus or the other, and thus they spend equal time between nuclei or not. These electrons neither contribute nor detract from bond strength.
Molecular orbitals are further divided according to the types of atomic orbitals combining to form a bond. These orbitals are results of electron-nucleus interactions that are caused by the fundamental force of electromagnetism. Chemical substances will form a bond if their orbitals become lower in energy when they interact with each other. Different chemical bonds are distinguished that differ by electron cloud shape and by energy levels.
MO theory provides a global, delocalized perspective on chemical bonding. For example, in the MO theory for hypervalent molecules, it is no longer necessary to invoke a major role for d-orbitals. In MO theory, any electron in a molecule may be found anywhere in the molecule, since quantum conditions allow electrons to travel under the influence of an arbitrarily large number of nuclei, so long as permitted by certain quantum rules. Although in MO theory some molecular orbitals may hold electrons which are more localized between specific pairs of molecular atoms, other orbitals may hold electrons which are spread more uniformly over the molecule. Thus, overall, bonding (and electrons) are far more delocalized (spread out) in MO theory, than is implied in VB theory. This makes MO theory more useful for the description of extended systems.
An example is that in the MO picture of benzene, composed of a hexagonal ring of 6 carbon atoms. In this molecule, 24 of the 30 total valence bonding electrons are located in 12 σ (sigma) bonding orbitals which are mostly located between pairs of atoms (C-C or C-H), similar to the valence bond picture. However, in benzene the remaining 6 bonding electrons are located in 3 π (pi) molecular bonding orbitals that are delocalized around the ring. Two are in an MO which has equal contributions from all 6 atoms. The other two have a vertical nodes at right angles to each other. As in the VB theory, all of these 6 delocalized pi electrons reside in a larger space which exists above and below the ring plane. All carbon-carbon bonds in benzene are chemically equivalent. In MO theory this is a direct consequence of the fact that the 3 molecular pi orbitals form a combination which evenly spreads the extra 6 electrons over 6 carbon atoms.[9]
In molecules such as methane, the 8 valence electrons are in 4 MOs that are spread out over all 5 atoms. However, it is possible to transform this picture, without altering the total wavefunction and energy, to one with 8 electrons in 4 localized orbitals that are similar to the normal bonding picture of four two-electron covalent bonds. This is what has been done above for the σ (sigma) bonds of benzene, but it is not possible for the π (pi) orbitals. The delocalised picture is more appropriate for ionisation and spectroscopic properties. Upon ionization, a single electron is taken from the whole molecule. The resulting ion does not have one bond different from the other three Similarly for electronic excitations, the electron that is excited is found over the whole molecule and not in one bond.
As in benzene, in substances such as beta carotene, chlorophyll or heme, some electrons the π (pi) orbitals are spread out in molecular orbitals over long distances in a molecule, giving rise to light absorption in lower energies (visible colors), a fact which is observed. This and other spectroscopic data for molecules are better explained in MO theory, with an emphasis on electronic states associated with multicenter orbitals, including mixing of orbitals premised on principles of orbital symmetry matching. The same MO principles also more naturally explain some electrical phenomena, such as high electrical conductivity in the planar direction of the hexagonal atomic sheets that exist in graphite. In MO theory, "resonance" (a mixing and blending of VB bond states) is a natural consequence of symmetry. For example, in graphite, as in benzene, it is not necessary to invoke the sp2 hybridization and resonance of VB theory, in order to explain electrical conduction. Instead, MO theory simply recognizes that some electrons in the graphite atomic sheets are completely delocalized over arbitrary distances, and reside in very large molecular orbitals that cover an entire graphite sheet, and some electrons are thus as free to move and conduct electricity in the sheet plane, as if they resided in a metal.
1. Daintith, J. (2004). Oxford Dictionary of Chemistry. New York: Oxford University Press. ISBN 0-19-860918-3.
2. Licker, Mark, J. (2004). McGraw-Hill Concise Encyclopedia of Chemistry. New York: McGraw-Hill. ISBN 0-07-143953-6.
3. Coulson, Charles, A. (1952). Valence. Oxford at the Clarendon Press.
4. a b Spectroscopy, Molecular Orbitals, and Chemical Bonding - Robert Mulliken's 1966 Nobel Lecture
5. Lennard-Jones Paper of 1929 - Foundations of Molecular Orbital Theory.
6. Hückel, E. (1934). Trans. Faraday Soc. 30, 59.
7. Coulson, C.A. (1938). Proc. Camb. Phil. Soc. 34, 204.
8. Hall, G.G. Lennard-Jones, Sir John. (1950). Proc. Roy. Soc. A202, 155.
9. Introduction to Molecular Orbital Theory - Imperial College London
Last modified on 25 May 2011, at 08:37 |
4796dbe08f1afef6 | Mathematical Physics
1009 Submissions
[2] viXra:1009.0047 [pdf] replaced on 2013-05-03 15:35:33
Authors: Jose Javier Garcia Moreta
Comments: 20 Pages.
Category: Mathematical Physics
[1] viXra:1009.0007 [pdf] replaced on 2012-03-21 15:41:57
A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations
Authors: DT Froedge
Comments: 36 Pages. V032112 ongoing
The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are, and single particle special cases. The basic concept is that there is a broader picture, based on a more general equation that includes the entire system of particles. The first part will be to postulate an equation, and then, by then by defining an action field based on the endpoint action of the particles in the system, develop a solution which properly illustrates internal dynamics as well as particle interactions. The complete function has both real, and imaginary, as well as timelike and spacelike parts, each of which are separable into independent expressions that define particle properties. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an atomic system, particle masses are eigenvalues in an interacting universe of particles. The Dirac massive and massless equation and solution will be shown as factorable independent parts of the Systemfunction. A clear relation between the classical and quantum properties of particles is made, increasing the scope of QM.
Category: Mathematical Physics |
ee6178b32d3179a5 | Maximal intensity higher-order Akhmediev breathers of the nonlinear Schrödinger equation and their systematic generation
It is well known that Akhmediev breathers of the nonlinear cubic Schrödinger equation can be superposed nonlinearly via the Darboux transformation to yield breathers of higher order. Surprisingly, we find that the peak height of each Akhmediev breather only adds linearly to form the peak height of the final breather. Using this peak-height formula, we show that at any given periodicity, there exists a unique high-order breather of maximal intensity. Moreover, these high-order breathers form a continuous hierarchy, growing in intensity with increasing periodicity. For any such higher-order breather, a simple initial wave function can be extracted from the Darboux transformation to dynamically generate that breather from the nonlinear Schrödinger equation.
Physics Letters A 380 (2016) 3625–3629 |
094e8119700af60d | Data Piques
Mar 20, 2017
From Analytical to Numerical to Universal Solutions
I've been making my way through the recently released Deep Learning textbook (which is absolutely excellent), and I came upon the section on Universal Approximation Properties. The Universal Approximation Theorem (UAT) essentially proves that neural networks are capable of approximating any continuous function (subject to some constraints and with upper bounds on compute).
Meanwhile, I have been thinking about the modern successes of deep learning and how many computer vision researchers resisted the movement away from hand-defined features towards deep, uninterpretable neural networks. By no means is computer vision the first field to experience such existential angst. Coming from a physics background, I recall many areas that slowly moved away from expert knowledge towards less-understood, numerical methods. I wonder how physicists dealt with such Sartrean dilemmas?
I think deep learning might be different, though.
To illustrate my point, it is helpful to think of the various methods for solving scientific, mathematical problems as existing on a spectrum. On one side is a closed-form, analytical solution. We express a scientific model with mathematical equations and then solve that problem analytically. Should these solutions resemble reality, then we have simultaneously solved the problem and helped to confirm our scientific understanding. Pretty much any introductory physics problem, like the kinematic equations, falls on this side of the spectrum.
On the other side of the spectrum, we have a complete black box solution. We put our inputs into the box, and we get some outputs back out which have fit our function or solved our problem. We know nothing about what is going on inside.
What falls towards the middle of the spectrum? There are many areas of science and applied math which live here, many of which consist of models expressed as differential equations with no analytical solution. With the lack of analytical solution, one must resort to numerical methods for solving these problems. Examples here abound: Density Functional Theory, Finite Element Analysis, Reflection Seismology, etc...
What is intersting about deep learning is that it is now being used to tackle these middle-ground, numerical problems. Hell, there is a paper on the arxiv using deep learning to solve the Schrödinger equation!
At least with the original numerical methods for solving differential equations, one had to do a decent job of modeling (and presumably understanding) the system before using a computer to solve the problem. What remains to be seen is whether or not this will still be true with deep learning. If not, then it feels like we are entering true black box territory. What I wonder is if the analytical side of the spectrum will even be necessary? Is intuition and domain knowledge necessary for innovation and pushing the boundaries of problem solving? Or, can this Universal Approximator break away from the pack and go on crushing records indefinitely?
Ideally, I would like to believe that we need both. Insights from domain knowledge help drive new approaches (e.g. neural networks were originally inspired by the brain), and breakthrough, black box results help us to understand the domain (not as many examples of these, yet...).
In the meantime, while everybody grapples with this new technology and publishes marginal increases in MNIST accuracy, it seems like the old, differential equation-heavy fields are up for grabs for anybody who can suitably aim the deep learning hammer. |
50eaef6e4697aa4d | Quantum harmonic oscillator
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Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (A–B), and according to the Schrödinger equation of quantum mechanics (C–H). In A–B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C–H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C, D, E, F, but not G, H, are energy eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory.
One-dimensional harmonic oscillator[edit]
Hamiltonian and energy eigenstates[edit]
Corresponding probability densities.
The Hamiltonian of the particle is:
where m is the particle's mass, k is the force constant, is the angular frequency of the oscillator, is the position operator (given by x in the coordinate basis), and is the momentum operator (given by in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.
One may write the time-independent Schrödinger equation,
where E denotes a to-be-determined real number that will specify a time-independent energy level, or eigenvalue, and the solution |ψ denotes that level's energy eigenstate.
One may solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function x|ψ⟩ = ψ(x), using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,
The functions Hn are the physicists' Hermite polynomials,
The corresponding energy levels are
The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.
Ladder operator method[edit]
The "ladder operator" method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a,
Note these operators classically are exactly the generators of normalized rotation in the phase space of and , i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator.
These operators lead to the useful representation of and ,
The operator a is not Hermitian, since itself and its adjoint a are not equal. The energy eigenstates |n (also known as Fock states), when operated on by these ladder operators, give
And the Hamilton operator can be expressed as
so the eigenstate of N is also the eigenstate of energy.
The commutation property yields
and similarly,
This means that a acts on |n to produce, up to a multiplicative constant, |n–1⟩, and a acts on |n to produce |n+1⟩. For this reason, a is called a annihilation operator ("lowering operator"), and a a creation operator ("raising operator"). The two operators together are called ladder operators. In quantum field theory, a and a are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.
the smallest eigen-number is 0, and
such that
which matches the energy spectrum given in the preceding section.
Arbitrary eigenstates can be expressed in terms of |0⟩,
Analytical questions[edit]
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation . In the position representation, this is the first-order differential equation
whose solution is easily found to be the Gaussian[4]
Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates constructed by the ladder method form a complete orthonormal set of functions.[5]
Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined by ,
so that , and so on.
Natural length and energy scales[edit]
The result is that, if energy is measured in units of ħω and distance in units of ħ/(), then the Hamiltonian simplifies to
while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,
where Hn(x) are the Hermite polynomials.
To avoid confusion, these "natural units" will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.
For example, the fundamental solution (propagator) of Hi∂t, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel,[6][7]
where K(x,y;0) = δ(xy). The most general solution for a given initial configuration ψ(x,0) then is simply
Coherent states[edit]
Time evolution of the probability distribution (and phase, shown as color) of a coherent state with |α|=3.
The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty σx σp = 2, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.
The coherent states are indexed by αC and expressed in the |n basis as
Because and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state: . The position space wave functions are
Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter α instead: .
Highly excited states[edit]
Wavefunction (top) and probability density (bottom) for the n = 30 excited state of the quantum harmonic oscillator. Vertical dashed lines indicate the classical turning points, while the dotted line represents the classical probability density.
When n is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy En can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.
The frequency of oscillation at x is proportional to the momentum p(x) of a classical particle of energy En and position x. Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately
This is also given, asymptotically, by the integral
Phase space solutions[edit]
In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution.
The Wigner quasiprobability distribution for the energy eigenstate |n is, in the natural units described above,[citation needed]
where Ln are the Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map.
Meanwhile, the Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have
This claim can be verified using the Segal–Bargmann transform. Specifically, since the raising operator in the Segal–Bargmann representation is simply multiplication by and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply . At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann transform.
N-dimensional isotropic harmonic oscillator[edit]
The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, …. In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x1, …, xN. Corresponding to each position coordinate is a momentum; we label these p1, …, pN. The canonical commutation relations between these operators are
The Hamiltonian for this system is
This observation makes the solution straightforward. For a particular set of quantum numbers the energy eigenfunctions for the N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:
where is an element in the defining matrix representation of U(N).
The energy levels of the system are
The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define n = n1 + n2 + n3. All states with the same n will have the same energy. For a given n, we choose a particular n1. Then n2 + n3 = nn1. There are nn1 + 1 possible pairs {n2, n3}. n2 can take on the values 0 to nn1, and for each n2 the value of n3 is fixed. The degree of degeneracy therefore is:
Formula for general N and n [gn being the dimension of the symmetric irreducible n-th power representation of the unitary group U(N)]:
This arises due to the constraint of putting N quanta into a state ket where and , which are the same constraints as in integer partition.
Example: 3D isotropic harmonic oscillator[edit]
Schrödinger 3D spherical harmonic orbital solutions in 2D density plots; the Mathematica source code that used for generating the plots is at the top
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables; see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with a different spherically symmetric potential
where μ is the mass of the particle. Because m will be used below for the magnetic quantum number, mass is indicated by μ, instead of m, as earlier in this article.
The solution reads[8]
is a normalization constant; ;
The energy eigenvalue is
The energy is usually described by the single quantum number
Because k is a non-negative integer, for every even n we have = 0, 2, …, n − 2, n and for every odd n we have = 1, 3, …, n − 2, n . The magnetic quantum number m is an integer satisfying m, so for every n and there are 2 + 1 different quantum states, labeled by m . Thus, the degeneracy at level n is
where the sum starts from 0 or 1, according to whether n is even or odd. This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of SU(3),[9] the relevant degeneracy group.
Harmonic oscillators lattice: phonons[edit]
As in the previous section, we denote the positions of the masses by x1, x2, …, as measured from their equilibrium positions (i.e. xi = 0 if the particle i is at its equilibrium position). In two or more dimensions, the xi are vector quantities. The Hamiltonian for this system is
Superposition of three oscillating dipoles- illustrate the time propagation of the common wave function for different n,l,m
Another illustration of the time propagation of the common wave function for three different atoms emphasizes the effect of the angular momentum on the distribution behavior
From the general result
The Hamiltonian may be written in wave vector space as
The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the (N + 1)-th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
The harmonic oscillator eigenvalues or energy levels for the mode ωk are
So an exact amount of energy ħω, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In analogy to the photon case when the electromagnetic field is quantised, the quantum of vibrational energy is called a phonon.
In the continuum limit, a→0, N→∞, while Na is held fixed. The canonical coordinates Qk devolve to the decoupled momentum modes of a scalar field, , whilst the location index i (not the displacement dynamical variable) becomes the parameter x argument of the scalar field, .
Molecular vibrations[edit]
where is the reduced mass and and are the masses of the two atoms.[11]
• Modelling phonons, as discussed above.
• A charge with mass in a uniform magnetic field is an example of a one-dimensional quantum harmonic oscillator: Landau quantization.
See also[edit]
1. ^ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-805326-0.
2. ^ Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison–Wesley. ISBN 978-0-8053-8714-8.
3. ^ Rashid, Muneer A. (2006). "Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian" (PDF-Microsoft PowerPoint). M.A. Rashid – Center for Advanced Mathematics and Physics. National Center for Physics. Retrieved 19 October 2010.
4. ^ The normalization constant is , and satisfies the normalization condition .
5. ^ See Theorem 11.4 in Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158
7. ^ Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, 158–164. online
8. ^ Albert Messiah, Quantum Mechanics, 1967, North-Holland, Ch XII, § 15, p 456.online
9. ^ Fradkin, D. M. "Three-dimensional isotropic harmonic oscillator and SU3." American Journal of Physics 33 (3) (1965) 207–211.
10. ^ Mahan, GD (1981). Many particle physics. New York: Springer. ISBN 978-0306463389.
11. ^ "Quantum Harmonic Oscillator". Hyperphysics. Retrieved 24 September 2009.
External links[edit] |
27d349513a0034ac | Erwin Schrödinger Nobel Prize in Physics 1933
Quantum Leap in Physics
Erwin Schrödinger proved that electrons could have the properties of either waves or particles, but are neither the one nor the other – a discovery that revolutionized physics.
In the fall of 1921, Erwin Schrödinger was appointed to the chair for theoretical physics at the University of Zurich, a position that had been vacant since 1914. At that time no one imagined that six years later he would leave the University and the city hailed as a genius by luminary figures such as Albert Einstein and Max Planck, and celebrated as a star.
While at the University, Erwin Schrödinger revolutionized physics by creating a new atomic theory, a scientific breakthrough he achieved in the winter of 1925/26. In the summer semester of 1925, Erwin Schrödinger had read the doctoral thesis of a young Frenchman, Louis de Broglie, who proposed that matter – such as electrons – also possessed wave properties. This contradicted the prevailing opinion of leading physicists of the time, who assumed that electrons were particles.
Schrödinger focused intensively on Broglie’s proposition that all matter has wave properties. What were the properties of such waves of matter? Schrödinger spent Christmas and New Year 1925/26 studying the matter while on holiday in Arosa. This vacation was the beginning of his annus mirabilis, a phase, lasting some twelve months, of concentrated, creative work.
These efforts resulted in his first article, “Quantisierung als Eigenwertproblem. Erste Mitteilung” (Quantization as a problem of proper values, part one), which he sent to the Annalen der Physik on 26 January 1926. In this paper, he first formulated his famous wave equation, which has gone down in the annals of physics as the “Schrödinger equation.” The wave equation makes it possible to calculate the energy levels of electrons in an atom, thus solving one of the great problems in quantum physics.
After Schrödinger’s wave equation, nothing in the world of physics was the same again. The dispute as to whether quantum objects such as electrons, atoms or molecules were waves or particles was settled. In a surprising fashion, however: Schrödinger demonstrated that electrons could have the properties of either waves or particles, but are neither the one nor the other; their state can be calculated only with a degree of probability. For this discovery, Erwin Schrödinger was awarded the 1933 Nobel Prize in Physics. |
0b2b69fa6d22005e | The periodic table
Original post:
This post is, in essence, a continuation of my series on electron orbitals. I’ll just further tie up some loose ends and then – hopefully – have some time to show how we get the electron orbitals for other atoms than hydrogen. So we’ll sort of build up the periodic table. Sort of. 🙂
We should first review a bit. The illustration below copies the energy level diagram from Feynman’s Lecture on the hydrogen wave function. Note he uses √E for the energy scale because… Well… I’ve copied the En values for n = 1, 2, 3,… 7 next to it: the value for E(-13.6 eV) is four times the value of E(-3.4 eV).
exponential scale
How do we know those values? We discussed that before – long time back: we have the so-called gross structure of the hydrogen spectrum here. The table below gives the energy values for the first seven levels, and you can calculate an example for yourself: the difference between E2 (-3.4 eV) and E(-0.85 eV) is 2.55 eV, so that’s 4.08555×10−19 J, which corresponds to a frequency equal to = E/h = (4.08555×10−19 J)/(6.626×10−34 J·s) ≈ 0.6165872×1015 Hz. Now that frequency corresponds to a wavelength that’s equal to λ = c/= (299,792,458 m/s)/0.6165872×1015/s) ≈ 486×10−9 m. So that’s the 486 nano-meter line the so-called Balmer series, as shown in the illustration next to the table with the energy values.
So far, so good. An interesting point to note is that we only have one solution for = 1. To be precise, we have one spherical solution only: the 1s solution. Now, for n = 2, we have one 2s solution but also three 2solutions (remember the stands for principal lines). In the simplified model we’re using (we’re not discussing the fine or hyperfine structure here), these three solutions are referred to as ‘degenerate states’: they are different states with the same energy. Now, we know that any linear combination of the solutions for a differential equation must also be a solution. Therefore, any linear combination of the 2solutions will also be a stationary state of the same energy. In fact, a superposition of the 2s and one or more of the 2p states should also be a solution. There is an interesting app which visualizes how such superimposed states look like. I copy three illustrations below, but I recommend you google for stuff like this yourself: it’s really fascinating! You should, once again, pay attention to the symmetries planes and/or symmetry axes.
But we’ve written enough about the orbital of one electron now. What if there are two electrons, or three, or more. In other word, how does it work for heliumlithium, and so on? Feynman gives us a bit of an intuitive explanation here – nothing analytical, really. First, he notes Schrödinger’s equation for two electrons would look as follows:
two electronsSecond, the ψ(x) function in the ψ(x, t) = ei·(E/ħ)·t·ψ(x) function now becomes a function in six variables, which he – curiously enough – now no longer writes as ψ but as f:formulaThe rest of the text speaks for itself, although you might be disappointed by what he writes (the bold-face and/or italics are mine):
“The geometrical dependence is contained in f, which is a function of six variables—the simultaneous positions of the two electrons. No one has found an analytic solution, although solutions for the lowest energy states have been obtained by numerical methods. With 34, or 5 electrons it is hopeless to try to obtain exact solutions, and it is going too far to say that quantum mechanics has given a precise understanding of the periodic tableIt is possible, however, even with a sloppy approximation—and some fixing—to understand, at least qualitatively, many chemical properties which show up in the periodic table.
The chemical properties of atoms are determined primarily by their lowest energy states. We can use the following approximate theory to find these states and their energies. First, we neglect the electron spin, except that we adopt the exclusion principle and say that any particular electronic state can be occupied by only one electron. This means that any particular orbital configuration can have up to two electrons—one with spin up, the other with spin down.
Next we disregard the details of the interactions between the electrons in our first approximation, and say that each electron moves in a central field which is the combined field of the nucleus and all the other electrons. For neon, which has 10 electrons, we say that one electron sees an average potential due to the nucleus plus the other nine electrons. We imagine then that in the Schrödinger equation for each electron we put a V(r) which is a 1/r field modified by a spherically symmetric charge density coming from the other electrons.
In this model each electron acts like an independent particle. The angular dependence of its wave function will be just the same as the ones we had for the hydrogen atom. There will be s-states, p-states, and so on; and they will have the various possible m-values. Since V(r) no longer goes as 1/r, the radial part of the wave functions will be somewhat different, but it will be qualitatively the same, so we will have the same radial quantum numbers, n. The energies of the states will also be somewhat different.”
So that’s rather disappointing, isn’t it? We can only get some approximate – or qualitative – understanding of the periodic table from quantum mechanics – because the math is too complex: only numerical methods can give us those orbitals! Wow! Let me list some of the salient points in Feynman’s treatment of the matter:
• For helium (He), we have two electrons in the lowest state (i.e. the 1s state): one has its spin ‘up’ and the other is ‘down’. Because the shell is filled, the ionization energy (to remove one electron) has an even larger value than the ionization energy for hydrogen: 24.6 eV! That’s why there is “practically no tendency” for the electron to be attracted by some other atom: helium is chemically inert – which explains it being part of the group of noble or inert gases.
• For lithium (Li), two electrons will occupy the 1s orbital, and the third should go to an = 2 state. But which one? With = 0, or = 1? A 2s state or a 2p state? In hydrogen, these two = 2 states have the same energy, but in other atoms they don’t. Why not? That’s a complicated story, but the gist of the argument is as follows: a 2s state has some amplitude to be near the nucleus, while the 2p state does not. That means that a 2electron will feel some of the triple electric charge of the Li nucleus, and this extra attraction lowers the energy of the 2state relative to the 2state.
To make a long story short, the energy levels will be roughly as shown in the table below. For example, the energy that’s needed to remove the 2s electron of the lithium – i.e. the ionization energy of lithium – is only 5.4 eV because… Well… As you can see, it has a higher energy (less negative, that is) than the 1s state (−13.6 eV for hydrogen and, as mentioned above, −24.6 eV for helium). So lithium is chemically active – as opposed to helium. energy values more electrons
You should compare the table below with the table above. If you do, you’ll understand how electrons ‘fill up’ those electron shells. Note, for example, that the energy of the 4s state is slightly lower than the energy of the 3d state, so it fills up before the 3shell does. [I know the table is hard to read – just check out the original text if you want to see it better.]
periodic table
This, then, is what you learnt in high school and, of course, there are 94 naturally occurring elements – and another 24 heavier elements that have been produced in labs, so we’d need to go all the way to no. 118. Now, Feynman doesn’t do that, and so I won’t do that either. 🙂
Well… That’s it, folks. We’re done with Feynman. It’s time to move to a physics grad course now! Talk stuff like quantum field theory, for example. Or string theory. 🙂 Stay tuned!
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ce9c08418800645b | Is There a Quantum Trajectory?
Heisenberg’s uncertainty principle is a law of physics – it cannot be violated under any circumstances, no matter how much we may want it to yield or how hard we try to bend it. Heisenberg, as he developed his ideas after his lone epiphany like a monk on the isolated island of Helgoland off the north coast of Germany in 1925, became a bit of a zealot, like a religious convert, convinced that all we can say about reality is a measurement outcome. In his view, there was no independent existence of an electron other than what emerged from a measuring apparatus. Reality, to Heisenberg, was just a list of numbers in a spread sheet—matrix elements. He took this line of reasoning so far that he stated without exception that there could be no such thing as a trajectory in a quantum system. When the great battle commenced between Heisenberg’s matrix mechanics against Schrödinger’s wave mechanics, Heisenberg was relentless, denying any reality to Schrödinger’s wavefunction other than as a calculation tool. He was so strident that even Bohr, who was on Heisenberg’s side in the argument, advised Heisenberg to relent [1]. Eventually a compromise was struck, as Heisenberg’s uncertainty principle allowed Schrödinger’s wave functions to exist within limits—his uncertainty limits.
Disaster in the Poconos
Yet the idea of an actual trajectory of a quantum particle remained a type of heresy within the close quantum circles. Years later in 1948, when a young Richard Feynman took the stage at a conference in the Poconos, he almost sabotaged his career in front of Bohr and Dirac—two of the giants who had invented quantum mechanics—by having the audacity to talk about particle trajectories in spacetime diagrams.
Feynman was making his first presentation of a new approach to quantum mechanics that he had developed based on path integrals. The challenge was that his method relied on space-time graphs in which “unphysical” things were allowed to occur. In fact, unphysical things were required to occur, as part of the sum over many histories of his path integrals. For instance, a key element in the approach was allowing electrons to travel backwards in time as positrons, or a process in which the electron and positron annihilate into a single photon, and then the photon decays back into an electron-positron pair—a process that is not allowed by mass and energy conservation. But this is a possible history that must be added to Feynman’s sum.
It all looked like nonsense to the audience, and the talk quickly derailed. Dirac pestered him with questions that he tried to deflect, but Dirac persisted like a raven. A question was raised about the Pauli exclusion principle, about whether an orbital could have three electrons instead of the required two, and Feynman said that it could—all histories were possible and had to be summed over—an answer that dismayed the audience. Finally, as Feynman was drawing another of his space-time graphs showing electrons as lines, Bohr rose to his feet and asked derisively whether Feynman had forgotten Heisenberg’s uncertainty principle that made it impossible to even talk about an electron trajectory.
It was hopeless. The audience gave up and so did Feynman as the talk just fizzled out. It was a disaster. What had been meant to be Feynman’s crowning achievement and his entry to the highest levels of theoretical physics, had been a terrible embarrassment. He slunk home to Cornell where he sank into one of his depressions. At the close of the Pocono conference, Oppenheimer, the reigning king of physics, former head of the successful Manhattan Project and newly selected to head the prestigious Institute for Advanced Study at Princeton, had been thoroughly disappointed by Feynman.
But what Bohr and Dirac and Oppenheimer had failed to understand was that as long as the duration of unphysical processes was shorter than the energy differences involved, then it was literally obeying Heisenberg’s uncertainty principle. Furthermore, Feynman’s trajectories—what became his famous “Feynman Diagrams”—were meant to be merely cartoons—a shorthand way to keep track of lots of different contributions to a scattering process. The quantum processes certainly took place in space and time, conceptually like a trajectory, but only so far as time durations, and energy differences and locations and momentum changes were all within the bounds of the uncertainty principle. Feynman had invented a bold new tool for quantum field theory, able to supply deep results quickly. But no one at the Poconos could see it.
Fig. 1 The first Feynman diagram.
Coherent States
When Feynman had failed so miserably at the Pocono conference, he had taken the stage after Julian Schwinger, who had dazzled everyone with his perfectly scripted presentation of quantum field theory—the competing theory to Feynman’s. Schwinger emerged the clear winner of the contest. At that time, Roy Glauber (1925 – 2018) was a young physicist just taking his PhD from Schwinger at Harvard, and he later received a post-doc position at Princeton’s Institute for Advanced Study where he became part of a miniature revolution in quantum field theory that revolved around—not Schwinger’s difficult mathematics—but Feynman’s diagrammatic method. So Feynman won in the end. Glauber then went on to Caltech, where he filled in for Feynman’s lectures when Feynman was off in Brazil playing the bongos. Glauber eventually returned to Harvard where he was already thinking about the quantum aspects of photons in 1956 when news of the photon correlations in the Hanbury-Brown Twiss (HBT) experiment were published. Three years later, when the laser was invented, he began developing a theory of photon correlations in laser light that he suspected would be fundamentally different than in natural chaotic light.
Because of his background in quantum field theory, and especially quantum electrodynamics, it was fairly easy to couch the quantum optical properties of coherent light in terms of Dirac’s creation and annihilation operators of the electromagnetic field. Glauber developed a “coherent state” operator that was a minimum uncertainty state of the quantized electromagnetic field, related to the minimum-uncertainty wave functions derived initially by Schrödinger in the late 1920’s. The coherent state represents a laser operating well above the lasing threshold and behaved as “the most classical” wavepacket that can be constructed. Glauber was awarded the Nobel Prize in Physics in 2005 for his work on such “Glauber states” in quantum optics.
Fig. 2 Roy Glauber
Quantum Trajectories
Glauber’s coherent states are built up from the natural modes of a harmonic oscillator. Therefore, it should come as no surprise that these coherent-state wavefunctions in a harmonic potential behave just like classical particles with well-defined trajectories. The quadratic potential matches the quadratic argument of the the Gaussian wavepacket, and the pulses propagate within the potential without broadening, as in Fig. 3, showing a snapshot of two wavepackets propagating in a two-dimensional harmonic potential. This is a somewhat radical situation, because most wavepackets in most potentials (or even in free space) broaden as they propagate. The quadratic potential is a special case that is generally not representative of how quantum systems behave.
Fig. 3 Harmonic potential in 2D and two examples of pairs of pulses propagating without broadening. The wavepackets in the center are oscillating in line, and the wavepackets on the right are orbiting the center of the potential in opposite directions. (Movies of the quantum trajectories can be viewed at Physics Unbound.)
To illustrate this special status for the quadratic potential, the wavepackets can be launched in a potential with a quartic perturbation. The quartic potential is anharmonic—the frequency of oscillation depends on the amplitude of oscillation unlike for the harmonic oscillator, where amplitude and frequency are independent. The quartic potential is integrable, like the harmonic oscillator, and there is no avenue for chaos in the classical analog. Nonetheless, wavepackets broaden as they propagate in the quartic potential, eventually spread out into a ring in the configuration space, as in Fig. 4.
Fig. 4 Potential with a quartic corrections. The initial gaussian pulses spread into a “ring” orbiting the center of the potential.
A potential with integrability has as many conserved quantities to the motion as there are degrees of freedom. Because the quartic potential is integrable, the quantum wavefunction may spread, but it remains highly regular, as in the “ring” that eventually forms over time. However, integrable potentials are the exception rather than the rule. Most potentials lead to nonintegrable motion that opens the door to chaos.
A classic (and classical) potential that exhibits chaos in a two-dimensional configuration space is the famous Henon-Heiles potential. This has a four-dimensional phase space which admits classical chaos. The potential has a three-fold symmetry which is one reason it is non-integral, since a particle must “decide” which way to go when it approaches a saddle point. In the quantum regime, wavepackets face the same decision, leading to a breakup of the wavepacket on top of a general broadening. This allows the wavefunction eventually to distribute across the entire configuration space, as in Fig. 5.
Fig. 5 The Henon-Heiles two-dimensional potential supports Hamiltonian chaos in the classical regime. In the quantum regime, the wavefunction spreads to eventually fill the accessible configuration space (for constant energy).
Youtube Video
Movies of quantum trajectories can be viewed at my Youtube Channel, Physics Unbound. The answer to the question “Is there a quantum trajectory?” can be seen visually as the movies run—they do exist in a very clear sense under special conditions, especially coherent states in a harmonic oscillator. And the concept of a quantum trajectory also carries over from a classical trajectory in cases when the classical motion is integrable, even in cases when the wavefunction spreads over time. However, for classical systems that display chaotic motion, wavefunctions that begin as coherent states break up into chaotic wavefunctions that fill the accessible configuration space for a given energy. The character of quantum evolution of coherent states—the most classical of quantum wavefunctions—in these cases reflects the underlying character of chaotic motion in the classical analogs. This process can be seen directly watching the movies as a wavepacket approaches a saddle point in the potential and is split. Successive splits of the multiple wavepackets as they interact with the saddle points is what eventually distributes the full wavefunction into its chaotic form.
Therefore, the idea of a “quantum trajectory”, so thoroughly dismissed by Heisenberg, remains a phenomenological guide that can help give insight into the behavior of quantum systems—both integrable and chaotic.
As a side note, the laws of quantum physics obey time-reversal symmetry just as the classical equations do. In the third movie of “A Quantum Ballet“, wavefunctions in a double-well potential are tracked in time as they start from coherent states that break up into chaotic wavefunctions. It is like watching entropy in action as an ordered state devolves into a disordered state. But at the half-way point of the movie, the imaginary part of the wavefunction has its sign flipped, and the dynamics continue. But now the wavefunctions move from disorder into an ordered state, seemingly going against the second law of thermodynamics. Flipping the sign of the imaginary part of the wavefunction at just one instant in time plays the role of a time-reversal operation, and there is no violation of the second law.
[1] See Chapter 8 , On the Quantum Footpath, in Galileo Unbound, D. D. Nolte (Oxford University Press, 2018)
[2] J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)
Quantum Chaos and the Cheshire Cat
Alice’s disturbing adventures in Wonderland tumbled upon her like a string of accidents as she wandered a world of chaos. Rules were never what they seemed and shifted whenever they wanted. She even met a cat who grinned ear-to-ear and could disappear entirely, or almost entirely, leaving only its grin.
The vanishing Cheshire Cat reminds us of another famous cat—Arnold’s Cat—that introduced the ideas of stretching and folding of phase-space volumes in non-integrable Hamiltonian systems. But when Arnold’s Cat becomes a Quantum Cat, a central question remains: What happens to the chaotic behavior of the classical system … does it survive the transition to quantum mechanics? The answer is surprisingly like the grin of the Cheshire Cat—the cat vanishes, but the grin remains. In the quantum world of the Cheshire Cat, the grin of the classical cat remains even after the rest of the cat vanished.
The Cheshire Cat fades away, leaving only its grin, like a fine filament, as classical chaos fades into quantum, leaving behind a quantum scar.
The Quantum Mechanics of Classically Chaotic Systems
The simplest Hamiltonian systems are integrable—they have as many constants of the motion as degrees of freedom. This holds for quantum systems as well as for classical. There is also a strong correspondence between classical and quantum systems for the integrable cases—literally the Correspondence Principle—that states that quantum systems at high quantum number approach classical behavior. Even at low quantum numbers, classical resonances are mirrored by quantum eigenfrequencies that can show highly regular spectra.
But integrable systems are rare—surprisingly rare. Almost no real-world Hamiltonian system is integrable, because the real world warps the ideal. No spring can displace indefinitely, and no potential is perfectly quadratic. There are always real-world non-idealities that destroy one constant of the motion or another, opening the door to chaos.
When classical Hamiltonian systems become chaotic, they don’t do it suddenly. Almost all transitions to chaos in Hamiltonian systems are gradual. One of the best examples of this is the KAM theory that starts with invariant action integrals that generate invariant tori in phase space. As nonintegrable perturbations increase, the tori break up slowly into island chains of stability as chaos infiltrates the separatrixes—first as thin filaments of chaos surrounding the islands—then growing in width to take up more and more of phase space. Even when chaos is fully developed, small islands of stability can remain—the remnants of stable orbits of the unperturbed system.
When the classical becomes quantum, chaos softens. Quantum wave functions don’t like to be confined—they spread and they tunnel. The separatrix of classical chaos—that barrier between regions of phase space—cannot constrain the exponential tails of wave functions. And the origin of chaos itself—the homoclinic point of the separatrix—gets washed out. Then the regular orbits of the classical system reassert themselves, and they appear, like the vestige of the Cheshire Cat, as a grin.
The Quantum Circus
The empty stadium is a surprisingly rich dynamical system that has unexpected structure in both the classical and the quantum domain. Its importance in classical dynamics comes from the fact that its periodic orbits are unstable and its non-periodic orbits are ergodic (filling all available space if given long enough). The stadium itself is empty so that particles (classical or quantum) are free to propagate between reflections from the perfectly-reflecting walls of the stadium. The ergodicity comes from the fact that the stadium—like a classic Roman chariot-race stadium, also known as a circus—is not a circle, but has a straight stretch between two half circles. This simple modification takes the stable orbits of the circle into the unstable orbits of the stadium.
A single classical orbit in a stadium is shown in Fig 1. This is an ergodic orbit that is non-periodic and eventually would fill the entire stadium space. There are other orbits that are nearly periodic, such as one that bounces back and forth vertically between the linear portions, but even this orbit will eventually wander into the circular part of the stadium and then become ergodic. The big quantum-classical question is what happens to these classical orbits when the stadium is shrunk to the nanoscale?
Fig. 1 A classical trajectory in a stadium. It will eventually visit every point, a property known as ergodicity.
Simulating an evolving quantum wavefunction in free space is surprisingly simple. Given a beginning quantum wavefunction A(x,y,t0), the discrete update equation is
Perfect reflection from the boundaries of the stadium are incorporated through imposing a boundary condition that sends the wavefunction to zero. Simple!
A snap-shot of a wavefunction evolving in the stadium is shown in Fig. 2. To see a movie of the time evolution, see my YouTube episode.
Fig. 2 Snapshot of a quantum wavefunction in the stadium. (From YouTube)
The time average of the wavefunction after a long time has passed is shown in Fig. 3. Other than the horizontal nodal line down the center of the stadium, there is little discernible structure or symmetry. This is also true for the mean squared wavefunction shown in Fig. 4, although there is some structure that may be emerging in the semi-circular regions.
Fig. 3 Time-average wavefunction after a long time.
Fig. 4 Time-average of the squared wavefunction after a long time.
On the other hand, for special initial conditions that have a lot of symmetry, something remarkable happens. Fig. 5 shows several mean-squared results for special initial conditions. There is definite structure in these cases that were given the somewhat ugly name “quantum scars” in the 1980’s by Eric Heller who was one of the first to study this phenomenon [1].
Fig. 5 Quantum scars reflect periodic (but unstable) orbits of the classical system. Quantum effects tend to quench chaos and favor regular motion.
One can superpose highly-symmetric classical trajectories onto the figures, as shown in the bottom row. All of these classical orbits go through a high-symmetry point, such as the center of the stadium (on the left image) and through the focal point of the circular mirrors (in the other two images). The astonishing conclusion of this exercise is that the highly-symmetric periodic classical orbits remain behind as quantum scars—like the Cheshire Cat’s grin—when the system is in the quantum realm. The classical orbits that produce quantum scars have the important property of being periodic but unstable. A slight perturbation from the symmetric trajectory causes it to eventually become ergodic (chaotic). These scars are regions with enhanced probability density, what might be termed “quantum trajectories”, but do not show strong interference patterns.
It is important to make the distinction that it is also possible to construct special wavefunctions that are strictly periodic, such as a wave bouncing perfectly vertically between the straight portions. This leads to large-scale interference patterns that are not the same as the quantum scars.
Quantum Chaos versus Laser Speckle
In addition to the bouncing-wave cases that do not strictly produce quantum scars, there is another “neutral” phenomenon that produces interference patterns that look a lot like scars, but are simply the random addition of lots of plane waves with the same wavelength [2]. A snapshot in time of one of these superpositions is shown in Fig. 6. To see how the waves add together, see the YouTube channel episode.
Fig. 6 The sum of 100 randomly oriented plane waves of constant wavelength. (A snapshot from YouTube.)
[1] Heller E J, Bound-state eigenfunctions of classically chaotic hamiltonian-systems – scars of periodic-orbits, Physical Review Letters 53 ,1515 (1984)
[2] Gutzwiller M C, Chaos in classical and quantum mechanics (New York: New York : Springer-Verlag, 1990)
The Solvay Debates: Einstein versus Bohr
Einstein is the alpha of the quantum. Einstein is also the omega. Although he was the one who established the quantum of energy and matter (see my Blog Einstein vs Planck), Einstein pitted himself in a running debate against Niels Bohr’s emerging interpretation of quantum physics that had, in Einstein’s opinion, severe deficiencies. Between sessions during a series of conferences known as the Solvay Congresses over a period of eight years from 1927 to 1935, Einstein constructed a challenges of increasing sophistication to confront Bohr and his quasi-voodoo attitudes about wave-function collapse. To meet the challenge, Bohr sharpened his arguments and bested Einstein, who ultimately withdrew from the field of battle. Einstein, as quantum physics’ harshest critic, played a pivotal role, almost against his will, establishing the Copenhagen interpretation of quantum physics that rules to this day, and also inventing the principle of entanglement which lies at the core of almost all quantum information technology today.
Debate Timeline
• Fifth Solvay Congress: 1927 October Brussels: Debate Round 1
• Einstein and ensembles
• Sixth Solvay Congress: 1930 Debate Round 2
• Photon in a box
• Seventh Solvay Congress: 1933
• Einstein absent (visiting the US when Hitler takes power…decides not to return to Germany.)
• Physical Review 1935: Debate Round 3
• EPR paper and Bohr’s response
• Schrödinger’s Cat
• Notable Nobel Prizes
• 1918 Planck
• 1921 Einstein
• 1922 Bohr
• 1932 Heisenberg
• 1933 Dirac and Schrödinger
The Solvay Conferences
The Solvay congresses were unparalleled scientific meetings of their day. They were attended by invitation only, and invitations were offered only to the top physicists concerned with the selected topic of each meeting. The Solvay congresses were held about every three years always in Belgium, supported by the Belgian chemical industrialist Ernest Solvay. The first meeting, held in 1911, was on the topic of radiation and quanta.
Fig. 1 First Solvay Congress (1911). Einstein (standing second from right) was one of the youngest attendees.
The fifth meeting, held in 1927, was on electrons and photons and focused on the recent rapid advances in quantum theory. The old quantum guard was invited—Planck, Bohr and Einstein. The new quantum guard was invited as well—Heisenberg, de Broglie, Schrödinger, Born, Pauli, and Dirac. Heisenberg and Bohr joined forces to present a united front meant to solidify what later became known as the Copenhagen interpretation of quantum physics. The basic principles of the interpretation include the wavefunction of Schrödinger, the probabilistic interpretation of Born, the uncertainty principle of Heisenberg, the complementarity principle of Bohr and the collapse of the wavefunction during measurement. The chief conclusion that Heisenberg and Bohr sought to impress on the assembled attendees was that the theory of quantum processes was complete, meaning that unknown or uncertain characteristics of measurements could not be attributed to lack of knowledge or understanding, but were fundamental and permanently inaccessible.
Fig. 2 Fifth Solvay Congress (1927). Einstein front and center. Bohr on the far right middle row.
Einstein was not convinced with that argument, and he rose to his feet to object after Bohr’s informal presentation of his complementarity principle. Einstein insisted that uncertainties in measurement were not fundamental, but were caused by incomplete information, that , if known, would accurately account for the measurement results. Bohr was not prepared for Einstein’s critique and brushed it off, but what ensued in the dining hall and the hallways of the Hotel Metropole in Brussels over the next several days has become one of the most famous scientific debates of the modern era, known as the Bohr-Einstein debate on the meaning of quantum theory. The debate gently raged night and day through the fifth congress, and was renewed three years later at the 1930 congress. It finished, in a final flurry of published papers in 1935 that launched some of the central concepts of quantum theory, including the idea of quantum entanglement and, of course, Schrödinger’s cat.
Einstein’s strategy, to refute Bohr, was to construct careful thought experiments that envisioned perfect experiments, without errors, that measured properties of ideal quantum systems. His aim was to paint Bohr into a corner from which he could not escape, caught by what Einstein assumed was the inconsistency of complementarity. Einstein’s “thought experiments” used electrons passing through slits, diffracting as required by Schrödinger’s theory, but being detected by classical measurements. Einstein would present a thought experiment to Bohr, who would then retreat to consider the way around Einstein’s arguments, returning the next hour or the next day with his answer, only to be confronted by yet another clever device of Einstein’s clever imagination that would force Bohr to retreat again. The spirit of this back and forth encounter between Bohr and Einstein is caught dramatically in the words of Paul Ehrenfest who witnessed the debate first hand, partially mediating between Bohr and Einstein, both of whom he respected deeply.
“Brussels-Solvay was fine!… BOHR towering over everybody. At first not understood at all … , then step by step defeating everybody. Naturally, once again the awful Bohr incantation terminology. Impossible for anyone else to summarise … (Every night at 1 a.m., Bohr came into my room just to say ONE SINGLE WORD to me, until three a.m.) It was delightful for me to be present during the conversation between Bohr and Einstein. Like a game of chess, Einstein all the time with new examples. In a certain sense a sort of Perpetuum Mobile of the second kind to break the UNCERTAINTY RELATION. Bohr from out of philosophical smoke clouds constantly searching for the tools to crush one example after the other. Einstein like a jack-in-the-box; jumping out fresh every morning. Oh, that was priceless. But I am almost without reservation pro Bohr and contra Einstein. His attitude to Bohr is now exacly like the attitude of the defenders of absolute simultaneity towards him …” [1]
The most difficult example that Einstein constructed during the fifth Solvary Congress involved an electron double-slit apparatus that could measure, in principle, the momentum imparted to the slit by the passing electron, as shown in Fig.3. The electron gun is a point source that emits the electrons in a range of angles that illuminates the two slits. The slits are small relative to a de Broglie wavelength, so the electron wavefunctions diffract according to Schrödinger’s wave mechanics to illuminate the detection plate. Because of the interference of the electron waves from the two slits, electrons are detected clustered in intense fringes separated by dark fringes.
So far, everyone was in agreement with these suggested results. The key next step is the assumption that the electron gun emits only a single electron at a time, so that only one electron is present in the system at any given time. Furthermore, the screen with the double slit is suspended on a spring, and the position of the screen is measured with complete accuracy by a displacement meter. When the single electron passes through the entire system, it imparts a momentum kick to the screen, which is measured by the meter. It is also detected at a specific location on the detection plate. Knowing the position of the electron detection, and the momentum kick to the screen, provides information about which slit the electron passed through, and gives simultaneous position and momentum values to the electron that have no uncertainty, apparently rebutting the uncertainty principle.
Fig. 3 Einstein’s single-electron thought experiment in which the recoil of the screen holding the slits can be measured to tell which way the electron went. Bohr showed that the more “which way” information is obtained, the more washed-out the interference pattern becomes.
This challenge by Einstein was the culmination of successively more sophisticated examples that he had to pose to combat Bohr, and Bohr was not going to let it pass unanswered. With ingenious insight, Bohr recognized that the key element in the apparatus was the fact that the screen with the slits must have finite mass if the momentum kick by the electron were to produce a measurable displacement. But if the screen has finite mass, and hence a finite momentum kick from the electron, then there must be an uncertainty in the position of the slits. This uncertainty immediately translates into a washout of the interference fringes. In fact the more information that is obtained about which slit the electron passed through, the more the interference is washed out. It was a perfect example of Bohr’s own complementarity principle. The more the apparatus measures particle properties, the less it measures wave properties, and vice versa, in a perfect balance between waves and particles.
Einstein grudgingly admitted defeat at the end of the first round, but he was not defeated. Three years later he came back armed with more clever thought experiments, ready for the second round in the debate.
The Sixth Solvay Conference: 1930
At the Solvay Congress of 1930, Einstein was ready with even more difficult challenges. His ultimate idea was to construct a box containing photons, just like the original black bodies that launched Planck’s quantum hypothesis thirty years before. The box is attached to a weighing scale so that the weight of the box plus the photons inside can be measured with arbitrarily accuracy. A shutter over a hole in the box is opened for a time T, and a photon is emitted. Because the photon has energy, it has an equivalent weight (Einstein’s own famous E = mc2), and the mass of the box changes by an amount equal to the photon energy divided by the speed of light squared: m = E/c2. If the scale has arbitrary accuracy, then the energy of the photon has no uncertainty. In addition, because the shutter was open for only a time T, the time of emission similarly has no uncertainty. Therefore, the product of the energy uncertainty and the time uncertainty is much smaller than Planck’s constant, apparently violating Heisenberg’s precious uncertainty principle.
Bohr was stopped in his tracks with this challenge. Although he sensed immediately that Einstein had missed something (because Bohr had complete confidence in the uncertainty principle), he could not put his finger immediately on what it was. That evening he wandered from one attendee to another, very unhappy, trying to persuade them and saying that Einstein could not be right because it would be the end of physics. At the end of the evening, Bohr was no closer to a solution, and Einstein was looking smug. However, by the next morning Bohr reappeared tired but in high spirits, and he delivered a master stroke. Where Einstein had used special relaitivity against Bohr, Bohr now used Einstein’s own general relativity against him.
The key insight was that the weight of the box must be measured, and the process of measurement was just as important as the quantum process being measured—this was one of the cornerstones of the Copenhagen interpretation. So Bohr envisioned a measuring apparatus composed of a spring and a scale with the box suspended in gravity from the spring. As the photon leaves the box, the weight of the box changes, and so does the deflection of the spring, changing the height of the box. This change in height, in a gravitational potential, causes the timing of the shutter to change according to the law of gravitational time dilation in general relativity. By calculating the the general relativistic uncertainty in the time, coupled with the special relativistic uncertainty in the weight of the box, produced a product that was at least as big as Planck’s constant—Heisenberg’s uncertainty principle was saved!
Fig. 4 Einstein’s thought experiment that uses special relativity to refute quantum mechanics. Bohr then invoked Einstein’s own general relativity to refute him.
Entanglement and Schrödinger’s Cat
Einstein ceded the point to Bohr but was not convinced. He still believed that quantum mechanics was not a “complete” theory of quantum physics and he continued to search for the perfect thought experiment that Bohr could not escape. Even today when we have become so familiar with quantum phenomena, the Copenhagen interpretation of quantum mechanics has weird consequences that seem to defy common sense, so it is understandable that Einstein had his reservations.
After the sixth Solvay congress Einstein and Schrödinger exchanged many letters complaining to each other about Bohr’s increasing strangle-hold on the interpretation of quantum mechanics. Egging each other on, they both constructed their own final assault on Bohr. The irony is that the concepts they devised to throw down quantum mechanics have today become cornerstones of the theory. For Einstein, his final salvo was “Entanglement”. For Schrödinger, his final salvo was his “cat”. Today, Entanglement and Schrödinger’s Cat have become enshrined on the alter of quantum interpretation even though their original function was to thwart that interpretation.
The final round of the debate was carried out, not at a Solvay congress, but in the Physical review journal by Einstein [2] and Bohr [3], and in the Naturwissenshaften by Schrödinger [4].
In 1969, Heisenberg looked back on these years and said,
[1] A. Whitaker, Einstein, Bohr, and the quantum dilemma : from quantum theory to quantum information, 2nd ed. Cambridge University Press, 2006. (pg. 210)
[2] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 47, no. 10, pp. 0777-0780, May (1935)
[3] N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?,” Physical Review, vol. 48, no. 8, pp. 696-702, Oct (1935)
[4] E. Schrodinger, “The current situation in quantum mechanics,” Naturwissenschaften, vol. 23, pp. 807-812, (1935)
[5] W Heisenberg, Physics and beyond : Encounters and conversations (Harper, New York, 1971)
Galileo: A New Scientist
1543 Copernicus dies, publishes posthumously De Revolutionibus
1564 Galileo born
1581 Enters University of Pisa
1585 Leaves Pisa without a degree
1586 Invents hydrostatic balance
1588 Receives lecturship in mathematics at Pisa
1592 Chair of mathematics at Univeristy of Padua
1595 Theory of the tides
1595 Invents military and geometric compass
1596 Le Meccaniche and the principle of horizontal inertia
1600 Bruno Giordano burned at the stake
1601 Death of Tycho Brahe
1609 Galileo constructs his first telescope, makes observations of the moon
1611 Scheiner discovers sunspots
1611 Galileo meets Barberini, a cardinal
1611 Johannes Kepler, Dioptrice
1613 Letters on sunspots published by Lincean Academy in Rome
1614 Galileo denounced from the pulpit
1615 (April) Bellarmine writes an essay against Coperinicus
1615 Galileo investigated by the Inquisition
1615 Writes Letter to Christina, but does not publish it
1615 (December) travels to Rome and stays at Tuscan embassy
1616 (January) Francesco Ingoli publishes essay against Copernicus
1616 (March) Decree against copernicanism
1618 Galileo, through Mario Guiducci, publishes scathing attack on Grassi
1619 Marina Gamba dies, Galileo legitimizes his son Vinczenzio
1619 Kepler’s Laws, Epitome astronomiae Copernicanae.
1624 Galileo visits Rome and Urban VIII
1629 Birth of his grandson Galileo
1630 Death of Johanes Kepler
1633 (February) Travels to Rome
1638 Blind, publication of Two New Sciences
1642 Galileo dies (77 years old)
Galileo’s Trajectory
1583 Galileo Notices isochronism of the pendulum
1588 Receives lecturship in mathematics at Pisa
1589 – 1592 Work on projectile motion in Pisa
1592 Chair of mathematics at Univeristy of Padua
1596 Le Meccaniche and the principle of horizontal inertia
1600 Guidobaldo shares technique of colored ball
1602 Proves isochronism of the pendulum (experimentally)
1604 First experiments on uniformly accelerated motion
1607-1608 Identified trajectory as parabolic
1609 Velocity proportional to time
1636 Letter to Christina published in Augsburg in Latin and Italian
1638 Blind, publication of Two New Sciences
1641 Invented pendulum clock (in theory)
1642 Dies (77 years old)
On the Shoulders of Giants
1644 Descartes’ vortex theory of gravitation
1662 Fermat’s principle
1669 – 1690 Huygens expands on Descartes’ vortex theory
1687 Newton’s Principia
1698 Maupertuis born
1729 Maupertuis entered University in Basel. Studied under Johann Bernoulli
1736 Euler publishes Mechanica sive motus scientia analytice exposita
1746 Maupertuis principle of Least Action for mass
1751 Samuel König disputes Maupertuis’ priority
1756 Cassini dies. Maupertuis reinstated in the French Academy
1759 Maupertuis dies
1759 du Chatelet’s French translation of Newton’s Principia published posthumously
1762 Beginning of the reign of Catherine the Great of Russia
1763 Euler colinear 3-body problem
1765 Euler publishes Theoria motus corporum solidorum on rotational mechanics
1766 Euler returns to St. Petersburg
1766 Lagrange arrives in Berlin
1775 Beginning of the American War of Independence
1776 Adam Smith Wealth of Nations
1781 William Herschel discovers Uranus
1783 Euler dies in St. Petersburg
1787 United States Constitution written
1787 Lagrange moves from Berlin to Paris
1788 Lagrange, Méchanique analytique
1789 Beginning of the French Revolution
1799 Pierre-Simon Laplace Mécanique Céleste (1799-1825)
Geometry on My Mind
1629 Fermat described higher-dim loci
1637 Descarte’s Geometry
1649 van Schooten’s commentary on Descartes Geometry
1694 Leibniz uses word “coordinate” in its modern usage
1697 Johann Bernoulli shortest distance between two points on convex surface
1732 Euler geodesic equations for implicit surfaces
1748 Euler defines modern usage of function
1801 Gauss calculates orbit of Ceres
1807 Fourier analysis (published in 1822(
1807 Gauss arrives in Göttingen
1830 Bolyai and Lobachevsky publish on hyperbolic geometry
1836 Liouville-Sturm theorem
1838 Liouville’s theorem
1841 Jacobi determinants
1843 Arthur Cayley systems of n-variables
1843 Hamilton discovers quaternions
1844 Hermann Grassman n-dim vector spaces, Die Lineale Ausdehnungslehr
1846 Julius Plücker System der Geometrie des Raumes in neuer analytischer Behandlungsweise
1848 Jacobi Vorlesungen über Dynamik
1848 “Vector” coined by Hamilton
1854 Riemann’s habilitation lecture
1861 Riemann n-dim solution of heat conduction
1868 Publication of Riemann’s Habilitation
1869 Christoffel and Lipschitz work on multiple dimensional analysis
1871 Klein publishes on non-euclidean geometry
1872 Boltzmann distribution
1872 Jordan Essay on the geometry of n-dimensions
1872 Felix Klein’s “Erlangen Programme”
1872 Weierstrass’ Monster
1872 Dedekind cut
1872 Cantor paper on irrational numbers
1872 Cantor meets Dedekind
1872 Lipschitz derives mechanical motion as a geodesic on a manifold
1874 Cantor beginning of set theory
1881 Gibbs codifies vector analysis
1883 Cantor set and staircase Grundlagen einer allgemeinen Mannigfaltigkeitslehre
1884 Abbott publishes Flatland
1887 Peano vector methods in differential geometry
1890 Peano space filling curve
1891 Hilbert space filling curve
1898 Ricci-Curbastro Lesons on the Theory of Surfaces
1902 Lebesgue integral
1904 Hilbert studies integral equations
1904 von Koch snowflake
1906 Frechet thesis on square summable sequences as infinite dimensional space
1908 Schmidt Geometry in a Function Space
1910 Brouwer proof of dimensional invariance
1913 Hilbert space named by Riesz
1914 Hilbert space used by Hausdorff
1915 Sierpinski fractal triangle
1918 Hausdorff non-integer dimensions
1918 Weyl’s book Space, Time, Matter
1918 Fatou and Julia fractals
1920 Banach space
1927 von Neumann axiomatic form of Hilbert Space
1935 Frechet full form of Hilbert Space
1967 Mandelbrot coast of Britain
1982 Mandelbrot’s book The Fractal Geometry of Nature
The Tangled Tale of Phase Space
1804 Jacobi born (1904 – 1851) in Potsdam
1804 Napoleon I Emperor of France
1806 William Rowan Hamilton born (1805 – 1865)
1808 Bethoven performs his Fifth Symphony
1809 Joseph Liouville born (1809 – 1882)
1821 Hermann Ludwig Ferdinand von Helmholtz born (1821 – 1894)
1824 Carnot published Reflections on the Motive Power of Fire
1836 Liouville-Sturm theorem
1837 Queen Victoria begins her reign as Queen of England
1847 Helmholtz Conservation of Energy (force)
1851 Thomson names Clausius’ First and Second laws of Thermodynamics
1854 Clausius stated Second Law of Thermodynamics as inequality
1857 Clausius constructs kinetic theory, Mean molecular speeds
1865 Loschmidt size of molecules
1865 Clausius names entropy
1868 Boltzmann adds (Boltzmann) factor to Maxwell distribution
1872 Boltzmann transport equation and H-theorem
1876 Loschmidt reversibility paradox
1877 Boltzmann S = k logW
1896 Zermelo criticizes Boltzmann
1896 Boltzmann posits direction of time to save his H-theorem
1898 Boltzmann Vorlesungen über Gas Theorie
1905 Boltzmann kinetic theory of matter in Encyklopädie der mathematischen Wissenschaften
1906 Boltzmann dies
1910 Paul Hertz uses “Phase Space” (Phasenraum)
1911 Ehrenfest’s article in Encyklopädie der mathematischen Wissenschaften
The Lens of Gravity
1728 Euler found the geodesic equation.
1827 Gauss curvature Theoriem Egregum
1844 The name “geodesic line” is attributed to Liouville.
1854 Riemann’s habilitationsschrift
1862 Discovery of Sirius B (a white dwarf)
1868 Darboux suggested motions in n-dimensions
1895 Hilbert arrives in Göttingen
1902 Minkowski arrives in Göttingen
1905 Einstein’s miracle year
1906 Poincaré describes Lorentz transformations as rotations in 4D
1907 Einstein has “happiest thought” in November
1907 Einstein’s relativity review in Jahrbuch
1908 Minkowski’s Space and Time lecture
1908 Einstein appointed to unpaid position at University of Bern
1909 Minkowski dies
1909 Einstein appointed associate professor of theoretical physics at U of Zürich
1911 Laue publishes first textbook on relativity theory
1911 Einstein accepts position at Prague
1912 Einstein’s two papers establish a scalar field theory of gravitation
1913 Einstein EG paper
1914 Adams publishes spectrum of 40 Eridani B
1915 Einstein Completes paper
1916 Density of 40 Eridani B by Ernst Öpik
1916 Schwarzschild paper
1916 Einstein’s publishes theory of gravitational waves
1919 Eddington expedition to Principe
1920 Eddington paper on deflection of light by the sun
1922 Willem Luyten coins phrase “white dwarf”
1933 Georges Lemaitre states the coordinate singularity was an artefact
1958 David Finkelstein paper
1967 Wheeler’s “black hole” talk
2017 LIGO detects the merger of two neutron stars
On the Quantum Footpath
1885 Balmer Theory
1897 J. J. Thomson discovered the electron
1904 Thomson plum pudding model of the atom
1911 Rutherford nuclear model
1911 First Solvay conference
1911 “ultraviolet catastrophe” coined by Ehrenfest
1913 Ehrenfest adiabatic hypothesis
1914-1916 Bohr at Manchester with Rutherford
1916 Schwarzschild and Epstein introduce action-angle coordinates into quantum theory
1920 Heisenberg enters University of Munich to obtain his doctorate
1920 Bohr’s Correspondence principle: Classical physics for large quantum numbers
1921 Bohr Founded Institute of Theoretical Physics (Copenhagen)
1924 Heisenberg Habilitation at Göttingen on anomalous Zeeman
1924 Pauli exclusion principle and state occupancy
1924 de Broglie hypothesis extended wave-particle duality to matter
1924 Bohr Predicted Halfnium (72)
1924 Kronig’s proposal for electron self spin
1924 Bose (Einstein)
1925 Heisenberg paper on quantum mechanics
1925 Uhlenbeck and Goudschmidt: spin
1926 Schrödinger wave mechanics
1927 de Broglie hypotehsis confirmed by Davisson and Germer
1927 Solvay Conference in Brussels
1928 Heisenberg to University of Leipzig
1928 Dirac relativistic QM equation
1929 de Broglie Nobel Prize
1930 Solvay Conference
1932 Heisenberg Nobel Prize
1932 von Neumann operator algebra
1933 Schrödinger and Dirac Nobel Prize
1935 Einstein, Poldolsky and Rosen EPR paper
1935 Bohr’s response to Einsteins “EPR” paradox
1935 Schrodinger’s cat
1939 Feynman graduates from MIT
1945 Pauli Nobel Prize
1945 Death of Feynman’s wife Arline (married 4 years)
1945 Fall, Feynman arrives at Cornell ahead of Hans Bethe
1947 Fall, Dyson arrives at Cornell
1948 Feynman and Dirac. Summer drive across the US with Dyson
1949 Karplus and Kroll first g-factor calculation
1950 Feynman moves to Cal Tech
1965 Schwinger, Tomonaga and Feynman Nobel Prize
1967 Hans Bethe Nobel Prize
From Butterflies to Hurricanes
1763 Euler colinear 3-body problem
1772 Lagrange equilateral 3-body problem
1892 – 1899 Poincare New Methods in Celestial Mechanics
1892 Lyapunov The General Problem of the Stability of Motion
1899 Poincare homoclinic trajectory
1927 van der Pol and van der Mark
1937 Coarse systems, Andronov and Pontryagin
1938 Morse theory
1942 Hopf bifurcation
1960 Lorenz: 12 equations
1963 Lorenz: 3 equations
1964 Arnold diffusion
1965 Smale’s horseshoe
1969 Chirikov standard map
1975 Gollub-Swinney observe route to turbulence along lines of Ruelle
1975 Yorke coins “chaos theory”
1976 Robert May writes review article of the logistic map
1977 New York conference on bifurcation theory
1987 James Gleick Chaos: Making a New Science
Darwin in the Clockworks
1202 Fibonacci
1766 Thomas Robert Malthus born
1776 Adam Smith The Wealth of Nations
1798 Malthus “An Essay on the Principle of Population
1817 Ricardo Principles of Political Economy and Taxation
1838 Cournot early equilibrium theory in duopoly
1848 John Stuart Mill
1848 Karl Marx Communist Manifesto
1859 Darwin Origin of Species
1867 Karl Marx Das Kapital
1871 Darwin Descent of Man, and Selection in Relation to Sex
1871 Jevons Theory of Political Economy
1871 Menger Principles of Economics
1890 Marshall Principles of Economics
1908 Hardy constant genetic variance
1910 Brouwer fixed point theorem
1910 Alfred J. Lotka autocatylitic chemical reactions
1913 Zermelo determinancy in chess
1922 Fisher dominance ratio
1922 Fisher mutations
1925 Lotka predator-prey in biomathematics
1926 Vita Volterra published same equations independently
1927 JBS Haldane (1892—1964) mutations
1928 von Neumann proves the minimax theorem
1930 Fisher ratio of sexes
1932 Wright Adaptive Landscape
1932 Haldane The Causes of Evolution
1933 Kolmogorov Foundations of the Theory of Probability
1934 Rudolph Carnap The Logical Syntax of Language
1936 Kolmogorov generalized predator-prey systems
1938 Borel symmetric payoff matrix
1942 Sewall Wright Statistical Genetics and Evolution
1944 von Neumann and Morgenstern Theory of Games and Economic Behavior
1950 Prisoner’s Dilemma simulated at Rand Corportation
1951 John Nash Non-cooperative Games
1952 McKinsey Introduction to the Theory of Games (first textbook)
1953 John Nash Two-Person Cooperative Games
1953 Watson and Crick DNA
1955 Braithwaite’s Theory of Games as a Tool for the Moral Philosopher
1961 Lewontin Evolution and the Theory of Games
1962 Patrick Moran The Statistical Processes of Evolutionary Theory
1962 Linus Pauling molecular clock
1968 Motoo Kimura neutral theory of molecular evolution
1972 Maynard Smith introduces the evolutionary stable solution (ESS)
1972 Gould and Eldridge Punctuated equilibrium
1973 Maynard Smith and Price The Logic of Animal Conflict
1973 Black Scholes
1977 Eigen and Schuster The Hypercycle
1978 Replicator equation (Taylor and Jonker)
1982 Hopfield network
1982 John Maynard Smith Evolution and the Theory of Games
1984 R. Axelrod The Evolution of Cooperation
The Measure of Life
1642 Galileo dies
1656 Huygens invents pendulum clock
1665 Huygens observes “odd kind of sympathy” in synchronized clocks
1673 Huygens publishes Horologium Oscillatorium sive de motu pendulorum
1736 Euler Seven Bridges of Königsberg
1845 Kirchhoff’s circuit laws
1852 Guthrie four color problem
1857 Cayley trees
1858 Hamiltonian cycles
1887 Cajal neural staining microscopy
1913 Michaelis Menten dynamics of enzymes
1926 van der Pol dimensioness form of equation
1927 van der Pol periodic forcing
1943 McCulloch and Pits mathematical model of neural nets
1948 Wiener cybernetics
1952 Hodgkin and Huxley action potential model
1952 Turing instability model
1956 Sutherland cyclic AMP
1957 Broadbent and Hammersley bond percolation
1958 Rosenblatt perceptron
1959 Erdös and Renyi random graphs
1962 Cohen EGF discovered
1965 Sebeok coined zoosemiotics
1966 Mesarovich systems biology
1967 Winfree biological rythms and coupled oscillators
1969 Glass Moire patterns in perception
1970 Rodbell G-protein
1971 phrase “strange attractor” coined (Ruelle)
1972 phrase “signal transduction” coined (Rensing)
1975 phrase “chaos theory” coined (Yorke)
1975 Werbos backpropagation
1975 Kuramoto transition
1976 Robert May logistic map
1977 Mackey-Glass equation and dynamical disease
1982 Hopfield network
1990 Strogatz and Murillo pulse-coupled oscillators
1997 Tomita systems biology of a cell
1998 Strogatz and Watts Small World network
1999 Barabasi Scale Free networks
2000 Sequencing of the human genome
Who Invented the Quantum? Einstein vs. Planck
Max Planck’s Discontinuity
Einstein’s Quantum
The Stimulated Emission of Light
Derivation of the Einstein A and B Coefficients
The Planck density of photons for ΔE = hf is
The total emission rate is
Einstein’s Quantum Legacy
Einstein’s Quantum Timeline
1913 – Bohr’s quantum theory of hydrogen.
1915 – Millikan measurement of the photoelectric effect.
1916 – Einstein proposes stimulated emission.
Selected Einstein Quantum Papers
Science 1916: Schwarzschild, Einstein, Planck, Born, Frobenius et al.
In one of my previous blog posts, as I was searching for Schwarzschild’s original papers on Einstein’s field equations and quantum theory, I obtained a copy of the January 1916 – June 1916 volume of the Proceedings of the Royal Prussian Academy of Sciences through interlibrary loan. The extremely thick volume arrived at Purdue about a week after I ordered it online. It arrived from Oberlin College in Ohio that had received it as a gift in 1928 from the library of Professor Friedrich Loofs of the University of Halle in Germany. Loofs had been the Haskell Lecturer at Oberlin for the 1911-1912 semesters.
As I browsed through the volume looking for Schwarzschild’s papers, I was amused to find a cornucopia of turn-of-the-century science topics recorded in its pages. There were papers on the overbite and lips of marsupials. There were papers on forgotten languages. There were papers on ancient Greek texts. On the origins of religion. On the philosophy of abstraction. Histories of Indian dramas. Reflections on cancer. But what I found most amazing was a snapshot of the field of physics and mathematics in 1916, with historic papers by historic scientists who changed how we view the world. Here is a snapshot in time and in space, a period of only six months from a single journal, containing papers from authors that reads like a who’s who of physics.
In 1916 there were three major centers of science in the world with leading science publications: London with the Philosophical Magazine and Proceedings of the Royal Society; Paris with the Comptes Rendus of the Académie des Sciences; and Berlin with the Proceedings of the Royal Prussian Academy of Sciences and Annalen der Physik. In Russia, there were the scientific Journals of St. Petersburg, but the Bolshevik Revolution was brewing that would overwhelm that country for decades. And in 1916 the academic life of the United States was barely worth noticing except for a few points of light at Yale and Johns Hopkins.
Berlin in 1916 was embroiled in war, but science proceeded relatively unmolested. The six-month volume of the Proceedings of the Royal Prussian Academy of Sciences contains a number of gems. Schwarzschild was one of the most prolific contributors, publishing three papers in just this half-year volume, plus his obituary written by Einstein. But joining Schwarzschild in this volume were Einstein, Planck, Born, Warburg, Frobenious, and Rubens among others—a pantheon of German scientists mostly cut off from the rest of the world at that time, but single-mindedly following their individual threads woven deep into the fabric of the physical world.
Karl Schwarzschild (1873 – 1916)
Schwarzschild had the unenviable yet effective motivation of his impending death to spur him to complete several projects that he must have known would make his name immortal. In this six-month volume he published his three most important papers. The first (pg. 189) was on the exact solution to Einstein’s field equations to general relativity. The solution was for the restricted case of a point mass, yet the derivation yielded the Schwarzschild radius that later became known as the event horizon of a non-roatating black hole. The second paper (pg. 424) expanded the general relativity solutions to a spherically symmetric incompressible liquid mass.
Schwarzschild’s solution to Einstein’s field equations for a point mass.
Schwarzschild’s extension of the field equation solutions to a finite incompressible fluid.
The subject, content and success of these two papers was wholly unexpected from this observational astronomer stationed on the Russian Front during WWI calculating trajectories for German bombardments. He would not have been considered a theoretical physicist but for the importance of his results and the sophistication of his methods. Within only a year after Einstein published his general theory, based as it was on the complicated tensor calculus of Levi-Civita, Christoffel and Ricci-Curbastro that had taken him years to master, Schwarzschild found a solution that evaded even Einstein.
Schwarzschild’s third and final paper (pg. 548) was on an entirely different topic, still not in his official field of astronomy, that positioned all future theoretical work in quantum physics to be phrased in the language of Hamiltonian dynamics and phase space. He proved that action-angle coordinates were the only acceptable canonical coordinates to be used when quantizing dynamical systems. This paper answered a central question that had been nagging Bohr and Einstein and Ehrenfest for years—how to quantize dynamical coordinates. Despite the simple way that Bohr’s quantized hydrogen atom is taught in modern physics, there was an ambiguity in the quantization conditions even for this simple single-electron atom. The ambiguity arose from the numerous possible canonical coordinate transformations that were admissible, yet which led to different forms of quantized motion.
Schwarzschild’s proposal of action-angle variables for quantization of dynamical systems.
Schwarzschild’s doctoral thesis had been a theoretical topic in astrophysics that applied the celestial mechanics theories of Henri Poincaré to binary star systems. Within Poincaré’s theory were integral invariants that were conserved quantities of the motion. When a dynamical system had as many constraints as degrees of freedom, then every coordinate had an integral invariant. In this unexpected last paper from Schwarzschild, he showed how canonical transformation to action-angle coordinates produced a unique representation in terms of action variables (whose dimensions are the same as Planck’s constant). These action coordinates, with their associated cyclical angle variables, are the only unambiguous representations that can be quantized. The important points of this paper were amplified a few months later in a publication by Schwarzschild’s friend Paul Epstein (1871 – 1939), solidifying this approach to quantum mechanics. Paul Ehrenfest (1880 – 1933) continued this work later in 1916 by defining adiabatic invariants whose quantum numbers remain unchanged under slowly varying conditions, and the program started by Schwarzschild was definitively completed by Paul Dirac (1902 – 1984) at the dawn of quantum mechanics in Göttingen in 1925.
Albert Einstein (1879 – 1955)
In 1916 Einstein was mopping up after publishing his definitive field equations of general relativity the year before. His interests were still cast wide, not restricted only to this latest project. In the 1916 Jan. to June volume of the Prussian Academy Einstein published two papers. Each is remarkably short relative to the other papers in the volume, yet the importance of the papers may stand in inverse proportion to their length.
The first paper (pg. 184) is placed right before Schwarzschild’s first paper on February 3. The subject of the paper is the expression of Maxwell’s equations in four-dimensional space time. It is notable and ironic that Einstein mentions Hermann Minkowski (1864 – 1909) in the first sentence of the paper. When Minkowski proposed his bold structure of spacetime in 1908, Einstein had been one of his harshest critics, writing letters to the editor about the absurdity of thinking of space and time as a single interchangeable coordinate system. This is ironic, because Einstein today is perhaps best known for the special relativity properties of spacetime, yet he was slow to adopt the spacetime viewpoint. Einstein only came around to spacetime when he realized around 1910 that a general approach to relativity required the mathematical structure of tensor manifolds, and Minkowski had provided just such a manifold—the pseudo-Riemannian manifold of space time. Einstein subsequently adopted spacetime with a passion and became its greatest champion, calling out Minkowski where possible to give him his due, although he had already died tragically of a burst appendix in 1909.
Relativistic energy density of electromagnetic fields.
The importance of Einstein’s paper hinges on his derivation of the electromagnetic field energy density using electromagnetic four vectors. The energy density is part of the source term for his general relativity field equations. Any form of energy density can warp spacetime, including electromagnetic field energy. Furthermore, the Einstein field equations of general relativity are nonlinear as gravitational fields modify space and space modifies electromagnetic fields, producing a coupling between gravity and electromagnetism. This coupling is implicit in the case of the bending of light by gravity, but Einstein’s paper from 1916 makes the connection explicit.
Einstein’s second paper (pg. 688) is even shorter and hence one of the most daring publications of his career. Because the field equations of general relativity are nonlinear, they are not easy to solve exactly, and Einstein was exploring approximate solutions under conditions of slow speeds and weak fields. In this “non-relativistic” limit the metric tensor separates into a Minkowski metric as a background on which a small metric perturbation remains. This small perturbation has the properties of a wave equation for a disturbance of the gravitational field that propagates at the speed of light. Hence, in the June 22 issue of the Prussian Academy in 1916, Einstein predicts the existence and the properties of gravitational waves. Exactly one hundred years later in 2016, the LIGO collaboration announced the detection of gravitational waves generated by the merger of two black holes.
Einstein’s weak-field low-velocity approximation solutions of his field equations.
Einstein’s prediction of gravitational waves.
Max Planck (1858 – 1947)
Max Planck was active as the secretary of the Prussian Academy in 1916 yet was still fully active in his research. Although he had launched the quantum revolution with his quantum hypothesis of 1900, he was not a major proponent of quantum theory even as late as 1916. His primary interests lay in thermodynamics and the origins of entropy, following the theoretical approaches of Ludwig Boltzmann (1844 – 1906). In 1916 he was interested in how to best partition phase space as a way to count states and calculate entropy from first principles. His paper in the 1916 volume (pg. 653) calculated the entropy for single-atom solids.
Counting microstates by Planck.
Max Born (1882 – 1970)
Max Born was to be one of the leading champions of the quantum mechanical revolution based at the University of Göttingen in the 1920’s. But in 1916 he was on leave from the University of Berlin working on ranging for artillery. Yet he still pursued his academic interests, like Schwarzschild. On pg. 614 in the Proceedings of the Prussian Academy, Born published a paper on anisotropic liquids, such as liquid crystals and the effect of electric fields on them. It is astonishing to think that so many of the flat-panel displays we have today, whether on our watches or smart phones, are technological descendants of work by Born at the beginning of his career.
Born on liquid crystals.
Ferdinand Frobenius (1849 – 1917)
Like Schwarzschild, Frobenius was at the end of his career in 1916 and would pass away one year later, but unlike Schwarzschild, his career had been a long one, receiving his doctorate under Weierstrass and exploring elliptic functions, differential equations, number theory and group theory. One of the papers that established him in group theory appears in the May 4th issue on page 542 where he explores the series expansion of a group.
Frobenious on groups.
Heinrich Rubens (1865 – 1922)
Max Planck owed his quantum breakthrough in part to the exquisitely accurate experimental measurements made by Heinrich Rubens on black body radiation. It was only by the precise shape of what came to be called the Planck spectrum that Planck could say with such confidence that his theory of quantized radiation interactions fit Rubens spectrum so perfectly. In 1916 Rubens was at the University of Berlin, having taken the position vacated by Paul Drude in 1906. He was a specialist in infrared spectroscopy, and on page 167 of the Proceedings he describes the spectrum of steam and its consequences for the quantum theory.
Rubens and the infrared spectrum of steam.
Emil Warburg (1946 – 1931)
Emil Warburg’s fame is primarily as the father of Otto Warburg who won the 1931 Nobel prize in physiology. On page 314 Warburg reports on photochemical processes in BrH gases. In an obscure and very indirect way, I am an academic descendant of Emil Warburg. One of his students was Robert Pohl who was a famous early researcher in solid state physics, sometimes called the “father of solid state physics”. Pohl was at the physics department in Göttingen in the 1920’s along with Born and Franck during the golden age of quantum mechanics. Robert Pohl’s son, Robert Otto Pohl, was my professor when I was a sophomore at Cornell University in 1978 for the course on introductory electromagnetism using a textbook by the Nobel laureate Edward Purcell, a quirky volume of the Berkeley Series of physics textbooks. This makes Emil Warburg my professor’s father’s professor.
Warburg on photochemistry.
Papers in the 1916 Vol. 1 of the Prussian Academy of Sciences
Schulze, Alt– und Neuindisches
Orth, Zur Frage nach den Beziehungen des Alkoholismus zur Tuberkulose
Schulze, Die Erhabunen auf der Lippin- und Wangenschleimhaut der Säugetiere
von Wilamwitz-Moellendorff, Die Samie des Menandros
Engler, Bericht über das >>Pflanzenreich<<
von Harnack, Bericht über die Ausgabe der griechischen Kirchenväter der dri ersten Jahrhunderte
Meinecke, Germanischer und romanischer Geist im Wandel der deutschen Geschichtsauffassung
Rubens und Hettner, Das langwellige Wasserdampfspektrum und seine Deutung durch die Quantentheorie
Einstein, Eine neue formale Deutung der Maxwellschen Feldgleichungen der Electrodynamic
Schwarschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
Helmreich, Handschriftliche Verbesserungen zu dem Hippokratesglossar des Galen
Prager, Über die Periode des veränderlichen Sterns RR Lyrae
Holl, Die Zeitfolge des ersten origenistischen Streits
Lüders, Zu den Upanisads. I. Die Samvargavidya
Warburg, Über den Energieumsatz bei photochemischen Vorgängen in Gasen. VI.
Hellman, Über die ägyptischen Witterungsangaben im Kalender von Claudius Ptolemaeus
Meyer-Lübke, Die Diphthonge im Provenzaslischen
Diels, Über die Schrift Antipocras des Nikolaus von Polen
Müller und Sieg, Maitrisimit und >>Tocharisch<<
Meyer, Ein altirischer Heilsegen
Schwarzschild, Über das Gravitationasfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie
Brauer, Die Verbreitung der Hyracoiden
Correns, Untersuchungen über Geschlechtsbestimmung bei Distelarten
Brahn, Weitere Untersuchungen über Fermente in der Lever von Krebskranken
Erdmann, Methodologische Konsequenzen aus der Theorie der Abstraktion
Bang, Studien zur vergleichenden Grammatik der Türksprachen. I.
Frobenius, Über die Kompositionsreihe einer Gruppe
Schwarzschild, Zur Quantenhypothese
Fischer und Bergmann, Über neue Galloylderivate des Traubenzuckers und ihren Vergleich mit der Chebulinsäure
Schuchhardt, Der starke Wall und die breite, zuweilen erhöhte Berme bei frügeschichtlichen Burgen in Norddeutschland
Born, Über anisotrope Flüssigkeiten
Planck, Über die absolute Entropie einatomiger Körper
Haberlandt, Blattepidermis und Lichtperzeption
Einstein, Näherungsweise Integration der Feldgleichungen der Gravitation
Lüders, Die Saubhikas. Ein Beitrag zur Gecschichte des indischen Dramas
Dirac: From Quantum Field Theory to Antimatter
Paul Adrian Maurice Dirac (1902 – 1984) was given the moniker of “the strangest man” by Niels Bohr while he was reminiscing about the many great scientists with whom he had worked over the years [1]. It is a moniker that resonates with the innumerable “Dirac stories” that abound in the mythology of the hallways of physics departments around the world. Dirac was awkward, shy, a loner, rarely said anything, was completely literal, had not the slightest comprehension of art or poetry, nor any clear understanding of human interpersonal interaction. Dirac was also brilliant, providing the theoretical foundation for the central paradigm of modern physics—quantum field theory. The discovery of the Higgs boson in 2012, a human achievement that capped nearly a century of scientific endeavor, rests solidly on the theory of quantum fields that permeate space. The Higgs particle, when it pops into existence at the Large Hadron Collider in Geneva, is a singular quantum excitation of the Higgs field, a field that usually resides in a vacuum state, frothing with quantum fluctuations that imbue all particles—and you and me—with mass. The Higgs field is Dirac’s legacy.
… all of a sudden he had a new equation with four-dimensional space-time symmetry.
Copenhagen and Bohr
Although Dirac as a young scientist was initially enthralled with relativity theory, he was working under Ralph Fowler (1889 – 1944) in the physics department at Cambridge in 1923 when he had the chance to read advanced proofs of Heisenberg’s matrix mechanics paper. This chance event launched him on his own trajectory in quantum theory. After Dirac was awarded his doctorate from Cambridge in 1926, he received a stipend that sent him to work with Niels Bohr (1885 – 1962) in Copenhagen—ground zero of the new physics. During his time there, Dirac became famous for taking long walks across Copenhagen as he played about with things in his mind, performing mental juggling of abstract symbols, envisioning how they would permute and act. His attention was focused on the electromagnetic field and how it interacted with the quantized states of atoms. Although the electromagnetic field was the classical field of light, it was also the quantum field of Einstein’s photon, and he wondered how the quantized harmonic oscillators of the electromagnetic field could be generated by quantum wavefunctions acting as operators. But acting on what? He decided that, to generate a photon, the wavefunction must operate on a state that had no photons—the ground state of the electromagnetic field known as the vacuum state.
In late 1926, nearing the end of his stay in Copenhagen with Bohr, Dirac put these thoughts into their appropriate mathematical form and began work on two successive manuscripts. The first manuscript contained the theoretical details of the non-commuting electromagnetic field operators. He called the process of generating photons out of the vacuum “second quantization”. This phrase is a bit of a misnomer, because there is no specific “first quantization” per se, although he was probably thinking of the quantized energy levels of Schrödinger and Heisenberg. In second quantization, the classical field of electromagnetism is converted to an operator that generates quanta of the associated quantum field out of the vacuum (and also annihilates photons back into the vacuum). The creation operators can be applied again and again to build up an N-photon state containing N photons that obey Bose-Einstein statistics, as they must, as required by their integer spin, agreeing with Planck’s blackbody radiation.
Dirac then went further to show how an interaction of the quantized electromagnetic field with quantized energy levels involved the annihilation and creation of photons as they promoted electrons to higher atomic energy levels, or demoted them through stimulated emission. Very significantly, Dirac’s new theory explained the spontaneous emission of light from an excited electron level as a direct physical process that creates a photon carrying away the energy as the electron falls to a lower energy level. Spontaneous emission had been explained first by Einstein more than ten years earlier when he derived the famous A and B coefficients, but Einstein’s arguments were based on the principle of detailed balance, which is a thermodynamic argument. It is impressive that Einstein’s deep understanding of thermodynamics and statistical mechanics could allow him to derive the necessity of both spontaneous and stimulated emission, but the physical mechanism for these processes was inferred rather than derived. Dirac, in late 1926, had produced the first direct theory of photon exchange with matter. This was the birth of quantum electrodynamics, known as QED, and the birth of quantum field theory [2].
Fig. 1 Paul Dirac in his early days.
Göttingen and Born
Dirac’s next stop on his postodctoral fellowship was in Göttingen to work with Max Born (1882 – 1970) and the large group of theoreticians and mathematicians who were like electrons in a cloud orbiting around the nucleus represented by the new quantum theory. Göttingen was second only to Copenhagen as the Mecca for quantum theorists. Hilbert was there and von Neumann too, as well as the brash American J. Robert Oppenheimer (1904 – 1967) who was finishing his PhD with Born. Dirac and Oppenheimer struck up an awkward friendship. Oppenheimer was considered arrogant by many others in the group, but he was in awe of Dirac who arrived with his manuscript on quantum electrodynamics ready for submission. Oppenheimer struggled at first to understand Dirac’s new approach to quantizing fields, but he quickly grasped the importance, as did Pascual Jordan (1902 – 1980), who was also in Göttingen.
Jordan had already worked on ideas very close to Dirac’s on the quantization of fields. He and Dirac seemed to be going down the same path, independently arriving at very similar conclusions around the same time. In fact, Jordan was often a step ahead of Dirac, tending to publish just before Dirac, as with non-commuting matrices, transformation theory and the relationship of canonical transformations to second quantization. However, Dirac’s paper on quantum electrodynamics was a masterpiece in clarity and comprehensiveness, launching a new field in a way that Jordan had not yet achieved with his own work. But because of the closeness of Jordan’s thinking to Dirac’s, he was able to see immediately how to extend Dirac’s approach. Within the year, he published a series of papers that established the formalism of quantum electrodynamics as well as quantum field theory. With Pauli, he systematized the operators for creation and annihilation of photons [3]. With Wigner, he developed second quantization for de Broglie matter waves, defining creation and annihilation operators that obeyed the Pauli exclusion principle of electrons[4]. Jordan was on a roll, forging ahead of Dirac on extensions of quantum electrodynamics and field theory, but Dirac was about to eclipse Jordan once and for all.
St. John’s at Cambridge
At the end of the Spring semester in 1927, Dirac was offered a position as a fellow of St. John’s College at Cambridge, which he accepted, returning to England to begin his life as a college professor. During the summer and into the Fall, Dirac returned to his first passion in physics, relativity, which had yet to be successfully incorporated into quantum physics. Oskar Klein and Walter Gordon had made initial attempts at formulating relativistic quantum theory, but they could not correctly incorporate the spin properties of the electron, and their wave equation had the bad habit of producing negative probabilities. Probabilities went negative because the Klein-Gordon equation had two time derivatives instead of one. The reason it had two (while the non-relativistic Schrödinger equation has only one) is because space-time symmetry required the double space derivative of the Schrödinger equation to be paired with a double time derivative. Dirac, with creative insight, realized that the problem could be flipped by requiring the single time derivative to be paired with a single space derivative. The problem was that a single space derivative did not seem to make any sense [5].
St. John’s College at Cambridge
As Dirac puzzled how to get an equation with only single derivatives, he was playing around with Pauli spin matrices and hit on a simple identity that related the spin matrices to the electron momentum. At first he could not get the identity to apply to four-dimensional relativistic momenta using the usual 2×2 spin matrices. Then he realized that four-dimensional space-time could be captured if he expanded Pauli’s 2×2 spin matrices to 4×4 spin matrices, and all of a sudden he had a new equation with four-dimensional space-time symmetry with single derivatives on space and time. As a test of his new equation, he calculated fine details of the experimentally-measured hydrogen spectrum, known as the fine structure, which had resisted theoretical explanation, and he derived answers in close agreement with experiment. He also showed that the electron had spin-1/2, and he calculated its magnetic moment. He finished his manuscript at the end of the Fall semester in 1927, and the paper was published in early 1928[6]. His relativistic quantum wave equation was an instant sensation, becoming known for all time as “the Dirac Equation”. He had succeeded at finding a correct and long-sought relativistic quantum theory where many others had failed, such as Oskar Klein and Paul Gordon. It was a crowning achievement, placing Dirac firmly in the firmament of the quantum theorists.
Fig. 1 The relativistic Dirac equation. The wavefunction is a four-component spinor. The gamma-del product is a 4×4 matrix operator. The time and space derivatives are both first-order operators.
In the process of ridding the Klein-Gordon equation of negative probability, which Dirac found abhorent, his new equation created an infinite number of negative energy states, which he did not find abhorent. It is perhaps a matter of taste what one theoriest is willing to accept over another, and for Dirac, negative energies were better than negative probabilities. Even so, one needed to deal with an infinite number of negative energy states in quantum theory, because they are available to quantum transitions. In 1929 and 1930, as Dirac was writing his famous textbook on quantum theory, he became intrigued by the similarity between the positive and negative electron states of the vacuum and the energy levels of valence electrons on atoms. An electron in a state outside a filled electron shell behaves very much like a single-electron atom, like sodium and lithium with their single valence electrons. Conversely, an atomic shell that has one electron less than a full complement can be described as having a “hole” that behaves “as if” it were a positive particle. It is like a bubble in water. As water sinks, the bubble rises to the top of the water level. For electrons, if all the electrons go one way in an electric field, then the hole goes the opposite direction, like a positive charge.
Dirac took this analogy of nearly-filled atomic shells and applied it to the vacuum states of the electron, viewing the filled negative energy states like the filled electron shells of atoms. If there is a missing electron, a hole in this infinite sea, then it would behave as if it had positive charge. Initially, Dirac speculated that the “hole” was the proton, and he even wrote a paper on that possibility. But Oppenheimer pointed out that the idea was inconsistent with observations, especially the inability of the electron and proton to annihilate, and that the ground state of the infinite electron sea must be completely filled. Hermann Weyl further pointed out that the electron-proton theory did not have the correct symmetry, and Dirac had to rethink. In early 1931 he hit on an audacious solution to the puzzle. What if the hole in the infinite negative energy sea did not just behave like a positive particle, but actually was a positive particle, a new particle that Dirac dubbed the “anti-electron”? The anti-electron would have the same mass as the electron, but would have positive charge. He suggested that such particles might be generated in high-energy collisions in vacuum, and he finished his paper with the suggestion that there also could be an anti-proton with the mass of the proton but with negative charge. In this singular paper, titled “Quantized Singularities of the Electromagnetic Field” published in 1931, Dirac predicted the existence of antimatter. A year later the positron was discovered by Carl David Anderson at Cal Tech. Anderson had originally called the particle the positive electron, but a journal editor of the Physical Review changed it to positron, and the new name stuck.
Fig. 3 An electron-positron pair is created by the absorption of a photon (gamma ray). Positrons have negative energy and can be viewed as a hole in a sea of filled electron states. (Momentum conservation is satisfied if a near-by heavy particle takes up the recoil momentum.)
The prediction and subsequent experimental validation of antmatter stands out in the history of physics in the 20th Century. In previous centuries, theory was performed mainly in the service of experiment, explaining interesting new observed phenomena either as consequences of known physics, or creating new physics to explain the observations. Quantum theory, revolutionary as a way of understanding nature, was developed to explain spectroscopic observations of atoms and molecules and gases. Similarly, the precession of the perihelion of Mercury was a well-known phenomenon when Einstein used his newly developed general relativity to explain it. As a counter example, Einstein’s prediction of the deflection of light by the Sun was something new that emerged from theory. This is one reason why Einstein became so famous after Eddington’s expedition to observe the deflection of apparent star locations during the total eclipse. Einstein had predicted something that had never been seen before. Dirac’s prediction of the existence of antimatter similarly is a triumph of rational thought, following the mathematical representation of reality to an inevitable conclusion that cannot be ignored, no matter how wild and initially unimaginable it is. Dirac went on to receive the Nobel prize in Physics in 1933, sharing the prize that year with Schrödinger (Heisenberg won it the previous year in 1932).
[1] Framelo, “The Strangest Man: The Hidden Life of Paul Dirac” (Basic Books, 2011)
[2] Dirac, P. A. M. (1927). “The quantum theory of the emission and absorption of radiation.” Proceedings of the Royal Society of London Series A114(767): 243-265.; Dirac, P. A. M. (1927). “The quantum theory of dispersion.” Proceedings of the Royal Society of London Series A114(769): 710-728.
[3] Jordan, P. and W. Pauli, Jr. (1928). “To quantum electrodynamics of free charge fields.” Zeitschrift Fur Physik 47(3-4): 151-173.
[4] Jordan, P. and E. Wigner (1928). “About the Pauli’s equivalence prohibited.” Zeitschrift Fur Physik 47(9-10): 631-651.
[5] This is because two space derivatives measure the curvative of the wavefunction which is related to the kinetic energy of the electron.
[6] Dirac, P. A. M. (1928). “The quantum theory of the electron.” Proceedings of the Royal Society of London Series A 117(778): 610-624.; Dirac, P. A. M. (1928). “The quantum theory of the electron – Part II.” Proceedings of the Royal Society of London Series A118(779): 351-361. |
15bedb8491d2b568 |
This section is more precisely about classical mechanics.
Basically the same as classical mechanics.
The idea tha taking the limit of the non-classical theories for certain parameters (relativity and quantum mechanics) should lead to the classical theory.
It appears that classical limit is only very strict for relativity. For quantum mechanics it is much more hand-wavy thing. See also: Subtle is the Lord by Abraham Pais (1982) page 55.
Basically the same as classical limit, but more for quantum mechanics.
Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of global x, y, z. Classical examples that you must have in mind include:
• compound Atwood machine. Here, we can use the coordinates as the heights of masses relative to the axles rather than absolute heights relative to the ground
• double pendulum, using two angles. The Lagrangian approach is simpler than using Newton's laws
• two-body problem, use the distance between the bodies
lagrangian mechanics lectures by Michel van Biezen (2017) is a good starting point.
When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector that maps time to the full state:
where each component can be anything, either the x/y/z coordinates relative to the ground of different particles, or angles, or nay other crazy thing we want.
The Lagrangian is a function that maps:
to a real number.
Then, the stationary action principle says that the actual path taken obeys the Euler-Lagrange equation:
This produces a system of partial differential equations with:
• equations
• unknown functions
• at most second order derivatives of . Those appear because of the chain rule on the second term.
The mixture of so many derivatives is a bit mind mending, so we can clarify them a bit further. At:
the is just identifying which argument of the Lagrangian we are differentiating by: the i-th according to the order of our definition of the Lagrangian. It is not the actual function, just a mnemonic.
Then at:
• the part is just like the previous term, just identifies the argument with index ( because we have the non derivative arguments)
• after the partial derivative is taken and returns a new function , then the multivariable chain rule comes in and expands everything into terms
However, people later noticed that the Lagrangian had some nice properties related to Lie group continuous symmetries.
Basically it seems that the easiest way to come up with new quantum field theory models is to first find the Lagrangian, and then derive the equations of motion from them.
For every continuous symmetry in the system (modelled by a Lie group), there is a corresponding conservation law: local symmetries of the Lagrangian imply conserved currents. Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "The Best Path" mentions that Richard Feynman didn't like the Lagrangian mechanics approach when he started university at MIT, because he felt it was too magical. The reason is that the Lagrangian approach basically starts from the principle that "nature minimizes the action across time globally". This implies that things that will happen in the future are also taken into consideration when deciding what has to happen before them! Much like the lifeguard in the lifegard problem making global decisions about the future. However, chapter "Least Action in Quantum Mechanics" comments that Feynman later notice that this was indeed necessary while developping Wheeler-Feynman absorber theory into quantum electrodynamics, because they felt that it would make more sense to consider things that way while playing with ideas such as positrons are electrons travelling back in time. This is in contrast with Hamiltonian mechanics, where the idea of time moving foward is more directly present, e.g. as in the Schrödinger equation.
Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.
And partly due to the above observations, it was noticed that the easiest way to describe the fundamental laws of particle physics and make calculations with them is to first formulate their Lagrangian somehow: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?s.
Video 1. Euler-Lagrange equation explained intuitively - Lagrangian Mechanics by Physics Videos by Eugene Khutoryansky (2018) Source. Well, unsurprisingly, it is exactly what you can expect from an Eugene Khutoryansky video.
Author: Michel van Biezen.
High school classical mechanics material, no mention of the key continuous symmetry part.
But does have a few classic pendulum/pulley/spring worked out examples that would be really wise to get under your belt first.
As mentioned on the Wikipedia page, "principle of least action" is not accurate since it could not necessarily be a minima, we could just be in a saddle-point.
Calculus of variations is the field that searches for maxima and minima of Functionals, rather than the more elementary case of functions from to .
A function that takes input function and outputs a real number.
• the term is a function of
or just omit the arguments of entirely:
These are the final equations that you derive from the Lagrangian via the Euler-Lagrange equation which specify how the system evolves with time.
The function that fully describes a physical system in Lagrangian mechanics.
When we particles particles, the action is obtained by integrating the Lagrangian over time:
In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.
Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.
E.g. for a 2-dimensional field :
Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with :
And in the context of special relativity, people condense that even further by adding to the spacetime Four-vector as well, so you don't even need to write that separate pesky .
The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.
The variables of the Lagrangian, e.g. the angles of a double pendulum. From that example it is clear that these variables don't need to be simple things like cartesian coordinates or polar coordinates (although these tend to be the overwhelming majority of simple case encountered): any way to describe the system is perfectly valid.
In quantum field theory, those variables are actually fields.
For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.
As mentioned at, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.
Video 3. The most beautiful idea in physics - Noether's Theorem by Looking Glass Universe (2015) Source. One sentence stands out: the generated quantities are called the generators of the transforms.
Video 4. The Biggest Ideas in the Universe | 15. Gauge Theory by Sean Carroll (2020) Source. This attempts a one hour hand wave explanation of it. It is a noble attempt and gives some key ideas, but it falls a bit short of Ciro's desires (as would anything that fit into one hour?)
Video 5. The Symmetries of the universe by ScienceClic English (2021) Source. explains intuitively why symmetry implies consevation!
Equivalent to Lagrangian mechanics but formulated in a different way.
TODO understand original historical motivation, says it is from optics.
Intuitively, the Hamiltonian is the total energy of the system in terms of arbitrary parameters, a bit like Lagrangian mechanics.
The key difference from Lagrangian mechanics is that the Hamiltonian approach groups variables into pairs of coordinates called the phase space coordinates:
• generalized coordinates, generally positions or angles
• their corresponding conjugate momenta, generally velocities, or angular velocities
This leads to having two times more unknown functions than in the Lagrangian. However, it also leads to a system of partial differential equations with only first order derivatives, which is nicer. Notably, it can be more clearly seen in phase space.
Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:
So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.
This is how you transform the Lagrangian into the Hamiltonian.
Video 6. Lagrangian Mechanics Example: The Compound Atwood Machine by Michel van Biezen (2017) Source. Part of lagrangian mechanics lectures by Michel van Biezen (2017).
The simplest harmonic oscillator system.
Video 7. Pendulum Waves by Harvard Natural Sciences Lecture Demonstrations (2010) Source. Holy crap. compares Lagrangian mechanics equation vs the direct x/y coordinate equation.
resonance in a mechanical system.
This idealization does not seems to be possible at all in the context of Maxwell's equations with pointlike particles. |
8579364d2d067671 | Most Important Scientific Discoveries of All Time
I collected more than 17 lists of the greatest or most important scientific discoveries of all time and combined them into one list – here are the results. The numbers in bold and underlined indicate the number of lists the scientific discovery was on. You may notice there is some overlap with the Best Inventions lists – it appears that the line between ‘invention’ and ‘discovery’ is often a blurry one. I have provided some information on the nature of the discovery and the identities of the discovers. As with inventions, the discovery is often one link in a chain of scientific work that extends before and after the discovery in time, or is a collaboration (sometimes rivalry) among multiple discoverers. Also, for some reason, history sometimes identifies the discoverer as the person who first hypothesized the correct answer to a question, while in other cases, the credit goes to the person who confirmed the hypothesis by experiments or observations. I have also provided images of the scientists or their discoveries where available and where the narrative for one discovery mentions another discovery, I have placed it in boldface. This list includes every discovery on three or more of the 17+ lists. For a chronological timeline of every discovery on two or more lists, go here.
17 Lists
Electricity is the name for a set of physical phenomena associated with the presence and flow of electric charge. One of the first to examine the phenomenon was Thales of Miletus (Ancient Greece), who studied static electricity in 600 BCE. It was not until the careful research of William Gilbert (England) in 1600 that electricity became a subject of scientific study. Gilbert also coined the Latin term ‘electricus’ from the Greek word for amber, which he rubbed to produce static electricity. The English words ‘electric’ and ‘electricity’ were derived by Thomas Browne in 1646. Otto von Guericke (Germany) made the first static electricity generator in 1660. Stephen Gray (England) discovered the conduction of electricity in 1729. The Leyden Jar, the first capacitor, was invented independently in 1745 in Germany and The Netherlands. Henry Cavendish (England) measured conductivity of materials in 1747. Benjamin Franklin (US) discovered that lightning is a form of electricity in 1752. Luigi Galvani (Italy) discovered the electrical basis of nerve impulses in 1786. Alessandro Volta (Italy) invented the electric battery in 1800. Hans Christian Ørsted (Denmark) noticed an interaction between electricity and magnetism in 1820, but it was French scientist André-Marie Ampère’s follow-up experiments that demonstrated the unity of electricity and magnetism. Beginning in 1831, Michael Faraday (England) discovered electromagnetic induction, diamagnetism and electrolysis and invented the first current-generating electric generator, or dynamo. Joseph Henry (US) discovered induction at about the same time. James Clerk Maxwell (England) linked electricity, magnetism and light in 1861-1862 in a series of mathematical equations. In 1866, Werner von Siemens (Germany) invented an industrial generator that did not need external magnetic power. In 1882, Thomas Edison (US) built the first large-scale electrical supply network, which provided 110 volts of direct current (DC) to 59 homes in Manhattan. In the late 1880s, George Westinghouse (US) set up a rival system using alternating current (AC), using an induction motor and transformer invented by Nikola Tesla (Serbia/US). AC eventually prevailed over DC. Another key invention was Sir Charles Parsons’ steam turbine, from 1884, which provides the mechanical power for most of the world’s electric power.
William Gllbert Demonstrates His Experiment on Electricity to Queen Elizabeth I and Her Court, a 19th Century painting by Arthur Ackland Hunt [~late 1800s]
A 19th Century painting by Arthur Ackland Hunt entitled, ‘William Gilbert Demonstrates His Experiment on Electricity to Queen Elizabeth I and Her Court.’
André-Marie Ampère (1775-1836).
André-Marie Ampère (1775-1836).
15 Lists
The law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Sir Isaac Newton (England) articulated the law in the first book of his Philosophiae Naturalis Principia Mathematica, which was presented to the Royal Society in 1686. The law was based in part on Galileo Galilei’s law of falling bodies, the result of experiments in 1589-1590. Newton was not the first to recognize the existence of gravity and the famous falling apple story is probably apocryphal. Previous theories of gravity contained some of the elements of Newton’s law, particularly the theory proposed by Robert Hooke (England) in 1674-1679, who accused Newton of stealing his idea. Newton’s law was amended (some would say superseded) by Einstein’s general theory of relativity in 1916.
Portrait of Sir Isaac Newton (1643-1727) painted in 1689, when Newton was 46, by Sir Godfrey Kneller.
Friedrich Miescher (Switzerland) first isolated deoxyribonucleic acid (DNA) in 1869. In 1910, Thomas Hunt Morgan determined that genes are located on chromosomes. In 1928, experiments by Frederick Griffith (UK) showed that traits could be transferred from one type of organism to another. In 1943, Oswald Avery (Canada/US), Colin MacLeod (Canada/US) and Maclyn McCarty (US) identified that genes are made of DNA. In 1952, Rosalind Franklin (UK) and Raymond Gosling (UK) created an x-ray diffraction image of DNA that was then used by James Watson (US) and Francis Crick (UK) to determine the double-helical structure of DNA in 1953. Experiments in 1953 by Maurice Wilkins (NZ/UK) confirmed the structure. The double helix structure, with paired nucleotide bases forming the rungs between the two strands, perfectly explains how DNA replicates during mitosis.
James Watson (b. 1928) (left) and Francis Crick (1916-2004) with a model of the DNA molecule.
14 Lists
With some exceptions, man believed that he was the center of the universe for most of history. (Archimedes suggested a heliocentric universe in 260 BCE.) In the 16th Century, it was a tenet of Christian doctrine that the Earth was a stationary globe, around which the sun, the planets and the stars revolved, and it was heresy to say otherwise. As early as 1514, astronomer and mathematician Nicolaus Copernicus (Poland) became convinced through his observations and mathematical calculations that the Earth and other planets revolved around the sun, not the other way around. He held off publishing his results until just before his death in 1543, for fear of reprisals. The Copernican model, which posited circular orbits, was revised by Johannes Kepler (Germany), who discovered that the orbits of the planets were ellipses, in his 1609 laws of planetary motion. Later in the 17th Century, Galileo Galilei (Italy) publicized telescopic observations that confirmed the heliocentric model and popularized the new view in his 1632 book Dialogue Concerning the Two Chief World Systems, the book that led to Galileo’s arrest and imprisonment by the Roman Catholic Church.
Copernicus's sun-centered model of the solar system.Copernicus’s sun-centered model of the solar system.
Dmitri Mendeleev (Russia) discovered in 1869 that the elements could be arranged according to their atomic weights and chemical properties into a table. It was then possible to derive relationships between the properties of the elements and to predict the existence, nature and properties of then-unknown elements. Mendeleev’s periodic table is essentially the same as the one in use today. Prior to Mendeleev’s discovery, other scientists made attempts to define the nature of an element, and to catalogue and categorize the known elements. These scientists included: Robert Boyle (UK) who defined an element in 1661 as “a substance that cannot be broken down into a simpler substance by a chemical reaction”; Antoine-Laurent Lavoisier (France), who made a list of elements in 1789; Johann Wolfgang Döbereiner (Germany), who made one of the first attempts to classify the elements into groups in 1829; geologist Alexandre-Emile Béguyer de Chancourtois (France), who first noticed that similar elements occur at regular intervals when ordered by their atomic weights and made an early version of the periodic table in 1862-1863; and chemist John Newlands (UK), who classified the 56 known elements into 11 groups based on their physical properties in 1865.
Mendeleev's 1871 version of the periodic table.
Mendeleev’s 1871 version of the periodic table.
In his 1687 book Philosophiae Naturalis Principia Mathematica, Sir Isaac Newton (England) established the three laws of motion – (1) the law of inertia; (2) the law of acceleration; and (3) the law of action and reaction – and derived the mathematical basis for the laws. These laws and Newton’s law of universal gravitation formed the basis for the science of physics for more than 200 years. In the early 20th Century, classical mechanics was displaced by relativity and quantum mechanics, but Newton’s laws still accurately explain the behavior of most objects in environments familiar to human life.
Newton's laws of motion, in graphic form.Newton’s laws of motion, in graphic form.
While most 19th Century scientists believed that biological organisms had undergone evolution over time, no one had been able to provide a convincing evolutionary mechanism. After returning from a voyage to South America in 1836, and reading Thomas Malthus’ works on population growth, Charles Darwin (UK) came to believe that (1) all species contained individual variations; (2) some of the variations were more advantageous than others and (3) given limits on population growth, those individuals with the more advantageous variations would be more likely to survive and reproduce. The result of such a system, over a long period of time, would be the generation of new species. Although Darwin first formulated the theory in 1839, he was afraid to publish, fearing the reaction to a theory based essentially on chance. Instead, he spent the next 20 years collecting evidence to support his conclusions. He drafted a comprehensive essay on the matter in 1844, but did not publish it. In 1858, Darwin learned that another biologist, Alfred Russell Wallace (UK), had reached nearly identical conclusions. Wallace’s paper was presented to the Royal Society in 1858 along with excerpts from Darwin’s 1844 essay. In 1859, Darwin published The Origin of Species, which set out the evidence behind his theory. The theory of evolution by means of natural selection is now the fundamental premise of the science of biology.
Photograph of Charles Darwin (1809-1882) in 1859.
An 1857 photograph of Charles Darwin (1809-1882).
X-RAYS (1895)
Researchers first noticed unidentified rays emanating from experimental discharge tubes called Crookes tubes around 1875. In 1886, Ivan Pulyui (Ukraine/Germany) discovered that sealed photographic plates darkened when exposed to Crookes tubes. Nikola Tesla (Serbia/US) began experimenting with the rays in 1887. Fernando Sanford (US) generated and detected the rays in 1891. Wilhelm Röntgen (Germany) began studying the rays in 1895 and announced their existence (coining the term ‘X-rays’) in a scientific paper. Röntgen was the first to recognize the medical use of X-rays when he X-rayed his wife’s hand. In 1896, Thomas Edison (US) invented the flouroscope for X-ray examinations. In the same year, John Hall-Edwards (UK) was the first physician to use X-rays under clinical conditions. Problems with the cold cathode tubes used to generate X-rays led to the invention of the Coolidge tube by William D. Coolidge (US) in 1913.
A photograph of Wilhelm Röntgen (1845-1923).
A photograph of Wilhelm Röntgen (1845-1923).
Albert Einstein (Germany) developed the special theory of relativity in 1905 to correct Newton’s laws of classical mechanics, which do not accurately explain phenomena at velocities near the speed of light. The theory explains how objects behave when moving at a constant speed relative to each other. Einstein relied on the principles that (a) the law of physics remain the same despite your frame of reference; and (b) the speed of light is the same to all observers. Under the theory, space and time are two aspects of the same phenomenon, meaning that reality has four dimensions instead of three. A key implication of the special theory of relativity is that time slows down as acceleration increases, a fact that has been proven many times by experiment.
A photograph of Albert Einstein (1879-1955) in about 1905.
On January 6, 1912, Alfred Wegener (Germany) proposed that the continents had once formed a single landmass and had drifted to their current positions, a theory he called ‘continental drift’. The idea that the continents moved was not new and had been suggested by Abraham Ortelius (Flanders) in 1596, Theodor Christoph Lilienthal (Germany) in 1756; Alexander von Humboldt (Germany) in 1801, Antonio Snider-Pellegrini (France) in 1858; Franklin Coxworthy in (1848-1890); Roberto Mantovani (Italy) in 1889-1909; William Henry Pickering (US) in 1907; and Frank Taylor (US) in 1908. Most scientists rejected Wegener’s hypothesis because, although there was fossil and glacial evidence to support the idea, he proposed no mechanism to explain the movements. In 1956 the discovery by Keith Runcorn (UK) and Warren Carey (Australia) that paleomagnetic stripes on the seafloor emanated from the mid-ocean ridges provided a clue to a continental drift mechanism. In 1963, Lawrence Morley (Canada), Fred Vine (UK), and Drummond Matthews (UK) independently proposed that Runcorn’s and Carey’s discovery was evidence that the seafloor was spreading, as predicted by Harry Hess in 1960-1962, and that was itself the mechanism for continental drift. Further support for the theory was found in the 1961 work of Allan Cox (US) on the magnetization of lava; W.C. Pitman’s discovery of similar patterns in the mid-Pacific ridge in 1966, and historical seismographic data analyzed by Jack Oliver (US) in 1968. Since the mid-1960s, continental drift has been subsumed within the more comprehensive plate tectonics theory.
Fossil evidence supported Wegener's theory.
A graphic depiction of some of the fossil evidence supporting the continental drift theory.
Albert Einstein’s general theory of relativity amended Newton’s law of universal gravitation to explain that the gravitational ‘pull’ of an object is best understood not as a force but as a warp in the curvature of space-time caused by the object’s mass. In 1919, Arthur Eddington and Frank W. Dyson (UK) measured the bending of starlight by the gravitational pull of the sun, thus confirming Einstein’s general theory of relativity. The general theory of relativity makes many predictions, including the expanding universe, and the existence of black holes and gravitational waves.
A 1921 photograph of Albert Einstein (1879-1955).
A 1921 photograph of Albert Einstein (1879-1955).
In 1928, Alexander Fleming (UK) discovered that a mold, Penicillium notatum, destroyed bacterial colonies. After years of research following up on Fleming’s discovery, Howard Florey (Australia/UK), Norman Heatley (UK), Ernst Chain (Germany/UK) and Andrew J. Moyer (US) developed a method of manufacturing penicillin as a drug in 1942. Dorothy Hodgkin (UK) discovered the structure of the penicillin molecule in 1943. The first antibiotic, penicillin proved to be effective against many serious diseases caused by bacterial infections.
Photograph of Alexander Fleming (1881-1955).
A photograph of Alexander Fleming (1881-1955).
In the mid-16th Century, European scientists began to challenge Aristotle’s claim that heavier objects fall faster than light ones through experimentation. Simon Stevin (Flanders), for example, showed in 1586 that two balls – one ten times heavier than the other – hit the ground at the same time when dropped 30 feet from a Delft church tower. In 1589-1590, while teaching at the University of Pisa, Galileo Galilei (Italy) not only performed similar experiments, but he also derived the mathematical equations to explain the phenomenon, as well as the acceleration of falling bodies and the phenomena of inertia and friction. He elaborated on his theories in 1634 and 1638 publications. The story that Galileo proved the theory by dropping balls from the Leaning Tower of Pisa is told by his pupil Vincenzo Viviani but may not be true, as Galileo preferred to experiment by rolling balls down an inclined board to reduce air resistance and simplify measurements. Galileo’s findings led to Isaac Newton’s law of universal gravitation.
Portrait of Galileo Galilei (1564-1642) by Giusto Sustermans in 1636.
A 1636 portrait of Galileo Galilei (1564-1642) by Giusto Sustermans.
After studying the detailed astronomical observations of astronomer Tycho Brahe (Denmark), Johannes Kepler (Germany) derived three laws that determine the motion of the planets. He devised the first two laws in 1609: (1) The orbit of every planet is an ellipse with the sun at one of the two foci; and (2) A line joining a planet and the sun sweeps out equal areas during equal time intervals. In 1619, Kepler discovered a third law: (3) The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In 1687, Sir Isaac Newton showed that Kepler’s laws were consistent with classical mechanics.
A 1610 portrait of Johannes Kepler (1571-1630).
A 1610 portrait of Johannes Kepler (1571-1630).
After years of careful study, physician William Harvey (England) published De Motu Cordis (On the Motion of the Heart and Blood), a 1628 book in which he described the entire system by which the heart distributes the blood through the arteries and the blood returns to the heart via the veins as well as many other details of the circulatory system of humans and animals. Prior discoveries about the circulatory system had been made by Galen (Ancient Greece/Ancient Rome) in the 2nd and 3rd centuries CE and Ibn al-Nafis (Syria) in 1242. Michael Servetus (Spain) published important discoveries about pulmonary circulation in 1553.
A portrait of William Harvey (1578-1657).
A portrait of William Harvey (1578-1657).
OXYGEN (1772)
Carl Wilhelm Scheele (Sweden) was the first to create oxygen gas and identify it as a separate element in 1772, although he did not publish his discovery until 1777. Joseph Priestley (England) isolated oxygen in 1774; because he published his findings in 1775, he is generally acknowledged as the discoverer. Prior to Scheele and Priestley, 16th Century scientist Robert Boyle (Ireland) determined that air was necessary for combustion and John Mayow (England) discovered that only a portion of the air was necessary for combustion and respiration. Research in the 17th and 18th centuries was slowed by the erroneous phlogiston theory, which held that when a substance burned it released phlogiston into the air, and the reason some substances burned more completely than others was that they consisted of a higher proportion of phlogiston. Although Antoine Laurent Lavoisier (France) claimed that he also discovered oxygen in 1774, most historians dispute it. Lavoisier did discover the nature of combustion and conducted important experiments on oxidation. His work also definitively disproved the phlogiston theory.
An engraved portrait of Carl Wilhelm Scheele (1742-1786).
The idea of infecting healthy individuals with some form of the same or a similar disease in order to create an immunity has a long history in China, Africa, and India. There is also evidence that inoculation was practiced in Turkey in the early 18th Century. Although there is some evidence that vaccination for smallpox occurred in England in the 1770s, Dr. Edward Jenner’s 1796 experiments with cowpox are usually identified as the first vaccinations. Jenner took pus from the blisters of farm workers infected with cowpox, a disease similar to but less lethal than smallpox, and exposed uninfected patients to it, making them immune to smallpox. He published his results in 1796, coining the phrase ‘vaccine’ to describe the method. In 1885, Louis Pasteur weakened or killed anthrax and rabies pathogens and vaccinated French citizens with great success. Pasteur adopted Jenner’s term ‘vaccination’ to describe his treatments. Vaccination became a common form of disease prevention, and vaccines have been developed for numerous other diseases, such as the polio vaccine. American microbiologist Maurice Hilleman developed 36 successful vaccines in the 1950s and 1960s for such diseases as measles, mumps, hepatitis A and B, chicken pox, meningitis and pneumonia.
An engraving taken from an 1833 portrait of Edward Jenner (1749-1823).
Radioactivity, also known as radioactive decay or nuclear decay, occurs when unstable atoms emit either alpha particles, beta particles or gamma rays from their nuclei. In the process of emitting radiation, the atom changes from one element to another. Henri Becquerel (France) discovered the radioactivity of uranium in 1896; he recognized the phenomenon was different from the recently discovered X-rays. In 1898, Marie and Pierre Curie (France) identified radium and polonium, two more radioactive elements. Ernest Rutherford (NZ/UK) identified two types of radiation – the alpha and beta rays – in 1899. Pierre Curie classified alpha and beta particle radiation in 1900. Paul Ulrich Villard (France) discovered a third type of radiation in 1900, which Rutherford called gamma rays. The dangerous effects of radiation exposure to humans were not identified until much later. Marie Curie herself died of an illness that was probably related to her frequent exposure to radioactivity.
A photograph of Henri Becquerel (1852-1908).
A photograph of Henri Becquerel (1852-1908).
Around the beginning of the 20th Century, physicists began to explore certain phenomena that did not appear to follow the rules of Newton’s classical mechanics, leading to the development of quantum theory, also known as old quantum theory, which was superseded by the more systematic quantum mechanics in about 1925. In 1900, German physicist Max Planck explained the results of his studies of light emission and absorption by theorizing that light and other forms of electromagnetic energy could only be emitted in quantized form, or quanta, which would later be renamed photons. In 1905, Albert Einstein (Germany) explained the photoelectric effect (identified by Heinrich Hertz in 1887) by postulating that light is made of individual quantum particles. Einstein also used quantum principles to explain the specific heat of solids. In 1913, Niels Bohr (Denmark) revised the model of atomic structure to explain the atomic spectra by incorporating quantum energy states into the electron orbits. In the following years Arnold Sommerfeld (Germany) further developed quantum theory.
A photograph of Max Planck (1858-1947) from 1915.
A 1915 photograph of Max Planck (1858-1947).
In 1911, Ernest Rutherford (NZ/UK) rejected J.J. Thompson’s ‘plum pudding’ model of the atom, and proposed instead what some referred to the solar system model, with a sun-like nucleus orbited by planet-like electrons. The negatively-charged electrons, which had very low mass, orbited a very small positively-charged nucleus, which contained most of the atom’s mass. Rutherford’s proposal was based in part on the 1909 experiments by Hans Geiger (Germany) and E. Marsden (UK), who scattered alpha particles using thin films of heavy metals, providing evidence that atoms possessed a discrete nucleus. Niels Bohr (Denmark) revised the model in 1913 to make it consistent with quantum theory. His electrons had fixed orbits and could only jump from one orbit to another. Arnold Sommerfeld (Germany) further revised the model to incorporate elliptical (instead of circular) electron orbits about 1920.
A photograph of Ernest Rutherford (1871-1937).
A photograph of Ernest Rutherford (1871-1937).
In 1924, Louis de Broglie (France) used Einstein’s special theory of relativity as the basis for a theory that particles can exhibit the characteristics of waves, and vice versa. De Broglie’s theory of matter waves set off a chain reaction of discoveries in 1925 setting out the principles of quantum mechanics: German physicists Werner Heisenberg, Max Born and Pascual Jordan created matrix mechanics; and Austrian physicist Erwin Schrödinger developed the Schrödinger equation, which allowed scientists to determine the likelihood that a particle would be in a particular place at a particular time, thus giving birth to wave mechanics. Further developments included Heisenberg’s uncertainty principle in 1927, and British physicist Paul Dirac’s 1928 equation, which describes the electorn’s wave function and predicted electron spin and the positron. John von Neumann (Hungary) formulated the mathematical basis for quantum mechanics in 1932.
A photograph of Erwin Schrödinger (1887-1962).
A photograph of Erwin Schrödinger (1887-1962).
In 1912, Vesto Slipher (US) became the first astronomer to discover and measure the Doppler redshifts of nebulae (later found to be distant galaxies), which provided the observational basis for the theory that the universe is expanding. In a 1924 paper, Alexander Friedmann (USSR) developed the mathematical basis for a number of possible universes, including an expanding universe. Georges LeMaitre (Belgium) first proposed that the universe was expanding in 1927. Edwin Hubble (US) obtained the first direct evidence that the universe is expanding in 1929 by comparing the distances to other galaxies with their redshifts. Hubble also devised the Hubble constant – a measure of the rate at which the universe is expanding. Recently, scientists have discovered that the expansion of the universe is accelerating.
A photograph of Edwin Hubble (1889-1953).
A photograph of Edwin Hubble (1889-1953).
Drawing on the findings of Slipher, Friedmann, Hubble and others, Georges Lemaître (Belgium) proposed in 1931 that the expanding universe, projected back in time, must have begun at a point when all the mass of the universe was concentrated at a single point, which he termed ‘the primeval atom.’ In the 1940s, George A. Gamow (USSR/US) was a stalwart proponent of Lemaître’s theory, which acquired the name ‘Big Bang’ in 1949 from steady state advocate and ‘Big Bang’ theory opponent Fred Hoyle (UK). Gamow developed aspects of the Big Bang theory, including a 1948 paper with Ralph Alpher (US) showing how the Big Bang explained current levels of hydrogen and helium in the universe through Big Bang nucleosynthesis. In addition, Alpher predicted in 1948 that cosmic microwave background radiation generated by the Big Bang should be detectable. In 1964, Arno A. Penzias and Robert W. Wilson (US) accidentally discovered the cosmic microwave background radiation. NASA’s Cosmic Microwave Background Explorer, launched in 1989, has since provided much more accurate and complex data which, as analyzed by John C. Mather and George Smoot (US) in 1992, showed fluctuations in the Big Bang that explain the large-scale structure of the universe.
A photograph of Georges Lemaître (1894-1966).
A photograph of Georges Lemaître (1894-1966).
Beginning in 1859, Louis Pasteur (France) conducted a series of experiments that proved the connection between disease and microorganisms, or germs, the results of which were published in 1862. This discovery revolutionized medicine and eventually had a significant impact on human mortality rates. Scientists whose prior work led to Pasteur’s discovery include: Girolamo Fracastoro (Italy), who proposed a germ theory in 1546; Agostino Bassi (Italy) who conducted crucial experiments in 1808-1813; Ignaz Semmelweis (Hungary) who conducted clinical studies of disease in 1847; and John Snow (UK), who studied public health response to disease outbreaks in 1854-1855. Following up on Pasteur’s findings in 1884, Robert Koch (Germany) articulated a four-part test for determining if disease is caused by microorganisms and also identified the bacteria that cause cholera, tuberculosis and anthrax.
A photograph of Louis Pasteur (1822-1895) by Nadar.
In the early 19th Century, a number of scientists working with pea plants noticed the segregation of a recessive trait, one of the key elements of the laws of heredity, but unfortunately none of these early scientists kept records of later generations, severely limiting the benefit of their work. Augustinian friar Gregor Mendel (Silesia) performed a comprehensive series of experiments on numerous generations of pea plants between 1856 and 1863 that allowed him to develop the basic rules of heredity and inheritance, including the existence of dominant and recessive traits, which would form the basis of the modern science of genetics. Mendel presented the results of his work in a paper he read at meetings of the Natural History Society of Brno, Moravia in 1865, which was published in 1866. Because the work was perceived to be about hybridization and not inheritance, it did not receive wide distribution, and most scientists, including Charles Darwin (UK), never learned of it. It was only after 1900, when other scientists, particularly Hugo de Vries (The Netherlands), Carl Correns (Germany), Erich von Tschermak (Austria) and William Jasper Spillman (US), independently rediscovered Mendel’s work, that his importance to science was appreciated.
A photograph of Gregor Mendel (1822-1884).
A photograph of Gregor Mendel (1822-1884).
A transistor is a device made of semiconductor material that amplifies and switches electronic signals and electrical power. The precursor to the transistor was the vacuum-tube triode, or thermionic valve, first created in 1907 by Lee De Forest (US). Julius Edgar Lilienfeld (Austria/Hungary) patented a field-effect transistor in 1925, but his work was ignored at the time. (Years later, William Shockley and Gerald Pearson (US) at Bell Labs made a functional device using Lilienfeld’s design.) German physicist Oskar Heil patented a field-effect transistor in 1934. In the mid-1940s, John Bardeen and Walter Brattain (US) built a semiconducting triode for use in military radar equipment. After the end of World War II, Schockley, Bardeen and Walter Brattain (US) worked on using semiconductors to replace vacuum tubes in electrical systems. In December 1947, they created a germanium point-contact transistor – the first solid-state electronic transistor. In June 1948, Shockley designed a grown-junction transistor; a prototype was built in 1949. German physicists Herbert F. Mataré and Heinrich Welker invented a transistor they called the transistron in August 1948. In 1950, Shockley developed a bipolar junction transistor. Morgan Sparks (US) at Bell Labs made the new transistor into a useful device. General Electric and RCA produced an alloy-junction transistor – a type of bipolar junction transistor – in 1951. By 1953, transistors were being used in products such as hearing aids and telephone exchanges. Dick Grimsdale (UK) built the first transistor computer in 1953. Also in 1953, Philco (US) invented the first surface-barrier transistor. In the early 1950s, Bell Labs also produced the first tetrode and pentode transistors. Around the same time, the spacistor was created, but it was soon obsolete. In 1954, teams led by Morris Tanenbaum (US) at Bell Labs and Gordon Teal (US) at Texas Instruments, working independently, invented the first silicon transistor. Also in 1954, Bell Labs produced the first diffusion transistor, while in 1955, Bell made the first diffused silicon mesa transistor, which was developed commercially by Fairchild Semiconductor (US) in 1958. Also in 1955, Tanenbaum and Calvin Fuller (US) invented a much improved silicon transistor. The first gallium-arsenide Schottky-gate field-effect transistor was invented by Carver Mead (US) in 1966.
From left: John Bardeen (1908-1991), William Shockley (1910-1989) and Walter Brattain (1902-1987) at Bell Labs in 1948.
The modern study of human anatomy began with physician Andreas Vesalius (Belgium), whose seven-volume 1543 treatise, De fabrica corporis humani, provided a detailed, well-researched and systematic study of the human body that corrected many errors of the past. Anatomical study has a long history before and after Vesalius. Ancient Egyptian treatises on anatomy date to 1600 BCE. Ancient Greek anatomists include Alcmaeon, Acron (480 BCE), Pausanias (480 BCE), Empedocles (480 BCE), Praxagoras (300 BCE?), Herophilus (280 BCE?); and Erasistratus (260 BCE?). Aristotle conducted empirical studies in the 4th Century BCE and began the study of comparative anatomy. Galen, a Greek living in the Roman Empire in the 2nd Century CE, was the first major anatomist. He was highly influential into the modern era, but performed few human dissections and propagated some serious errors. Italian physician Mondino de Luzzi performed the first human dissections since Ancient Greece between 1275 and 1326. In the late 15th Century, Leonardo da Vinci dissected approximately 30 human bodies and made detailed drawings, until the Pope ordered him to stop. In 1541, Giambattista Canano (Italy) published illustrations of each muscle and its relation with the bones.
An engraved portrait of Andreas Vesalius (1514-1564) taken from his treatise.
An engraved portrait of Andreas Vesalius (1514-1564) taken from his 1543 treatise.
In 1609 and 1610, Galileo Galilei built a series of progressively more powerful telescopes and began making detailed scientific observations of the heavens. In addition to providing support for the Copernican/Keplerian heliocentric model, Galileo discovered four of the moons of Jupiter; the phases of Venus; sunspots; lunar mountains and craters; and masses of stars in the Milky Way, which was thought to be made of clouds. Galileo published his discoveries in Sidereus Nuncius (Starry Messenger) in 1610, which became a bestseller.
Two of Galileo's original telescopes from the early 1600s.
Two of Galileo Galilei’s original telescopes from the early 1600s.
CELLS (1665)
Seventeenth Century scientist Robert Hooke (England) used the newly-invented microscope to make detailed observations of biological organisms and other materials. He published his results in 1665 in a book titled Micrographia. In the book, Hooke coined the term ‘cell’ to describe the small compartments he observed in plant tissues, including cork. One theory is that the term came from the resemblance of the plant cells to the cells of a honeycomb; others say they reminded Hooke of monk’s living quarters, known as cells. The importance of cells in biological systems would not be fully recognized until Theodor Schwann and Matthias Schleiden (Germany) proposed their cell theory in 1838-1839.
This microscope was built by Christopher Cook for Robert Hooke in the 17th Century.
A microscope built by Christopher Cook for Robert Hooke (1635-1703) in the 17th Century.
Antonie van Leeuwenhoek was a 17th Century Dutch amateur scientist and inventor who was fascinated by the microscope, and built at least 25 microscopes in his lifetime. In 1674 and 1675, van Leeuwenhoek turned his lens on pond water and was surprised to find an entire universe of tiny living creatures that humans could not see with the naked eye. Van Leeuwenhoek called his discoveries ‘animalcules’ but we now refer to them as microorganisms. Most of the microorganisms van Leeuwenhoek described belonged to the Protista, a group one-celled creatures, although some were probably multi-celled larvae of larger animals, such as insects and crustaceans. In 1676, van Leeuwenhoek was the first to see and describe bacteria. Science historians recognize van Leeuwenhoek as the first microbiologist.
A portrait of Antonie van Leeuwenhoek by Jan Verkolje from between 1670 and 1693. It is located in the Museum Boerhaave in Leiden.
A portrait of Antonie van Leeuwenhoek (1632-1723) by Jan Verkolje from between 1670 and 1693. It is located in the Museum Boerhaave in Leiden.
Although there is some evidence of primitive batteries from the first centuries of the Common Era in Mesopotamia and India, the modern precursor to the electric battery was the Leyden Jar, which was invented in 1745-1746. Benjamin Franklin coined the term ‘battery’ to describe a set of linked Leyden jars because of its resemblance to a battery of artillery pieces. Then, in 1791, Italian scientist Alessandro Volta published the results of experiments showing that two metals joined by a moist intermediary could create electric energy. In 1800, Volta used this principle to create the voltaic pile, the first true battery. Over the next century, many scientists developed Volta’s invention further: William Cruickshank (UK) invented the trough battery in 1800; William Sturgeon (UK) improved upon the design in 1835; John Daniell (UK) invented the Daneill cell in 1836; Golding Bird (UK) invented the Bird cell in 1837; John Dancer (UK) invented the porous pot Daniell cell in 1838; William Grove (UK) invented the Grove cell in 1844; Gaston Planté (France) invented the lead-acid battery in 1859; Callaud (France) created the gravity cell in the 1860s; Johann Poggendorff (Germany) created the Poggendorff cell; Georges Leclanché (France) invented the Lelanché cell in 1866; and the first dry cells were invented independently by Carl Gassner (Germany), Frederick Hellesen (Denmark) and Yai Sakizo (Japan) in 1886-1887.
One of Allessandro Volta's early voltaic piles on display at his museum in Como, Italy.
One of Allessandro Volta’s (1745-1827) early voltaic piles on display at his museum in Como, Italy.
In 1808, John Dalton (UK) theorized that all matter was made of very small indivisible particles called atoms, that each element is made of different atoms, that each element’s atoms are identical, that atoms combine to make chemical compounds and are combined, separated or rearranged in chemical reactions. The individual elements of this atomic theory were confirmed experimentally over the next two centuries.
An 1834 portrait of John Dalton by Charles Turner.
An 1834 portrait of John Dalton (1766-1844) by Charles Turner.
In the mid-19th Century, Richard Laming suggested that atoms consist of a core surrounded by small charged particles. George Johnstone Stoney (Ireland) proposed in 1874 that electricity consisted of charged ions that had a measurable charge. Hermann von Helmholz (Germany) suggested in 1881 that the positive and negative charges were divided into basic parts and both were atoms of electricity. In 1891, Stoney coined the name ‘electron’ for the fundamental unit of electricity. Experiments leading up to the discovery of the electron began with German physicist Johann Wilhelm Hittorf’s conductivity work in 1869; the discovery of cathode rays by Eugen Goldstein (Germany) in 1876; and the development of a high vacuum cathode ray tube by Sir William Crookes (UK) in the 1870s. Arthur Schuster (Germany/UK) performed cathode ray experiments that allowed him to estimate the charge-to-mass ratio of the electron. In 1896-1897, J.J. Thomson, assisted by John S. Townsend and H.A. Wilson (UK), performed a series of experiments that conclusively identified the cathode ray emissions as particles with a definite mass and a negative charge. They also showed that these particles were identical even when produced in different contexts (heating, illumination, radioactivity). George Fitzgerald (Ireland) proposed the name ‘electron’ for Thomson’s particle. In 1900, Henri Becquerel (France) showed that beta rays emitted by radioactive elements were electrons. The charge of the electron was measured more carefully by Robert Millikan and Harvey Fletcher (US) in a 1909 experiment, the results of which were published in 1911.
J.J. Thomson.
J.J. Thomson (1856-1940).
In a groundbreaking 1928 experiment, Frederick Griffith (UK) found a ‘transforming’ principle that could change one type of bacteria to another. Over the next 15 years, scientists at the Rockefeller Institute for Medical Research in New York sought to isolate the transformative substance by working with bacteria and bacteriophage viruses. In 1944, Oswald Avery, Colin MacLeod and Maclyn McCarty (US) published their surprising results: the substance that contained the genetic information was a nucleic acid, DNA (deoxyribonucleic acid), not a protein as supposed. The scientific community was reluctant to accept the results of the Avery-MacLeod-McCarty experiment. In 1952, Alfred Hershey and Martha Chase (US) followed up with conclusive proof that DNA is the substance of genes, which led to general acceptance by scientists.
From left: Oswald Avery, Colin Macleod and Maclyn McCarty.
From left: Oswald Avery (1877-1955), Colin MacLeod (1909-1972) and Maclyn McCarty (1911-2005).
Ole Rømer (Denmark) determined in 1676 that light travels at a finite speed, contrary to the common belief that light traveled infinitely fast. Christiaan Huygens (The Netherlands) used Rømer’s results to calculate the speed of light to be 220,000 kilometers/second. In 1704, Isaac Newton (England) calculated the time for light to travel from the sun to the Earth as “seven or eight minutes” (the actual time is 8 minutes, 19 seconds). James Bradley (England) discovered the phenomenon known as ‘aberration of light’ in 1729 and adjusted the calculation of the sun-earth time to 8 minutes, 12 seconds. In the 19th Century, James Clerk Maxwell (UK) proposed that light was a type of electromagnetic wave, and that all such waves traveled at the same speed. Hippolyte Fizeau (France) made a calculation of 313,300 km/sec in 1849 without using astronomical measurements. Albert Michelson and Edward Morley (US) conducted an experiment in 1887 that measured light at 185,000 miles per second. A 1928 experiment by Michelson refined the speed of light to 186,284 miles per second. The current estimate for the speed of light is 186,282 miles per second (officially 299,792,458 meters per second).
Ole Rømer.
Ole Rømer (1644-1710).
Humans have been using the steam from boiling water to do mechanical work since ancient times, but practical designs only arrived in the 17th Century. Jerónimo de Ayanz y Beaumont (Spain) patented a steam engine in 1606 for removing water from mines. In 1679, Denis Papin (France/England) developed a steam digester, a precursor to the steam engine. British engineer Thomas Savery’s pistonless steam pump of 1698 was the first practical design based on Papin’s ideas. Thomas Newcomen’s (UK) 1712 piston-driven “atmospheric-engine” proved to be the first commercially viable steam engine. In 1725, Savery and Newcomen built a steam engine for pumping water from collieries. Between 1765 and 1774, James Watt (UK) improved on the Newcomen engine by making it condensing and double acting, which hugely increased its efficiency. A high pressure steam engine was developed by Oliver Evans (US) in 1804. Further improvements followed throughout the 19th Century.
Thomas Savery.
Thomas Savery (c. 1650-1715).
One of the first steps towards an electrical telegraph was taken in 1750 by Benjamin Franklin (US), who created a device that sent an electrical signal across a conductive wire that was registered at a remote location. An electrochemical telegraph was created by Francisco Salva Campillo (Spain) in 1804; Samuel von Sömmering (Germany) made an improved version in 1809. The messages could be transmitted a few kilometers and would release a stream of bubbles in a tube of acid, which had to be read to determine the letter or number. In 1823, Francis Ronalds (UK) created the first working electrostatic telegraph using eight miles of wire in insulated glass tubing attached to clocks marked with letters of the alphabet. Baron Pavel Schilling von Canstatt (Estonia) created an electromagnetic telegraph in 1832, but it was Carl Friedrich Gauss and Wilhelm Weber (Germany) who built the first electromagnetic telegraph used for regular communication, in 1833. David Alter (US) invented the first American electric telegraph in 1836 but never used it to make a practical system. The first commercial electrical telegraph was created by William Cooke and Charles Wheatstone (UK); it was patented in May 1837 and successfully demonstrated in July 1837; they installed the system between two railway stations 13 miles apart in 1838. Edward Davy (UK) also demonstrated a telegraph system in 1837 and patented it in 1838 although he did not pursue it. Samuel Morse (US) independently invented his own electrical telegraph in 1837, while his assistant Alfred Vail developed Morse code. Morse sent the first telegram using his system on January 11, 1838, but it was not until 1844 that he sent his famous message, “What hath God wrought” from Washington, D.C. to Baltimore, Maryland. Telegraph lines connected the east and west coasts of the US by 1861 and by 1866, a trans-Atlantic telegraph cable linked Europe and the US.
A telegraph key designed by Samuel Morse and Alfred Vail, from 1844-1845.
A telegraph key designed by Samuel Morse (1791-1872) and Alfred Vail (1807-1859), from 1844-1845.
According to the law of conservation of energy, energy can change form, but it cannot be created or destroyed; because the total energy of a system does not change over time, the energy is said to be conserved. German chemist Karl Friedrich Mohr gave one of the first statements of the law in 1837. The key concept that heat and mechanical work are equivalent was first stated by Julius Robert von Mayer (Germany) in 1842. James Prescott Joule (UK) reached the same conclusion independently in 1843, as did Ludwig A. Colding (Denmark). In 1844, William Robert Grove (UK) suggested that mechanics, heat, light, electricity and magnetism were all manifestations of a single force, a notion he published in 1846. Drawing on the work of Joule and others, Hermann von Helmholtz (Germany) reached conclusions similar to Grove’s in an 1847 book, which brought about wide acceptance of the idea. In 1850, William Rankine (UK) first coined the phrase ‘law of conservation of energy’ to describe the principle.
Hermann von Helmholtz (1821-1894).
Hermann von Helmholtz (1821-1894).
Ancient physicians used various herbs, including Solanum, opium and coca to induce unconsciousness and/or relieve pain in their patients, as well as alcohol. There is some evidence that Medieval Arabs used an inhaled anesthetic. In the late 12th Century, in Salerno, Italy, physicians used a ‘sleep sponge’ soaked in a solution of opium and various herbs, which was held under the patient’s nose. The sleep sponge was used by Ugo Borgognoni and his son Theodoric (Italy) in the 13th Century. In 1275, Spanish physician Raymond Lullus invented what would later be called ether. He and Swiss physician Paracelsus experimented with animals but not humans. In 1772, Joseph Priestley (England) discovered nitrous oxide, or laughing gas, and in 1799, British chemist Humphry Davy discovered the gas’s anesthetic properties by experimenting on himself and his friends. Morphine was discovered in 1804 by Friedrich Sertürner (Germany) but it was only widely used as an anesthetic after the invention of the hypodermic syringe. In 1842, American physician Crawford Long became the first to use ether as an anesthetic for human surgery when he removed two small tumors from James Venable, one of his students, in a painless procedure, but the operation was not publicized until 1849. In a widely publicized 1846 event, Boston dentist William Morton administered inhaled ether to a patient in Massachusetts General Hospital, after which a surgeon painlessly removed a tumor. Ether was eventually replaced by other chemicals due to its flammability. Cocaine, which was first identified in 1859, became the first effective local anesthetic in 1884 when Austrian physician Karl Koller used it during eye surgery.
An artist's rendering of William Morton's 1846 use of general anesthesia, an event that became much more well known than Crawford's 1842 breakthrough.
An artist’s rendering of William Morton’s 1846 use of general anesthesia, an event that became much more well known than Crawford Long’s 1842 breakthrough.
Through his experiments with fruit flies (Drosophila melanogaster), biologist Thomas Hunt Morgan (US) proved that genes are carried on chromosomes and are the mechanical basis for heredity. In so doing, Morgan established the modern science of genetics.
Thomas Hunt Morgan in the fly room at Columbia University.
Thomas Hunt Morgan (1866-1945) in the fly room at Columbia University.
Block printing was first invented in Japan in about 700 CE. Bi Sheng (China) invented movable type printing in 1040. He made the characters from wood at first but found that ceramics worked better. Choe Yun-ui (Korea) was the first to use metal for the type, in 1234. The technology did not spread to Europe. Johannes Gutenberg (Germany) invented movable type printing independently between 1440 and 1450. Gutenberg’s major innovation was to adapt the already-existing screw press to print his pages. He also created a special metal alloy for the type; invented a device for moving type quickly; and developed a new, superior ink. The result was the production of higher quality printing at a much faster pace. Offset printing was invented by Aloys Senefelder (Germany) in 1796. The cast iron printing press, which reduced the force needed and doubled the size of the printed area, was invented by Lord Stanhope (UK) in 1800. Between 1802 and 1818, Friedrich Koenig (Germany) created a steam-powered press with rotary cylinders instead of a flatbed. In 1843, Richard M. Hoe (US) invented a steam-powered rotary printing press. Linotype printing was invented by Ottmar Mergenthaler (US) in 1884.
This replica of Gutenberg's printing press (left) and workshop is located in St. George's, Bermuda.
A replica of Johannes Gutenberg’s (1398-1468) printing press (left) and workshop located in St. George’s, Bermuda.
BOYLE’S LAW (1662)
Boyle’s Law states that as the volume of a gas increases, the pressure of the gas decreases according to an inverse mathematical proportion. The relationship between pressure and volume of gases was first recognized by Richard Towneley and Henry Power (UK) in 1661, but Irish scientist Robert Boyle confirmed the relationship and published his results in 1662 with a mathematical formula, the first to accompany a natural law. Boyle’s assistant Robert Hooke (UK) built the experimental apparatus. Edme Mariotte (France) independently reached the same result in 1676.
Johann Kerseboom's 1689 portrait of Robert Boyle.
Johann Kerseboom’s 1689 portrait of Robert Boyle (1627-1691).
THE CALCULUS (1666, 1674)
Sir Isaac Newton (England) and Gottfried Leibniz (Germany) independently invented the infinitesimal calculus in the mid-17th Century. Newton appears to have priority over Leibniz, although the question of who was the inventor was the subject of much controversy at the time. An unpublished manuscript of Newton’s supports his claim to have been working on ‘fluxions and fluents’ as early as 1666. Leibniz began his work in 1674 and first introduced the concept of differentials in 1675, as he explained to Newton in a 1677 letter. Leibniz’s first publication on calculus using differentials was in 1684. Newton explained his geometrical form of calculus in his Principia of 1687, but did not publish his fluxional notation until 1693 and not fully until 1704. Precursors to Newton and Leibniz included Pierre de Fermat (France) in 1636, René Descartes (France) in 1637, Blaise Pascal (France) in 1654, John Wallis (England) in 1656, and Newton’s teacher Isaac Barrow (England) in 1669. Bonaventura Cavalieri (Italy) developed his method of indivisibles in the 1630s and 1640s, and computed Cavalieri’s quadrature formula. Evangelista Torricelli (Italy) extended this work to other curves such as the cycloid in the 1640s, and the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. In a 1668 book, James Gregory (England) published the first statement and proof of the fundamental theorem of the calculus, stated geometrically, and only for a subset of curves. Mathematical developments after Newton and Leibniz include those of Augustin Louis Cauchy (France) in 1821; Karl Weierstrauss (Germany) in the 1850s; and Bernhard Riemann (Germany) in the 1850s.
A portrait of Gottfried Leibniz by Christoph Bernhard Francke.
A portrait of Gottfried Leibniz (1646-1716) by Christoph Bernhard Francke.
The Leyden jar is the prototype electrical condenser and the first capacitor; it was capable of storing static electric charge. The Leyden jar was invented independently in 1745 by German cleric Ewald Georg von Kleist and in 1746 by Dutch scientists Pieter van Musschenbroek and Andreas Cunaeus at the University of Leyden in The Netherlands. Leyden jars were used in many early experiments on electricity. Daniel Gralath (Poland) was the first to join multiple Leyden jars to each other in parallel to increase the stored charge, a formation for which Benjamin Franklin (US) coined the term “battery.”
An artist's imagining of the discovery of the Leyden jar by Andreas Cuneus in the laboratory of Pieter van Musschenbroek.
An artist’s imagining of the discovery of the Leyden jar by Andreas Cunaeus (1743-1797) in the laboratory of Pieter van Musschenbroek (1692-1761).
URANUS (1781)
Uranus, the seventh planet from the sun, had been recognized possibly as early as 188 BCE, and again by John Flamsteed (England) in 1690 and Pierre Lemonnier (France) between 1750 and 1769, but it was not identified as a planet due to its dimness and slow orbit. William Herschel (England) first observed Uranus in March 1781, although he originally identified it as a comet. When Anders Johan Lexell (Finland/Sweden) computed the object’s orbit in 1781, he concluded it was a planet, not a comet, the same conclusion reached by Johann Elert Bode (Germany) about the same time. Herschel acknowledged that he had discovered a new planet in 1783. Herschel suggested the name Georgium Sidus, after King George III, but it was Bode’s suggestion of Uranus, the father of Saturn, that eventually won out.
A replica of the telescope that William Herschel used to discover Uranus.
A replica of the telescope that William Herschel (1738-1822) used to discover Uranus.
In electromagnetic induction, an electromotive force is produced across a conductor when it is exposed to an electromagnetic field. Michael Faraday (UK) and Joseph Henry (US) independently discovered this phenomenon in 1831. Because Faraday published his results first, he is usually credited with the discovery. James Clerk Maxwell (UK) later devised the mathematical principle underlying electromagnetic induction, naming it Faraday’s Law. Electromagnetic induction is the principle underlying electrical generators, transformers and many other electrical machines.
A sketch of Michael Faraday's electromagnetic induction experiment. The battery (right) sends a current to the coil (A), which creates a magnetic field but no current. When the small coil is moved in and out of the large coil (B) the magnetic field changes and a current flows.
Cell theory holds that (1) All living organisms are composed of one or more cells or the products of cells; (2) The cell is the most basic unit of life; and (3) All cells arise from pre-existing, living cells. German botanist Matthias Schleiden proposed the first two premises of cell theory in 1838 to describe the plant kingdom. In 1839, Theodor Schwann (Germany) extended Schleiden’s theory to animals. Barthelemy Dumortier (Belgium) had proposed a similar theory in 1832, and Schleiden’s theory adopted Dumortier’s erroneous belief that cells were created by a crystallization process either from other cells or from outside. This portion of the theory was refuted by Robert Remak (Poland/Germany), Rudolf Virchow (Germany), and Albert Kolliker (Switzerland) in the 1850s. In 1855, Virchow proposed the third premise of cell theory, that all cells arise only from pre-existing cells.
Matthias Schleiden.
Matthias Schleiden (1804-1881).
The first law of thermodynamics is derived from the law of conservation of energy, as applied to thermodynamic systems. The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but cannot be created or destroyed. The first law of thermodynamics states that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings. The first law of thermodynamics was stated by Rudolf Clausius (Germany) in 1850. The principles were developed by William Rankine (UK) in the 1850s. The law was conceptually revised by George H. Bryan (UK) in 1907 to state, “When energy flows from one system or part of a system to another otherwise than by the performance of mechanical work, the energy so transferred is called heat.” Max Born (Germany/UK) revised this reformulation in 1921 and 1949.
Rudolf Clausius.
Rudolf Clausius (1822-1888).
The first automobiles powered by internal combustion engines used gases instead of gasoline. Samuel Brown (UK) used hydrogen to fuel his vehicle in 1826. John Joseph Etienne Lenoir (Belgium) also used hydrogen, then coal gas, to power his Hippomobile in 1860. In 1870, Siegfried Marcus (Austria) used liquid fuel to propel a handcart, known as “the Marcus car.” He developed a more sophisticated four-seat vehicle in 1888-1889. Edouard Delamare-Debouttevile (France) built a gas-powered automobile in 1884. German inventor Karl Benz made his first automobile in 1885 – the first with a practical high-speed internal combustion engine – and started production in 1888. Gottlieb Daimler and Wilhelm Maybach (Germany) designed and built the first true automobile (not a carriage with a motor) from scratch in 1885. John William Lambert (US) built a three-wheeler in 1891, the same year that Henry Nadiq (US) built a four-wheeler. René Panhard and Emile Lavassor (France) built the first automobile with a spray carburetor in 1891. Charles Duryea tested the first US gasoline-powered automobile in Massachusetts in 1893. Frederick William Lanchester (UK) built an early automobile in 1895. Canadian George Foss built a single-cylinder gasoline car in 1896. In 1908, Ford Motor Company (US) introduced the mass-produced Model T, which was considered the first affordable automobile for the middle class. More than 15 million Model T Fords were produced in factories in the US and Europe between 1908 and 1927.
An 1886 photograph of a gasoline-powered automobile designed by Gottfried Daimler.
An 1886 photograph of a gasoline-powered automobile designed by Gottfried Daimler (1834-1900).
INSULIN (1921)
In a series of experiments beginning in 1869 in Germany, scientists identified the islets of Langerhans in the pancreas and determined that these islets secreted a substance that controlled blood sugar levels. Absence of this secretion caused diabetes mellitus. Early attempts to treat diabetes with general pancreatic fluids had mixed results. Frederick Banting (Canada), working with medical student Charles Best (Canada), finally isolated and extracted the substance, now known as insulin, in 1921. James Collip (Canada) was instrumental in developing a purified extract. The first successful treatment of a human diabetic occurred in 1922. Later the same year, Eli Lilly and Co. developed a method for producing large quantities of insulin. In 1923, Banting and John MacLeod (UK) were able to purify insulin for use in humans. Frederick Sanger (UK) identified the molecular structure of insulin in the 1950s. In the early 1960s, Panayotis Katsoyannis (US) and Helmut Zahn (Germany) independently invented the first synthetic insulin, but it was not specifically designed for humans. Scientists in China synthesized insulin in 1966. In 1977, a team of scientists (Arthur Riggs, US; Keiichi Itakura, Japan/US; and Herbert Boyer, US) created the first genetically engineered synthetic ‘human’ insulin. It went on the market in 1982 as Humulin.
Banting and Best with one of the diabetic dogs they used to test insulin.
Frederick Banting (1891-1941) (right) and Charles Best (1899-1978) with one of the diabetic dogs they used to test insulin.
In 1914, Arthur Stanley Eddington (UK) hypothesized that what were called spiral nebula were actually distant galaxies. In 1924, Edwin Hubble (US) conclusively proved that the Milky Way is just one of many millions of galaxies in the universe. Precursors to Hubble included Thomas Wright (UK), who speculated in 1750 that the Milky Way was a flattened disk of stars (a galaxy) and that some nebulae might be separate galaxies. Lord Rosse (Ireland/UK) in 1845 detected individual stars in some nebulae. Vesto Slipher (US) studied nebulae and detected red shifts in 1912. Heber Curtis (US) found evidence to support independent galaxies in 1917. In 1922, Ernst Öpik (Estonia) proved that the Andromeda Galaxy is separate from the Milky Way.
The Andromeda Galaxy. At just over 2.5 million light years from Earth, it is one of the closest galaxies to the Milky Way.
At just over 2.5 million light years from Earth, the Andromeda Galaxy is one of the closest galaxies to the Milky Way.
Modern rocketry was born in 1926 when Robert H. Goddard (US) launched the first liquid fuel rocket in Auburn, Massachusetts. His invention led to the V-2 and the ICBM missiles as well as the rockets that sent satellites into orbit, men to the moon, and probes into deep space. The first rockets, fueled by gunpowder, were made by the Chinese in the 13th Century for war and fireworks. They spread to the Mongols, who then brought them to Europe and the Muslim world, including the Ottoman Empire, in the 13th, 14th and 15th centuries. The Kingdom of Mysore in southern India in the 1780s and 1790s developed an artillery rocket that used iron cylinders to contain the combustible element, which significantly improved range. William Congreve (UK) adapted the Mysore rocket to create the Congreve rocket. In 1844, William Hale (UK) altered the design of the Congreve rocket to improve its accuracy significantly.
Robert Goddard with his first liquid fueled rocket in 1926.
Robert Goddard (1882-1945) with his first liquid fueled rocket in 1926.
Ernest Rutherford proposed the existence of the neutron in 1920 to explain the disparity between the atomic number of an atom’s nucleus (i.e., the number of positively-charged protons) and the atomic mass. Some scientists believed that all the atomic mass came from protons, but that the negative charge of electons present in the nucleus canceled out some of the protons’ positive charge. But Viktor Ambartsumian and Dmitri Ivanenko (USSR) proved in 1930 that electrons could not exist in the nucleus and there must be neutral particles present. Walther Bothe and Herbert Becker (Germany) discovered unusual radiation in 1931, a result that was pursued in 1932 by Irène Joliot-Curie and Frédéric Joliot (France). Following up on the strange radiation found by the German and French scientists, James Chadwick (UK) in 1932 definitively identified the neutron, an uncharged particle approximately the same mass as the proton. The discovery of the neutron was a key in the development of nuclear reactors and atomic weapons.
James Chadwick (1891-1974).
James Chadwick (1891-1974).
The Ancient Greeks made analog computing machines to perform astronomical calculations, including the Antikythera mechanism and astrolabe (c. 150-100 BCE) and Hero of Alexandria’s automata and programmable cart (c. 10-70 CE). Abu Rayhan al-Biruni (Persia) invented the planisphere in 1000 CE; Abu Ishaq Ibrahim al-Zarqali (Moorish Spain) invented an equatorium and latitude-independent astrolabe about 1015 CE. In China, Su Song created an astronomical clock in 1090 CE. John Napier (Scotland) invented Napier’s Bones, an abacus-like device, in 1617. William Oughtred (UK) and others invented the slide rule in 1622. In 1623, Wilhelm Schickard (Germany) invented a calculating clock that was destroyed in a fire in 1624. Blaise Pascal (France) created a mechanical calculator (the Pascaline) in 1642 and built 20 copies, nine of which survive. Gottfried Wilhelm von Leibniz (Germany) invented the Stepped Reckoner in 1672; he also described the binary number system. In 1801, Josephe-Marie Jacquard (France) used punch cards to control a loom weaving a pattern. Charles Xavier Thomas de Colmar (France) made the first successful mass-produced mechanical calculator – the Thomas Arithmometer – in 1820. Between 1833 and 1837, Charles Babbage (UK) used a punch card system to design an analytical engine that, if ever completed, would have been the first programmable computer. (In 1843, Per Georg and Edward Schulz of Sweden built a working model of an older, less sophisticated Babbage design – the 1822 difference engine.) Beginning in the 1880s, a number of other mechanical calculators arrived that were based on Colmar’s Arithmometer, such as: the comptometer (Dorr Felt, US, 1887); the Addiator (Louis Troncet, France, 1889); the Yazu Arithmometer (Ryoichi Yazu, Japan, 1903); the Monroe (Jay R. Monroe, US, 1912); the Addo-X (AB Addo, Sweden, 1918); and the Curta (Curt Herzstark, Austria, 1948). Late in the 1880s, Herman Hollerith (US) used punch cards on a machine that could store and read the data contained on them by using a tabulator and a key punch machine. The machine was used to tabulate the 1890 U.S. Census. Hollerith’s company eventually became IBM. In the first half of the 20th Century, a number of analog computers were developed, usually for specific purposes. These include the Dumaresq (John Dumaresq, UK, 1902); Arthur Pollen’s fire-control system (UK, 1912); the differential analyzer (H.L. Hazen and Vannevar Bush/MIT, US, 1927); the FERMIAC (Enrico Fermi, Italy/US, 1947); MONIAC (US, 1949); Project Cyclone (Reeves, US, 1950); Project Typhoon (RCA, US, 1952); and the AKAT-1 (Jacek Karpiński, Poland, 1959). In 1909, Percy Ludgate, of Ireland, apparently unaware of Babbage’s work, independently designed a programmable mechanical computer. In 1936, Alan Turing (UK) published a paper that described the Turing Machine – the theoretical basis for all modern computers. John von Neumann (Hungary/US) invented a computer architecture based on Turing’s theory. In a 1937 MIT master’s thesis, Claude Shannon (US) showed how electronic relays and switches can realize the expressions of Boolean algebra. In 1937, George Stibitz (US), of Bell Labs, invented and built the first relay-based calculator to use binary form – the Model K. Starting in 1936, Konrad Zuse (Germany) built a series of progressively more complex programmable binary computers with memory: the Z1 (1938) never worked reliably, but the Z3 (May 1941) is considered by some the first working programmable fully automatic modern computer that meets the criteria for Alan Turing’s “universal machine.” In 1939, John V. Atanasoff and Clifford E. Berry (US) at Iowa State created the Atanasoff-Berry Computer, which was electronic and digital but not programmable. In 1940, George Stibitz and his team produced and demonstrated their Complex Number Calculator. In 1943, Max Newman, Tommy Flowers and others (UK) built the Mk I Colossus, a computer designed to break the German encryption system, building on 1941 work by Britons Turing and Gordon Welchman (who in turn built on 1938 work by Marian Rejewski, of Poland). Some consider Colossus to be the world’s first electronic programmable computing device. The improved Mk II Colossus followed in 1944. Also in 1944, the Harvard Mark I began operation, after being built at IBM’s Endicott labs by a team headed by Howard Aiken, starting in 1939. Beginning in 1943, the U.S. Government sponsored the development of ENIAC under the lead of John Mauchly and J. Presper Eckert (US) at the University of Pennsylvania. When it began operating at the end of 1945, ENIAC met all of Alan Turing’s criteria for a true computer. Also in 1945, Konrad Zuse developed the Z4, which also met Turing’s criteria. Improvements to ENIAC in 1948 made it possible to execute stored programs set in function table memory. Frederic C. Williams, Tom Kilburn and Geoff Tootill (UK) at Victoria University of Manchester built the Manchester Small-Scale Experimental Machine, or “Baby” in 1948, the first stored-program computer. Baby led to the Manchester Mark 1, which became operational in 1949. The Mark 1, in turn, led to the first commercial computer, the Ferranti Mark 1, in 1951. Maurice Wilkes (UK) at Cambridge developed the EDSAC in 1949. Not to be outdone, Australians Trevor Pearcey and Maston Beard built CSIRAC in 1949. Another commercial computer was the LEO I, made by J. Lyons & Co. (UK) in 1951. Also in 1951, the U.S. Census Bureau purchased a UNIVAC I (essentially a variation of ENIAC using a new metal magnetic tape) from Remington Rand. After years of delays, EDVAC, Eckert and Mauchly’s follow-up to ENIAC, began operations in 1951 at the Ballistics Research Lab. In 1952, IBM began marketing the 701, its first mainframe computer. In 1954, IBM released the IBM 650, a smaller, more affordable computer. Maurice Wilkes (UK) invented microprogramming in 1955. In 1956, IBM introduced the first hard disk drive – it could store five megabytes of data. Beginning about 1953, transistors began replacing vacuum tubes in computers. The invention of the integrated circuit, or microchip, led to the invention of the microprocessor in the late 1960s.
The ENIAC Computer.
J. Presper Eckert (1919-1995) (left) and John Mauchly (1907-1980) with the ENIAC computer in the 1940s.
Civilizations in Mesopotamia, the Indus Valley, the Northern Caucasus and Central Europe all invented vehicles with wheels of solid wood between 4000 and 3500 BCE. The earliest clear depiction of a wheeled vehicle was found in Poland and dates to 3500-3350 BCE. The oldest surviving wheel was found in the Ljubljana Marshes in Slovenia and dates to approximately 3250 BCE. Wheeled vehicles are found in the Indus Valley by 3000-2000 BCE. The spoke-wheeled chariot was invented in Russia and Kazakhstan some time between 2200 and 1550 BCE, and reached China and Scandanavia by 1200 BCE. Wire wheels and pneumatic tires were invented in the mid-19th Century.
The remains of the oldest existing wheel and axle, dating to 3000 BCE.
These wooden fragments from 3000 BCE are thought to be the oldest existing remains of a wheel and axle.
Greek mathematician Euclid, who lived in Alexandria, Egypt, published his Elements in about 300 BCE, setting out the fundamentals of what is now called Euclidean geometry. Many of the axioms, postulates and proofs in the Elements were originally discovered by others, but Euclid fit them all into a single comprehensive system. After Euclid, Archimedes (3rd Century BCE) developed equations for volumes and areas of various figures and Apollonius of Perga (late 3rd Century-early 2nd Century BCE) investigated conic sections. In the 17th Century, René Descartes and Pierre de Fermat (France) developed analytic geometry, an alternative method that focused on turning geometry into algebra. Also in the 17th Century, Girard Desargues (France) invented projective geometry.
This fragment of a copy of Euclid's Elements dating to c. 100 CE was found at Oxyrhynchus in Egypt.
This fragment of a copy of Euclid’s Elements dating to c. 100 CE was found at Oxyrhynchus in Egypt.
PAPER (200-100 BCE)
Although the invention of paper is traditionally attributed to Ts’ai Lun (China) in 105 CE, strong evidence indicates that the pulp process was developed in China some time earlier in the 2nd Century BCE, during the Han Dynasty. The first recipe may have included tree bark, cloth rags, hemp and fishing nets. The earliest use of paper was to wrap and pad delicate objects such as mirrors. The use of paper for writing is first seen in the 3rd Century CE. Paper was used as toilet tissue from at least the 6th Century CE. In the Tang Dynasty (618-907 CE), paper was used to make tea bags, paper cups and paper napkins. In the Song Dynasty (960-1279 CE), paper was used to make bank notes, or currency. Paper was introduced into Japan between 280 and 610 CE. In America, the Mayans developed a type of paper called amatl, made from tree bark, beginning in 5th Century CE. The Islamic world obtained the secret of papermaking by the 6th Century CE, when it was being made in Pakistan. The knowledge had spread to Baghdad by 793 CE, to Egypt by 900 CE and to Morocco by 1100 CE. In Baghdad, an inventor discovered a way to make thicker sheets of paper, a crucial development. The first water-powered pulp mills were built in 8th Century Samarkand (modern-day Uzbekistan). In 1035, a traveler noted that Cairo market sellers were wrapping customers’ purchases in paper. The first European papermaking occurred in Toledo, Spain in 1085 CE. The first French paper mill was established by 1190 CE. Arab merchants introduced paper into India in the 13th Century. The first definitive reference to a water-powered paper mill in Europe is from 1282 in Spain. Paper was expensive to make until after 1844, when Charles Fenerty (Canada) and F.G. Keller (Germany) independently developed processes for using wood pulp to make paper, instead of recycled fibers.
These scraps of hemp paper, made in China about 100 BCE, were used for wrapping.
Prior to the invention of the mechanical clock, humans kept time using sundials (a type of shadow clock), hourglasses, water clocks and candle clocks. Chinese inventors improved on the water clock by adding escapements. Liang Lingzan and Yi Xing designed and built a mechanized water clock with the first known escapement mechanism in 725 CE. Islamic scientists had also made improvements on the water clock, including a clock given as a gift to Charlemagne in 797 CE by Harun al-Rashid of Baghdad. In 976 CE, Zhang Sixun (China) was the first to replace the water in his clock tower with mercury. In 1000 CE, Pope Sylvester brought water clocks to Europe. In 1088, Su Song (China) further improved on Zhang’s design in his astronomical clock tower, nicknamed ‘Cosmic Engine.’ The first geared water clock was invented by Arab engineer Ibn Khalaf al-Muradi in Spain in the 11th Century. There is some evidence of mechanical clocks that used falling weights instead of water in France in 1176 and England in 1198. Al-Jazari (Mesopotamia) built numerous clocks in the early 13th Century; there is evidence of an Arabic mechanical clock in a 1277 Spanish book. There is also evidence of mechanical clocks in England in 1283 and 1292, as well as Italy and France. The oldest surviving mechanical clock is at Salisbury Cathedral (UK) and dates to 1386. Spring-driven clocks first appeared in the 15th Century. Clocks indicating minutes and seconds also begin to appear in the 15th Century. Jost Bürgi (Switzerland) invented the cross-beat escapement in 1584. Around the same time, the first alarm clocks were invented. The first pendulum clock was invented by Christiaan Huygens (The Netherlands) in 1656. A pendulum clock uses a weight that swings back and forth in a precise time interval, thus making this type of clock much more precise than previous designs. Galileo Galilei (Italy) had been exploring the properties of pendulums since 1602 and he designed a pendulum clock in 1637, but died without completing it. With the assistance of clockmaker Salomon Coster (The Netherlands), Huygens designed and built a pendulum clock that realized Galileo’s dream.
An illustration from a book by Su Song showing his 1088 clock tower.
An illustration from a book by Su Song (1021-1101) showing his 1088 Cosmic Engine clock tower.
The Chinese were aware by 1 CE that a magnet will align with north and south directions. About 200 CE, Chinese scientists discovered that magnetic north and true north were different. In the 16th Century, Georg Hartmann (Germany) and Robert Norman (England) independently discovered magnetic inclination, the angle between the magnetic field and the horizontal. In 1600, William Gilbert (England) published the results of his experiments using a small model of Earth, which led to his discovery that the Earth is a giant magnet, thus explaining why compasses point north. He also predicted accurately that the Earth has an iron core. Carl Friedrich Gauss (Germany) was the first to measure the Earth’s magnetic field in 1835. The true cause of the magnetic field was only discovered in the 20th Century, after the dominant theory – that the Earth is made of magnetic rocks – was disproved. In 1919, Sir Joseph Larmor (UK) proposed that a self-exciting dynamo could be the mechanism. W.M. Elsasser and Edward Bullard (UK) showed in the 1940s that the motion of a liquid core could produce a self-sustaining magnetic field.
The title page from William Gilbert's De Magnete, in which he first proposed that the Earth had a magnetic field.
The title page from the 1600 book De Magnete, by William Gilbert (1544-1603), in which it was first proposed that the Earth had a magnetic field.
The invention of logarithms by John Napier (Scotland) in 1614 made multiplying easier and thus made calculators practical. In 1632, William Oughtred (England) invented the slide rule. The first mechanical calculator, the Pascaline, was invented by Blaise Pascal (France) in 1642. Gottfried Leibniz (Germany) made a multiplication machine in 1671, but it did not improve on Pascal’s. Several machines were made in the 18th Century, including that of Poleni (Italy). The first commercial mechanical calculator was the Arithmometer of Thomas de Colmar (France), which was invented in 1820 but not marketed until 1851. Charles Babbage (UK) invented the difference machine in 1822 and a calculating machine in 1834-1835, which were programmable and precursors to the computer but were never built. Americans Frank S. Baldwin, Jay R. Monroe, and W. T. Ohdner all produced calculators in the second half of the 19th Century. Other machines included the 1886 calculating machine of William Seward Burroughs (US), the comptometer of Dorr E. Felt (US) from 1887, and Swiss inventor Otto Steiger’s “Millionaire” in 1894. James Dalton (US) introduced the Dalton Adding Machine in 1902, the first with push buttons. The Curta calculator, invented by Curt Herzstark (Austria) in 1948, was the last popular mechanical calculator. Casio (Japan) introduced the first all-electric calculator, the Model 14-A, in 1957, which was built into a desk. British Bell announced its all-electronic desktop calculators – the ANITA Mk VII and Mk VIII – in 1961. The ANITAs were among the last to use vacuum tubes. The 1963 Friden EC-130 (US) used transistors. In 1964, Sharp (US) produced the CS-10A and Industria Macchine Elettroniche (Italy) announced the IME 84. Similar models followed from these and other companies, including Canon, Olivetti, SCM, Sony, Toshiba and Wang. The next development was the hand-held pocket calculator. In 1967, Jack Kilby, Jerry Merryman and James Van Tassel (US) at Texas Instruments made a prototype of the Cal Tech, although it was still too large to fit in a pocket. In the 1970s, manufacturers reduced size by switching from transistors to integrated circuits. The first microchip pocket calculators were the Sanyo Mini Calculator, the Canon Pocketronic (based on Kilby’s Cal Tech) and the Sharp micro Compet, all in 1970. Sharp brought out the EL-8 in 1971. Mostek (US) made the MK6010 the same year. Also in 1971, Pico Electronics and General Instrument collaborated on the Monroe Digital III, a single chip calculator. Busicom (Japan) made the first truly pocket-sized calculator, the 1971 LE-120A “Handy”, at 4.9 X 2.8 X 0.9 inches. The first US pocket-sized device was the Bowmar Brain from late 1971.
The Pascaline, Blaise Pascal's calculator, from 1652.
The Pascaline, from 1652, a calculator invented by Blaise Pascal (1623-1662).
Aristotle (Ancient Greece) was the first to systematically classify living things into categories in the 4th Century BCE; he introduced the concepts of genus and species. In 1552, Konrad Gesner (Switzerland) developed a system that distinguished genus from species and order from class. Other early classification systems were developed by Andrea Caesalpino (Italy) in 1583, John Ray (UK) in 1686, Augustus Quirinus Rivinus (Germany) in 1690, and Joseph Pitton de Tournefort (France) in 1694. Beginning in 1735, Carolus Linnaeus (Sweden) developed the modern system of taxonomy for living organisms by establishing three kingdoms divided into classes, orders, families, genera and species. Classification was based on physical characteristics, often the sexual organs. He also adopted the binomial system of naming by using the genus and species name. Since the 1960s, biologists have adapted Linnaean taxonomy to include evolutionary relationships, looking at the DNA of the organism rather than relying on physical characteristics only.
A 1775 portrait of Carl Linnaeus by Alexander Roslin. It is now in Gripsholm Castle in Sweden.
A 1775 portrait of Carl Linnaeus (1707-1778) by Alexander Roslin. It is now in Gripsholm Castle in Sweden.
The spinning jenny is a multi-spindle spinning frame that was invented by James Hargreaves (England) in 1764. Dependent on the recently-invented flying shuttle, the spinning jenny held more than one ball of yarn so it could make more yarn in a shorter time, thus reducing cost and increasing productivity. The technology was replaced in about 1810 by the spinning mule.
A spinning jenny, now located at the North Hill Museum (UK).
A spinning jenny at Belper North Mill in Derbyshire, UK.
Although the notion that biological organisms change over time had ancient roots, Jean-Baptiste Lamarck (France) proposed the first fully-developed theory of evolution, or transmutation of species, in his Zoological Philosophy in 1809. Early formulations of the idea of evolution come from Epicurus (Ancient Greece) in the 3rd Century BCE; Lucretius (Ancient Rome) in the 1st Century BCE; Augustine of Hippo (Ancient Rome/Algeria) in the 4th Century CE; and Ibn Khaldun (Tunisia) in 1377. More sophisticated concepts of evolution, with or without divine intervention, came from Gottfried Leibniz (Germany) in the early 18th Century; Benoît de Maillet (France) in 1748; and Pierre Louis Maupertuis (France) in 1751. Charles Bonnet (Switzerland) first used the term evolution to refer to species development in 1762. Between 1749 and 1788, G.L.L. Buffon (France) suggested that each species is just a well-marked variety that was modified from an original form by environmental factors. In 1753, Denis Diderot (France) wrote that species were always changing through a constant process of experiment where new forms arose and survived or not based on trial and error. James Burnett, Lord Monboddo (England), suggested between 1767 and 1792 that man had descended from apes and that organisms had transformed their characteristics over long periods of time in response to their environments. In 1796, Charles Darwin’s grandfather, Erasmus Darwin, published Zoönomia, which proposed that “all warm-blooded animals have arisen from one living filament”, a theme he developed in his 1802 poem Temple of Nature. The mechanism for evolution was a source of much controversy. Lamarck proposed that organisms acquired new characteristics during their lifespans (such as longer necks from stretching to reach food on trees), which they then passed down to their offspring. He also believed in spontaneous generation of species. Many scientists rejected these ideas. Prominent evolutionists in the years after Lamarck included Étienne Geoffroy Saint-Hilaire (France), Robert Grant (UK) (whose pupils included a young Charles Darwin), Robert Jameson (UK), and Robert Chambers (UK), whose anonymous Vestiges of the Natural History of Creation proposed that evolution was progressively leading to better and better organisms. It was not until 1858 that Charles Darwin and Alfred Russell Wallace provided a convincing mechanism for evolution: natural selection. In the years after Darwin, developments in genetics, molecular biology and paleontology have brought about many changes to the field now known as evolutionary biology.
Jean-Baptiste Lamarck (1744-1829).
Jean-Baptiste Lamarck (1744-1829).
The second law of thermodynamics states that the entropy of an isolated system never decreases, because isolated systems always evolve toward thermodynamic equilibrium, which is a state with maximum entropy. The earliest statement of the law was by Sadi Carnot (France) in 1824, who, while studying steam engines, postulated that no reversible processes exist in nature. Beginning in 1850, Rudolph Clausius (Germany) set out the first and second laws of thermodynamics, although it is his 1854 formulation that was most highly regarded: “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.” William Thomson, Lord Kelvin (UK) reformulated the second law in 1851 as: “It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.”
A graphic description of the Second Law of Thermodynamics.
A graphic description of the Second Law of Thermodynamics.
Michael Faraday (UK) created the first disk generator in 1831. Hippolyte Pixii (France) made the first alternating current generator in 1832 and the first oscillating direct current generator in 1833. Charles Wheatstone (UK) created a magneto-electric generator in 1840. Anyos Jedlik (Hungary) created electromagnetic rotating devices between 1852-1854. Werner von Siemens (Germany) made a generator with double-T armature and slots windings in 1856. Wheatstone, von Siemens and Samuel Alfred Varley (UK) independently invented the dynamo-electric machine (dynamo) in 1866-1867. Zénobe Gramme (Belgium) made the first anchor ring motor in 1871. J.E.H. Gordon (UK) invented an alternating current generator in 1882. William Stanley, Jr. (US) of Westinghouse Electric demonstrated an alternating current generator in 1886. In 1891, Sebastian Ziani de Ferranti and Lord Kelvin (UK) invented the Ferranti-Thompson alternator. Also in 1891, Nikola Tesla (Serbia/US) patented a high-frequency alternator.
A drawing of Michael Faraday’s original disk generator. The horseshoe-shaped magnet (A) created a magnetic field through the disk (D). The turning of the disk induced an electric current, which traveled radially from the center toward the rim. The current then flowed through the sliding spring contact (m), through the external circuit, and back into the center of the disk through the axle.
Humphry Davy (UK) invented an arc lamp in 1801 but it was not very bright and did not last very long. James Bowman Lindsay (Scotland) invented an incandescent electric light in 1835 but failed to pursue it. Others who produced light bulbs were: Warren de la Rue (UK) in 1840; Frederick de Moleyns (UK) in 1841; John W. Starr (US) in 1845; Jean Eugène Robert-Houdin (France) in 1851; Joseph Swan (UK) in 1860; Alexander Lodygin (Russia) in 1872; and Henry Woodward and Mathew Evans (Canada) in 1874. A.E. Becquerel (France) invented a fluorescent lamp in 1867. In 1878, Joseph Swan and Charles Stearn (UK) developed an effective light bulb using a carbon rod from an arc lamp, but it was not commercially viable due to the high current required and short lifetime. Swan switched to a carbon filament by 1880 and began installing light bulbs in British homes. Thomas Edison (US) began experimenting with light bulbs in 1878 and tested a long-lasting carbon filament bulb in 1879. He began installing light bulbs in 1880. Lewis Latimer, an Edison employee, made further improvements on the Edison bulb between 1880 and 1882. Meanwhile Hiram Maxim and William Sawyer (US) set up a competitor to Edison. In 1897, Walther Nernst (Germany) made an incandescent bulb that did not require a vacuum. Carl Auer von Welsbach (Austria) made the first commercial metal filament lamp in 1898. Frank Poor (US) also made improvements in 1901. In 1903, Willis Whitney (US) made a metal-coated carbon filament that did not blacken the bulb. In 1915 Irving Langmuir invented a tungsten filament. Peter Cooper Hewitt (US) made the first mercury vapor lamp in 1903 and Georges Claude (France) invented the neon light bulb in 1911.
Light bulbs from 1878-1880 from Joseph Swan (left) and Thomas Edison (right).
Light bulbs from 1878-1880 from Joseph Swan (1828-1914) (left) and Thomas Edison (1847-1931).
RADIO (1895)
James Clerk Maxwell (Scotland) established the mathematical basis for propagating electromagnetic waves through space in a paper published in 1873. David E. Hughes (Wales/US) was probably the first to intentionally send a radio signal through space in 1879 using his spark-gap transmitter, although the achievement was misunderstood at the time. In 1880, Alexander Graham Bell and Charles Sumner (US) invented the photophone, a wireless telephone that transmitted sound on a beam of light. In 1885, Thomas Edison (US) invented a method of electric wireless communication between ships at sea. In 1886, Heinrich Hertz (Germany) conclusively demonstrated the transmission of electromagnetic waves through space to a receiver. Édouard Branly (France) improved the receiver device in 1890. In 1892, Nikola Tesla (Serbia/US) invented the Tesla coil, which generated alternating current electricity; in 1893 Tesla developed a wireless lighting device and in 1898 he demonstrated a remote controlled boat. In 1894, Sir Oliver Lodge (UK) improved Branly’s receiver, calling it a coherer, and demonstrated a radio transmission in 1894. In the same year, Lodge showed the reception of Morse code signals by a wireless receiver. Also in 1894, Jagadish Chandra Bose (India) demonstrated transmission of radio waves over distance; Bose developed an improved transmitter and receiver in 1899. Guglielmo Marconi (Italy/UK) read Lodge’s and Tesla’s papers in 1894 and built his first radio devices in early 1895. By the end of 1895, he had developed a device that could transmit radio waves 1.5 miles. In 1896, Marconi moved to England, where he presented his device to Sir William Preece at the British Telegraph Service. By 1897, Marconi had patented his device and started his own wireless business, which established radio stations at various locations. In 1899, Marconi sent radio waves across the English Channel; he sent the first transatlantic message, possibly as early as 1901. Alexander Popov (Russia) built and demonstrated improved versions of both the transmitter and receiver, first in May 1895 for a scientific group and then a public display in March 1896. There is some evidence that Popov set up a radio transmitter with two-way communication between a naval base and a battleship in 1900. Beginning in 1899, Ferdinand Braun (Germany) made significant improvements to the design of wireless devices, including inventing the closed circuit system and increasing the distance the signals would carry. Roberto Landell de Moura, a Brazilian priest and scientist, invented a radio in 1900 that could transmit a distance of eight kilometers. In 1904, Sir John Fleming (UK) invented the vacuum electron tube, which became the basis for radio telephony. Lee de Forest (US) invented the triode amplifying tube in 1906. In 1912, Edwin H. Armstrong (US) invented the regenerative circuit, which allowed long-distance sound reception. Armstrong also discovered frequency modulation, or FM radio, in 1933.
Guglielmo Marconi is shown with an early radio shortly after his arrival in England in 1896.
A photograph of Guglielmo Marconi (1874-1937) with his radio shortly after his 1896 arrival in England.
The Earth’s atmosphere is made up of the following layers: (1) troposphere (0 to 7 miles) (the top of the troposphere is called the tropopause); (2) stratosphere (7-31 miles); (3) mesophere (31-50 miles); (4) thermosphere (50-440 miles) and (5) exosphere (440 miles and up). The ozone layer is located in the stratosphere, usually between 9.3-21.7 miles. The ionosphere includes the mesosphere, the thermosphere and part of the exosphere (31-621 miles). In 1902, Léon Philippe Teisserenc de Bort (France) and German scientist Richard Assmann both discovered independently the atmosphere is divided into troposphere and stratosphere. Oliver Heaviside (UK) proposed the existence of the conducting layer known as the ionosphere, in 1902. Also in 1902, Arthur Edwin Kennelly (Ireland/US) discovered some of the radio and electrical properties of the ionosphere. Robert Watson-Watt (UK) coined the term ‘ionosphere’ in 1926. Edward V. Appleton (UK) experimentally confirmed the existence of the ionosphere in 1927. Lloyd Berkner (US) measured the ionosphere’s height and density in the 1950s. Charles Fabry and Henri Buisson (France) discovered the ozone layer in 1913. G.M.B. Dobson (UK) studied the ozone layer and set up a worldwide network of ozone monitoring stations between 1928 and 1958.A diagram of the Earth's atmosphere.
A diagram of the Earth’s atmosphere.
Humans have tried to fly since ancient times. Abbas Ibn Firnas (Berber/Andalusia) built a glider in the 9th Century; Eilmer of Malmesbury (UK) tried it in the 11th Century; and Leonardo da Vinci (Italy) designed a man-powered aircraft in 1502. Sir George Cayley (UK) designed fixed-wing airplanes from 1799 and built models from 1803. He built a successful glider in 1853. In 1856, Jean-Marie Le Bris (France) took the first powered flight when a horse pulled his glider, the Albatross, across a beach. John J. Montgomery (US) made a controlled flight in a glider in 1883, as did Otto Lilienthal (Germany), Percy Pilcher (UK) and Octave Chanute (France/US) about the same time. Between 1867 and 1896, Lilienthal made numerous heavier-than-air glider flights. Clément Ader (France) built a steam-powered airplane in 1890 and may have flown 50 meters in it. Hiram Maxim (US/UK) built an airplane powered by steam engines in 1894 that had enough lift to fly, but was uncontrollable and never actually flew. Lawrence Hargrave (Australia) experimented with box kites and rotary aircraft engines in the 1890s. In 1896, American Samuel Pierpont Langley’s Aerodrome No. 5 made the first successful sustained flight of an unmanned, engine-driven heavier-than-air craft, but his attempts at manned flight in 1903 did not succeed. There is some evidence that Gustave Whitehead (Germany/US) flew his Number 21 powered monoplane at Fairfield, Connecticut (US) in 1901, two and a half years before the Wright Brothers, but the matter is subject to debate. Most believe that Orville and Wilbur Wright (US) accomplished “the first sustained and controlled heavier-than-air powered flight” (FAI) on December 17, 1903 at Kill Devil Hills, North Carolina. By 1905, the third version of the Wright Brothers’ airplane was capable of fully controllable, stable flight for substantial periods. Traian Vuia (France) flew in a self-designed, fully self-propelled, fixed wing aircraft with a wheeled undercarriage in 1906. Jacob Ellehammer (Denmark) also flew a monoplane in 1906. In 1906, Alberto Santos Dumont (Brazil) flew 220 meters in less than 22 seconds, without the assistance of a catapult. In 1908-1910, Dumont designed a number of Demoiselle airplanes that were well received. In 1908 and 1909, Louis Blériot (France) designed airplanes that were improvements over earlier models. The first jet aircraft was the German Heinkel He 178, first tested in 1939, followed by the Messerschmitt Me 262 in 1943. The first aircraft to break the sound barrier was the Bell X-1, in 1947. The first jet airliner was the de Havilland Comet, introduced in 1952. The first widely successful commercial jet was the Boeing 707, which arrived in 1958. The Boeing 747 was the largest passenger jet from 1970 until 2005, when it was surpassed by the Airbus A380.
The Wright Brothers' first powered flight, December, 1903.
Orville Wright (1871-1948) observes as his brother Wilbur Wright (1867-1912) pilots an airplane in the first powered flight on December 17, 1903 at Kill Devil Hills in North Carolina. Photograph by John T. Daniels.
Superconductivity is a phenomenon in which certain materials experience zero electrical resistance and expulsion of magnetic fields when cooled below a critical temperature. Heike Kamerlingh Onnes (The Netherlands) discovered lack of electrical resistance in liquid helium in 1911. In 1933, Fritz Walther Meissner and Robert Ochsenfeld (Germany) discovered that substances undergoing superconductivity expelled their magnetic fields, which became known as the Meissner Effect. In 1935, Fritz and Heinz London (Germany) developed a mathematical explanation for superconductivity. Lev Landau and Vitaly Ginzburg (USSR) proposed a phenomenological theory of superconductivity in 1950. John Bardeen, Leon Cooper and John Scheiffer (US) developed a complete microscopic theory of superconductivity (the BCS theory) in 1957. The Landau-Ginzburg and BCS models were reconciled through the work of N.N. Bogolyubov (USSR) in 1958 and Lev Gor’kov (USSR) in 1959.
Heike Kamerlingh Onnes (1853-1926).
Heike Kamerlingh Onnes (1853-1926).
That certain diseases were caused by the lack of particular nutrients was known by the Ancient Egyptians. In 1747, Scottish physician James Lind discovered that citrus fruits prevented scurvy. Deprivation experiments allowed late 18th and early 19th Century scientists to identify a lipid from fish oil, then called ‘antirachitic A’ (later identified as vitamin D), that cured rickets. In a series of experiments with mice in 1881, Nikolai Lunin (Russia) found that an unidentified natural component of milk prevented scurvy. Takaki Kanehiro (Japan) performed an experiment on Japanese naval crews showing that a diet of only white rice lacked a nutrient that prevented beriberi. In 1897, Christiaan Eijkman (The Netherlands) showed that a diet of unpolished white rice led to beriberi in chickens, while polished rice prevented it. In 1907, Norwegian physicians Axel Holst and Theodor Frølich conducted a series of experiments with guinea pigs that set the stage for the discovery of ascorbic acid, or vitamin C. In 1910, Umetaro Suzuki (Japan) became the first scientist to isolate a vitamin complex, which he called aberic acid (later Orizanin and ultimately identified as vitamin B1, or thiamin) but the discovery received little attention. In 1912, Frederick Hopkins (UK) conducted a series of experiments that led him to the conclusion that some foods contained what he called ‘accessory factors’ that were necessary for functioning. Casimir Funk (Poland) independently repeated Suzuki’s results in 1912, calling the micronutrients a “vitamine” for vital amines, although the name was shortened to vitamin when it became clear that not all vitamins were amines. Elmer V. McCollum and M. Davis (US), discovered vitamin A in 1912–1914. McCollum also discovered vitamin B in 1915-1916. Sir Edward Mellanby (US) discovered vitamin D in 1920; McCollum also isolated vitamin D in 1922. Also in 1922 Sir Herbert McLean Evans (US) discovered vitamin E. D.T. Smith and E.G. Hendrick (US) discovered vitamin B2 (riboflavin) in 1926. Henrik Dam (Denmark) and Edward Adelbert Doisy (US) discovered vitamin K in 1929. Paul Karrer (Switzerland) determined the structure for beta-carotene, the precursor of vitamin A, in 1930. Between 1928 and 1932, a Hungarian team led by Albert Szent-Györgyi and Joseph L. Svirbely, and an American team led by Charles Glen King, first identified and isolated vitamin C (ascorbic acid). The discovery was confirmed by Karrer and Norman Haworth (UK). Vitamin C was the first vitamin to be synthesized in the laboratory, by Haworth and Edmund Hirst in 1933-1934, and independently by Tadeus Reichstein (Poland) in 1933.
Frederick Gowland Hopkins (1861-1947).
Frederick Gowland Hopkins (1861-1947).
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electrostatic force of attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction. In 1704, Sir Isaac Newton (England) proposed that “particles attract one another by some force, which in immediate contact is exceedingly strong, at small distances performs the chemical operations, and reaches not far from the particles with any sensible effect.” In 1801, Jöns Jakob Berzelius (Sweden) developed a theory of chemical bonding that emphasized the electronegative and electropositive character of the combining atoms. By the mid-19th century, Edward Frankland (UK), F.A. Kekulé (Germany), A.S. Couper (UK), Alexander Butlerov (Russia), and Hermann Kolbe (Germany), developed the theory of valency (originally called ‘combining power’), which held that compounds joined due to an attraction of positive and negative poles. In 1916, American chemist Gilbert N. Lewis developed the modern concept of the electron-pair bond, in which two atoms may share one to six electrons, thus forming the single electron bond, a single bond, a double bond, or a triple bond. According to Lewis, “An electron may form a part of the shell of two different atoms and cannot be said to belong to either one exclusively.” Also in 1916, Walther Kossel (Germany) put forward a theory that assumed complete transfers of electrons between atoms, and was thus a model of ionic bonds. Both Lewis and Kossel structured their bonding models on that of Abegg’s rule of 1904. In 1927, Danish physicist Oyvind Burrau was the first to describe a simple chemical bond in mathematically complete quantum terms. Walter Heitler (Germany) and Fritz London (Germany/US) invented a more practical approach in 1927, which is now called valence bond theory. In 1929, Sir John Lennard-Jones (UK) introduced the linear combination of atomic orbitals molecular orbital method (LCAO) approximation.
Notes of chemical bonds made by Gilbert Lewis (1875-1946) in 1902.
Notes of cubic chemical bonds made by Gilbert Lewis (1875-1946) in 1902.
Even before publication of Charles Darwin’s The Origin of Species, skulls of Neanderthals, a close relative of modern man, had been discovered in Belgium (1829), Gibraltar (1848) and the Neander Valley in Germany (1856). Eugène Dubois (The Netherlands) discovered a fossil skeleton of “Java Man”, now called Homo erectus, in Java in 1891. It was Australian scientist Raymond Dart’s 1924 discovery (published in 1925) of a fossilized skull of a new species of hominid, Australopithecus africanus, in Taung, South Africa, that convinced many in the scientific community that humans had evolved from other species in Africa. (Dart’s find became known as the Taung Child.) Subsequent discoveries have included British scientist Louis Leakey’s 1964 discovery of Homo habilis in Tanzania; American Donald Johanson’s discovery of an almost complete skeleton of Australopithecus afarensis, known as “Lucy”, in Ethiopia in 1974; British scientist Mary Leakey’s 1978 discovery of 3.5 million year old fossilized human footprints in Tanzania; and the discovery of a 1.6 million year old Homo erectus skeleton in Kenya in 1984 by Richard Leakey (UK) and Alan Walker (UK). In 1994, Meave Leakey (UK) discovered Australopithecus anamensis, which lived in Kenya and Ethiopia about 4 million years ago. Tim White (US) discovered the 4.2 million year old Ardipithecus ramidus in Ethiopia in 1995. In 2000, Martin Pickford (UK) and Brigitte Senut (France) found a bipedal hominid in Kenya from six million years ago that they named Orrorin tugenensis. In 2001, Michel Brunet (France) found a skull of a 7.2 million year old bipedal hominid in Chad, which he named Sahelanthropus tchadensis. In addition to the fossil record, since the 1960s, much of the study of human evolution has been conducted through analysis of the DNA of living humans and apes.
Raymond Dart (1893-1988) with the Taung Child skull.
While no one has yet definitively determined how life began, a number of theories have been proposed and at least one famous experiment conducted. In a famous 1871 letter, Charles Darwin speculated:
In 1922 and 1924, Alexander Oparin (USSR) suggested that life could have arisen from basic organic chemicals in the Earth’s primordial ocean given a strongly reducing atmosphere (methane, ammonia, hydrogen and water vapor) and the forces of natural selection. J.B.S. Haldane (UK) made similar proposals in 1926 and 1929, in which he suggested that an ‘oily film’ would have enclosed self-reproducing molecules, creating the first cells. Both Oparin and Haldane suggested that complex organic molecules might begin to self-reproduce while still inanimate. The first experiment to test the Oparin-Haldane theory was conducted by Stanley Miller and Harold Urey (US) in 1953. They simulated an early Earth atmosphere and ocean by placing liquid water, methane, ammonia and hydrogen in a sealed container with a pair of electrodes. They heated the water to induce evaporation, and fired sparks between the electrodes to simulate lightning, then cooled the environment to allow the products in the atmosphere to condense. The result was the production of many organic compounds, including all the amino acids needed to make proteins, and sugars. No nucleic acids were created. Many others have followed up the experiment. In 1961, Joan Oró (Spain) was able to create a nucleotide base from hydrogen cyanide and ammonia in water. As scientists have learned more about the early Earth’s atmosphere and other conditions, revised experiments have been conducted.
A diagram of the Miller-Urey experiment.
A diagram of the Miller-Urey experiment.
QUARKS (1964)
Murray Gell-Mann and George Zweig (US) independently proposed the quark model, also known as quantum chromodynamics, in 1964. They suggested that there were three types of quarks (up, down and strange) and that all hadrons (including protons and neutrons) were composed of combinations of quarks and antiquarks. In 1965, Sheldon Lee Glashow and James Bjorken (US) proposed charm, the fourth quark. Experiments by Jerome Friedman, Henry Kendall, and Richard Taylor (US) in 1968 using the Stanford Linear Accelerator eventually revealed the existence of the up, down and strange quarks. In 1973, Makoto Kobayashi and Toshihide Maskawa (Japan) proposed two more quarks: top and bottom. The charm quark was observed in 1974 by Burton Richter and Samuel Ting (US). In 1977, the bottom quark was observed by Leon Lederman (US). A team at Fermilab found the top quark in 1995.
Murray Gell-Mann (1929- ).
Murray Gell-Mann (1929- ).
A 2011 photograph of George Zweig (1937- ).
A 2011 photograph of George Zweig (1937- ).
The modification of living organisms has been part of human culture since the domestication of plants and animals through selective breeding and hybridization, which dates back many thousands of years. After the discoveries of genetics and the chemistry of DNA, scientists began to learn how to modify and engineer biological organisms through manipulation of their genes. In 1927, H.J. Muller (US) first used x-rays to create genetic mutations in plants. Barbara McClintock and Harriet Creighton (US) showed direct physical recombination in DNA in 1931. In 1967, scientists discovered DNA ligases, which could join pieces of DNA together. In the late 1960s, Stewart Linn (US) and Werner Arber (Switzerland) discovered restriction enzymes. In 1970, Hamilton Smith (US) used restriction enzymes to target DNA at a specific location and separate the pieces. Also in 1970, Morton Mandel and Akiko Higo (US) inserted a bacteriophage virus into the DNA of the E. coli bacteria. In 1972, Paul Berg (US) created the first recombinant DNA molecules. Also in 1972, Herbert Boyer and Stanley Cohen (US) inserted recombinant DNA into bacterial cells using a technique called DNA cloning. They then created the first genetically modified organism by inserting a gene for resistance to an antibiotic into bacteria that had no such gene, making the bacteria resistant. Later, they placed a frog gene into a bacterial cell. In 1973, Rudolf Jaenisch (Germany/US) inserted foreign DNA into a mouse. In 1974, Cohen, Annie Chang and Herbert Boyer (US) created a genetically modified DNA organism. Beginning in 1976, recombinant DNA research has been subject to regulation in the US. Frederick Sanger (UK) developed a way to sequence DNA in 1977. In 1979, scientists were able to modify bacteria to produce human insulin. In 1981, Frank Ruddle (US), Frank Constantini and Elizabeth Lacy (UK) were able to pass new genes into subsequent generations by inserting foreign DNA into a mouse embryo. In 1983, Michael Bevan, Richard Flavell (UK) and Mary-Dell Chilton (US) inserted new genetic material into a tobacco plant – the first genetically modified plant. In 1983, Kary Mullis (US) identified the polymerase chain reaction, which amplified small sections of DNA. In 1984, mice were genetically modified to predispose them to cancer. In the late 1980s, electroporation – the use of electricity to make a cell membrane more porous – increased scientists’ ability to insert foreign DNA into cells. In 1989, Mario Capecchi (US), Martin Evans (UK) and Oliver Smithies (UK/US) were the first to manipulate a mouse’s DNA to turn off a gene. After the discovery of microRNA in 1993, Craig Mello and Andrew Fire (US) were able to silence genes in mammalian cells in 2002 and in an entire mouse in 2005. The first of many commercial enterprises featuring genetic engineering was Genentech, founded by Boyer and Robert Swanson (US) in 1976. The release of GMOs into the environment has been a source of controversy and has generated protests around the world.
A diagram of one form of genetic engineering.
A diagram of one form of genetic engineering.
The US (with assistance from Europe) launched the Hubble Space Telescope and its 2.4 meter (7.9 ft.) mirror into low Earth orbit in April 1990. After an adjustment to its mirror in 1993, the telescope has been able to observe distant space objects in the ultraviolet, visible and infrared spectra. Its images have helped scientists: (1) determine the rate of expansion of the universe (the Hubble constant); (2) accurately measure the age of the universe; (3) identify that the expansion of the universe in accelerating; (4) locate black holes in the center of galaxies; (5) create deep field images of distant galaxies; (6) understand the nature of the early universe; (7) identify and measure the effects of dark energy; and (8) measure the atmospheres of extrasolar planets. Over 9,000 papers based on Hubble data have been published in peer-reviewed journals. As of September 2014, the Hubble Space Telescope was still operating.
A photo of the Hubble Space Telescope taken by the crew of the Space Shuttle Columbia in 2002.
(c. 500 BCE)
The idea that all matter is composed of tiny particles called atoms, known as the atomic theory, or atomism, was proposed in India by Jain philosophers of the Ajivika and Carvaka schools in the 6th Century BCE. Ancient Greek philosophers Leucippus and Democritus advocated atomism in c. 500 BCE. Epicurus adopted a form of atomism in the 3rd Century BCE, and his ideas were promoted by Roman philosopher-poet Lucretius in the 1st Century BCE. In the 2nd Century BCE, Kanada (India) founder of the Vaisheshika philosophy, held that that the world was composed of atoms, but that there were different kinds of atoms for each element, while the Jains and the Greeks believed that all atoms were alike. It is not clear if the Indian and Greek atomistic philosophies developed independently or whether one influenced the other.
A bust of Democritus (460-370 BCE).
GUNPOWDER (800-900 CE)
Historians believe that Chinese alchemists invented gunpowder in the 9th Century CE while they were looking for a chemical that would make them immortal. They soon found out the explosive potential for their discovery and it was used in creating many weapons, including fireworks (10th Century CE), flamethrowers (1000) and bombs (1220). The Chinese had perfected the recipe by the mid-14th Century. The Mongols learned about gunpowder when they conquered China in the mid-13th Century and spread it throughout the world during their subsequent invasions. The Arab world obtained gunpowder in the mid-13th Century. The Mamluks used cannons against the Mongols in 1260. In 1270, Syrian chemist Hasan al-Rammah described a method for purifying saltpeter in making gunpowder. Europeans first saw gunpowder used by the Mongols at the Battle of Mohi in 1241. Roger Bacon (UK) referred to gunpowder in a 1267 book. The first known use of gunpowder used by Europeans in battle was during the 1262 siege of the Spanish city of Niebla by Castilian King Alfonso X. By 1350, cannons were a common sight in European wars. India had gunpowder technology from at least 1366 CE, if not earlier. In the late 14th Century, European powdermakers began adding liquid and ‘corning’ the powder, which improved performance significantly.
A 14th Century illustration of a phalanx-charging fire-gourd, a type of Chinese fire lance.
A 14th Century illustration of a phalanx-charging fire-gourd, a type of Chinese fire lance that was powered by gunpowder.
The scientific method is the set of techniques and principles used in investigating phenomena, obtaining new knowledge and correcting or assimilating prior knowledge. The scientific method is based on empirical and measurable evidence and rests on certain rational principles. According to the Oxford English Dictionary, the scientific method involves “systematic observation, measurement and experiment, and the formulation, testing and modification of hypotheses.” The scientific method contrasts with the very influential method proposed by Aristotle of reasoning from first principles. Muslim scientists such as Jabir ibn Hayyan (721-815 CE) and Alkindus (801-873 CE) were among the first to use experiment and quantification to test theories. Use of the scientific method is clear in Arab Iraqi scientist Ibn al-Haytham’s Book of Optics (1021) and Persian scholar Kamal al-Din al-Farisi’s early 14th Century revision of the Optics. Abu Rayhan al-Biruni (Persia) used a quantitative scientific method in studying mineralogy, sociology and mechanics in the 1020s and 1030s. Persian scientist and physician Ibn Sina (Avicenna) set out a method using hypotheses in The Book of Healing, from 1027. In the 1220s, Robert Grosseteste (England) published a commentary of Aristotle’s Posterior Analytics in which he set out some aspects of the scientific method, including (1) take particular observations to create a universal law, and then use the universal law to predict particular observations; and (2) verify scientific principles through experimentation. Roger Bacon (England) followed up on Grosseteste’s work in his 1267 Opus Majus, which systematically set out the principles of the scientific method. The scientific method’s next champion did not arise for almost 400 years. Francis Bacon (England) sought to overturn the Aristotelian methods used in science education and practice by focusing on inductive reasoning and experimentation, especially in his Novum Organum of 1620, which reintroduced the scientific method to the modern world. In the first half of the 17th Century, Galileo Galilei (Italy) promoted the scientific method in the face of Aristotelianism, by using observation, experiment, and inductive reasoning, and by changing his views based on the empirical findings. René Descartes (France) provided philosophical premises for the scientific method in 1637. In 1687, Sir Isaac Newton (England) set out four rules of reasoning in science that embodied principles of the new scientific method. After philosopher David Hume (Scotland) attacked inductive reasoning beginning in 1738, scientists and philosophers sought to rehabilitate scientific knowledge. These included Hans Christian Ørsted (Denmark) in 1811, John Herschel (UK) in 1831, William Whewell (UK) in 1837 and 1840, and John Stuart Mill (UK) in 1843, and William Stanley Jevons (UK) in 1873 and 1877. Claude Bernard (France) applied the scientific method to medicine in 1865. Charles Sanders Peirce (US) articulated the modern scheme for testing hypotheses and the importance of statistical knowledge in science in 1878. Karl Popper proposed a revision of the scientific method in 1934 by stating that a scientific hypothesis must be falsifiable. Not all scientists agreed with Popper, including Thomas Kuhn, in 1962, who noted that different scientists worked differently and falsifiability was not a methodology that scientists actually follow.
A statue of Roger Bacon at the Oxford University Museum of Natural History.
A statue of Roger Bacon (1219-1294) at the Oxford University Museum of Natural History, UK.
The first telescopes were known as refractors because they used lenses to collect and magnify light. The earliest versions were made in 1608 by Hans Lippershey, Zacharias Jansen and Jacob Metius (The Netherlands). Galileo Galilei (Italy) built a series of improved refractor telescopes beginning in 1609. In 1655, Christiaan Huygens (The Netherlands) developed a compound eyepiece refractor based on a theory by Johannes Kepler (Germany). In 1668, Isaac Newton (England) invented the first reflector telescope, which used a mirror instead of a lens to collect light. Laurent Cassegrain (France) improved on the reflector in 1672. Further improvements were made throughout the 18th Century.
Isaac Newton's first reflecting telescope. Photograph by Peter Macdiarmid/Getty Images.
Sir Isaac Newton’s first reflecting telescope. Photograph by Peter Macdiarmid/Getty Images.
Atmospheric pressure, also known as air pressure, is the force exerted on a surface by the weight of the air above that surface in the atmosphere of the Earth. A barometer measures atmospheric pressure, which can forecast short-term changes in the weather. Evangelista Torricelli (Italy), a student of Galileo’s, discovered atmospheric pressure and invented the mercury barometer in 1643. Torricelli built on previous discoveries. In 1630, Giovanni Battista Baliani (Italy) conducted an experiment in which a siphon failed to work. Galileo Galilei (Italy) explained the result by noting that power of a vacuum held up the water, but that at a certain point the weight of the water was too much for the vacuum. René Descartes (France) designed an experiment to determine atmospheric pressure in 1631. Having read of Galileo’s ideas, Raffaele Magiotti and Gasparo Berti (Italy) devised an experiment between 1639 and 1641 in which Berti filled a long tube with water, plugged both ends, and stood the tube in a basin of water. Berti then unplugged the bottom of the tube. The result was that only some of the water flowed out, and the water in the tube leveled off at 10.3 meters, the same height Baliani observed in the siphon. Above the water in the tube was a space that appeared to be a vacuum. Torricelli analyzed the results from a different angle: instead of explaining the phenomenon with a vacuum, he chose to challenge common understanding and claim that the air itself had weight, and exerted pressure on the water. From this, he concluded that he could create a device that would measure the pressure of the atmosphere. By using mercury, which is 14 times heavier than water, he could use a tube only 80 centimeters long instead of 10.5 meters. He also discovered that the barometer measured different pressures on rainy days and sunny days. Blaise Pascal and Pierre Petit (France) repeated and perfected Torricelli’s experiment in 1646, showing that the liquid used did not change the results. Pascal had his brother-in-law, Florin Perier (France) perform another experiment which showed that the barometer (and therefore the air pressure) became lower as one increased in altitude, thus proving that the weight of the air was the cause of the barometer’s movements. In 1654, Otto von Guericke (Germany) demonstrated that a vacuum could exist, and he invented a pump that could create a vacuum. In 1661, Robert Boyle (Ireland) took advantage of the vacuum pump to discover Boyle’s Law.
A diagram of Torricelli's mercury barometer.
A diagram of the mercury barometer invented by Evangelista Torricelli (1608-1647).
Probability theory is a branch of mathematics that analyzes random phenomena. In the 16th Century, Gerolamo Cardano (Italy) took the first steps toward probability theory in his attempts to analyze games of chance. The next developments came from Pierre de Fermat and Blaise Pascal (France), who originated probability theory in 1654. Christiaan Huygens (The Netherlands) published a book on probability in 1657. Books by Jacob Bernoulli (Switzerland) in 1713 and Abraham de Moivre (France) in 1718 developed the mathematical basis for probability theory. The fundamentals of probability and statistics were set down by Pierre-Simon Laplace (France) in a 1812 treatise. Richard von Mises (Austria-Hungary) made advances in the 20th Century, and modern probability theory was established by Andrey Nikolaevich Kolmogorov (USSR) and later Bruno de Finetti (Italy).
Daniel Bernoulli (1700-1782).
Daniel Bernoulli (1700-1782).
FOSSILS (1669)
While some early scientists, such as Leonardo da Vinci (Italy) in c. 1500, hypothesized that fossils were the remains of living things, this notion did not gain wide acceptance for many centuries. In 1665, Athanasius Kircher (Germany) suggested that giant fossil bones belonged to a race of extinct giant humans. When Robert Hooke (England) looked at petrified wood through a microscope in 1665, he suggested that it and fossil seashells were formed when living trees and shells were filled with water containing “stony and earthy particles.” In 1668, however, Hooke proposed that fossils told us about the history of life on Earth, a radical idea at the time. Danish cleric Nicholas Steno is credited with first identifying the true nature of fossils. In 1667, he dissected a shark’s head and noticed that common fossils called tongue stones were actually shark’s teeth. Steno then began studying rock strata and published in 1669 a work that systematically disproved many of the prior theories about fossils (such as the theory that they grew inside of rocks like crystals). He proposed that fossils were the remains of living organisms that had become buried in layers of sediment, which had then hardened and formed horizontal layers of rock. One of the obstacles to acceptance of Steno’s theory was the existence of fossils of organisms that did not resemble any living creatures. More than a century later, in 1796, Georges Cuvier (France) definitively proved that some creatures that had lived on Earth in the past were now extinct. The next advances came from William Smith (UK), who studied fossils in the different layers of rock and, between 1799 and 1819, proposed the law of superposition (younger rocks lay atop older rocks) and the principle of faunal succession, which would allow scientists to compare fossils from different areas.
In a 1667 book, Nicholas Steno compared the head of a contemporary shark with fossil shark's teeth.
In a 1667 book, Nicolas Steno (1638-1686) compared the head of a contemporary shark with fossil shark’s teeth.
Denis Papin (France) made a ship powered by his steam engine, mechanically linked to paddles in 1704, although it did not create sufficient pressure to be practical. Jonathan Hulls (England) received a patent for a Newcomen steamboat in 1736, but there is little evidence of any real success. William Henry (US) built several steamboats in 1763 and after but had little success with them. Marquis Claude de Jouffroy (France) made a steam-powered ship in 1783, the paddle steamer Pyroscaphe, which worked for 15 minutes and then stopped. John Fitch (US) and William Symington (Scotland) made similar boats in 1785. Symington and Patrick Miller (Scotland) made a boat with manually-cranked paddle wheels between double hulls in 1785, with a successful try-out in 1788. Using Symington’s design, Alexander Hart (UK) built and launched a successful steamboat in 1801. The same year, Symington designed a second steamboat with a horizontal steam engine linked directly to a crank, the Charlotte Dundas, which was built by John Allan (UK) and the Carron Company. Its maiden voyage was in 1803. The same year, Robert Fulton (US) observed the Charlotte Dundas and, with engineer Henry Bell (UK), designed his own steamboat, which he sailed on the Seine in 1803. Fulton then brought the boat to the US where as the North River Steamboat (later the Clermont), it carried passengers between New York City and Albany, New York in 1807. Other names in the steamboat saga include: J.C. Perier (France), 1775; James Rumsey (US), 1787; and Oliver Evans (US), 1804.
An artist's depiction of the Charlotte Dundas under way.
An artist’s depiction of the Charlotte Dundas under way. The steamboat, which was designed by William Symington (1764-1831), provided the inspiration for Robert Fulton (1765-1815).
The atomists of Ancient Greece theorized that different atoms connected to one another in different ways depending on the substance involved. Iron atoms, they supposed, had hooks to connect to other iron atoms, while water atoms were slippery. When the atom theory saw a resurgence in the 17th Century, Pierre Gassendi (France) adopted some of the Ancient Greek ideas. Sir Isaac Newton (England), on the other hand, suggested in 1704 that particles attract one another by a force that is strong at short distances. Irish chemist Robert Boyle first discussed the concept of the molecule in his 1661 treatise, The Sceptical Chymist, in which he suggested that matter is made of clusters of particles or corpuscles of various shapes and sizes and that chemical reactions rearrange those clusters. In 1680, Nicolas Lemery (France) hypothesized that acidic substances had points, while alkalis had pores, and the points locked into the pores to create Boyle’s clusters. In 1738, Daniel Bernoulli (Switzerland) proposed his kinetic theory of gases, which presumed that gases consist of great numbers of clusters of atoms. William Higgins (Ireland) proposed a theory describing the behavior of clusters of ultimate particles in 1789. John Dalton (UK) published in 1803 the atomic weight of the smallest amount of certain compounds. Italian chemist Amedeo Avogadro published a paper in 1811 that coined the word ‘molecule’, although he uses it to refer to both molecules and atoms. Later, in setting out Avogadro’s Law, Avogadro distinguished between atoms and molecules for the first time. Jean-Baptiste Dumas (France) built on Avogadro’s findings in 1826 and Marc Antoine Auguste Gaudin (France) clearly stated the implications of the molecular hypothesis in 1833, where he suggests molecular geometries and molecular formulas that are consistent with atomic weights. In 1857-1858, German chemist Friedrich August Kekulé proposed that every atom in an organic molecule was bonded to every other atom, and he showed how carbon skeletons could form in organic molecules. At about the same time, Archibald Couper (UK) developed a theory of molecular structure complete with a new form of notation very similar to that used today. In 1861, Joseph Loschmidt (Austria) self published a booklet with a number of new molecular structures. August Wilhelm von Hofmann (Germany) made the first stick and ball models of molecules in 1865. Summing up the knowledge gained so far, James Clerk Maxwell (UK) published an article in 1873 entitled, ‘Molecules’ in which he defined a molecule as “the smallest possible portion of a particular substance.”
A model of two water molecules.
A model of two water molecules.
In 1811, Amedeo Avogadro (Italy) first stated the law that equal volumes of all gases at the same temperature and pressure contain the same number of molecules. This law had the effect of reconciling French chemist Joseph Louis Gay-Lussac’s 1808 law on volumes and combining gases with British physicist John Dalton’s atomic theory.
A graphic illustration of Avogadro's Law.
A graphic illustration of Avogadro’s Law.
Before the 18th Century, most scientists believed that the Earth was very young and that its features were the result of sudden, catastrophic processes. Beginning in the mid-1700s, some geologists challenged the prevailing theory. In 1759, Mikhail Vasilevich Leomonsov (Russia) suggested that the Earth’s topography is result of very slow natural activity, including uplift and erosion. Beginning in 1785, James Hutton (Scotland) proposed that the Earth formed through the gradual solidification of molten rock at a slow rate, by the same processes, particularly erosion and vulcanism, that occur today. Hutton called this process Plutonism to contrast with Neptunists, who believed that the Biblical Flood was the cause of much geology. The implication of this theory was the then-shocking notion that the Earth was millions of years old. The study of fossils in rock layers by William Smith (UK) in the 1790s and independently by French scientists Georges Cuvier and Alexandre Brogniart in 1811 introduced the notion of stratigraphy for determining the relative age of different rocks. In his Principles of Geology, the first volume of which was published in 1830, Scottish geologist Charles Lyell set out the voluminous evidence for uniformitarianism (a form of Hutton’s plutonism) over catastrophism. Charles Darwin read the volumes of Lyell’s Principles while serving as naturalist on the Beagle in the 1830s and uniformitarianism ultimately provided the geological basis for his theory of evolution by natural selection.
A diagram of some of the processes of uniformitarianism.A diagram of some of the processes of uniformitarianism.
Choosing a single inventor for the electric motor would ignore the complexity of the machine’s development, but there is a reasonable argument that the first true electric motor or motors were invented in 1828, 1833 or 1834 (see below). Andrew Gordon (Scotland) created a simple electrostatic motor as early as the 1740s. André-Marie Ampère (France) invented the solenoid in 1820. Peter Barlow (UK) invented Barlow’s wheel, an early homopolar motor, in 1822. Ányos István Jedlik (Hungary) made the first commutated rotary electromagnetical engine in 1828. William Sturgeon (UK) made a commutated rotating electric machine in 1833, the same year that Joseph Saxton (US) made a magneto-electric machine. Thomas Davenport (US) created a battery-powered direct current (DC) motor in 1834 and obtained a patent for a motor in 1837 but high battery power cost made the invention impractical. Moritz von Jacobi (Germany/Russia) made a 15-watt rotating motor in 1834 and the first useful rotary electrical motor in 1838. Sibrandus Stratingh and Christopher Becker (The Netherlands) built an electrical motor in 1835 that powered a small model car. Between 1837 and 1842, British railway pioneer Robert Davidson made electric motors for a lathe and a locomotive. Solomon Stimpson (US) made a 12-pole electric motor with segmental commutator in 1838. Truman Cook (US) made the first electric motor with PM armature in 1840. Paul-Gustave Froment (France) made the first motor that translated a linear electromagnetic piston’s energy to a wheel’s rotary motion in 1845. Zénobe Gramme (Belgium) made the first anchor ring motor in 1871. Galileo Ferraris (Italy) made the first alternating current (AC) commutatorless induction motor with two-phase AC windings in space quadrature in 1885. Nikola Tesla (Serbia/US) made three different two-phase four-stator-pole motors, including a synchronous motor with separately excited DC supply to rotor winding in 1886-1889. Frank Sprague (US) built a constant-speed DC motor in 1886. The three-phase cage induction motor, the most frequently produced machine for 1 kW and above, was first built by Michael Dolivo-Dobrowolsky (Russia) in 1889.
An artist's rendering of Davenport's first electric motor, from 1835.
An artist’s rendering of the electric motor invented by Thomas Davenport (1802-1851) in 1834.
As with many scientific discoveries, the concept of ice ages developed slowly over time. The first inklings of a theory were provided by scientists and others seeking to explain the presence of large erratic boulders and moraines, who suggested that glaciers had placed them in their current locations in the past. These included: Pierre Martel (Switzerland) in 1744; James Hutton (Scotland) in 1795; Jean-Pierre Perraudin (France) in 1815; Göran Wahlenberg (Sweden) in 1818; Johann Wolfgang von Goethe (Germany) in 1820; Ignaz Venetz (Switzerland) in 1829; and Ernst von Bibra (Germany) in 1849-1850. In 1824, Jens Esmark (Denmark/Norway) proposed that changes in climate caused a sequence of worldwide ice ages. Robert Jameson (Scotland) accepted and promoted Esmark’s ideas, as did Albrecht Reinhard Bernhardi (Germany), who speculated in 1832 that former polar ice caps may have reached the temperate zones. Momentum began to build when Venetz convinced Jean de Charpentier (Switzerland/Germany) of his glaciation theory and de Charpentier presented a paper on the subject in 1834. German botanist Karl Friedrich Schimper gave lectures in Munich in 1835-1836 in which he proposed that erratic boulders were the result of global times of obliteration, when the climate was cold and water was frozen. Schimper spent the summer of 1836 in the Swiss Alps with de Charpentier and Louis Agassiz (Switzerland), during which time Agassiz became convinced of the glaciation theory. Agassiz and Schimper developed a theory of a sequence of glaciations in 1836-1837. Schimper coined the term ‘ice age’ in 1837. The reception from the scientific community was cool, so Agassiz set out to collect more data to support the theory, which he published in 1840. Widespread acceptance of the theory did not come until 1875, when James Croll (UK) became the first to propose a convincing mechanism to explain the ice ages. In his book Climate and Time, in their Geological Relations, Croll hypothesized that cyclical changes in the Earth’s orbit could have triggered the growth of the glaciers. The orbital changes were later proven experimentally.
An illustration of the extent of glaciation at the height of the last ice age, about 20,000 years ago.An illustration of the extent of glaciation at the height of the last ice age, about 20,000 years ago.
Absolute zero is the lower limit of the thermodynamic temperature scale. It is the state at which the enthalpy and entropy of a cooled ideal gas reaches its minimum value of zero. Absolute zero equals −273.15° Celsius, −459.67° Fahrenheit and 0° Kelvin. Robert Boyle (Ireland) was one of the first to propose the idea of an absolute zero, or primum frigidum, in 1665. Eighteenth Century scientists accepted the idea of absolute zero and tried to calculate it. Some calculations were more accurate than others. While the calcuations of Guillaume Amontons (France) (–240° C) in 1702 and Johann Heinrich Lambert (Switzerland) (-270° C) in 1779 were relatively close to the actual figure, Pierre-Simon Laplace and Antoine Lavoisier (France) put the number at –600° C or colder, while in 1808, John Dalton (UK) suggested a value of –3000° C. In 1848, William Thomson, Lord Kelvin (UK), arrived at -273.15° C, the temperature for absolute zero that is still recognized today. Kelvin’s scale is based on Carnot’s theory of the motive power of heat and is independent of the properties of any particular substance.
William Thomson, Lord Kelvin (1824-1907).
Humans had been making steel since at least 2000 BCE, but before the 19th Century, steel manufacturing was a slow, expensive process that required the use of carbon-free wrought iron; as a result it was impossible to produce steel in mass quantities. In 1740, Benjamin Hunstman (UK) developed the crucible technique, which increased the cost and duration of the process but increased quality. Beginning in 1847, William Kelly (US) began to experiment with reducing carbon content by blowing air through the molten iron and by 1851, he had developed a process that greatly improved the purity of the finished product and allowed for production of mass quantities of steel. In the UK, Henry Bessemer independently invented a similar process, which he patented in 1855 and which bears his name. Shortly afterwards, Robert Mushet (UK) improved the Bessemer process, creating a more malleable final product. In 1878, Sidney Thomas (UK) designed a way to reduce phosphorus residue in the Bessemer process, increasing the quality of the steel. In the late 20th Century, the Bessemer process was replaced by the basic oxygen process, which allowed better control of the chemistry.
This Bessemer converter, now located at Kelham Island Museum, UK, stopped operating in 1978.
A Bessemer converter that operated until 1978. It is now located at Kelham Island Museum, UK.
While fermentation has been used by humans to make fermented beverages and foods since at least 7000 BCE, the scientific explanation for the process only became understood in the 19th Century. In 1837 and 1838, Theodor Schwann (Germany), Charles Cagniard de la Tour (France) and Friedrich Traugott Kützing (Germany), working independently, concluded that fermentation was caused by yeast, a living organism. In 1857, Louis Pasteur (France) demonstrated that lactic acid fermentation is carried out by living bacteria. In 1897, Eduard Buechner (German) isolated the enzyme in yeast that caused fermentation.
A diagram of the chemical reactions leading to lactic acid fermentation.
A diagram of the chemical reactions leading to lactic acid fermentation.
The history of the internal combustion engine (ICE) is long and complex. Components of the system were invented as long ago as the 3rd Century CE. In the 17th Century, Christiaan Huygens (The Netherlands) created a rudimentary ICE piston engine when he used gunpowder to drive water pumps for the Versailles palace gardens. In the 1780s, Alessandro Volta (Italy) built a toy pistol, in which an electric spark exploded a mix of air and hydrogen, firing a cork. In 1791, John Barber (UK) received a patent for a turbine. In 1794, Robert Street (UK) built the first compressionless engine. In 1807, Nicéphore Niépce (France) powered a boat, the Pyréolophore, with an ICE, fueling it with moss, coal dust and resin. In 1807, Swiss engineer François Isaac de Rivaz built an ICE powered by a mix of hydrogen and oxygen, and ignited by an electric spark. In 1823, Samuel Brown patented the first industrial ICE, a compressionless model. Nicolas Léonard Sadi Carnot (France) established the theoretical basis for idealized heat engines in 1824. In 1826, Samuel Morey (US) received a patent for a compressionless ICE. In 1833, Lemuel Wellman Wright (UK) invented a table-type gas engine with a double acting gas engine and, for the first time, a water-jacketed cylinder. In 1838, William Barnett (UK) received a patent for the first machine with in-cylinder compression. Between 1853 and 1857, Eugenio Barsanti and Felice Matteucci (Italy) invented and patented an engine using the free-piston principle that was possibly the first 4-cycle engine. In 1856, Pietro Benini (Italy) built an engine that supplied five horsepower. Later, he developed more powerful engines with one or two pistons. In 1860, Jean Joseph Etienne Lenoir (Belgium) produced and sold the first two-stroke gas-fired ICE with cylinders, pistons, connecting rods, and flywheel – Lenoir is generally recognized as the inventor of the ICE. In 1861, Alphonse Beau de Rochas (France) received the first patent for a four-cycle engine. In 1862, German inventor Nikolaus Otto built and sold a four-cycle free-piston engine that was indirect-acting and compressionless. Alphonse Beau de Roche, France set out the ideal operating cycle for a four-stroke ICE in 1862. In 1865, Pierre Hugon (France) created the Hugon engine, similar to the Lenoir engine, but with better economy, and more reliable flame ignition. In 1867, Nikolaus Otto and Eugen Langen (Germany) introduced a free piston engine with less than half the gas consumption of the Lenoir or Hugon engines. In 1870, Siegfried Marcus (Austria) put the first mobile gasoline engine on a handcart. In 1872, American George Brayton invented Brayton’s Ready Motor, which used constant pressure combustion, and was the first commercial liquid fueled ICE. In 1876, Nikolaus Otto, working with Gottlieb Daimler and Wilhelm Maybach (Germany), began developing and patenting the four-cycle engine. In 1878, Dugald Clerk (UK) designed the first two-stroke engine with in-cylinder compression. In 1879, Karl Benz (Germany), working independently, received a patent for a two-stroke gas ICE using De Rochas’s four-stroke design. In 1885, Benz designed and built a four-stroke engine to use in an automobile. In 1882 James Atkinson (UK) invented the Atkinson cycle engine, which had one power phase per revolution together with different intake and expansion volumes. In 1884, British engineer Edward Butler constructed the first gasoline ICE. Butler also invented the spark plug, ignition magneto, coil ignition and spray jet carburetor. Rudolf Diesel (Germany) invented the diesel engine in 1892 and Felix Wankel (Germany) invented the rotary engine in 1956.
Internal combustion engine invented by Jean Joseph Lenoir.
The internal combustion engine invented by Jean Joseph Etienne Lenoir (1822-1900) in 1860.
The telephone evolved from the telegraph. Numerous inventors sought to develop acoustic telegraphy, to send sound waves over the electrical wires. Antonio Meucci (US), an Italian immigrant, created a voice communication device about 1854 that he described to the US Patent Office in an 1871 patent caveat. Johann Philipp Reis (Germany) created a device in 1860 that could transmit music and speech, although usually indistinctly. There is some evidence that Innocenzo Manzetti (Italy) may have created a telephone in 1864. In 1870, Cromwell Varley (UK) created a machine that could transmit sounds, but not distinct speech. Poul la Cour (Denmark) made a similar machine in 1874. In 1875, Elisha Gray (US) invented a tone telegraph that could transmit musical notes. Gray filed a patent caveat for a true telephone with a water transmitter on the same day in 1876 that Alexander Graham Bell (Scotland/Canada/US) filed a patent application for his telephone. In future models, however, Bell did not use the water transmitter. The invention of the carbon microphone in 1877 by Thomas Edison and Emile Berliner (US) and independently by David Hughes in the UK, further improved the telephone.
Replicas of Alexander Graham Bell's original 1876 telephone - transmitter (left) and receiver.
A replica of the transmitter component of the original 1876 telephone made by Alexander Graham Bell (1847-1922).
Bell's original receiver.
A replica of Bell’s original receiver.
MITOSIS (1879)
Hugo von Mohl (Germany) described the splitting of one cell into two (mitosis) in the cells of living organisms in 1839, including the appearance of cell plate between daughter cells during cell division. Carl Nageli (Germany) observed cell division and chromosomes in 1842, but he thought what he was seeing was an anomaly. Walther Flemming (Germany) used aniline dyes to study salamander embryos beginning in 1879. Flemming made the first accurate counts of chromosomes and observed longitudinal splitting of chromosomes. His 1882 book on cell division was seminal. Additional work was done by Edouard Van Beneden (Belgium) and Eduard Strasburger (Poland/Germany), who identified chromosome distribution during mitosis. In 1888, Heinrich Wilhelm Gottfried von Waldeyer-Hartz (German) coined the term ‘chromosome’ to name what Flemming had described.
A whitefish blastula undergoing mitosis.
A whitefish blastula cell undergoing mitosis.
In 1867, James Clerk Maxwell (Scotland) predicted the existence of radio waves, electromagnetic waves that are radiated by charged particles as they accelerate. Heinrich Hertz (Germany) proved the existence of radio waves by generating them experimentally in his laboratory in 1887. He also showed that the radio waves traveled at the speed of light.
A replica of Hertz's 1887 radio wave experiment.
A replica of the 1887 radio wave experiment by Heinrich Hertz (1857-1894).
During experiments with blood transfusion, Karl Landsteiner (Austria) identified types A, B and O blood (the ABO blood group) in 1901. Alfred von Decastello and Adriano Sturli (Austria) identified the AB blood type in 1902. Czech physician Jan Jansky discovered the four basic blood groups independently and published the finding in a little-noticed 1907 paper. William Lorenzo Moss (US) made similar discoveries, which were published in 1910. In 1910-1911, Ludwik Hirszfeld (Poland) and Emil von Durgern (Germany) discovered that ABO blood groups are inherited. Felix Bernstein (Germany) determined the chromosomal basis for blood groups in 1924. In 1937, Landsteiner, together with Alexander Wiener (US), identified the Rhesus group. In 1945, Robin Coombs, Arthur Mourant and Rob Race (UK) developed the Coombs blood test. At present, 33 human blood group systems have been identified, and more than 600 blood group antigens.
Karl Landsteiner (1868-1943).
Karl Landsteiner (1868-1943).
The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero Kelvin is exactly equal to zero. Walter Nernst (Germany) first formulated the law in 1906; in 1912, Nernst stated the law as follows: “It is impossible for any procedure to lead to the isotherm T = 0 in a finite number of steps.” Gilbert N. Lewis and Merle Randall (US) proposed an alternative version of the law in 1923: “If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.” A later formulation of the third law, known as the Nernst-Simon statement, is: “The entropy change associated with any condensed system undergoing a reversible isothermal process approaches zero as temperature approaches 0 K, where condensed system refers to liquids and solids.”
Walther Nernst (1864-1941).
Walther Nernst (1864-1941).
PLASTIC (1907)
Prior to the invention of Bakelite, the first completely synthetic plastic, chemists made artificial plastics from naturally-occurring nitrocellulose mixed with other materials. Alexander Parkes (UK) invented Parkesine in 1856, a thermoplastic celluloid based on nitrocellulose treated with solvents. John W. Hyatt modified Parkesine to create Celluloid in 1869. In an effort to find a substitute for shellac, Leo Baekeland, a Belgian-born chemist working in the US, invented Bakelite, which contains no naturally-occurring ingredients, in 1907. The chemical name of Bakelite is polyoxybenzylmethylenglycolanhydride. In 1922, Hermann Staudinger (Germany) set out the theoretical background of macromolecules and polymerization on which the modern plastics industry rests. In 1958, Robert Banks and Paul Hogan (US) invented polypropylene and devised a low-pressure method for producing high-density polyethylene.
This bakelite radio was sold by General Electric in Australia in 1932.
This Bakelite radio was sold by General Electric in Australia in 1932.
The uncertainty principle holds that there is a mathematically-determined fundamental limit to knowing precisely and simultaneously certain pairs of physical properties (known as complementary variables) of a particle, such as position and momentum. Werner Heisenberg (Germany) first articulated the uncertainty principle in 1927 by stating that the more precisely a particle’s position is determined, the less precisely its momentum can be known, and vice versa. The uncertainty principle is sometimes confused with the observer effect, which states that measurements of certain systems cannot be made without affecting the system.
A diagram explaining the uncertainty principle.
A diagram explaining the uncertainty principle.
The weak interaction is the mechanism responsible for the weak nuclear force, one of the four basic forces of nature, along with electromagnetism, gravity and the strong nuclear force. The weak interaction is responsible for radioactive decay and nuclear fusion of subatomic particles; caused by emission or absorption of W and Z bosons. Fermions (which include quarks, leptons and certain particles made from them) also interact through weak interaction. Ernest Rutherford (NZ/UK) proposed the weak nuclear force in 1899 to explain beta decay of radioactive elements. Enrico Fermi (Italy) first suggested the existence of the weak interaction in 1933 in explaining beta decay. Fermi thought it was a force with no range, dependent on contact. (It is now believed that the weak force is a non-contact force with a finite range.) In 1956, Clyde Cowan and Frederick Reines (US) showed that electrons and antineutrinos were released in beta decay. The same year, Tsung-Dao Lee and Chen Ning Yang (China/US) predicted that the weak force did not follow parity, the symmetry of the other forces. In 1968, Sheldon Glashow (US), Abdus Salam (Pakistan) and Steven Weinberg (US) proved that the weak interaction and electromagnetism were two aspects of the same force, now known as the electroweak force. W and Z bosons were first experimentally detected by Carlo Rubbia (Italy) and Simon van der Meer (The Netherlands) in 1983.
Carlo Rubbia (1934- ) (right) and Simon van der Meer (1925-2011), who jointly won the Nobel Prize in Physics in 1984 for discovering the W and Z bosons.
The road to the atom bomb began in 1934, when Hungarian scientist Leó Szilárd proposed bombarding radioactive atoms with neutrons to form a nuclear chain reaction, an idea he patented and then transferred to the British Admiralty so it would be kept secret. In 1938, Otto Hahn and Fritz Strassmann (Germany) split the uranium atom, a fact explained and confirmed by Lise Meitner and Otto Robert Frisch (Germany) in January 1939. Meitner and Frisch named the process ‘fission’ by analogy to biological processes. Scientists at Columbia University repeated the experiment in January 1939. In August 1939, fearing that the Germany would produce a fission-based weapon, Szilárd wrote and Albert Einstein (Germany) signed a letter of warning to US President Franklin Roosevelt, who responded by setting up a committee to study the matter, which only received significant funding after the US entered World War II in December 1941. In 1940 and 1941, the British took the lead in conducting research into uranium and potential weapons. The US research did not begin in earnest until September 1942, with the start of the Manhattan Project, led by General Leslie Groves, which took over the British research. Robert Oppenheimer (US) led the Manhattan Project’s team of physicists. In addition to the Los Alamos laboratory, an Oak Ridge, Tennessee facility produced the rare uranium-235 isotope needed for a chain reaction. The project also used plutonium-239, a byproduct of a uranium-238 reaction, as a basis for a fission weapon. The Manhattan Project ultimately produced two types of fission bombs: a uranium-235 gun-type weapon (“Little Boy”) and a plutonium-239 implosion-type bomb (“Fat Man”). The first atomic weapon – a plutonium implosion bomb – was detonated at Los Alamos, New Mexico on July 16, 1945, releasing the equivalent of 19 kilotons of TNT. On August 6, 1945, the US dropped a uranium gun-type bomb on Hiroshima, Japan. On August 9, 1945, the US dropped a plutonium implosion-type bomb on Nagasaki, Japan. The two bombings resulted in the deaths of approximately 220,000 people, mostly civilians. The USSR tested its first fission bomb on August 29, 1949. In 1950, the US began developing the much more powerful thermonuclear or hydrogen bomb, which uses fission to create a fusion reaction; the first bomb was tested in 1952, releasing energy equal to 10.4 megatons of TNT. The USSR followed with its first thermonuclear bomb test on August 12, 1953.
The first atomic bomb explodes on July 16, 1945 in New Mexico.
The first atomic bomb explodes on July 16, 1945 at Almogordo, New Mexico.
Information theory is a branch of applied mathematics, electrical engineering, and computer science that involves the quantification of information. Information theory was developed by Claude E. Shannon (US) in 1948 to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception, information theory has expanded and has been applied in numerous contexts.
Claude E. Shannon (1916-2001).
Claude E. Shannon (1916-2001).
THE LASER (1958-1960)
A laser (an acronym for ‘light amplification by stimulated emission of radiation’) emits light through optical amplification based on the stimulated emission of electromagnetic radiation. Albert Einstein (Germany) established the theoretical basis for lasers and masers in a 1917 paper. Other aspects of the science were developed by Rudolf Ladenburg (Germany) in 1928; Valentin Fabrikant (USSR) in 1939; Willis E. Lamb and R.C. Retherford (US) in 1947 (stimulated emission) and Alfred Kastler (France) in 1950 (optical pumping). Charles Hard Townes, with students James Gordon and Herbert Zeiger (US) created the first microwave amplifier, or maser, in 1953, although it was incapable of continuous output. In the USSR, Nikolay Basov and Aleksandr Prokhorov had solved the continuous output problem using a quantum oscillator in 1952, but results were not published until 1954-1955. In 1957, Townes and Arthur Leonard Schawlow (US), at Bell Labs, began working on an infrared laser, but soon changed to visible light, for which they sought a patent in 1958. Also in 1957, Columbia University grad student Gordon Gould, after meeting with Townes, began working on the idea for a ‘laser’ using an open resonator. Prokhorov independently proposed the open resonator in 1958. In 1959, Gould published the first paper using the term ‘laser’ and filed for a patent the same year. In 1960, the US Patent Office granted Townes’ and Schawlow’s patent and denied Gould’s. The first working laser was created by Theodore Maiman (US) in 1960, but it was only capable of pulsed operation. Also in 1960, Ali Javan (Iran/US), William Bennett and Donald Herriott (US) made the first gas laser. In 1962, Robert N. Hall (US) invented the first laser diode device. The same year, Nick Holonyak, Jr. (US) made the first semiconductor laser with a visible emission, although it could only be used in pulsed-beam operation. In 1970, Zhores Alferov (USSR), Izuo Hayashi (Japan) and Morton Panish (US) independently developed room-temperature, continual-operation diode lasers. In 1987, following years of patent litigation, a Federal judge ordered the US Patent Office to issue patents to Gordon Gould for the optical pump and gas discharge lasers.
Ted Maiman's first laser, from 1960.
The first working laser, which was made by Theodore Maiman (1927-2007) in 1960.
PULSARS (1967)
A pulsar (short for ‘pulsating star’) is a highly magnetized, rotating neutron star that emits a beam of electromagnetic radiation. On November 28, 1967, Antony Hewish and Jocelyn Bell Burnell (UK) became the first scientists to observe a pulsar, which had a pulse period of 1.33 seconds. Walter Baade (Germany/US) and Fritz Zwicky (Switzerland/US) had predicted neutron stars in 1934, and in early 1967, Franco Pacini (Italy) suggested that a rotating neutron star with a magnetic field would emit radiation. In 1968, David Staelin, Edward C. Reifenstein III and Richard Lovelace (US) discovered a pulsar in the Crab Nebula with a 33 millisecond pulse period and a rotation speed of 1,980 revolutions per minute. Joseph Hooton Taylor, Jr. and Russell Hulse (US) discovered the first pulsar in a binary system in 1974. A team led by Don Backer (US) discovered the first millisecond pulsar, with a rotation period of 1.6 milliseconds, in 1982.
The Vela Pulsar.
A Chandra X-ray image of the Vela Pulsar, which is located inside the Milky Way galaxy about 950 light years from Earth and has a pulse period of 89 milliseconds.
The development of the Internet was a complex, many-faceted process. It is impossible to identify one person who invented the Internet, and it is very difficult to choose a point in time when the Internet was invented, but there is a rational basis for choosing 1969, as shown below. Leonard Kleinrock’s (US) July 1961 paper on packet switching theory at MIT was an early precursor to the Internet, as was a series of “Galactic Network” memos by J.C.R. Licklider (US), also at MIT, in August 1962. In October 1962, Licklider became the first computer research program head at DARPA (Defense Advanced Research Projects Agency), where he convinced Ivan Sutherland, Robert Taylor and MIT’s Lawrence G. Roberts (US) of the importance of networking. Kleinrock convinced Roberts to use packets rather than circuits. The first wide-area computer network was built in 1965 when Roberts and Thomas Merrill (US) connected the TX-2 computer in Massachusetts to the Q-32 in California. In 1966, Roberts went to DARPA, where he developed the computer network concept and put together a plan for the ARPANET, which he published in 1967. Parallel research on networks had been going on at RAND (1962-1965) (esp. Paul Baran (US)) and National Physical Laboratory (NPL) (1964-1967) (esp. Donald Davies and Roger Scantlebury (UK)). After Roberts and DARPA refined the ARPANET’s specifications, in 1968, they chose Frank Heart’s (US) team at Bolt Beranek and Newman (BBN) to built the packet switches, called Interface Message Processors (IMPs). Robert Kahn (US) at BBN; Howard Frank (US) at Network Analysis Corp.; and Kleinrock, played significant roles. In September 1969, BBN installed the first IMP at UCLA, which became the first node. Doug Engelbart’s Stanford Research Institute (SRI) provided the second node. The first message was sent between UCLA and SRI in October 1969. Four computers were linked by the end of 1969 and many more joined in the next few years. In December 1970, S. Crocker (US) and his Network Working Group finished the ARPANET’s initial host-to-host protocol, the Network Control Protocol (NCP). Also in 1970, NPL started the Mark I network. In 1971, the Merit Network and Tymnet networks became operational. Kahn successfully demonstrated the ARPANET at a conference in October 1972. Also in 1972, Louis Pouzin in France began an Internet-like project called Cyclades, that was based on the notion that the host computer, not the network, should be responsible for data transmission. Cyclades was eventually shut down, but the Internet eventually adopted its basic principle. The first trans-Atlantic transmission occurred in 1973, to University College of London. In 1974, a proposal was made to link ARPA-like networks into a larger inter-network that would have no central control. Also in 1974, the International Telecommunication Union developed X.25 packet switching network standards. The PC modem was invented by Dennis Hayes and Dale Heatherington in 1977. The first bulletin board system was invented in 1978. Usenet was invented in 1979 by Tom Truscott and Jim Ellis (US) and CompuServe was launched the same year. In 1981, the National Science Foundation created CSNET, the Computer Science Network, which linked to ARPANET. In 1982, the TCP/IP protocol suite, invented by Vinton Cerf (US), was formalized. ARPANET computers were required to switch from the NCP protocol to the TCP/IP protocols by January 1, 1983. In 1984, the system of domain names was adopted – the first .COM domain name was registered in 1985. In 1986, NSF created NSFNET, which was linked with ARPANET. In 1988, Internet Relay Chat was first introduced. America Online (AOL) was launched in 1989. In 1990, ARPANET was decommissioned in favor of NSFNET. NSFNET was decommissioned in 1995 when it was replaced by networks operated by several commercial Internet Service Providers.
A 1969 diagram of the Arpanet network.
A 1969 diagram of the Arpanet network.
A visualization of routing paths through a portion of the Internet.
A recent visualization of routing paths through a portion of the Internet.
According to string theory, all elementary particles are actually made of vibrating one-dimensional objects called strings. String theory purports to unite all four basic forces in one explanatory framework. String theory requires multiple spatial dimensions; one version of the theory requires 11 dimensions. While some physicists have embraced string theory, others have criticized it because it is difficult (some say impossible) to test its predictions. A precursor to string theory was S-matrix theory, proposed by Werner Heisenberg (Germany) in 1943. Some physicists expanded on the theory in the 1950s, particularly Tullio Regge (Italy), Geoffrey Chew and Steven Frautschi (US). The theory eventually developed into the dual resonance model of Gabriele Veneziano in 1968. The scattering amplitude that Veneziano predicted was essentially a closed vibrating string. Then in 1970, Yochiro Nambu (Japan/US), Holger Bech Nielsen (Denmark) and Leonard Susskind (US) proposed a theory that represented nuclear forces as one-dimensional vibrating strings. John H. Schwarz (US) and Joel Scherk (France) proposed bosonic string theory in 1974. Michael Green (UK) and John Schwarz proposed the existence of supersymmetric strings, or superstrings, in the early 1980s. Between 1984 and 1986, a number of scientific discoveries occurred that have been termed the first superstring revolution. These discoveries resulted in a number of rival versions of the theory. In 1994, Edward Witten (US) suggested that the five different versions of string theory were all different limits on an 11-dimensional theory he called M-theory, an announcement that led to the second superstring revolution between 1994 and 1997. Chris Hull and Paul Townsend (UK) played important roles in this phase. Some scientists believe that the Large Hadron Collider at CERN may be able to produce enough energy to provide experimental evidence for string theory.
Edward Witten.
Edward Witten (1951- ).
Discussions began in the US in 1984 to sequence the entire human genome. The Human Genome Project began in 1986 through the US National Institutes of Health and the Department of Energy, but the actual project did not begin until 1990. Researchers from all over the world identified the genetic sequencing of human DNA using samples from approximately 270 individuals. A first draft of the genome was announced in 2000, and the project was declared finished in 2003. Celera Genomics undertook a parallel human genome project in the private sector in 1990s, which was much faster and less expensive than the government’s Human Genome Project. (Some pointed out that Celera was able to finish so quickly in part because it was able to freely obtain all the Human Genome Project’s results daily as they were placed online for the public, while Celera refused to share its results on proprietary grounds.) In 2001, Craig Venter of Celera Genomics and Francis Collins of the Human Genome Project, jointly published their decoding of the human genome.
A graphic depiction of the human genome.
A graphic depiction of the human genome.
(2600 BCE)
A lever is a simple machine consisting of a beam or rigid rod that pivots at a fixed hinge, or fulcrum, thereby amplifying an input force to provide a greater output force. Greek scientist philosopher Archimedes first correctly stated the mathematical principle behind the lever in the 3rd Century BCE. Pappus of Alexandria quotes Archimedes as saying of the lever, “Give me a place to stand, and I shall move the Earth with it.” Although there is no written evidence of levers prior to Archimedes, historians believe that the Ancient Egyptians must have had levers in order to construct the pyramids and other massive monuments weighing more than 100 tons in the 3rd Millenium BCE.
types of levers
Three types of levers.
Also known as the Indo-Arabic Counting System, the Hindu-Arabic Numeral System was the first counting system to include a zero and is the basis for most of subsequent mathematics. This positional decimal numeral system was invented in India, but there is much debate about the date. Some scholars believe there is evidence for a 1st Century CE date, while others say the earliest evidence is from the 3rd or 4th Century CE. All agree that the system was in use by 600 CE. The system began to spread elsewhere: Severus Sebokht (Syria) mentions it in 662 CE and Muslim scholar al-Qifti (Egypt) cites an encounter between a Caliph and an Indian mathematics book in 776 CE. Persian mathematician Al-Khwarizmi gave a treatise on the system in an 825 CE book and Arab mathematician Al-Kindi did the same in 830 CE. Arabic numerals first appear in Europe in a 976 CE Spanish text. Italian mathematician Fibonacci sought to promote the system in a book published in 1202, but the system did not become standard in Europe until after the printing press was invented after 1540.
This chart shows the changes in numerals from Hindu India, to the Islamic world, and then to Europe.
The Canon of Medicine, a five-book encyclopedia by Persian scientist, physician and philosopher Ibn Sina (often referred to by the Latinate form of his name, Avicenna) set out in a systematic way the medical knowledge and procedures known in the 11th Century. While the Canon relies primarily on Galenic medical theories, Ibn Sina also adopts Aristotle’s explanations in some cases and drew from many other sources, including Chinese texts from 310 CE and 610 CE. In his introduction, Ibn Sina sets out his belief in medicine as a science, that the physician must determine the causes of both health and disease before the body can be restored to health. The book contains specific instructions on diagnosis and treatments, including surgical procedures, and analyzes the efficacy of over 600 different drugs or herbal remedies. Originally written in Arabic, the Canon was translated into Latin by Gerard of Cremona (Italy) in the 13th Century, allowing it to become the premier textbook for European medical education in the medieval period.
A page from a 1597 Arabic copy of Ibn Sina's Canon of Medicine.
A page from a 1597 Arabic copy of the Canon of Medicine, by Ibn Sina (c. 980-1037 CE).
A supernova occurs when a star suffers a catastrophic explosion, causing it to increase greatly in brightness. The explosions of supernovae radiate enormous amounts of energy and normally expel most or all of the star’s contents at a velocity of 30,000 km/s, which sends a shock wave and an expanding shell of gas and dust (called a supernova remnant) into interstellar space. Supernovae generate much more energy than novae. There are two types of supernova: the first occurs when nuclear fusion suddenly reignites in a degenerate star due to accumulation of material from a companion star; the second occurs when a massive star undergoes sudden gravitational collapse. The first supernovae to be observed were those occurring in the Milky Way galaxy that were visible to the naked eye. Chinese astronomers observed a supernova in 185 CE. Chinese and Islamic astronomers described a supernova in 1006. A widely-seen supernova in 1054 created the Crab Nebula. Tycho Brahe (Denmark) described a supernova in Cassiopeia 1572 and Johannes Kepler (Germany) described one in 1604. The first supernova in another galaxy was seen in the Andromeda galaxy in 1885. Prior to 1931, supernovae were not distinguished from ordinary novae. Based on observations at Mt. Wilson Observatory, Walter Baade (Germany/US) and Fritz Zwicky (Switzerland/US) created a new category for supernovae, a term they began using in a series of 1931 lectures and announced publicly in 1933. In 1941, Zwicky and Rudolph Minkowski (Germany/US) developed the modern supernova classification scheme. In the 1960s, astronomers began to use supernova explosions as ‘standard candles’ to measure astronomical distances. More recently, scientists have been able to determine the dates and locations of supernovae that occurred in the past based on their aftereffects.
A multiwavelength X-ray, infrared, and optical compilation image of Kepler's supernova remnant, SN 1604.
A multiwavelength X-ray, infrared, and optical compilation image of the supernova remnant of the supernova observed by Johannes Kepler in 1604 (SN 1604).
Roger Bacon first proposed the idea of a microscope in 1267, but it was not until about 1590 that two Dutch eyeglass makers, Hans Lippershey & Zacharias Jansenmade the first compound optical microscope. (‘Optical’ because it used visible light and lenses to magnify objects and ‘compound’ because it used multiple lenses, allowing for much greater magnification than the single lens, or simple optical microscope.) Galileo Galilei (Italy) developed a compound microscope in 1609, and Cornelius Drebbel (The Netherlands) created one in 1619. Early microscope researchers were Robert Hooke (UK), who published a book of drawings from the microscope entitled Micrographia in 1665, containing numerous scientific discoveries, including the first description of a biological cell, and Antonie van Leeuwenhoek (The Netherlands), who made many discoveries in the 1670s.
A replica of the microscope used by Antonie van Leeuwenhoek, which magnified objects 270 times.
A replica of the microscope used by Antonie van Leeuwenhoek (1632-1723), which magnified objects 270 times.
Chinese astronomer Gan De reportedly observed a moon orbiting Jupiter about 364 BCE. It is Galileo Galilei (Italy), however who is credited with discovering the four largest moons of Jupiter – Ganymede, Callisto, Io and Europa – by making observations using progressively stronger telescopes in 1609 and 1610. E.E. Barnard (US) discovered a fifth moon, Amalthea, in 1892. Using photographic telescopes, additional moons were discovered in 1904, 1905, 1908, 1914, 1938, 1951, and 1974. A 14th moon was discovered in 1975. The Voyager space probes found three more moons in 1979. Between 1999 and 2003, a team led by Scott S. Sheppard and David C. Jewitt (US) found 34 additional moons, most of them very small (averaging 1.9 miles in diameter) with eccentric orbits. Between 2003 and 2014, scientists have discovered 17 additional moons, bringing the total to 67.
A view of Jupiter and the four moons discovered by Galileo, as seen through a 10" Meade LX200 telescope.
A view of Jupiter and the four moons discovered by Galileo, as seen through a 10″ Meade LX200 telescope.
In the 13th Century, Roger Bacon (England) suggested that rainbows were produced the same way that light produced colors when passed through a glass or crystal. In 1666, Isaac Newton (England) discovered that visible white light is composed of a spectrum of colors. He made this discovery by studying the passage of light through a dispersive prism, which refracted the light into the colors of the rainbow: red, orange, yellow, green, blue and violet. He also found that the multicolored spectrum could be recomposed into white light by a lens and a second prism. He published his results in 1671.
Light dispersion of a mercury-vapor lamp with a prism made of flint glass. D-Kuru Photo (2009).
Light dispersion of a mercury-vapor lamp with a prism made of flint glass. Photo by D-Kuru (2009).
LIGHT THEORY (1675 (particle); 1678 (wave); 1862 (electromagnetic); 1900 (quanta))
Light is electromagnetic radiation that is visible to the human eye. Scientists now accept that light has wavelike and particle-like qualities. Explanations for the nature of light began with the Ancient Greeks, including Empedocles in the 5th Century BCE, who believed that sight results from a beam of light emitted by the eye. Euclid questioned the ‘beam from the eye’ theory in 300 BCE with a thought experiment, although he supposed the theory could be true if the speed of light was infinite. Lucretius (Ancient Rome) in 55 BCE supposed that light consisted of atoms moving from the sun to the Earth. Ptolemy in the 2nd Century CE discussed refraction of light. Indian Hindu philosophers in the early centuries of the common era proposed a particle theory of light, but Indian Buddhists in the 5th and 7th centuries CE suggested that light was composed of flashes of energy. In 1604, Johannes Kepler found that the intensity of a light source varies inversely with the square of one’s distance from that source. René Descartes (France) theorized in 1637 that light was a mechanical property of the luminous body and the medium transmitting the light. The modern particle theory of light was proposed by Pierre Gassendi (France) and published after his death in the 1660s. Isaac Newton (England) adopted the particle theory in 1675 (with a final version published in 1704), stating that corpuscles of light were emitted from a source in all directions. He also explained diffraction, polarization and (incorrectly) refraction. Further work on polarization of light was done by Étienne-Louis Malus (France) in 1810 and Jean-Baptiste Biot (France) in 1812. Although Newton’s particle theory was dominant for at least a century, others found that light had wavelike properties. Robert Hooke (England) invoked a wave theory of light to explain the origin of colors in 1665 and expanded on the theory in 1672. Christiaan Huygens (The Netherlands) developed a mathematical wave theory of light in 1678. The theory predicted interference patterns, which were confirmed by Thomas Young (UK) in 1801. In 1746, Leonhard Euler (Switzerland) argued that wave theory provided a better explanation for diffraction than particle theory. Augustin-Jean Fresnel (France) developed a separate wave theory in 1817, which received support from Siméon Denis Poisson (France). Measurements of the speed of light in 1850 supported wave theory. Wave theory suffered a non-fatal blow in 1887. Huygens had proposed that waves were propagated by a luminiferous aether, but the Michelson-Morley experiment in 1887 proved that the aether did not exist. Meanwhile, as the result of experiments performed in 1845-1847, Michael Faraday (UK) suggested that light was a form of electromagnetic wave, which could be propagated in a vacuum. In 1862 and then in 1873, James Clerk Maxwell (UK) took the results of Faraday’s experiments and provided a mathematical basis for the conclusion that light, electricity and magnetism were all forms of the same wave force. Heinrich Hertz (Germany) provided experimental confirmation of Maxwell’s theory by propagating electromagnetic, or radio waves in his laboratory in 1886-1887. In 1900, Max Planck (Germany) proposed that light and other electromagnetic radiation consisted of waves that could gain and lose energy only in finite amounts or quanta. German physicist Albert Einstein’s 1905 paper on the photoelectric effect suggested that quanta were real, and Arthur Holly Compton (US) in 1923 showed that certain behavior of X-rays could be explained by particles, but not waves. In 1926, Gilbert N. Lewis (US) named the electromagnetic quanta ‘photons.’
A chart that explores the wave-particle duality of light and other electromagnetic radiation.
One of the earliest and most common forms of life on Earth, bacteria are usually a few micrometers long. Antonie van Leeuwenhoek (The Netherlands) first observed bacteria in 1676, using a single-lens microscope. After 1773, Otto Frederik Müller distinguished bacillum and spirillum bacteria. Christian Gottfried Ehrenberg (Germany) coined the term ‘bacterium’ in 1828 to describe certain rod-shaped bacteria. Robert Koch (Germany) identified the bacteria that cause anthrax (1875), tuberculosis (1882) and cholera (1883). In 1977, Carl Woese (US) recognized that, based on their ribosomal RNA, some organisms formerly considered bacteria belonged to another domain or kingdom, the Archaea.
A cross-section diagram of an average bacterium.
A cross-section diagram of an average bacterium.
COMETS (1705)
While comets have been observed since ancient times, the first modern scientific theory of comets was developed by Tycho Brahe (Denmark), who measured the parallax of the Great Comet of 1577 and determined that it must exist outside the Earth’s atmosphere. Isaac Newton (England) demonstrated the orbit of the comet of 1680 in his Principia of 1687. In 1705, Edmund Halley (England) analyzed 23 appearances of comets between 1337 and 1698 and concluded that three of the appearances were the same comet, which he predicted would return in 1758-1759. (Three French mathematicians further refined the date.) When the comet returned as scheduled, it was named Halley’s Comet. Scientists of the 17th, 18th and 19th centuries proposed various theories for the composition of comets, but it was Fred Lawrence Whipple (US) who suggested in 1950 that comets were made of ice mixed with dust and rock – the ‘dirty snowball’ theory. A number of observations appeared to confirm this view, but in 2001, high resolution images of Comet Borrelly showed no ice, only a hot, dry, dark surface. Another probe, which crashed into Comet Temple 1 in 2005, found that most of the ice is beneath the surface.
Comet McNaught in 2007.
Comet McNaught in 2007.
William Cullen (Scotland) invented artificial refrigeration at the University of Glasgow in 1748. Oliver Evans (US) created the vapor-compression refrigeration process in 1805. Jacob Perkins (US) took Evans’s process and built the first actual refrigerator in 1834. John Gorrie (US) invented the first mechanical refrigeration unit in 1841. Further improvements were made by Alexander Twining (US) in 1853; James Harrison (Scotland/Australia) in 1856; Ferdinand Carré (France) in 1859; Andrew Muhl (France/US) in 1867 and Carl von Linde (Germany) in 1895. Electrolux produced the first electric refrigerator in 1923.
This photo is said to show a Jacob Perkins refrigerator built in the early 19th Century.
This photo is said to show a refrigerator made by Jacob Perkins (1766-1849) built in the early 19th Century.
A lightning rod is a metal rod or other object mounted on top of a building or other elevated structure that is electrically bonded using a wire or electrical conductor to connect with a ground through an electrode, in order to protect the structure if lightning hits it. For thousands of years, builders in Sri Lanka have protected their buildings from lightning by installing metal tips made of silver or copper on the highest point. The Leaning Tower of Nevyansk in Russia, which was built between 1721 and 1745, is crowned with a metal rod that is grounded and pierces the entire building, but it is not known whether it was intended as a lightning rod. Benjamin Franklin (US) invented the lightning rod in 1749. Prokop Diviš (Bohemia) independently invented the grounded lightning rod in 1754.
A lightning rod at the Franklin Institute in Philadelphia, Pennsylvania, believed to be one of Benjamin Franklin's originals.
This lightning rod at the Franklin Institute in Philadelphia, Pennsylvania, is believed to have been made by Benjamin Franklin (1706-1790).
Lightning was in the air in the late 1740s and early 1750s. Benjamin Franklin (US) listed a dozen analogies between lightning and electricity in his notebooks in 1749. Similar speculation by Jean Antoine Nollet (France) led to a French essay contest on the topic, which was won in 1750 by Denis Barbaret (France), who said lightning was caused by the triboelectric effect. Jacques de Romas (France) proposed a similar theory in a 1750 memoir; he also claimed to have suggested a test of the theory using a kite. In 1752, Franklin proposed to test the theory by using rods to attract lightning to a Leyden jar. The experiment was carried out by Thomas-François Dalibard in May 1752 and by Franklin himself in June 1752, but using a kite instead of a rod. He attached a key to the kite string, which was connected to a Leyden jar. Although the kite was not struck by lightning, static electricity was conducted to the key, and Franklin felt a shock when he moved his hand near the key. Georg Wilhelm Richmann (Germany/Russia) was killed by electrocution while attempting to recreate the experiment in St. Petersburg in 1753.
An 1876 rendering of Benjamin Franklin's kite-flying experiment by Currier & Ives. © Museum of the City of New York/Corbis.
An 1876 rendering of Benjamin Franklin’s kite-flying experiment by Currier & Ives. © Museum of the City of New York/Corbis.
Combustion is a sequence of exothermic chemical reactions between a fuel and an oxidant that is accompanied by the production of heat and the conversion of chemical species. The release of heat can produce light in the form of glowing or flames. Modern scientific attempts to determine the nature of combustion began in 1620, when Francis Bacon (England) observed that a candle flame has a structure. At about the same time, Robert Fludd (England) described an experiment in a closed container in which he determined that a burning flame used up some of the air. Otto von Guericke (Germany) demonstrated in 1650 that a candle would not burn in a vacuum. Robert Hooke (England) suggested in 1665 that air had an active component that, combined with combustible substances when heated, caused flame. Antoine-Laurent Lavoisier (France) was the first to give an accurate account of combustion when in 1772 he found that the products of burned sulfur or phosphorus outweighed the initial substances, and he proposed that the additional weight was due to the combining of the substances with air. Later Lavoisier concluded that the part of the air that had combined with the sulfur was the same as the gas released when English chemist Joseph Priestley heated the metallic ash of mercury, which was the same as the gas described by Carl Wilhelm Scheele (Sweden) as the active fraction of air that sustained combustion. Lavoisier gave the name ‘oxygen’ to the gas found by Priestley and Scheele.
A drawing of one of the experiments Lavoisier conducted to discover the nature of combustion.
A drawing of one of the experiments Antoine Lavoisier (1743-1794) conducted to discover the nature of combustion.
The ability to raise small unmanned balloons into the air using hot air was known in China from the 3rd Century CE. French brothers Jacques and Joseph Montgolfier built the first hot-air balloons capable of carrying human passengers in the late 18th Century. They tested their design first with no passengers on June 4, 1783, then on September 19, 1783 with a sheep, a duck and a rooster, who survived an eight-minute flight. Then, on November 21, 1783, French scientist Pilâtre de Rozier and the Marquis d’Arlandes, an Army officer, climbed aboard a Montgolfier balloon to make the first untethered manned flight. They traveled for 25 minutes, covered a distance of five miles and attained an altitude of 3,000 feet before safely landing. Among those in the audience were King Louis XVI and Benjamin Franklin.
A 1786 illustration of the first manned balloon flight, just three years earlier.
A cotton gin separates the cotton seeds from the fibers, a task previously done by hand. Primitive labor-intensive gins had been invented in India (5th Century CE) and elsewhere, but American Eli Whitney’s 1793 hand-powered cotton gin was the first mechanical cotton gin that efficiently separated fibers and seeds from large amounts of cotton. Whitney’s invention revolutionized the U.S. cotton industry and led to the growth of slave labor in the South. Modern cotton gins are automated and much more productive than Whitney’s original.
Eli Whitney created this model of his cotton gin in 1800 to use in court while defending his patent against multiple infringers.
Eli Whitney (1765-1825) created this working model of his cotton gin in 1800 to use in court while defending his patent against multiple infringers.
In 1804, Richard Trevithick’s first steam locomotive pulled a train containing 10 tons of iron and 70 passengers in five cars approximately nine miles near Merthyr Tydfil in Wales. The first commercially successful steam locomotives were built by Matthew Murray (UK) in 1812 (Salamanca); and Christopher Blackett & William Hedley (UK) in 1813 (Puffing Billy). George Stephenson (UK) improved on Trevithick’s and Hedley’s designs by adding a multiple fire tube boiler in 1814 with the Blücher and again in 1825 with the Locomotion and in 1929 with The Rocket. The largest steam-powered locomotive was the Union Pacific’s Big Boy, (US) of 1941. Steam locomotives were gradually phased out in the first half of the 20th Century, to be replaced by diesel and electric locomotives.
An 1862 photo of the early steam locomotive Puffing Billy.
An 1862 photo of the 1813 steam locomotive Puffing Billy.
Joseph Nicéphore Niépce (France) created what was probably the first photograph on bitumen-covered pewter in 1826. His photographic method required an exposure of eight hours or more, and the final image was only viewable when held at an angle. In 1835 William Talbot (UK) created a method using a paper negative that allowed multiple positive prints from the same exposure. In 1837, Louis-Jacques-Mandé Daguerre (France) created a process with a much shorter exposure time and much clearer images, called daguerrotypes. Unfortunately, there was no way to make multiple copies of daguerrotypes, which were direct positive images on silver plate. Alexandre Becquerel and Claude Niepce de Saint-Victor (France) produced the first color images between 1848–1860. John Carbutt (US) produced the first commercially successful celluloid film in 1888. Also in 1888, George Eastman (US) introduced the hand-held Kodak camera with roll film. Kodak also introduced Kodachrome, the first commercial color film with three emulsion layers, in 1935.
The first photograph, from 1826, "View from the Window at Le Gras."
The first photograph, from 1826, “View from the Window at Le Gras.”
OHM’S LAW (1827)
Ohm’s Law states that the ratio of the potential difference between the ends of a conductor and the current flowing through it is constant, and that ratio equals the resistance of the conductor. (Alternately, the law states that the current through a conductor between two points is directly proportional to the potential difference across the two points.) Ohm’s Law establishes the relationship between strength of electric current, electromotive force, and circuit resistance. Henry Cavendish (England) arrived at a formulation of Ohm’s Law in 1781 but he did not communicate his results at the time. Georg Ohm conducted experiments on resistance in 1825 and 1826 and published his results, including a more complicated version of Ohm’s Law, in 1827.
A diagram of Ohm's Law.
A diagram of Ohm’s Law.
Dutch scientist Herman Boerhaave discovered urea in urine in 1727. In 1828, Friedrich Wöhler (Germany) synthesized urea by treating silver cyanate with ammonium chloride. This was the first artificial synthesis of an organic compound from inorganic materials. It had important consequences for organic chemistry and also provided evidence against vitalism, the notion that living organisms are fundamentally different from inanimate matter.
An 1856 lithograph of Friedrich Wohler by
An 1856 lithograph of Friedrich Wöhler (1800-1882).
In ancient times, dinosaur fossils were explained as the bones of a giant race of humans that have vanished from the Earth. More scientific approaches came in the 19th Century. In 1808, Georges Cuvier (France) identified a German fossil as a giant marine reptile that would later be named Mosasaurus. He also identified another German fossil as a flying reptile, which he named Pterodactylus. Cuvier speculated, based on the strata in which these fossils were found, that large reptiles had lived prior to what he called “the age of mammals.” Cuvier’s speculation was supported by a series of finds in Great Britain in the next two decades. Mary Anning (UK) collected the fossils of marine reptiles, including the first recognized ichthyosaur skeleton, in 1811, and the first two plesiosaur skeletons ever found, in 1821 and 1823. Many of Anning’s discoveries were described scientifically by the British geologists William Conybeare, Henry De la Beche, and William Buckland. Anning first observed that stony objects known as “bezoar stones”, which were often found in the abdominal region of ichthyosaur skeletons, often contained fossilized fish bones and scales when broken open, as well as sometimes bones from small ichthyosaurs. This led her to suggest to Buckland that they were fossilized feces, which he named coprolites. In 1824, Buckland found and described a lower jaw that belonged to a carnivorous land-dwelling reptile he called Megalosaurus. That same year Gideon Mantell (UK) realized that some large teeth he had found in 1822 belonged to a giant herbivorous land- dwelling reptile that he named Iguanodon because the teeth resembled those of an iguana. In 1831, Mantell published an influential paper entitled “The Age of Reptiles” in which he summarized the evidence for an extended time during which giant reptiles roamed the Earth. Based on the appearance of the different giant reptile fossils in the rock strata, Mantell divided the era into three intervals, which anticipated the modern division of the Mesosoic era into the Triassic, Jurassic, and Cretaceous periods. In 1832, Mantell found a partial skeleton of an armored reptile he called Hylaeosaurus. In 1841 the English anatomist Richard Owen created a new order of reptiles, which he called Dinosauria, to contain MegalosaurusIguanodon, and Hylaeosaurus.
A portrait of Mary Anning and her dog, painted before 1833.
A portrait of Mary Anning (1799-1847) and her dog, painted before 1833.
A stellar parallax is the apparent shift of position of a nearby star against the background of distant objects that is made possible by the movement of the Earth in its orbit. Once a stellar parallax is measured, the distance to the star can be determined using trigonometry. The distance of most stars from the Earth makes stellar parallax so difficult to detect that some scientists argued that it did not exist. For example, James Bradley tried but could not measure stellar parallaxes in 1729. Then, in 1838, Friedrich Bessel (Germany) measured the stellar parallax for the star 61 Cygni using a Fraunhofer heliometer. This discovery was closely followed by Thomas Henderson (Scotland) for the star Alpha Centuri in 1839, and Friedich von Struve (Germany) for the star Vega in 1840.
An 1839 portrait of Friedrich Bessel.
An 1839 portrait of Friedrich Bessel (1784-1846).
NEPTUNE (1846)
Neptune is the eighth and farthest planet from the sun. It is the fourth largest planet by diameter and the third largest by mass. In 1821, French astronomer Alexis Bouvard published tables of the orbit of Uranus that contained significant discrepancies, which led to the prediction of another planet. In 1835, Benjamin Valz (France), Friedrich Bernhard Gottfried Nicolai (German) and Niccolo Cacciatore (Italy) each independently conjectured that a trans-Uranian planet caused the otherwise inexplicable discrepancies in the historical record of the orbits of both Halley’s comet and Uranus. Using Bouvard’s tables, both Urbain Jean Joseph Le Verrier (France) and John Couch Adams (UK), working independently, calculated the location where the new planet should be found in 1846. On September 23, 1846, German astronomer Johann Gottfried Galle, with the assistance of Heinrich Louis d’Arrest, observed the new planet within one degree of the predicted location.
A 1989 photograph of Neptune taken by the Voyager 2 spacecraft.
A 1989 photograph of Neptune taken by the Voyager 2 spacecraft.
George Boole (UK/Ireland) developed Boolean algebra and Boolean logic in books published in 1847 and 1854. Boolean algebra has been fundamental in the development of digital electronics and is used in set theory and statistics. Many are familiar with it as the basis for computer database search engines.
George Boole (1915-1864).
George Boole (1815-1864).
ASPIRIN (1853)
Aspirin is acetylsalicylic acid. In ancient times, plants containing salicylate, such as willow, were used to prepare medicines. There are references to it in Egyptian manuscripts from between 2000 and 1000 BCE and Hippocrates mentions salicylic tea to reduce fever in 400 BCE. Willow bark extract was a common remedy in the 18th and early 19th centuries, after which pharmacists began to experiment with and prescribe chemicals related to salicylic acid. French chemist Charles Frédéric Gerhardt first produced acetylsalicylic acid in the lab in 1853. A pure form of the chemical was synthesized by Felix Hoffmann (Germany), a chemist with the Bayer Company, in 1897, and it was soon marketed all over the world. Sales rose after the flu epidemic of 1918, but dropped after the introduction of acetaminophen in 1956 and ibuprofen in 1962. Aspirin sales once again increased in the last decades of the 20th Century, when scientists discovered aspirin’s anti-clotting benefits.
Felix Hoffman
Felix Hoffmann (1868-1946).
It took a long time for the idea of spontaneous generation – that living things could arise from non-living matter – to die. Francesco Redi (Italy) proved in 1668 that maggots did not spontaneously generate from rotten meat but were hatched from tiny eggs laid by flies. Lazzaro Spallanzani (Italy) conducted an experiment in 1768 that supported Redi’s conclusion and contradicted the 1745 experiment of John Needham that seemed to support spontaneous generation. Louis Pasteur (France) put the final nails in spontaneous generation’s coffin in 1859 with an experiment in which no life grew in a sterile flask for a year until the neck of the flask was removed and microorganisms had access to the liquid inside. John Tyndall conducted further investigations in 1875-1876 to support Pasteur’s work and dispel any lingering objections to his conclusion, although his experiments were plagued by airborne bacterial spores.
A diagram of Louis Pasteur’s experiment disproving spontaneous generation.
Digging or drilling for underground oil dates back to the 4th Century CE in China, where drill bits were attached to bamboo poles to dig wells of up to 800 feet deep. People in Arabian countries and Persia dug for oil as far back as the 9th Century. Also from the 9th to the 16th centuries, those living near Baku, in modern-day Azerbaijan, hand dug holes of up to 115 feet. Also in Baku, the first offshore drilling began in 1846. The first recorded land-based commercial oil well was begun in Oil Springs, Ontario in 1858. But American Edwin Drake’s drilling operation in Titusville, Pennsylvania in 1859 was the first oil well using modern principles. One of Drake’s key innovations was the drive pipe – he drove a cast iron pipe into the ground and then lowered the drill through the pipe, thus preventing the hole from collapsing.
A replica of the engine house and derrick at Drake's Well in Titusville, PA.
A replica of the engine house and derrick at Drake’s Well in Titusville, Pennsylvania.
An antiseptic is a substance applied to living tissue or skin to kill microbes and reduce the possibility of infection or sepsis. Sumerian clay tablets from 2150 BCE and writings of Hippocrates (c. 400 BCE) and Galen (c. 130-200 CE) all advocate the use of antiseptic agents. In the early 13th Century, Italian surgeons Hugh of Lucca and Theodoric of Lucca disregarded Galen’s view that pus was good and cleaned pus from wounds, then used wine to clean the wound and prevent infection. In an 1843 paper that was reissued in 1855, Oliver Wendell Holmes (US) advocated cleaning of medical instruments to prevent the spread of puerperal fever. In 1861, Ignaz Semmelweis (Austria) recommended that physicians wash their hands in chlorine solution before assisting in childbirth. While serving in the Confederate Army in the American Civil War in the early 1860s, George H. Tichenor (US) used alcohol on wounds. The adoption of antiseptic practices only became mainstream after British surgeon Joseph Lister’s 1867 paper, Antiseptic Principle of the Practice of Surgery, in which he advised the use of carbolic acid to create a sterile surgical environment.
A 1902 photograph of Joseph Lister.
A 1902 photograph of Joseph Lister (1827-1912).
Hormones are signaling molecules produced by the glands of living organisms that are transported to distant target organs by the circulatory system in order to regulate physiology and behavior. In 1894, George Oliver and Eduard Albert Sharpey-Schaeffer (UK) demonstrated the effect of an extract of the adrenal gland (the hormone adrenaline), which contracted blood vessels and muscles and raised blood pressure. In 1902, Ernest Starling and William Bayliss (UK) discovered secretin, which upon stimulation was released from the duodenum and carried to the pancreas, where it stimulated the pancreas to release digestive juices into the intestine. In 1905, Starling and Bayliss coined the term ‘hormone’ to describe secretin and similar substances. Edward C. Kendall isolated the thyroid hormone thyroxin in 1915. The same year, Walter Bradford Cannon (US) demonstrated the close connection between endocrine glands and emotions.
A diagram showing the source of some human hormones.
A diagram showing the sources of some human hormones.
Albert Einstein’s famous equation ‘E = mc2’ states the physical law that matter and energy are two forms of the same substance, that one can be converted to the other, and that the amount of energy produced by converting (i.e., destroying) even a small amount of mass is enormous, as it is proportional to the square of the speed of light. A number of precursors led up to Albert Einstein’s revolutionary equation. In 1717, Isaac Newton wondered whether particles of mass and particles of light might be converted into one another. Emanuel Swedenborg (Sweden) speculated in 1734 that matter was made of points of potential motion. Numerous physicists at the end of the 19th and beginning of the 20th Century sought to understand how electromagnetic fields affect the mass of charged particles. Albert Einstein (Germany) first introduced a mass-energy equivalence equation in his 1905 paper on special relativity; it was later reduced to the famous form of E = mc2. The equivalence of mass and energy has been experimentally proven in both directions. In 1932, John Cockcroft and E.T.S. Walton (UK) broke apart an atom, releasing energy, and found that the total mass of the fragments had decreased slightly, proving the conversion of mass into energy. In 1933, Irène and Frédéric Joliot-Curie (France) detected the conversion of energy into mass when they photographed a photon (a quantum of electromagnetic energy) converting into two subatomic particles.
A 2006 sculpture entitled "Relativity", located in Berlin, Germany.
A sculpture entitled “The Theory of Relativity”, designed by Scholz & Friends, was displayed in Berlin, Germany in 2006 as part of the “Walk of Ideas.”
Many metals emit electrons when light shines on them, a phenomenon known as the photoelectric effect. Heinrich Hertz (Germany) discovered the photoelectric effect in 1887. In 1905, Albert Einstein (Germany) discovered that the results of experiments measuring the photoelectric effect could be explained if light energy was carried in discrete quantized packets, or quanta. Einstein’s explanation lent support to quantum theory.
A diagram of the photoelectric effect.
A diagram of the photoelectric effect.
Brownian motion refers to the random movements of particles suspended in a liquid or gas fluid that result from collisions with smaller atoms or molecules in the fluid. Dutch scientist Jan Ingenhousz described the irregular motion of coal dust particles on the surface of alcohol in 1785, an early example of Brownian motion. The official discovery of Brownian motion took place in 1827, when Scottish botanist Robert Brown noted the unusual random movements of pollen grains suspended in water. Thorvald N. Theile (Denmark) provided the mathematical underpinnings of Brownian motion in 1880, and Louis Bachelier (France) used the model of Brownian motion to explain the stochastic processes of economic markets in a 1900 thesis. In 1905, Albert Einstein (Germany) explained Brownian motion as the result of the larger particle (e.g., pollen grain) being moved by individual molecules of the fluid in which it is suspended (e.g., water). Einstein’s explanation proved definitively that atoms and molecules exist. The predictions of Einstein’s paper were verified experimentally in 1908 by Jean Perrin (France).
An animated example of Brownian motion.
An animated example of Brownian motion.
A black hole is a region of spacetime, the gravitational pull of which is so strong that nothing, not even electromagnetic radiation, can escape it. The boundary of the region from which there is no escape is called the event horizon. According to the general theory of relativity, a mass that is sufficiently compact will deform spacetime enough to form a black hole. Some scientists believe supermassive black holes lie at the center of many galaxies, including the Milky Way. John Michell (UK) in 1783 and Pierre-Simon Laplace (France) in 1796 both suggested that some objects might have such strong gravitational fields that light could not escape. In 1916, soon after Albert Einstein (Germany) published his general theory of relativity, Karl Schwarzschild (Germany) was the first to show mathematically that Einstein’s theory predicted black holes under certain conditions. Johannes Droste (The Netherlands?) followed up Schwarzschild’s findings in 1916-1917, finding that Schwarzschild’s solution to general relativity created a singularity (where some terms became infinite) at a point known as the Schwarzschild radius, which defines the event horizon. Arthur Eddington (UK) showed in 1924 that the singularity disappeared after a change of coordinates. Subrahmanyan Chandrasekhar (India) showed in 1931 that stars and other objects above a certain mass (1.4 solar masses) were inherently unstable and would eventually collapse. In 1939, Robert Oppenheimer (US) and others predicted that neutron stars larger than three suns would collapse into black holes. In 1958, David Finkelstein (US) was the first to describe a black hole as a region of space from which nothing could escape. Important theoretical discoveries about the nature of black holes were made by Roy Kerr (NZ) in 1963, Ezra Newman (US) in 1965, Werner Israel (Germany/South Africa/Canada), Brandon Carter (Australia) and David Robinson. The term ‘black hole’ was first used by journalist Ann Ewing in 1964; John Wheeler used the term in a 1967 lecture. Roger Penrose and Stephen Hawking (UK) showed in the late 1960s that singularities appear in generic solutions of general relativity. In the early 1970s, Hawking, Carter, James Bardeen (US) and Jacob Bekenstein (Mexico/Israel) formulated black hole thermodynamics and Hawking showed in 1974 that black holes should give off black body radiation. Black holes cannot be detected directly, but indirect evidence exists. The first indirect evidence of a black hole in an X-ray binary system, Cygnus X-1, was discovered by Charles Thomas Bolton, Louise Webster and Paul Murdin in 1972. Numerous other candidates have since been found.
An artist's depiction of the black hole near the star Cygnus X-1. It formed when a large star caved in. This black hole pulls matter from blue star beside it. Image Credit: NASA/CXC/M.Weiss
An artist’s depiction of Cygnus X-1, which scientists believe is a black hole. It formed when a large star collapsed on itself. This black hole pulls matter from the supergiant blue star beside it. Image Credit: NASA/CXC/M.Weiss
According to the Pauli exclusion principle, no two electrons in an atom can be in the same quantum state; in other words, two electrons must have opposite spin, thus cancelling each other, and there can be no more than two in the same orbital. A number of discoveries led up to the articulation of the principle by Austrian physicist Wolfgang Pauli in 1925. In 1916, for example, Gilbert N. Lewis (US) stated that the atom tends to hold an even number of electrons in the shell and especially to hold eight electrons that are normally arranged symmetrically at the eight corners of a cube. In 1919, Irving Langmuir (US) suggested that the periodic table could be explained if the electrons in an atom were connected or clustered in some manner. In 1922, Niels Bohr updated his model of the atom by assuming that certain numbers of electrons (for example 2, 8 and 18) corresponded to stable “closed shells.” Pauli tried to explain these empirical findings as well as the results of experiments on the Zeeman effect in atomic spectroscopy and in ferromagnetism. A 1924 paper by Edward Stoner (UK) pointed out that for a given value of the principal quantum number (n), the number of energy levels of a single electron in the alkali metal spectra in an external magnetic field, where all degenerate energy levels are separated, is equal to the number of electrons in the closed shell of the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state, if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck (The Netherlands/US) as electron spin.
A 1945 photograph of Wolfgang Pauli (1900-1958).
A 1945 photograph of Wolfgang Pauli (1900-1958).
TELEVISION (1925-1927)
As with so many technological advancements, choosing a specific date and a particular inventor fails to appreciate the contributions of so many people over a long period of time. Nevertheless, the chronology below supports the conclusion that the invention of television reached critical mass between 1925 and 1927, particularly with the work of Baird, Zworykin, Jenkins, Farnsworth and Bell Labs. An early milestone in the history of television was the first known transmission of a still image over an electronic wire by Abbe Giovanna Caselli (Italy) in 1862. In 1877, George Carey (US) designed a machine that would use selenium to allow people to see electrically-transmitted images; by 1880, he had built a primitive system with light-sensitive cells. In 1884, Paul Nipkow (Germany) sent images over wires with 18 lines of resolution using a rotating metal disk – this was the mechanical approach. In 1906, Lee De Forest (US) invented the Audion vacuum tube, which could amplify electronic signals. In the same year, Boris Rosing (Russia) combined a cathode ray tube with Nipkow’s disk to make a working television. In 1907, Rosing begin developing an electronic scanning method of reproducing images using a cathode ray tube – this was the electronic approach. In 1908, Alan Campbell-Swinton (UK) described how a cathode ray tube could be used as a transmitting and receiving device in a television system. In 1909, Georges Rignoux and A. Fournier (France) demonstrated instantaneous transmission of images in a mechanical system. In 1911, Rosing and Vladimir Zworykin (Russia) developed a mechanical/electronic system that transmitted crude images. In 1923, Zworykin (now in the US) patented a TV camera tube, the iconoscope, and later the kinescope, or receiver, although a 1925 demonstration was unimpressive. On March 25, 1925, John Logie Baird (Scotland) demonstrated transmission of silhouette images. In May, 1925, Bell Labs transmitted still images. On June 13, 1925, Charles Francis Jenkins (US) transmitted the silhouette image of a moving toy over a distance of five miles. Also in 1925, Zworykin patented a color TV system. On December 25, 1925, Kenjiro Takayanagi (Japan) demonstrated a mechanical/electronic system with 40 lines of resolution. In the USSR, Leon Theremin developed a series of increasingly higher resolution television systems, from 16 lines in 1925 to 100 lines in 1927. On January 26, 1926, Baird demonstrated a system with 30 lines of resolution, running at five frames per second, showing a recognizable human face. Also in 1926, Kálmán Tihanyi (Hungary) solved the problem of low sensitivity to light in television cameras through charge-storage. In 1927, Philo Farnsworth patented the Image Dissector, the first complete electronic television system. On April 7, 1927, Herbert Ives and Frank Gray of Bell Labs (US) demonstrated a mechanical television system that produced much higher-quality images than any prior system. Charles Jenkins (US) received the first television station license in 1928. In 1929, Zworykin demonstrated both transmission and reception of images in an electronic system. After a series of improvements to his design, Farnsworth transmitted live human images in 1929. In 1931, Jenkins invented the Radiovisor and began selling it as a do-it-yourself kit. Manfred von Ardenne (Germany) demonstrated a new type of system in 1931. Farnsworth gave a public demonstration of an all-electronic TV system, with a live camera, on August 25, 1934. The BBC began the first public television service on November 2, 1936 with 405 lines of resolution. In 1937, the BBC adopted new equipment that was far superior to prior systems. In 1940, Peter Goldmark invented a mechanical color TV system with 343 lines of resolution. In 1941, the US adopted a 525-line standard. In 1943, Zworykin developed an improved camera tube that allowed recording of night events. In 1948, USSR began broadcasting at 625-lines of resolution, which was eventually adopted throughout Europe. Cable television was introduced in 1948 to bring television to rural areas. Videotape broadcasting was introduced in 1956 by Ampex. In 1962, the launching of the Telstar satellite permitted international broadcasting. Color televisions began to outnumber black & white TVs in the 1970s. Satellite television began in 1983. High definition TV appeared in 1998. Analog broadcast TV ended on June 12, 2009, leaving digital television remaining.
John Logie Baird, with one of his earliest television systems. In the end, it was the electronic path of Zworkin and Farnsworth that carried the day.
John Logie Baird (1888-1946), with one of his earliest television systems. In the end, it was the electronic path of Zworkin and Farnsworth that carried the day.
People in ancient Egypt, China and Mesoamerica used molds to treat infected wounds. The Germ Theory of Disease, propagated by Louis Pasteur (France) in the mid-19th Century, sparked the search for antibiotic agents. In 1871, Joseph Lister discovered that bacteria would not grow in mold-infected urine. John Tyndall (Ireland/UK) noted fungal inhibition of bacteria in 1875. In 1887, Louis Pasteur and Jules-François Joubert (France) demonstrated the antibiotic effect. In 1895, Italian physician Vincenzo Tiberio noted that the Penicillum mold killed bacteria. In the 1890s, Rudolf Emmerich and Oscar Löw (Germany) created an antibiotic but it often failed. In 1904 Paul Ehrlich (Germany) sought the ‘magic bullet’ against syphilis and systematically tested hundreds of substances before finding Salvorsan in 1909. Alexander Fleming discovered penicillin in 1928. In 1932, German scientists at Bayer (Josef Klarer, Fritz Mietzsch and Gerhard Domagk) synthesized and tested the first sulfa drug, Prontosil. In 1939, Rene Dubos (France/US) created the first commercially manufactured antibiotic – tyrothricin – although it proved too toxic for systemic usage. In 1943, Selman Waksman (US), derived stretomycin from soil bacteria. In 1955, Lloyd Conover (US) patented tetracycline. In 1957, Nystatin was patented. SmithKline Beecham patented the semisynthetic antibiotic amoxicillin in 1981; it was first sold in 1998. A major concern throughout the history of antibiotics is the development of antibiotic resistant strains of bacteria, which is such a significant problem that researchers are looking for alternatives to antibiotic treatments.
A diagram showing the different mechanisms used by antibiotics to kill bacteria.A diagram showing the different mechanisms used by antibiotics to kill bacteria.
The positron (also known as the antielectron) is the antimatter counterpart of the electron that is part of the Standard Model. Paul Dirac (UK) suggested in 1928 that electrons can have both positive and negative charge. In a follow-up paper in 1929, Dirac suggested that the proton might be the negative energy electron. Robert Oppenheimer strongly disagreed with Dirac’s suggestion, which led Dirac in 1931 to predict the existence of an anti-electron with the same mass as an electron, which would be annihilated upon contact with an electron. Ernst Stueckelberg (Germany) and Richard Feynmann (US) developed the theory of the positron, while Yoichiro Nambu (Japan/US) applied the theory to all matter-antimatter pairs of particles. The first to observe the positron was Dmitri Skobelsyn (USSR) in 1929. The same year, Chung-Yao Chao (China/US) conducted similar experiments but results were inconclusive. Carl David Anderson (US) is acknowledged to have discovered the positron on August 2, 1932; he also coined the word.
Carl Anderson's 1932 photograph of a cloud chamber showing the trail of a positron.
Carl Anderson (1905-1991) was able to prove the existence of the positron from this 1932 photograph of a trail in a cloud chamber.
Sulfonamide (also known as sulphonamide) is the basis for several groups of drugs, some of which are antibacterial. The first antibacterial sulfa drug was Prontosil, which has the chemical name sulfonamidochrysoidine. Although Paul Gelmo (Austria) had synthesized the chemical in 1909, he did not pursue his findings. Josef Klarer and Fritz Mietzsch (Germany) synthesized it at Bayer, and in 1932, Gerhard Domagk (Germany) discovered it was effective in treating bacterial infections in mice. Results of clinical human studies were published in 1935, but it was treatment of Franklin Roosevelt’s son’s bacterial infection in 1936 that led to widespread acceptance of the drug. In 1935, scientists at the Pasteur Institute (France) discovered that Prontosil is metabolized to sulfanilamide, a much simpler molecule, which, as Prontalbin, soon replaced Prontosil. The chemical nature of sulfanilamide made it easy for chests to link it to other molecules, which led to hundreds of sulfa drugs.
Gerhard Domagk
Gerhard Domagk (1895-1964).
Dark matter is a substance that scientists have proposed to explain certain gravitational effects in the universe. Although there is significant indirect evidence for the existence of dark matter, it has not been directly observed or detected. According to the dark matter hypothesis, it cannot be seen with telescopes and does not appear to emit or absorb electromagnetic radiation (including light) at any significant level. Some have suggested that it may be composed of an undiscovered subatomic particle. According to the most recent estimate, the known universe contains 4.9% ordinary matter; 26.8% dark matter and 68.3% dark energy. Jan Oort (The Netherlands) first proposed the existence of unseen matter in 1932 to explain the orbital velocities of stars in the Milky Way. Fritz Zwicky (Switzerland/US) suggested the existence of what he called ‘dark matter’ in 1933 to explain what appeared to be missing mass in measuring the orbital velocities of galaxies in clusters. In 1973, Jeremiah Ostriker and James Peebles (US) calculated mathematically that galaxies would collapse if they only contained the mass we can see. They proposed that an additional mass of three to ten times the size of the visible mass was necessary to explain the observed shapes of the galaxies. At about the same time, Kent Ford and Vera Rubin (US), using new photon detectors, found that the movement of hydrogen clouds in the Andromeda galaxy could not be explained if the majority of the galaxy’s mass was contained in the visible matter, but only if it was contained in invisible matter that existed outside the visible edge of the galaxy. In 2013, a team of scientists said they had discovered a weakly-interacting massive particle (WIMP) that could make up dark matter. In 2014, NASA’s Fermi Gamma-ray Space Telescope recorded high-energy gamma-ray light emanating from the center of the Milky Way that confirmed a prediction about dark matter.
A pie chart showing the composition of the universe.
A pie chart showing the composition of the universe.
In 1865, an artificial fiber made from cellulose was used to make acetate. Sir Joseph Swan (UK) invented the first artificial fiber in about 1883 by chemically modifying fibers from tree bark to create a cellulose liquid, from which the fibers were drawn. Swan displayed fabrics made from his material at an 1885 exhibition. Hilaire de Chardonnet (France) produced an artificial silk in the late 1870s from nitrocellulose. He displayed products made from the artificial fiber at an 1889 exhibition, but the material was extremely flammable and was not successful. Arthur D. Little (US) reinvented acetate from cellulose in 1893. In 1894, Charles Frederick Cross, with Edward John Bevan and Clayton Beadle (UK), produced an artificial fiber from cellulose that they called viscose. Courtaulds Fibers (UK) produced viscose commercially in 1905 and in 1924 renamed it rayon. Camille and Henry Dreyfus (Switzerland) used acetate to make motion picture film and other products beginning in 1910. The Celanese Company (US), Camille Dreyfus founder, used acetate to make textiles beginning in 1924. The first completely synthetic fiber not based on naturally-occurring cellulose, was nylon, which was invented by Wallace Carothers (US) at DuPont in 1935. In 1938, Paul Schlack of I.G. Farben in Germany invented another form of nylon. DuPont began commercial production of nylon for use in women’s stockings as well as parachutes and ropes, among other things, in 1939. Polyester was invented by John Rex Whinfield and James Tennant Dickson (UK) at the Calico Printers’ Association in 1941; they patented their first polyester fiber as Dacron. Also in 1941, DuPont introduced acrylic, a new synthetic fiber that resembled wool, under the brand name Orlon.
Wallace Carothers demonstrates the strength of his invention, nylon.
Wallace Carothers (1896-1937) demonstrates the strength of his invention, nylon.
THE VIRUS (1935)
A virus is a very small infectious agent that replicates only inside the living cells of other organisms. In the mid-late 19th Century, when Louis Pasteur (France) could not find a bacterial cause for rabies, he hypothesized that the disease might be caused by a pathogen too small to be seen by a microscope. In 1884, Charles Chamberland (France) created a filter with holes smaller than bacteria, which would become essential for studying viruses. Dmitri Iosefovich Ivanovsky (Russia) used the Chamberland filter to determine that the cause of the tobacco mosaic disease was a pathogen smaller than a bacterium, which he announced in an 1882 article. Martinus Beijerinck (The Netherlands) arrived at similar results in 1898; he coined the term ‘virus’ to describe the unseen pathogen. Also in 1898, Friedrich Loeffler and Paul Frosch (Germany) determined that foot and mouth disease in animals was caused by a virus. In 1914, Frederick Twort (UK) discovered the first bacteriophage, a type of virus that infects bacteria; he published his results in 1915 but they were ignored. French-Canadian microbiologist Félix d’Herelle discovered bacteriophages indendently, and announced his discovery in 1917. Ernst Ruska and Max Knoll (Germany) made the first electron micrographs of viruses in 1931. In 1935, Wendell Stanley (US) succeeded in crystallizing the tobacco mosaic virus in 1935, proving it was a particle, not a fluid, and that it was made largely of protein. Electron micrographs of the tobacco mosaic virus were made in 1939 and X-ray crystallography was performed on it by Bernal and Fankuchen in 1941. Rosalind Franklin (UK) discovered the full structure of the tobacco mosaic virus in 1955. As of 2014, scientists have identified over 2,000 species of virus.
Transmission electron micrograph of multiple bacteriophage viruses attached to a bacterial cell wall.
A transmission electron micrograph of bacteriophage viruses attached to a bacterial cell wall.
Charles Babbage (UK) is the grandfather of computer science. Beginning in the 1810s, he developed a theory of computing machines, which he put into practice with progressively more complex designs. Babbage’s 1837 proposal for an Analytical Engine would possibly have been the first true computer, had it actually been built. It had expandable memory, an arithmetic unit, logical processing abilities, and the ability to interpret a complex programming language. Ada Lovelace (UK), who worked with Babbage, furthered his work by designing the first computer algorithm and by predicting that a computer would not only perform mathematical calculations but manipulate symbols of all kinds. Kurt Gödel (Austria/US) established the mathematical foundations of computer science in 1931 with his incompleteness theorem, which showed that every formal system contained limits to what could be proved or disproved within it. If Babbage was the grandfather, Alan Turing (UK) was the father of computer science. In 1936, Turing (along with American Alonzo Church), formalized an algorithm containing the limits of what can be computed, as well as a purely mechanical model for computing. The Church-Turing thesis of the same year states that, given sufficient time and storage space, a computer algorithm can perform any possible calculation. Turing introduced the ideas of the Turing machine and the Universal Turing machine (which can simulate any other Turing machine) in 1937. Turing machines are not real objects but mathematical constructs designed to determine what can be computed by any proposed computer.
A photograph of Alan Turing.
Alan Turing (1912-1954).
In induced nuclear fission, the nucleus of an atom is split by bombarding it with a subatomic particle, often a neutron. The fission process usually releases free neutrons and protons (in the form of gamma rays) and a very large amount of energy. In 1917, Ernest Rutherford (NZ/UK) used alpha particles to convert nitrogen into oxygen, the first nuclear reaction. Ernest Walton (Ireland) and John Cockcroft (UK) used artificially accelerated protons to split the nucleus of a lithium-7 atom into two alpha particles. In 1934, Enrico Fermi (Italy) and his team, bombarded uranium with neutrons, but concluded the experiments created new elements with atomic numbers higher than uranium. In 1934, Ida Noddack (Germany) suggested that Fermi’s experiments had actually broken the nucleus into several large fragments. After reading of Fermi’s results, Otto Hahn, Fritz Strassman (Germany) and Lise Meitner (Austria) began performing similar experiments until Meitner, a Jew, was forced to flee the Nazis to Sweden. In December 1938, Hahn and Strassman proved that bombarding uranium nuclei with neutrons had created barium, an element with 40% less atomic mass than uranium. In 1939, Meitner and Otto Robert Frisch (Austria) interpreted Hahn and Strassmann’s results as proof they had split the uranium nucleus; they coined the term ‘fission’ to describe the reaction. Firsch confirmed this theory experimentally in January 1939. Also in January 1939, a team at Columbia University, including Erico Fermi, replicated the nuclear fission experiment.
A replica of the nuclear fission experiment conducted by Hahn and Strassman in 1938, at at the Deutsches Museum in Munich.
A replica of the nuclear fission experiment conducted by Otto Hahn (1879-1968) and Fritz Strassmann (1902-1980) in 1938, located at the Deutsches Museum in Munich.
A nuclear reactor initiates and controls a sustained nuclear chain reaction. Heat from nuclear fission occurring in a nuclear reactor is used to generate electricity and propel ships. In 1933, Hungarian-American scientist Leó Szilárd recognized that neutron-caused nuclear reactions ould lead to a nuclear chain reaction. In 1934, Szilárd filed the first patent application for the idea of a nuclear chain reaction using neutrons bombarding light elements. After the discovery of nuclear fission of uranium in 1938, Szilárd and Enrico Fermi (Italy) confirmed experimentally in 1939 Otto Hahn and Fritz Strassmann’s prediction that nuclear fission released several neutrons, which were then available to bombard other nuclei. Also in 1939, Francis Perrin (France) and Rudolph Peierls (Germany/US) independently worked out the ‘critical mass’ of uranium needed to sustain the reaction. In 1939, Szilárd proposed that a nuclear chain reaction would work best by stacking alternate layers of graphite and uranium in a lattice, the geometry of which would define neutron scattering and subsequent fission events. In 1942, Ernico Fermi and his team at the University of Chicago (including Szilárd) created the first controlled, self-sustaining nuclear chain reaction (the first nuclear reactor) from ‘piles,’ using Szilárd’s lattice of uranium and graphite. (The term ‘reactor’ has since replaced ‘piles’.) A number of nuclear reactors were built by the US military beginning in 1943 as part of the Manhattan Project to build a nuclear weapon. The first nuclear reactor for civilian use was launched in June 1954 in the USSR.
Scientists observing the world’s first self-sustaining nuclear chain reaction, in the Chicago Pile No. 1, December 2, 1942. Photograph of an original painting by Gary Sheehan, 1957.A black and white photograph of Gary Sheehan’s 1957 painting showing Enrico Fermi (1901-1954) and other scientists observing the world’s first self-sustaining nuclear chain reaction, in the Chicago Pile No. 1 on December 2, 1942.
In the 1930s, while experimenting with the genes for eye color in fruit flies, George Beadle (US) and Boris Ephrussi (USSR/France) concluded that each gene was responsible for an enzyme (a type of protein) acting in the metabolic pathway of pigment synthesis. In 1941, Beadle and Edward Lawrie Tatum (US), using the bread mold Neurospora crassa, published their discovery that genes control cells by controlling the specificity of enzymes, i.e., one gene controls one enzyme so a mutation in a gene will change the enzymes available, causing the blockage of a metabolic step. With modifications, the one gene-one enzyme hypothesis remains essentially valid.
George Beadle and Edward Tatum won the 1958 Nobel Prize in __ for their genetic research.
George Beadle and Edward Tatum won the 1958 Nobel Prize in Physiology/Medicine for their genetic research.
Radiocarbon dating uses carbon-14, a radioactive isotope of carbon, to determine the age of organic materials. Radiocarbon dating works because radioactive carbon in an organism begins to decay at a predictable rate starting at the time of death. Radiocarbon dating is normally accurate for objects that are 50,000 years old or younger. In a series of experiments beginning in 1939, Willard F. Libby (US) investigated isotopes of elements in organic material, including carbon-14. A 1939 paper by W.E. Danforth and S.A. Korff (US) on carbon-14 sparked Libby’s idea that radiocarbon dating might be possible. In 1946, Libby proposed that living matter might contain carbon-14 and went on to discover carbon-14 in organic methane. The suggestion of using carbon-14 as a way to date organic materials came in a 1947 paper by Libby and others. Libby and James Arnold (US) announced in 1949 that they had used carbon-14 to date wood samples from the tombs of two Ancient Egyptian kings to 2800 BCE, plus or minus 250 years, which was consistent with independent dates of 2625 BCE, plus or minus 75 years. In the years following, scientists improved and refined the accuracy of radiocarbon dating.
Willard Libby (1908-1980).
Willard Libby (1908-1980).
Also known as transposons or jumping genes, transposable elements (TEs) are sequences of DNA that can change position within the genome. In a series of experiments with maize beginning in 1944, American biologist Barbara McClintock at Cold Spring Harbor Laboratory discovered in 1948 that certain parts of the chromosomes had switched positions, which disproved the common belief that genes had fixed positions. McClintock also showed that TEs sometimes reversed earlier mutations and may be responsible for turning genes on and off. McClintock reported her findings in a series of reports and articles between 1950 and 1953, but her work was largely ignored until after TEs were independently discovered in bacteria in 1967 and 1968 by E. Jordan, H. Saedler and P. Starlinger (US). In 1996, Philip SanMiguel (US) estimated that TEs make up a large proportion of the genome of eukaryotic organisms: 50% of human genes and up to 90% of maize genes.
Barbara McClintock (1902-1992).
Barbara McClintock (1902-1992).
Poliomyelitis (often called polio) is an acute infectious disease carried by a virus and spread from person to person. Polio, which can cause paralysis, was first identified by Jakob Heine (Germany) in 1840. Karl Landsteiner (Austria) identified the pathogen, poliovirus, in 1908. Failed early attempts to create a vaccine for polio were made by Maurice Brodie and John Kollmer (US), working independently, in 1936. The successful cultivation of human poliovirus in the laboratory in 1948 by Americans John Enders and Thomas Weller was a significant step for vaccine research. Jonas Salk (US) developed a vaccine using inactivated (i.e., dead) poliovirus in 1952, which was approved and released in 1955. Albert Sabin (Poland/US) used live but attenuated poliovirus to create a second, oral vaccine in 1957, which was licensed for use in 1962. The two vaccines have eliminated polio from most of the world.
Jonas Salk. Photo by Yousef Kauch.
Jonas Salk (1914-1995). Photo by Yousef Karsh.
Global warming refers to a rise in the average temperature of the Earth’s climate system in recent years. Because 90% of the recent increase in temperature has been absorbed by the oceans, global warming is often used to refer to the average temperature of the air and sea at the Earth’s surface. Climate change, in this context, refers to changes in the climate, including temperature, caused by human activities. The major human activities influencing climate change are fossil fuel combustion, which sends gaseous emissions into the atmosphere, aerosols, carbon dioxide released by cement manufacture, land use, ozone depletion, animal agriculture and deforestation. While humans have long speculated how their activity affects the climate (e.g., 19th Century Americans debated whether cutting down trees might affect rainfall), the modern science of climate change began in 1896, when Svante Arrhenius (Sweden) predicted the ‘greenhouse effect’ – as humans burned fossil fuels, they would add carbon dioxide to the atmosphere, which would raise the temperature. Arrhenius was not concerned about his conclusions, however, because he believed the warming would take thousands of years and would benefit humanity. Guy Steward Callendar, a British engineer and inventor, followed up on Arrhenius’s predictions in the 1930s, when he published a number of papers on the effects of human-caused carbon dioxide increases on global climate. His estimates of temperature increases in the half-century before 1938 have been confirmed with modern detectors. Scientists such as Canadian physicist Gilbert Plass followed up on Callendar’s work in the 1950s. Several important scientific discoveries in the 1950s increased concern in the scientific community about carbon dioxide. First, Hans Suess (Austria/US) found in 1955 that carbon dioxide released by burning of fossil fuels was not immediately absorbed by the ocean. Then, in 1957, work by Roger Revelle (US) showed that the ocean surface layer had a limited ability to absorb carbon dioxide. Finally, in 1958, Charles David Keeling (US) published detailed, comprehensive measurements showing that the amount of carbon dioxide in the atmosphere was rising. In 1967, Syukuro Manabe (Japan) and Richard Wetherald (US) developed a detailed computer model of the climate incorporating convection, the first of many computer models that scientists used to manage the huge amount of data and many different variables involved in predicting climate over time.
Charles David Keeling (1928-2005).
Charles David Keeling (1928-2005).
German physicist Werner Jacobi at Siemens AG designed the first integrated transistor amplifier in 1949. In 1952, Geoffrey Dummer (UK) suggested that a variety of standard electronic components could be integrated in a monolithic semiconductor crystal. In 1956, Dummer built a prototype integrated circuit. In 1952, American Bernard Oliver invented a method of manufacturing three electrically connected planar transistors on one semiconductor crystal. Also in 1952, Jewell James Ebers (US) at Bell Labs created a four-layer transistor, or thyristor. William Shockley (US) simplified Ebers’s design to a two-terminal, four-layer diode, but it proved unreliable. Harwick Johnson (US) at RCA patented a prototype integrated circuit in 1953. In 1957, Jean Hoerni (Switzerland/US) at Fairchild Semiconductor proposed a planar technology of bipolar transistors. Three breakthroughs occurred in 1958: (1) Jack Kilby (US) at Texas Instruments patented the principle of integration and created the first prototype integrated circuits; (2) Kurt Lehovec (Czech Republic/US) of Sprague Electric Co. invented a method of isolating components on a semiconductor crystal electrically; and (3) Robert Noyce (US) of Fairchild Semiconductor invented aluminum metallization – a method of connecting integrated circuit components. Noyce also adapted Hoerni’s planar technology as the basis for an improved version of insulation. Hoerni made the first prototype of a planar transistor in 1959. Jay Last and others at Fairchild built the first operational semiconductor integrated circuit on September 27, 1960. Texas Instruments announced its first integrated circuit in April 1960, but it was not marketed until 1961. Texas Instruments sued Fairchild in 1962 based on Kilby’s patent and the parties settled in 1966 with a cross-licensing agreement. The first integrated circuits with transistor-transistor logic instead of resistor-transistor logic were invented by Tom Long (US) at Sylvania in 1962. In 1964, both Texas Instruments and Fairchild replaced the resistor-transistor logic of their integrated circuits with diode-transistor logic, which was not vulnerable to electromagnetic interference. In 1968, Italian physicist Federico Faggin developed the first silicon gate integrated circuit with self-aligned gates. The same year, Robert H. Dennard (US) invented dynamic random-access memory, a specialized type of integrated circuit. Also in the late 1960s, medium scale integration (MSI), in which each chip contained hundreds of transistors, was introduced. The specialized integrated circuit known as a microprocessor was introduced by Intel in 1971. Large-scale integration (LSI), which arrived in the mid-1970s, brought chips with tens of thousands of transistors each. Ferranti (Italy) introduced the first gate-array, the Uncommitted Logic Array (ULA) in 1980, which led to the creation of application-specific integrated circuits (ASICs). Very large-scale integration (VLSI) brought chips with hundreds of thousands of transistors in the 1980s and several billion transistors as of 2009.
Jack Kilby's prototype integrated circuit, from 1959.
A 1959 prototype integrated circuit, made by Jack Kilby (1923-2005).
While man has been observing the Earth’s moon since ancient times, and Galileo Galilei made the first detailed telescopic observations in 1610-1612, physical exploration of the moon did not begin until September 14, 1959, when the USSR’s unmanned probe Luna 2 made a hard landing on the moon’s surface. Luna 3 photographed the far side of the moon for the first time on October 7, 1959. Luna 9 made a soft landing on the moon and sent the first pictures from the moon’s surface on February 3, 1966. Frank Borman, James Lovell and William Anders (US), in Apollo 8, became the first humans to enter lunar orbit and see the far side of the moon on December 24, 1968. Neil Armstrong and Edwin “Buzz” Aldrin in Apollo 11 (US) landed on the moon on July 20, 1969. The next day, Armstrong became the first man to walk on the moon. US astronauts Edwin Aldrin and Neil Armstrong land on the moon. The US sent a total of six manned missions to the moon between 1969 and 1972. There were 59 unmanned missions by the US or USSR between 1959 and 1976. Three Luna probes and six Apollo missions returned to Earth with moon rock samples. Japan sent probes into the moon’s orbit in 1990 and 2007. NASA and the Ballistic Missile Defense Organization launched orbiters in 1994 and 1998. A European Space Agency probe began orbiting the moon in 2004, then intentionally crashed in 2006. China send an orbital probe in 2007; it was intentionally crashed on the moon’s surface in 2009. A rover from a second Chinese orbiter soft-landed on the moon on December 14, 2013. India sent an orbiter to the moon in 2008 and landed an impact probe on November 14, 2008. Many other orbiters and landers are planned for the future.
American astronaut Buzz Aldrin on the surface of the moon on July 20, 1969. Photo by Neil Armstrong.
American astronaut Buzz Aldrin (1930- ) on the surface of the moon on July 20, 1969. Photo by Neil Armstrong (1930-2012).
QUASARS (1963)
Quasars (short for ‘quasi-stellar radio sources’) belong to a class of objects called active galactic nuclei: they are very luminous sources of electromagnetic energy with a high redshift. Most scientists now believe that a quasar is the compact region in the center of a galaxy that surrounds a supermassive black hole and that the quasar’s energy comes from the mass falling onto the accretion disc around the black hole. Allan Sandage (US) and others discovered the first quasars in the early 1960s. In 1960, a radio source named 3C 48 was tied to a visible object. John Bolton (UK/Australia) observed a very large redshift for the object but his claim was not accepted at the time. In 1962, Bolton and Cyril Hazard identified another such radio source, 3C 273. In 1963, Marten Schmidt (The Netherlands) used their measurements to identify the visible object associated with the radio source and obtain an optical spectrum, which showed a very high redshift (37% of the speed of light). Hong-Yee Chiu (China/US) first used the term ‘quasar’ to describe the new type of object in a 1964 article. Scientists debated the distance of quasars until the 1970s, when the mechanisms of black hole accretion discs were discovered. In 1979, images of a double quasar provided the first visual evidence of the gravitational lens effect predicted by Einstein’s general theory of relativity.
An X-ray photograph of quasar PKS 1127-145, taken at the Chandra X-ray Observatory in 2000. The quasar is a highly luminous source of X-rays and visible light that is located about 10 billion light years from Earth. The photo shows an X-ray jet a million light years long that probably resulted from the collision of a beam of high-energy electrons with microwave photons. Photo: NASA.
Also known as Standard Model of Quantum Field Theory and the Standard Model of Particle Physics, the Standard Model, which is the result of the work of many scientists over the period of 1970-1973 and after, summarizes the forces and particles that make up the universe. According to the Standard Model, there are three classes of elementary particles: fermions, gauge bosons, and the Higgs boson. There are 12 fermions, all of which have spin ½; they include six leptons (including electrons, muons, and tauons and their neutrino counterparts), and six quarks (including up, charm, and top and their charge complements, down, strange, and bottom). Leptons and quarks interact by exchanging generalized quanta, particles of spin 1. Bosons, which have spin 1, are particles involved in the transmission of forces and include gluons, which carry the strong force that binds quarks together. Thus bound together, the quarks form hadrons, including the protons and neutrons that make up atomic nuclei. Bosons also include photons, which carry the electroweak force and attract electrons to orbit the nuclei. Other weak interactions are carried by the W , W+, and Z particles. Additional forces are carried by gravitons and Higgs bosons. The combination of the electromagnetic force and the weak interaction into the electroweak force by Sheldon Glashow (US) and others in 1961 paved the way for the Standard Model. The muon neutrino was first detected in 1962. In 1964, Murray Gell-Mann and George Zweig propose that hadrons are made of quarks. Steven Weinberg (US) and Abdus Salam (Pakistan) incorporated the Higgs mechanism into the electroweak theory in 1967. Experimental confirmation of the electroweak theory came in 1973 when the CERN supercollider detected the neutral weak currents that were predicted to result from Z boson exchange. The Standard Model’s explanation of the strong interaction received experimental confirmation in 1973-1974 when it was shown that hadrons are composed of fractionally-charged quarks. In 1983, Carlo Rubbia discovered the W and Z bosons. In 1995, the final undiscovered quark, the top quark, was discovered. The tau neutrino was detected in 2000 at Fermilab. The Higgs boson was finally discovered in the Large Hadron Collider at CERN in 2012.A diagram of the Standard Model, courtesy of Fermilab.A diagram of the Standard Model, courtesy of Fermilab.
A hydrothermal vent is a fissure in the planet’s surface from which geothermally heated water issues. Hydrothermal vents are found near volcanic activity, in ocean basins and hotspots and in areas where tectonic plates are moving apart. Deep sea hydrothermal vents often form large features called black smokers. Although they have no access to sunlight, some hydrothermal vents are biologically active and host dense and complex communities based on chemosynthetic bacteria and archaea. A deep water survey of the Red Sea in 1949 revealed hot brines that could not be explained. In the 1960s, the hot brines and muds were confirmed and found to be coming from an active subseafloor rift. No biological activity was found in the highly saline environment. A team from Scripps Intitution of Oceanography led by Jack Corliss (US) found the first evidence of chemosynthetic biological activity surrounding underwater hydrothermal vents that formed black smokers along the Galapagos Rift in 1977; they returned in 1979 to use Alvin, a deep-water research submersible, to observe the hydrothermal vents directly. Peter Lonsdale published the first paper on hydrothermal vent biology in 1979. Neptune Resource NL discovered a hydrothermal vent off the coast of Costa Rica in 2005. Among the deepest hydrothermal vents are the Ashadze hydrothermal field on the Mid-Atlantic Ridge (-4200 meters), a vent at the Beebe site in the Cayman Trough (-5000 meters), discovered in 2010 by scientists from NASA and Woods Hole Oceanographic Institute; and a series of hydrothermal vents in the Caribbean found in 2014 (-5000 meters). By 1993, more than 100 species of gastropods had been found in hydrothermal vent communities. Scientists have discovered 300 new species at hydrothermal vents, including the Pompeii worm, discovered by Daniel Desbruyères and Lucien Laubier (France) in 1980 and Craig Gary (US) in 1997 and the scaly-foot gastropod in 2001.
A hydrothermal vent with black smokers and a biological community with large numbers of tube worms.
According to inflation theory, the universe underwent an exponential expansion of space from 10 ‾36 seconds after the Big Bang to between 10 ‾33 and 10 ‾32 seconds post-Big Bang. Inflation theory purports to explain the origin of the large-scale structure of the universe. The origins of inflation theory go back to 1917, when Albert Einstein invoked the cosmological constant to prove that the universe was static. At about the same time, Dutch scientist Willem de Sitter, analyzing general relativity, discovered a formula that described a highly symmetric inflating empty universe with a positive cosmological constant. Some believe that inflation theory was proposed by Erast Gliner (USSR) in 1965, who was not taken seriously at the time. In the early 1970s, Yakov Zeldovich (USSR) noted that the Big Bang model had serious problems with flatness and horizon. Vladimir Belinski (USSR), Isaak M. Khalatnikov (USSR), and Charles Misner (US) tried to solve the problems. American physicist Sidney Coleman’s study of false vacuums in the late 1970s raised important questions for cosmology. In 1978, Zeldovich drew attention to the monopole problem, a version of the horizon problem. In 1979, Alexei Starobinsky (USSR) predicted that the early universe went through a de Sitter phase, or inflationary era. In January 1980, Alan Guth (US) proposed scalar driven inflation to solve Zeldovich’s problem of the nonexistence of magnetic monopoles. In October 1980, Demosthenes Kazanas (Greece/US) suggested that exponential expansion might eliminate the particle horizon. Martin Einhorn (US) and Katsuhiko Sato (Japan) published a model similar to Guth’s in 1981. Guth’s theory and other early versions of inflation had a problem: bubble wall collisions. Andrei Linde (USSR/US) solved the problem in 1981, as did Andreas Albrecht and Paul Steinhardt (US), independently, with new inflation, or slow-roll inflation. Linde revised the model in 1983, calling the new version chaotic inflation. Numerous scientists worked on calculating the tiny quantum fluctuations in the inflationary universe that led to the structure we see today, particularly at a 1982 workshop at Cambridge University. Predictions of inflation theory were experimentally confirmed in 2003-2009 by the Wilkinson Microwave Anisotropy Probe’s findings of the flatness of the universe. The first direct evidence of gravitational waves, announced by Harvard-Smithsonian Center astronomers on March 17, 2014, provides additional support for inflation.
Alan Guth (1947- ).
Alan Guth (1947- ).
At the end of the Cretaceous Period 66 million years ago, a mass extinction eliminated 75% of all animal and plant species, including the dinosaurs. Although many hypotheses have been offered to explain this mass extinction (one of several in Earth’s history), the predominant theory is that of Luis Alvarez, who proposed in 1980 that an asteroid impact resulted in the extinctions. In 1980, Alvarez, an American physicist, his son geologist Walter Alvarez and chemists Frank Asaro and Helen Michel (US) reported that the sedimentary rocks at the border between the Cretaceous and Paleogene (formerly Tertiary) periods contained an abnormally high amount of the rare element iridium, which is common in asteroids and comets. They suggested an asteroid impact occurred about 66 million years ago. The theory has been supported by additional evidence, including the finding of rock spherules formed by the impact and shocked minerals from intense pressure. The presence of thicker sedimentary layers and giant tsunami beds in southern US and Central America supported the idea that the asteroid impact site was nearby, a prediction confirmed by the discovery of a giant crater (110 miles in diameter) at Chicxulub along the coast of the Yucatan in Mexico in 1990. Some scientists believe that the asteroid was only one of several factors in the mass extinction.
A view of the Chicxulub impact crater in the Yucatan based on seismic readings.
A view of the Chicxulub impact crater in the Yucatan based on seismic readings. Image courtesy of the Canada Geological Survey.
HIV (1983)
The human immunodeficiency virus (HIV) causes acquired immunodeficiency syndrome (AIDS), a highly fatal disease that cripples the immune system, allowing opportunistic infections and cancers to wreak havoc. AIDS was first observed in the US in 1981 in patients with a rare form of pneumonia and later a rare skin cancer called Kaposi’s sarcoma. In May 1983, a French research group led by Luc Montagnier (with Françoise Barré-Sinoussi) isolated a new retrovirus they called LAV (lymphadenopathy-associated virus), that appeared to be the cause of AIDS. In May 1984, an American team led by Robert Gallo discovered the same virus, which they named HTLV-III (human T lymphotropic virus type III). By March 1985, it was clear that LAV and HTLV-III were the same virus and in May 1986, the International Committee on Taxonomy of Viruses named the virus discovered by both groups HIV, for human immunodeficiency virus. Further study indicated that two types of HIV originated in primates in west-central Africa and transferred to humans in the early 20th Century.
Luc Montagnier (left) and Robert Gallo in 2000.
Luc Montagnier (1932- ) (left) and Robert Gallo (1937- ) in 2000.
A fullerene is a molecule made entirely of carbon in the form of a hollow sphere, ellipsoid, tube or certain other shapes. Spherical fullerenes are also known as buckyballs. Cylindrical fullerenes are called carbon nanotubes or buckytubes. Sumio Iijima (Japan) had predicted the existence of the C 60 molecule (which became the first fullerene) in 1970 and identified it in a electron micrograph in 1980. R.W. Henson (US) had proposed the structure of C 60 in 1970 and made a model of it, but his results were not accepted. In 1973, Professor Bochvar (USSR) made a quantum-chemical analysis of C 60’s stability and calculated its electronic structure, but the scientific community rejected his conclusions. In 1985, Harold Kroto (UK) and Americans Richard Smalley, Robert Curl, James Heath and Sean O’Brien and at Rice University, in the course of experiments designed to mimic carbon clusters, discovered and prepared C-60, the first fullerene, which they named buckminsterfullerene, by firing an intense pulse of laser light at a carbon surface in the presence of helium and then cooling the gaseous carbon to near absolute zero.
A diagram of a buckyball.
A diagram of a buckyball.
In 1980, Tim Berners-Lee (UK), working at CERN in Switzerland, built a personal database of people and software models called ENQUIRE, that used hypertext. In March 1989, Berners-Lee proposed a large hypertext database with typed links. He began implementing his proposal on a NeXT workstation, calling it the World Wide Web. Berners-Lee’s collaborator Robert Cailliau (Belgium) rewrote the proposal in 1990. By Christmas 1990, Berners-Lee had created the HyperText Transfer Protocol (HTTP), the Hypertext Markup Language (HTML), the first Web browser, the first HTTP server software, the first Web server and the first Web pages. Nicola Pellow (UK) created Line Mode Browser, that allowed the system to run on any computer. In January 1991, the first non-CERN servers came online. The Web became publicly available after August 23, 1991. The first American Web server was established at the Stanford Linear Accelerator Center by Paul Kunz and Louise Addis in September 1991. In 1993, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, under the lead of Marc Andreessen (US), introduced the Mosaic graphical web browser, which evolved into Netscape Navigator in 1994. Also in 1993, Microsoft also released Cello, written by Thomas R. Bruce (US) at Cornell, a browser for Microsoft Windows. As of 2008, there were one trillion public web pages on the World Wide Web.
Tim Berners-Lee in 2008.
Tim Berners-Lee (1955- ) in 2008.
In 1584, Giordano Bruno (Italy) speculated that other stars had planets circling them. Although a number of 19th and early 20th Century astronomers claimed to have discovered planets around other stars, they have all been discredited. A 1988 claim by Canadian astronomers Bruce Campbell, G. A. H. Walker, and Stephenson Yang that they had discovered a planet orbiting the star Gamma Cephei was tentative at the time but confirmed in 2003 after advances in technology. In 1992, Aleksander Wolszczan (Poland) and Dale Frail (Canada/US) discovered two Earth-sized planets orbiting the pulsar PSR 1257+12, which is generally considered the first definitive detection of exoplanets, or extrasolar planets. The team found a third planet in 1994. In 1995, Michel Mayor and Didier Queloz (Switzerland) observed a giant planet in a four-day orbit around 51 Pegasi, the first detection of a planet circling a standard, or main sequence star. In 2009, the US launched the Kepler Space Telescope (KST), which has the mission of discovering Earth-like extrasolar planets. As of August 14, 2014, KST had facilitated the discovery of 1815 confirmed exoplanets in 1130 planetary systems, of which 466 are multiple planetary systems. The smallest planet known is about twice the size of the moon, while the largest is 29 times the size of Jupiter. Some planets are so near their stars they their orbits take only a few hours, while some are so distant that they take thousands of years to complete one orbit. In March 2014, KST identified the first exoplanet that is similar to Earth in size with an orbit within the habitable zone (i.e., the area that would support life) of another star. The planet is Kepler-186f and the star is a red dwarf (Kepler 186) about 500 light years from Earth.
The first exoplanet to be photographed.
In 2005, scientists captured the first visual image (using false colors) of a planet outside our solar system. The large bluish-white object is a brown dwarf star called 2MASSWJ1207334-3932, which is 230 light years from Earth. The smaller red object is a planet called 2M1207 b. The exoplanet has five times the mass of Jupiter and travels in an orbit that is five billion miles from its star, about twice the distance of Neptune from the Sun.
The move from hunter-gatherer to agriculture was gradual and occurred at different times in different places. Evidence of humans exerting some control over wild grain is found in Israel in 20,000 BCE. There is evidence of planned cultivation and trait selection of rye at a Syrian site dating to 11,000 BCE. Domesticated lentils, vetch, pistchios and almonds were also found in Franchthi Cave in Greece at about 11,000 BCE. Humans domesticated the eight founder crops of agriculture (emmer wheat, einkorn wheat, barley, peas, lentils, bitter vetch, chickpeas and flax) some time after 9500 BCE at various sites throughout the Levant (Syria, Lebanon, Palestine, Jordan, Cyprus and part of Turkey). The oldest known settlement associated with human agriculture is in Cyprus, dating from 9100-8600 BCE. Rice and millet were domesticated in China by 8000 BCE. Farming was fully established along the Nile River by 8000 BCE (although there is evidence of agriculture in Egypt as early as 10,000 BCE) and in Mesopotamia by 7000 BCE. The first evidence of agriculture in the Indus Valley dates from 7000-6000 BCE. The first signs of agriculture in the Iberian peninsula date from 6000-4500 BCE. The oldest field systems, including stone walls, are found in Ireland and date to 5500 BCE. By 5500 BCE, the Sumerians had developed large-scale intensive cultivation of land, mono-cropping, organized irrigation and use of a specialized labor force. By 5000 BCE, humans in Africa’s Sahel region had domesticated rice and sorghum. Maize was domesticated in Mesoamerica around 3000-2700 BCE.
A map showing the origin of various domesticated crops.
A map showing the origin of various domesticated crops.
Written language evolved from pictures and other symbols into proto-writing, and then true writing. Sumerian archaic cuneiform script was invented independently around 3200 BCE, although the first true written texts do not appear until about 2600 BCE. The first Egyptian hieroglyphics date to 3400-3200 BCE; the Indus script of Ancient India dates to 3200 BCE; and Chinese characters date to 1600-1200 BCE, but there is debate about whether these were independent discoveries or derived from pre-existing scripts. The Phoenicians began to develop the first phonetic writing system between 2000 and 1000 BCE. Mesoamerican cultures developed writing systems independently. The Olmecs of Mexico developed the first Mesoamerican writings, beginning about 900-600 BCE.
An inscription on a clay tablet written in the archaic cuneiform script and dating to c. 26th Century BCE.
IRON WORKING (3000-2700 BCE)
Iron, a common element in the Earth’s crust and in meteorites, is easily corroded, so few ancient iron artifacts remain. The earliest man-made iron objects that still exist were found in Iran and date to 5000-4000 BCE. They were made from iron-nickel meteorites, as were the earliest iron artifacts from Egypt and Mesopotamia, dating from 4000-3000 BCE, and China, from 2000-1000 BCE. The earliest evidence of smelting of native iron ore to create wrought iron comes from Mesopotamia and Syria about 3000-2700 BCE and India in 1800-1200 BCE. The Hittites of Anatolia (present-day Turkey) created iron artifacts as early as 2500 BCE, but beginning in 1500-1200 BCE, they developed a sophisticated iron working industry that involved bellows-aided furnaces called bloomeries. The products of iron smelting also become more common in Mesopotamia, Egypt and Niger starting about 1500. By 1100-1000 BCE, the technology of smelting iron ore to make wrought iron had spread to Greece, sub-Saharan Africa and China and the Iron Age had begun. Wrought iron working reached central Europe in the 8th Century BCE and was common in Northern Europe and Britain after 500 BCE. Meanwhile in the 5th Century BCE, Chinese ironworkers produced the first cast iron, which was cheaper to produce than wrought iron. Cast iron production reached Europe in the Middle Ages. In 1709, Abraham Darby (UK) built a coke-fired blast furnace to make cast iron more efficiently. The next major improvement in wrought iron technology did not arrive until 1783, Englishman Henry Cort introduced the puddling process for refining iron ore.
The remains of one of the oldest bloomery iron smelting operation, located at Tell Hammeh in Jordan.
The remains of one of the oldest bloomery iron smelting operations, located at Tell Hammeh in Jordan.
The notion that humans believed the Earth was flat until Christopher Columbus’s voyages in the 1490s is simply untrue. The earliest suggestion that the Earth is a sphere comes from the Rig Veda, the ancient Hindu scripture, which was composed in India about 1500 BCE, although the oldest existing texts are much later. The Ancient Greeks also believed that the Earth was a sphere, but it is not clear who first made the discovery: Pythagoras in the 6th Century BCE or Parmenides or Empedocles in the 5th Century. Plato (Ancient Greece) asserted the roundness of the Earth in the early 4th Century BCE. Later in the 4th Century, Aristotle (Ancient Greece) reasoned that the Earth was a sphere because some stars are visible in the south that are not visible in the north, and vice versa. In 240 BCE, Eratosthenes (Ancient Greece) experimentally determined that the Earth was curved.
A diagram of Eratosthenes' measurements of the Earth's circumference.
A diagram of Eratosthenes’ measurements of the Earth’s circumference.
The alphabet may have its origins in the Proto-Sinaitic scripts that date to 1700 BCE, but there are not enough examples to be sure. The Ugarit writing of about 1400 BCE in Syria appears to use the first known alphabet. The Proto-Canaanite alphabet, precursor to the Phoenician alphabet, is known from about 1300 BCE. Scholars date the Phoenician alphabet to 1050 BCE. The Phoenician alphabet, which had no vowels, was the precursor to many of the writing systems in use today and throughout history. It led directly to Aramaic, which led to Arabic and Hebrew. When the Hellenic Greeks adopted the Phoenician alphabet in about 800 BCE, they converted some Phoenician letters to vowels, and the resulting language became the basis for the Latin, Cyrillic and Coptic scripts used in so many Western languages today.
The Phoenician alphabet and the alphabets derived from it.
The Phoenician alphabet and the alphabets derived from it.
Crossbows were first used in China as weapons of war between 600 and 500 BCE. Greek soldiers began using crossbows between 500-400 BCE. Romans used crossbows in war and hunting between 50 and 150 CE. There is evidence of crossbow use in Scotland between the 6th and 9th Centuries CE. In the early 11th Century, crossbows with sights and mechanical triggers were developed. The invention of pushlever and ratchet drawing mechanisms in the Middle Ages allowed the use of crossbows on horseback. The Saracens invented composite bows, made from layers of different material, and the Crusaders adopted the design upon their return to Europe. By 1525, the military crossbow had mostly been replaced by firearms.
This is a reconstruction of a Greek crossbow, or gastraphetes, from
A reconstructed Greek crossbow, or gastraphetes, from the 5th Century BCE.
Magnetic compasses were invented after humans discovered that iron could be magnetized by contact with lodestone and once magnetized, would always point north. There is some evidence that the Olmecs, in present-day Mexico, used compasses for geomancy (a type of divination) between 1400-1000 BCE. The first confirmed compasses, made in China about 206 BCE, were also used for divination and geomancy and used a lodestone or magnetized ladle. The first recorded use of a compass for navigation is 1040-1044 CE, but possibly as early as 850 CE, in China; 1187-1202 in Western Europe and 1232 in Persia. Later (by 1088 in China), iron needles that had been magnetized by a lodestone replaced the lodestone or other large object as directional arm of the compass. In many early compasses, the iron needle would float in water. The first dry needle compasses are described in Chinese documents dating from 1100-1250. Another form of dry compass, the dry mariner’s compass, was invented in Europe around 1300, possibly by Flavio Gioja (Italy). Further developments included the bearing compass and surveyor’s compass (early 18th Century); the prismatic compass (1885); the Bézard compass (1902); and the Silva orienteering compass (Sweden 1932). Liquid compasses returned in the 19th Century, with the first liquid mariner’s compass invented by Francis Crow (UK) in 1813. In 1860, Edward Samuel Ritchie invented an improved liquid marine compass that was adopted by the U.S. Navy. Finnish inventor Tuomas Vohlonen produced a much-improved liquid compass in 1936 that led to today’s models.
The first Chinese compasses used a spoon on a flat board. It is not clear if this image shows an actual Han Dynasty compass or a reproduction.
Precursors to the windmill include the windwheel of Heron of Alexandria, a Greek engineer, and the prayer wheels that have been used in Tibet and China since the 4th Century. A windmill uses the power of the wind to create energy; the first windmills were used to mill grain. The first practical windmills were made in Persia in the 9th Century and had ‘sails’ that rotated in a horizontal plane, not vertical as we normally see in the West. This technology spread throughout the Middle East and Central Asia and later to China and India. A visitor to China in 1219 remarked on a horizontal windmill he saw there. Vertical windmills first appeared in an area of Northern Europe (France, England and Flanders) beginning in about 1175. The earliest type of European windmill was probably a post mill. The oldest known post mill, dating to 1191, is located in Bury St. Edmunds, England. By the late 13th Century, masonry tower mills were introduced; the smock mill was a 17th Century variation. Hollow-post mills arose in the 14th Century.
Nashtifan windmills.
The horizontal windmills at Nashtifan in Iran, some of which are still working, were built during the Safavid Dynasty (1501–1736).
EYEGLASSES (c. 1286)
References to the use of lenses, jewels or water-filled globes to correct vision go back as far as the 5th Century BCE, but it was only after Arab scholar Ibn al-Haytham’s Book of Optics was translated into Latin between 1275 and 1280 that the stage was set for the invention of true eyeglasses in Italy by an unnamed individual in 1286 who was “unwilling to share them” according to a historian. Alessandro di Spina (Italy) followed soon afterwards and “shared them with everyone.” The first eyeglasses used convex lenses to correct both farsightedness (hyperopia) and presbyobia. They were designed to be held in the hand or pinched onto the nose (pince-nez). Some have speculated that eyeglasses originated in India prior to the 13th Century. The earliest depiction of eyeglasses is Tommaso da Modena’s 1352 portrait of Cardinal Hugh de Provence. A German altarpiece from 1403 also shows the invention. The first glasses that extended over the ears were made in the early 18th Century. Concave lenses to cure shortsightedness, or myopia, were not developed until c. 1450.
Detail of 1352 portrait of Hugh de Provence wearing eyeglasses, by Tommaso de Moderna.
Detail of Tommaso da Modena’s 1352 portrait of Cardinal Hugh de Provence wearing eyeglasses.
In 45 BCE, Sosigenes of Alexandria developed a calendar and presented it to Roman emperor Julius Caesar, who adopted it for the Roman Empire as the Julian Calendar. In the Julian calendar, each year consisted of 365 days divided into 12 months, with a leap year every four years, creating an average year of 365.25 days. Because a true solar year is slightly less than 365.25 days, the Julian calendar became out of sync with the seasons and religious holidays over the centuries. This led Pope Gregory XIII in 1582 to revise the calendar to skip three leap years every four centuries, which keeps the calendar in line with the seasons to this day. The average year is now 365.2425 days. Although some non-Western countries and religious groups maintain their own calendars, the Gregorian calendar is used universally for trade and international relations, including by the United Nations.
A copy of Pope Gregory's 1582 proclamation of the new calendar.
A copy of Pope Gregory’s 1582 proclamation of the new calendar.
Ancient Greek scientists had invented primitive thermoscopes, based on the principle that certain substances expanded when heated. Scientists in Europe, including Galileo Galilei (Italy) in c. 1593, developed more sophisticated thermoscopes in the 16th and 17th centuries. The thermometer was born in 1611-1613, when either Francesco Sagredo or Santorio Santorio (Italy) first added a scale to a thermoscope. Daniel Gabriel Fahrenheit (Netherlands/Germany/ Poland) invented an alcohol thermometer in 1709 and the first mercury thermometer in 1718. Each inventor used a different scale for his thermometer until Fahrenheit suggested the scale that bears his name in 1724. That scale was becoming the standard when, in 1742, Anders Celsius (Sweden) suggested a different scale. The two scales have been in competition ever since. William Thomson, Lord Kelvin (UK) developed the absolute zero scale, known as the Kelvin scale, in 1848.
One of the original thermometers made by Daniel Farenheit, dating to between 1714 and 1724.
One of the original thermometers made by Daniel Fahrenheit (1686-1736), dating to between 1714 and 1724.
The exponent that a fixed value, called the base, must be raised to in order to produce a number is called the logarithm of that number. (E.g., the logarithm of 1000 in base 10 is 3 because 1000 = 103.) Precursors to logarithms were invented by the Babylonians in 2000-1600 BCE, Indian mathematician Virasena in the 8th Century CE; and Michael Stifel (Germany) in a 1544 book. In 1614, John Napier of Merchiston (Scotland) announced the discovery of logarithms, which highly simplified multiplication and addition calculations. Henry Briggs (England) created the first table of logarithms in 1617. Joost Bürgi (Switzerland) discovered logarithms independently before Napier, but did not publish until 1620. Alphonse Antonio de Sarasa (Flanders) related logarithms to the hyperbola in 1649. Natural logarithms were first identified by Nicholas Mercator (Germany) in 1668, but John Speidell (England) had been using them since 1619. Swiss mathematician Leonhard Euler vastly expanded the theory and applications of logarithms in the 18th Century by using them in analytic proofs, expressing logarithmic functions using power series, and defining logarithms for negative and complex numbers.
John Napier.
John Napier (1550-1617).
Galileo Galilei (Italy) observed Saturn’s rings through a telescope in 1610, but did not identify them as rings, but as ears or a ‘triple form’. Galileo noted but could not explain the disappearance of the rings when Saturn was oriented directly at the Earth in 1612 and their reappearance in 1613. Christiaan Huygens (The Netherlands), using a 50-power refracting telescope, definitively identified Saturn’s rings in 1655. In 1666, Robert Hooke (England) also identified the rings and noted that Saturn cast a shadow on the rings. Giovanni Domenic Cassini (Italy) noted in 1675 that Saturn had multiple rings with gaps between them. In 1787, Pierre-Simon Laplace (France) suggested that the rings consisted of many solid ringlets, a theory that James Clerk Maxwell (UK) disproved in 1859 by showing that solid rings would become unstable and break apart. Maxwell proposed instead that the rings were made of numerous small particles, which was experimentally confirmed by James Keeler (US) and Aristarkh Belopolsky (Russia) in 1895 using spectroscopy.
A color enhanced photograph of Saturn's rings.
A color enhanced photograph of Saturn’s rings.
Binary numbers are numbers expressed in a binary or base-2 numeral system, which normally represents numeric values with the symbols zero and one. A binary code is text or computer processor instructions using the binary number system. An early form of binary system is used in the ancient Chinese book, the I Ching (2000 BCE?). Between the 5th and 2nd Centuries BCE, Indian scholar Pingala invented a binary system. Shao Yong (China) developed a binary system for arranging hexagrams in the 11th Century. Traditional African geomancy such as Ifá used binary systems and French Polynesians on the island of Magareva used a hybrid binary-decimal system before 1450. Francis Bacon invented an encoding system in 1605 that reduced the letters of the alphabet to binary digits. Gottfried Leibniz (Germany), who was aware of the I Ching, invented the modern binary number system in 1679 and presented it in his 1703 article Explication de l’Arithmétique Binaire. In 1875, Émile Baudot (France) added binary strings to his ciphering system. In 1937, Claude Shannon (US), in his MIT master’s thesis, first combined Boolean logic and binary arithmetic in the context of electronic relays and switches. He showed that relay circuits, being switches, resembled the operations of symbolic logic: two relays in series are and, two relays in parallel are or, and a circuit which can embody not and or can embody if/then. This last meant that a relay circuit could make a choice. Since switches are either on or off, binary mathematics was therefore possible. George Stibitz (US), at Bell Labs, demonstrated a relay-based computer in 1937 that calculated using binary addition. Stibitz and his team made a more complex version called the Complex Number Computer in 1940. Common binary coding systems include ASCII (American Standard Code for Information Interchange) and BCD (binary-coded decimal).
A page from Gottfried Leibniz's 1703 article Explication de l'Arithmétique Binaire.
A page from Gottfried Leibniz’s 1703 article Explication de l’Arithmétique Binaire.
According to the kinetic theory of gases, a gas consists of a large number of atoms or molecules in constant, random motion that constantly collide with each other and the walls of a container. Lucretius (Ancient Rome) proposed in 50 BCE that objects were composed of tiny rapidly moving atoms that bounced off each other. Daniel Bernoulli (Switzerland) proposed the kinetic theory of gases in 1738. He proposed that gas pressure is caused by the impact of gas molecules hitting a surface and heat is equivalent to the kinetic energy of the molecules’ motion. Other advocates of the kinetic theory included: Mikhail Lomonosov (Russia, 1747), Georges-Louis Le Sage (Switzerland, ca. 1780, published 1818), John Herapath (UK, 1816)John James Waterston (UK, 1843), August Krönig (Germany, 1856) and Rudolf Clausius (Germany, 1857). James Clerk Maxwell (UK) formulated the Maxwell distribution of molecular velocities in 1859, and Ludwig Boltzmann (Austria) formulated the Maxwell-Boltzmann distribution in 1871. In papers on Brownian motion, Albert Einstein (Germany, 1905) and Marian Smoluchowski (Poland, 1906) made testable predictions based on kinetic theory.
The kinetic theory of gases states that the average kinetic energy of the atoms in an ideal gas can be determined by the temperature of the gas, while pressure is related to the impacts of the atoms on the walls of their enclosure. This animation shows helium atoms under a pressure of 1950 atmospheres, drawn to scale, at room temperature. Their movements have been slowed down by two trillion times.
According to the law of conservation of mass, the mass of any system that is closed to all transfers of matter and energy must remain constant over time. The law has ancient roots. The Jains in 6th Century BCE India believed that the universe and its constituents cannot be created or destroyed. In Ancient Greece, Empedocles said in the 5th Century BCE that nothing can come from nothing and once something exists it can never be completely destroyed, a belief echoed by Epicurus in the 3rd Century BCE. Persian philosopher Nasir al-Din al-Tusi stated a version of the law in the mid-13th Century. The first modern scientific statement of the law came from Mikhail Lomonosov (Russia) in 1748. Although Antoine Lavoisier (France) is often credited with discovering the law in 1774, precursors include Jean Rey (France, 1583-1645), Joseph Black (Scotland, 1728-1799) and Henry Cavendish (UK, 1731-1810).
A diagram explaining the law of conservation of mass.
A diagram explaining the law of conservation of mass.
A marine chronometer is a clock that is accurate enough to be a portable time standard, which can be used to determine longitude by using celestial navigation. Until the 18th Centuries, navigators were able to determine the latitude of a ship at sea, but not its longitude. Gemma Frisius (The Netherlands) suggested in 1530 that a highly accurate clock could be used to calculate longitude. In the 17th Century, Galileo Galilei (Italy), Edmund Halley (England), Tobias Mayer (Germany) and Nevil Maskelyne (England) proposed observations of astronomical objects as the solution, but the deck of a ship at sea proved too unstable for accurate measurements. Recognizing that his pendulum clock would not be effective at sea, Christiaan Huygens (The Netherlands) invented a chronometer in 1675 with a balance wheel and a spiral spring, but it proved too inaccurate in nautical conditions. Similar problems plagued the chronometers made by Jeremy Thacker (England) in 1714 and Henry Sully (France) in 1716. In 1714, the British government offer a large cash reward for anyone who could invent an accurate chronometer. John Harrison (England) submitted versions in 1730, 1735 and 1741, although they were all sensitive to centrifugal force. A 1759 version, with a bi-metallic strip and caged roller bearings, was even more accurate, but it was the much smaller 1761 design that won Harrison the £20,000 prize in 1765. French clockmaker Pierre Le Roy’s 1766 chronometer, with a detent escapement, temperature-compensated balance and isochronous balance spring, was the first modern design. Thomas Earnshaw and John Arnold developed an improved version with Le Roy’s innovations in 1780, which led to the standard chronometer used for many years afterwards.
John Harrison's 1761 'sea watch.'
The 1761 ‘sea watch’ created by John Harrison (1693-1776) to solve the problem of determining longitude, which won him a prize.
In the mid-18th Century, the English textile industry was growing, and its machines were becoming faster. The flying shuttle had doubled loom speed, and the invention of the spinning jenny in 1764 had also increased speed and production. John Kay and Thomas Highs (England) had designed a new machine called the spinning frame, which produced a stronger thread than the spinning jenny. The spinning frame used the draw rollers invented by Lewis Paul to stretch the yarn. In 1769, Sir Richard Arkwright (England) asked John Kay to produce the spinning frame for him. Because the spinning frame was too large to be operated by hand, Arkwright experimented with other power sources, trying horses first and then switching to the water wheel. Unlike the spinning jenny, which was inexpensive but required skilled labor, the spinning frame required considerable capital outlay but little skill to operate.
A 1790 spinning frame made by Slater, now in the Smithsonian Institution.
Flemish scientist Jan van Helmont discovered in the mid-17th Century that the mass of the soil used by a plant changed very little as the plant’s mass increased. He hypothesized that the additional mass came from the added water. In 1774-1777, Joseph Priestley (England) published the results of experiments in which he burned a candle in a sealed jar, it quickly stopped burning, and that a mouse trapped in a jar would soon stop breathing, but he found that if he added a plant to the jar, both mouse and candle would continue to flourish. Priestley concluded that plants make and absorb gases. Following up on Priestley’s experiments, Jan Ingenhousz (The Netherlands) discovered that when light is present, plants give off bubbles from their green parts, which he identified as oxygen, but not in the shade, and that it was the oxygen that revived Priestley’s mouse. He also discovered that plants give off carbon dioxide in the dark, but that the amount of oxygen given off in the light is greater than the amount of carbon dioxide given off in the dark. In 1796, Jean Senebier (Switzerland) confirmed Ingenhousz’s finding that plants release oxygen in the light, and also found that they consume carbon dioxide in the light. Calculations by Nicolas-Théodore de Saussure (Switzerland) in the late 1790s showed that the increase in the plant’s mass was due to both carbon dioxide and water. Charles Reid Barnes (US) proposed the term ‘photosynthesis’ in 1893. In 1931, Cornelis Van Niel (The Netherlands/US) studied the chemistry of photosynthesis and demonstrated that photosynthesis is a light-dependent reaction in which hydrogen reduces carbon dioxide. Also in the 1930s, scientists proved that the oxygen liberated in photosynthesis comes from water. |
7459413f5f372133 | Section 13.6: Angular Solutions of the Schrödinger Equation
Please wait for the animation to completely load.
Most potential energy functions in three dimensions are not often rectangular in form. In fact, they are most often in spherical coordinates (due to a spherical symmetry) and occasionally in cylindrical coordinates due to a cylindrical symmetry. We begin by considering the generalization of the time-independent Schrödinger equation to three-dimensional spherical coordinates, which is1
−(ħ2/2μ)[(1/r2)∂/∂r(r2∂/∂r) + (1/r2sin(θ))(∂/∂θ)(sin(θ)∂/∂θ) + (1/r2sin2(θ))(∂2/∂φ2)]ψ(r) + V(r)ψ(r) = Eψ(r) . (13.19)
The probability per unit volume, the probability density, is ψ*(r)ψ(r) and therefore we require ∫ ψ*(r)ψ(r) d3r = 1 (where d3r = dV = r2sin(θ)drdθdφ) to maintain a probabilistic interpretation of the energy eigenfunction in three dimensions.
As in the two-dimensional case, we use separation of variables variables, but now using ψ(r) = R(r) Y(θ,φ), i.e., separate the radial part from the angular part. This substitution yields
[(1/R(r))d/dr(r2dR(r)/dr) + (1/Ysin(θ))(∂/∂θ)(sin(θ)∂Y/∂θ) + (1/Ysin2(θ))(∂2Y/∂φ2)] − (2μr2/ħ2)[V(r) − E] = 0 , (13.20)
as long as V(r) = V(r) only. Note that each term involves either r or θ and φ. We can separate these equations using the technique of separation of variables to give
(1/R(r)) d/dr (r2 dR(r)/dr) − (2μr2/ħ2)[V(r) − E] = l(l + 1) , (13.21)
(1/Ysin(θ)) (∂/∂θ)(sin(θ) ∂Y/∂θ) + (1/Ysin2(θ)) (∂2Y/∂φ2) = −l(l + 1) , (13.22)
for the radial and angular parts, respectively. The constant l(l + 1) is the separation constant that allows us to separate one differential equation into two. We can do so because the only way for preceding equation to be true for all r, θ, and φ is for the angular part and the radial part to each be equal to a constant, ± l(l + 1). Despite the seemingly odd form of the separation constant, it is completely general and can be made to equal any complex number. For the angular piece, we can again separate variables using the substitution Y(θ,φ) = Θ(θ)Φ(φ). This gives:
sin(θ)/Θ d/dθ(sin(θ) dΘ/dθ) + l(l + 1)sin2(θ) = m2 , (13.23)
1/Φ d2Φ/dφ2 = −m2 , (13.24)
where we have written the separation constant as ± m2, again without any loss of generality. The Φ(φ) part of the angular equation is a differential equation, d2Φ/dφ2 = −m2Φ, we have solved before. We get as its unnormalized solution
Φm(φ) = exp(imφ) , (13.25)
where m is the separation constant which can be both positive and negative. Since the angle φ ε {0, 2π}, we have that Φm(φ) = Φm(φ + 2π). Like the ring problem in Section 13.5, in order for Φm(φ) to be single valued means that m = 0, ±1, ±2, ±3,…. We show these solutions in Animation 1. The Θ(θ) part of the angular equation is harder to solve. It has the unnormalized solutions
Θlm(θ) = A Plm(cos(θ)) ,
where the Plm are the associated Legendre polynomials, where
Plm(x) = (1 − x2)|m|/2 (d/dx)|m| Pl(x),
are calculated from the Legendre polynomials
Pl(x) = (1/2ll!) (d/dx)l(x2 − 1)l . (Rodriques' formula)
The first few Legendre polynomials are
P0(x) = 1 , P1(x) = x , and P2(x) = (1/2) (3x2 1) ,
or in terms of cos(θ)
P0 = 1, P1 = cos(θ), and P2 = (1/2) (3cos2(θ)−1) .
We can also write the Plm(x) using the above formulas as:
P00 = 1, P11 = sin(θ), P01 = cos(θ) ,
P02 = (1/2)(3cos2(θ)-1), P12 = 3sin(θ)cos(θ), P22 = 3sin2(θ) .
We notice that l > 0 for Rodrigues' formula to be valid. In addition, |m| ≤ l since Pl|m|>l = 0. (For |m| > l, the power of the derivative is larger than the order of the polynomial and hence the result is zero.) We also note that there must be 2l + 1 values for m, given a particular value of l. Polar plots (zx plane) of associated Legendre polynomials are shown in Animation 2. A positive angle θ is defined to be the angle down from the z axis toward the positive x axis. The length of a vector from the origin to the wave function, Plm, is the magnitude of the wave function at that angle. You may vary l and m to see how Plm varies. We normalize Θlm(θ)Φm(φ) by normalizing the angular part separately from the radial part (which we have yet to consider):
∫∫ Ylm*(θ,φ)Ylm(θ,φ) sin (θ) dθdφ = 1 [θ integration from 0 to π, φ integration from 0 to 2π]
where Ylm(θ,φ) = Θlm(θ)Φm(φ). When the Ylm(θ,φ) are normalized, they are called the spherical harmonics.The first few are
Y00(θ,φ) = (1/4π)1/2 ,
Y ±11(θ,φ) = −/+ (3/8π)1/2 sin(θ) exp(±iφ) Y 01(θ ,φ) = (3/4π)1/2 cos(θ) ,
and in general for m > 0,
Ylm(θ,φ) = (−1)m [(2l + 1)(lm)!/(4π(l + m)!)]1/2 exp(imφ) Plm cos(θ) ,
and Yl−m(θ,φ) = (−1)mYlm*(θ,φ) for m < 0. When we represent the spherical harmonics this way, they are automatically orthogonal:
∫ Ylm*(θ,φ)Yl'm'(θ,φ) sin(θ) dθdφ = δm m' δl l' .
1To avoid future confusion, we hereafter use μ for mass, and reserve m for the azimuthal (or magnetic) quantum number.
2Classically, angular momentum is L = r × p. We can write L using quantum-mechanical operators in rectangular coordinates as Lx = ypzzpy, Ly = zpxxpz, and Lz = xpyypx. We find that if we write L2 and Lz in spherical coordinates,
L2 = −ħ2 [(1/sin(θ)) (∂/∂θ)(sin(θ) ∂/∂θ) + (1/sin2(θ)) (∂2/∂φ2) ,
Lz = − (∂/∂φ) .
To which we note L2Ylm = l(l + 1)ħ2Ylm and LzYl= Ylm; the spherical harmonics, the Ylm , are eigenstates of L2 and Lz.
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c4069f8281e24a20 | You are hereSZZKT
Course: Quantum Theory of Molecules
Department/Abbreviation: KBF/SZZKT
Year: 2021
Guarantee: 'prof. RNDr. Petr Ilík, Ph.D.'
• Basic terms and definitions of quantum mechanics: wave function, Schrödinger equation, stationary and non-stationary states, operators of physical quantities, basic conceptions of the quantum theory of systems with many particles, symmetric and antisymmetric wave functions, entire wave function, basics of theory of representations
• Elementary quantum theory of atoms with two electrons: helium atom, basic and excited states, parastates and orthostates of helium atom
• Quantum theory of atoms with more than two electrons: Hartree-Fock method of self-consistent field
• Basic approximation in the theory of chemical bond: Bohr-Oppenheimer approximation, adiabatic approximation, separation of vibrational and rotational degrees of freedom of diatomic molecule
• One-electron approximation, Hartree-Fock equations for calculation of one-electron functions and one-electron energies, molecules as systems with closed shells, Fock operator
• Approximation of n-electron function of a molecule, method of valence bands (VB), MO LCAO method, choice of base v MO LCAO method, VTO, STO, GTO orbitals, their properties, correlation problem, method of configuration interaction (CI)
• Quantum theory of chemical bond: quantitative description of covalent bond in homonuclear diatomic molecules, solution of hydrogen molecule by VB and MO LCAO methods
• Quantitative description of chemical bond: atomic and molecular orbitals in quantitative description of chemical bond, their representation and characteristics, hybrid atomic orbitals, construction of molecular orbitals, overlapping of atomic orbitals, characteristics of homonuclear diatomic molecules
• Covalent bond in heteronuclear diatomic molecules: ion bond, multi-atomic molecules, delocalized and localized molecular orbitals of multi-atomic molecules, hybridization in the bond theory
• Overview of calculating methods in the quantum theory of chemical bond: "ab initio" methods, semi-empirical and empirical methods, methods using valence electrons, pi-electron approximation
Course review:
Experimenty, které nelze vysvětlit klasicky. Postuláty kvantové mechaniky. Matematický aparát kvantové mechaniky. Harmonický oscilátor. Poruchová teorie. Atom vodíku. Born-Oppenheimerova a adiabatická aproximace. Metody VB a MO LCAO. Kvantový výklad chemické vazby molekul. Víceatomové molekuly. |
6064f93c1c66aeef | Deriving A Quaternion Analog to the Schrödinger Equation
The Schrödinger equation gives the kinetic energy plus the potential (a sum also known as the Hamiltonian H) of the wave function psi, which contains all the dynamical information about a system. Psi is a scalar function with complex values.
The hamiltonian operator acting on psi = -i h bar phi dot = -h bar squared
over 2 m Laplacian psi + the potential V\(0, X\) psi
For the time-independent case, energy is written at the operator -i hbar d/dt, and kinetic energy as the square of the momentum operator, i hbar Del, over 2m. Given the potential V(0, X) and suitable boundary conditions, solving this differential equation generates a wave function psi which contains all the properties of the system.
In this section, the quaternion analog to the Schrödinger equation will be derived from first principles. What is interesting are the constraint that are required for the quaternion analog. For example, there is a factor which might serve to damp runaway terms.
The Quaternion Wave Function
The derivation starts from a curious place :-) Write out classical angular momentum with quaternions.
\(0, L\) = \(0, R Cross P\) = the odd part of \(0, R\) times \(0, P\)
What makes this "classical" are the zeroes in the scalars. Make these into complete quaternions by bringing in time to go along with the space 3-vector R, and E with the 3-vector P.
\(t, R\) times \(E, P\) = \(E t - R dot P, E R + P t + R Cross P\)
Define a dimensionless quaternion psi that is this product over h bar.
psi is defined to be \(t, R\) times \(E, P\) over hbar = \(E t - R dot P, E
R + P t + R Cross P\) over h bar
The scalar part of psi is also seen in plane wave solutions of quantum mechanics. The complicated 3-vector is a new animal, but notice it is composed of all the parts seen in the scalar, just different permutations that evaluate to 3-vectors. One might argue that for completeness, all combinations of E, t, R and P should be involved in psi, as is the case here.
Any quaternion can be expressed in polar form:
q = the absolute value of q times e to the arc cosine of the scalar over the
absolute value of q times the 3-vector over its absolute
Express psi in polar form. To make things simpler, assume that psi is normalized, so |psi| = 1. The 3-vector of psi is quite complicated, so define one symbol to capture it:
I is defined to be E R + P t + R cross P over the absolute value of the
Now rewrite psi in polar form with these simplifications:
psi = e to the E t - R dot P time I over h
This is what I call the quaternion wave function. Unlike previous work with quaternionic quantum mechanics (see S. Adler's book "Quaternionic Quantum Mechanics"), I see no need to define a vector space with right-hand operator multiplication. As was shown in the section on bracket notation, the Euclidean product of psi (psi* psi) will have all the properties required to form a Hilbert space. The advantage of keeping both operators and the wave function as quaternions is that it will make sense to form an interacting field directly using a product such as psi psi'. That will not be done here. Another advantage is that all the equations will necessarily be invertible.
Changes in the Quaternion Wave Function
We cannot derive the Schrödinger equation per se, since that involves Hermitian operators that acting on a complex vector space. Instead, the operators here will be anti-Hermitian quaternions acting on quaternions. Still it will look very similar, down to the last h bar :-) All that needs to be done is to study how the quaternion wave function psi changes. Make the following assumptions.
1. Energy and Momentum are conserved.
d E by d t = 0 and d P by d t = 0
1. Energy is evenly distributed in space
The Gradient of E = 0
3. The system is isolated
The Curl of P = 0
4. The position 3-vector X is in the same direction as the momentum 3-vector P
X dot P over the absolute value of the two = 1 which implies d e to the I by
d t = 0 and the Curl of e to the I = 0
The implications of this last assumption are not obvious but can be computed directly by taking the appropriate derivative. Here is a verbal explanation. If energy and momentum are conserved, they will not change in time. If the position 3-vector which does change is always in the same direction as the momentum 3-vector, then I will remain constant in time. Since I is in the direction of X, its curl will be zero.
This last constraint may initially appear too confining. Contrast this with the typical classical quantum mechanics. In that case, there is an imaginary factor i which contains no information about the system. It is a mathematical tool tossed in so that the equation has the correct properties. With quaternions, I is determined directly from E, t, P and X. It must be richer in information content. This particular constraint is a reflection of that.
Now take the time derivative of psi.
d psi by dt = E I over h bar times psi over the square root of \(E t - R dot
P over h bar\) squared
The denominator must be at least 1, and can be greater that that. It can serve as a damper, a good thing to tame runaway terms. Unfortunately, it also makes solving explicitly for energy impossible unless Et - P.X equals zero. Since the goal is to make a direct connection to the Schrödinger equation, make one final assumption:
E t - R dot P = 0
There are several important cases when this will be true. In a vacuum, E and P are zero. If this is used to study photons, then t = |R| and E = |P|. If this number happens to be constant in time, then this equation will apply to the wave front.
if d E t - R dot P by d t = 0, then E = d R by d t dot P or d R by d t = E
over P
Now with these 5 assumptions in hand, energy can be defined with an operator.
d psi dt = E I over h bar psi
- I h bar d psi d t = E psi or E = - I h bar d by
The equivalence of the energy E and this operator is called the first quantization.
Take the spatial derivative of psi using the under the same assumptions:
Del psi = - P I over h bar times psi over the square root of \(E t - R dot P
over h bar\) squared
I h bar Del acting on psi = P acting on psi or P = I h bar Del
Square this operator.
P squared = m v squared = 2 m times m v squared over 2 = 2 m Kinetic Energy
= - h bar squared Del squared
The Hamiltonian equals the kinetic energy plus the potential energy.
The Hamiltonian acting on psi = - I hbar d psi by d t = - h bar squared Del
squared + the potential acting on psi
Typographically, this looks very similar to the Schrödinger equation. Capital I is a normalized 3-vector, and a very complicated one at that if you review the assumptions that got us here. phi is not a vector, but is a quaternion. This give the equation more, not less, analytical power. With all of the constraints in place, I expect that this equation will behave exactly like the Schrödinger equation. As the constraints are removed, this proposal becomes richer. There is a damper to quench runaway terms. The 3-vector I becomes quite the nightmare to deal with, but it should be possible, given we are dealing with a division algebra.
Any attempt to shift the meaning of an equation as central to modern physics had first be able to regenerate all of its results. I believe that the quaternion analog to Schrödinger equation under the listed constraints will do the task. These is an immense amount of work needed to see as the constraints are relaxed, whether the quaternion differential equations will behave better. My sense at this time is that first quaternion analysis as discussed earlier must be made as mathematically solid as complex analysis. At that point, it will be worth pushing the envelope with this quaternion equation. If it stands on a foundation as robust as complex analysis, the profound problems seen in quantum field theory stand a chance of fading away into the background. |
02a4f20e68fe4a58 | The amazing engineering of Fractons
fracton is a collective quantized vibration on a substrate with a fractal structure.[1][2]
Fractons are the fractal analog of phonons. Phonons are the result of applying translational symmetry to the potential in a Schrödinger equation. Fractal self-similarity can be thought of as a symmetry somewhat comparable to translational symmetry. Translational symmetry is symmetry under displacement or change of position, and fractal self-similarity is symmetry under change of scale. The quantum mechanical solutions to such a problem in general lead to a continuum of states with different frequencies. In other words, a fracton band is comparable to a phonon band. The vibrational modes are restricted to part of the substrate and are thus not fully delocalized, unlike phonon vibrational modes. Instead, there is a hierarchy of vibrational modes that encompass smaller and smaller parts of the substrate. Source Wiki
Theorists are in a frenzy over “fractons,” bizarre, but potentially useful, hypothetical particles that can only move in combination with one another.
The theoretical possibility of fractons surprised physicists in 2011 (;
). Recently, these strange states of matter have been leading physicists toward new theoretical frameworks that could help them tackle some of the grittiest problems in fundamental physics.
Partial Particles
In 2011, Jeongwan Haah, then a graduate student at Caltech, was searching for unusual phases of matter that were so stable they could be used to secure quantum memory, even at room temperature. Using a computer algorithm, he turned up a new theoretical phase that came to be called the Haah code. The phase quickly caught the attention of other physicists because of the strangely immovable quasiparticles that make it up.
They seemed, individually, like mere fractions of particles, only able to move in combination. Soon, more theoretical phases were found with similar characteristics, and so in 2015 Haah — along with Sagar Vijay and Liang Fu — coined the term “fractons” for the strange partial quasiparticles. (An earlier but overlooked paper by Claudio Chamon is now credited with the original discovery of fracton behavior.(This Letter presents solvable examples of quantum many-body Hamiltonians of systems that are unable to reach their ground states as the environment temperature is lowered to absolute zero. These examples, three-dimensional generalizations of quantum Hamiltonians proposed for topological quantum computing, (1) have no quenched disorder, (2) have solely local interactions, (3) have an exactly solvable spectrum, (4) have topologically ordered ground states, and (5) have slow dynamical relaxation rates akin to those of strong structural glasses.))
The resultant movement is that of a particle-antiparticle pair moving sideways in a straight line. In this world — an example of a fracton phase — a single particle’s movement is restricted, but a pair can move easily.
The immovability of fractons makes it very challenging to describe them as a smooth continuum from far away. Because particles can usually move freely, if you wait long enough they’ll jostle into a state of equilibrium, defined by bulk properties such as temperature or pressure. Particles’ initial locations cease to matter. But fractons are stuck at specific points or can only move in combination along certain lines or planes. Describing this motion requires keeping track of fractons’ distinct locations, and so the phases cannot shake off their microscopic character or submit to the usual continuum description.
“Without a continuous description, how do we define these states of matter?”
Fractons have yet to be made in the lab, but that will probably change. Certain crystals with immovable defects have been shown to be mathematically similar to fractons. And the theoretical fracton landscape has unfurled beyond what anyone anticipated, with new models popping up every month.
“Probably in the near future someone will take one of these proposals and say, ‘OK, let’s do some heroic experiment with cold atoms and exactly realize one of these fracton models,’” said Brian Skinner, a condensed matter physicist at Ohio State University who has devised fracton models.
Fractons do not fit into [the quantum field theory] framework. So my take is that the framework is incomplete.
Nathan Seiberg
Even without their experimental realization, the mere theoretical possibility of fractons rang alarm bells for Seiberg, a leading expert in quantum field theory, the theoretical framework in which almost all physical phenomena are currently described.
Quantum field theory depicts discrete particles as excitations in continuous fields that stretch across space and time. It’s the most successful physical theory ever discovered, and it encompasses the Standard Model of particle physics — the impressively accurate equation governing all known elementary particles.
“Fractons do not fit into this framework. So my take is that the framework is incomplete,” said Seiberg.
There are other good reasons for thinking that quantum field theory is incomplete — for one thing, it so far fails to account for the force of gravity. If they can figure out how to describe fractons in the quantum field theory framework, Seiberg and other theorists foresee new clues toward a viable quantum gravity theory.
“Fractons’ discreteness is potentially dangerous, as it can ruin the whole structure that we already have,” said Seiberg. “But either you say it’s a problem, or you say it’s an opportunity.”
He and his colleagues are developing novel quantum field theories that try to encompass the weirdness of fractons by allowing some discrete behavior on top of a bedrock of continuous space-time (We discuss nonstandard continuum quantum field theories in 2+1 dimensions. They exhibit exotic global symmetries, a subtle spectrum of charged excitations, and dualities similar to dualities of systems in 1+1 dimensions. These continuum models represent the low-energy limits of certain known lattice systems. One key aspect of these continuum field theories is the important role played by discontinuous field configurations. In two companion papers, we will present 3+1-dimensional versions of these systems. In particular, we will discuss continuum quantum field theories of some models of fractons.).
“Quantum field theory is a very delicate structure, so we would like to change the rules as little as possible,” he said. “We are walking on very thin ice, hoping to get to the other side.”
1. Alexander, S; C. Laermans; R. Orbach; H.M. Rosenberg (15 October 1983). “Fracton interpretation of vibrational properties of cross-linked polymers, glasses, and irradiated quartz”. Physical Review B28 (8): 4615–4619. Bibcode:1983PhRvB..28.4615Adoi:10.1103/physrevb.28.4615.
2. ^ Srivastava, G. P. (1990), The Physics of Phonons, CRC Press, pp. 328–329, ISBN 9780852741535.
1. Sounds a lot like phase prime metrics and meander flower garden of garden concept applied to smallest quanta we can in particle physics.
Also sounds like alpha and beta tubulin dimers creating topologies that lock their vibrations into a stable time crystal of information. Would need to read the original papers referenced on the “fracton” terminology, but sounds like they are proposing these fracton stable topologies are a method to create artificial stable topological qantum qubit technology with standard particle physics math models. Seems like a simplified lower complexity version of organic microtubule or the helical mw CNT synthetic nano brain systems.
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bdc489bf49ab1981 | Open access peer-reviewed chapter
Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables
Written By
Gabino Torres-Vega
Submitted: May 29th, 2012 Reviewed: September 19th, 2012 Published: April 3rd, 2013
DOI: 10.5772/53598
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1. Introduction
There are many proposals for writing Classical and Quantum Mechanics in the same language. Some approaches use complex functions for classical probability densities [1] and other define functions of two variables from single variable quantum wave functions [2,3]. Our approach is to use the same concepts in both types of dynamics but in their own realms, not using foreign unnatural objects. In this chapter, we derive many inter relationships between conjugate variables.
1.1. Conjugate variables
An important object in Quantum Mechanics is the eigenfunctions set |n>n=0of a Hermitian operator F^. These eigenfunctions belong to a Hilbert space and can have several representations, like the coordinate representation ψnq=qn. The basis vector used to provide the coordinate representation, |q>, of the wave function are themselves eigenfunctions of the coordinate operator Q^We proceed to define the classical analogue of both objects, the eigenfunction and its support.
Classical motion takes place on the associated cotangent space T*Qwith variable z=(q,p), where qand pare ndimensional vectors representing the coordinate and momentum of point particles. We can associate to a dynamical variable F(z)its eigensurface, i.e. the level set
Where fis a constant, one of the values that F(z)can take. This is the set of points in phase space such that when we evaluate F(z), we obtain the value f. Examples of these eigensurfaces are the constant coordinate surface, q=X, and the energy shell, Hz=E, the surface on which the evolution of classical systems take place. These level sets are the classical analogues of the support of quantum eigenfunctions in coordinate or momentum representations.
Many dynamical variables come in pairs. These pairs of dynamical variables are related through the Poisson bracket. For a pair of conjugate variables, the Poisson bracket is equal to one. This is the case for coordinate and momentum variables, as well as for energy and time. In fact, according to Hamilton’s equations of motion, and the chain rule, we have that
Now, a point in cotangent space can be specified as the intersection of 2nhypersurfaces. A set of 2nindependent, intersecting, hypersurfaces can be seen as a coordinate system in cotangent space, as is the case for the hyper surfaces obtained by fixing values of coordinate and momentum, i.e. the phase space coordinate system with an intersection at z=(q,p). We can think of alternative coordinate systems by considering another set of conjugate dynamical variables, as is the case of energy and time.
Thus, in general, the T*Qpoints can be represented as the intersection of the eigensurfaces of the pair of conjugate variables Fand G,
ΣFGf,g=zT*Q|Fz=f, Gz=g.E3
A point in this set will be denoted as an abstract bra (f,g|, such that (f,g|u)means the function u(f,g).
We can also have marginal representations of functions in phase space by using the eigensurfaces of only one of the functions,
ΣFf=zT*Q|Fz=f , and ΣGg=zT*QGz=g.
A point in the set ΣFf[ΣGg]will be denoted by the bra (f|[(g|]and an object like fu[(g|u)]will mean the f[g]dependent function uf[ug].
1.2. Conjugate coordinate systems
It is usual that the origin of one of the variables of a pair of conjugate variables is not well defined. This happens, for instance, with the pair of conjugate variables qand p. Even though the momentum can be well defined, the origin of the coordinate is arbitrary on the trajectory of a point particle, and it can be different for each trajectory. A coordinate system fixes the origin of coordinates for all of the momentum eigensurfaces.
A similar situation is found with the conjugate pair energy-time. Usually the energy is well defined in phase space but time is not. In a previous work, we have developed a method for defining a time coordinate in phase space [4]. The method takes the hypersurface q1=X, where Xis fixed, as the zero time eigensurface and propagates it forward and backward in time generating that way a coordinate system for time in phase space.
Now, recall that any phase space function G(z)generates a motion in phase space through a set of symplectic system of equations, a dynamical system,
dzdf=XG, XG=Gp,-Gq,E4
where fis a variable with the same units as the conjugate variable F(z). You can think of G(z)as the Hamiltonian for a mechanical system and that fis the time. For classical systems, we are considering conjugate pairs leading to conjugate motions associated to each variable with the conjugate variable serving as the evolution parameter (see below). This will be applied to the energy-time conjugate pair. Let us derive some properties in which the two conjugate variables participate.
1.3. The interplay between conjugate variables
Some relationships between a pair of conjugate variables are derived in this section. We will deal with general F(z)and G(z)conjugate variables, but the results can be applied to coordinate and momentum or energy and time or to any other conjugate pair.
The magnitude of the vector field |XG|is the change of length along the fdirection
XG=dqidfdqidf+dpidfdpidf=dlFdf ,E5
where dlF=dqi2+dpi2is the length element.
A unit density with the eigensurface ΣGgas support
(zg=δz-v, vΣGgE6
is the classical analogue of the corresponding quantum eigenstate in coordinate qgand momentum pgrepresentations. When G(z)is evaluated at the points of the support of (z|g), we get the value g. We use a bra-ket like notation to emphasise the similarity with the quantum concepts.
The overlap between a probability density with an eigenfunction of F^or G^provides marginal representations of a probability density,
ρffρfzzρdz=δz-fρzdz , fFf.E7
ρggρgzzρdz=δz-gρzdz , gGg.E8
But, a complete description of a function in T*Qis obtained by using the two dimensions unit density zf,g=δz-f,g, the eigenfunction of a location in phase space,
ρf,gf,gρf,gzzρdz=δz-f,gρzdz , (f,g)FGf,g.E9
In this way, we have the classical analogue of the quantum concepts of eigenfunctions of operators and the projection of vectors on them.
1.4. Conjugate motions
Two dynamical variables with a constant Poisson bracket between them induce two types of complementary motions in phase space. Let us consider two real functions F(z) and G(z) of points in cotangent space zT*Qof a mechanical system, and a unit Poisson bracket between them,
F,G=FqiGpi-GqiFpi=1 ,E10
valid on some domain D=DFqiDFqiDFqiDFqi, according to the considered functions Fand G. The application of the chain rule to functions of pand q, and Eq. (10), suggests two ways of defining dynamical systems for functions Fand Gthat comply with the unit Poisson bracket. One of these dynamical systems is
dpidF=-Gqi , dqidF=Gpi .E11
With these replacements, the Poisson bracket becomes the derivative of a function with respect to itself
F,G=FqiqiF+piFFpi=dFdF=1 .E12
Note that Fis at the same time a parameter in terms of which the motion of points in phase space is written, and also the conjugate variable to G.
We can also define other dynamical system as
dpidG=Fqi , dqidG=-Fpi .E13
Now, Gis the shift parameter besides of being the conjugate variable to F. This also renders the Poisson bracket to the identity
F,G=dpidGGpi+dqidGGqi=dGdG=1 .E14
The dynamical systems and vector fields for the motions just defined are
dzdG=XF , XF=-Fpi,Fqi , and dzdF=XG , XG=Gpi,-GqiE15
Then, the motion along one of the For Gdirections is determined by the corresponding conjugate variable. These vector fields in general are not orthogonal, nor parallel.
If the motion of phase space points is governed by the vector field (15), Fremains constant because
dFdG=FqiqiG+FpipiG=piGqiG-qiGpiG=0 .E16
In contrast, when motion occurs in the Fdirection, by means of Eq. (16), it is the Gvariable the one that remains constant because
dGdF=GqiqiF+GpipiF=-piFqiF+qiFpiF=0 . E17
Hence, motion originated by the conjugate variables F(z)and G(z)occurs on the shells of constant F(z)or of constant G(z), respectively.
The divergence of these vector fields is zero,
XF=-qiFpi+piFqi=0 , XG=qiGpi-piGqi=0 .E18
Thus, the motions associated to each of these conjugate variables preserve the phase space area.
A constant Poisson bracket is related to the constancy of a cross product because
XGXF=dzdFdzdG=q^p^n^Gp-Gq0-FpFq0=n^GpFq-GqFp=n^F,G .E19
where n^is the unit vector normal to the phase space plane. Then, the magnitudes of the vector fields and the angle between them changes in such a way that the cross product remains constant when the Poisson bracket is equal to one, i.e. the cross product between conjugate vector fields is a conserved quantity.
The Jacobian for transformations from phase space coordinates to (f,g)variables is one for each type of motion:
J=qfpfqgpg=Gp-Gqqgpg=Gppg+Gqqg=dGdg=1 ,E20
J=qfpfqgpg=qfpf-FpFq=Fqqf+Fppf=dFdf=1 .E21
We have seen some properties related to the motion of phase space points caused by conjugate variables.
1.5. Poisson brackets and commutators
We now consider the use of commutators in the classical realm.
The Poisson bracket can also be written in two ways involving a commutator.One form is
F,G=Gpq-GqpF=LG,F=1 ,E22
and the other is
F,G=Fqp-FpqG=LF,G=1 .E23
With these, we have introduced the Liouville type operators
LF=Fqp-Fpq=XF, and LG=Gpq-Gqp=XG . E24
These are Lie derivatives in the directions of XFand XG, respectively. These operators generate complementary motion of functions in phase space. Note that now, we also have operators and commutators as in Quantum Mechanics.
Conserved motion of phase space functions moving along the for gdirections can be achieved with the above Liouvillian operators as
f=-LG, and g=-LF .E25
Indeed, with the help these definitions and of the chain rule, we have that the total derivative of functions vanishes, i.e. the total amount of a function is conserved,
ddf=dqdfq+dpdfp+f=dzdf+f=XG+f=LG+f=-f+f=0 ,E26
ddg=dqdgq+dpdgp+g=dzdg+g=XF+g=LF+g=-g+g=0 .E27
Also, note that for any function u(z)of a phase space point z, we have that
LG,u(z)=LGuz=XGuz=dzdFuz=-fuz ,E29
which are the evolution equations for functions along the conjugate directions fand g. These are the classical analogues of the quantum evolution equation ddt=1i ,H^for time dependent operators. The formal solutions to these equations are
uz;g=e-gLFuz , and uz;f=e-fLGuz.E30
With these equations, we can now move a function u(z)on T*Qin such a way that the points of their support move according to the dynamical systems Eqs. (15) and the total amount of uis conserved.
1.6. The commutator as a derivation and its consequences
As in quantum theory, we have found commutators and there are many properties based on them, taking advantage of the fact that a commutator is a derivation.
Since the commutator is a derivation, for conjugate variables F(z)and G(z)we have that, for integer n,
LGn,F=n LGn-1 , LG,Fn=n Fn-1, LFn,G=n LFn-1 , LF,Gn=n Gn-1 .E31
Based on the above equalities, we can get translation relationships for functions on T*Q. We first note that, for a holomorphic function ux=n=0unxn,
uLG,F=n=0unLGn,F=n=0nunLGn-1=u'LG .E32
In particular, we have that
efLG,F=fefLG .E33
Then, efLGis the eigenfunction of the commutator ,Fwith eigenvalue f.
From Eq. (32), we find that
But, if we multiply by u-1(LG)from the right, we arrive to
This is a generalized version of a shift of F, and the classical analogue of a generalization of the quantum Weyl relationship. A simple form of the above equality, a familiar form, is obtained with the exponential function, i.e.
efLGFe-fLG=F+f .E36
This is a relationship that indicates how to translate the function F(z)as an operator. When this equality is acting on the number one, we arrive at the translation property for Fas a function
Fz;f=efLGF(z)=Fz efLG 1+f efLG1=Fz+f .E37
This implies that
ddfFz;f=1 ,E38
i.e., up to an additive constant, fis the value of F(z)itself, one can be replaced by the other and actually they are the same object, with fthe classical analogue of the spectrum of a quantum operator.
Continuing in a similar way, we can obtain the relationships shown in the following diagram
Diagram 1.
where the constant shas units of action, length times momentum, the same units as the quantum constant .
Some of the things to note are:
The operator egLFis the eigenoperator of the commutator ,Gand can be used to generate translations of G(z)as an operator or as a function. This operator is also the propagator for the evolution of functions along the gdirection. The variable gis more than just a shift parameter; it actually labels the values that G(z)takes, the classical analogue of the spectrum of a quantum operator.
The operators LFand G(z)are also a pair of conjugate operators, as well as the pair LGand F(z).
But LFcommutes with F(z)and then it cannot be used to translate functions of F(z), F(z)is a conserved quantity when motion occurs along the G(z)direction.
The eigenfunction of LF,and of sLFis efG(z)/sand this function can be used to shift LFas an operator or as a function.
The variable fis more than just a parameter in the shift of sLF, it actually is the value that sLFcan take, the classical analogue of the spectra of a quantum operator.
The steady state of LFis a function of F(z), but egF(z)/sis an eigenfunction of LGand of LG,and it can be used to translate LG.
These comments involve the left hand side of the above diagram. There are similar conclusions that can be drawn by considering the right hand side of the diagram.
Remember that the above are results valid for classical systems. Below we derive the corresponding results for quantum systems.
2. Quantum systems
We now derive the quantum analogues of the relationships found in previous section. We start with a Hilbert space Hof wave functions and two conjugate operators F^and G^acting on vectors in H, and with a constant commutator between them
F^,G^=i ,E39
together with the domain D=DF^G^D(G^F^)in which the commutator holds. Examples of these operators are coordinate Q^and momentum P^operators, energy H^and time T^operators, creation a^and annihilation a^operators.
The eigenvectors of the position, momentum and energy operators have been used to provide a representation of wave functions and of operators. So, in general, the eigenvectors |fand |gof the conjugate operators F^and G^provide with a set of vectors for a representation of dynamical quantities like the wave functions fψand gψ.
With the help of the properties of commutators between operators, we can see that
F^n,G^=iF^n-1, F^,G^n=iG^n-1.E40
Hence, for a holomorphic function uz=n=0unznwe have that
u^F^,G^=iu^'F^ , F^,u^G^=iu^'G^ , E41
i.e., the commutators behave as derivations with respect to operators. In an abuse of notation, we have that
1i,G^=ddF^ , 1iF^,=ddG^ .E42
We can take advantage of this fact and derive the quantum versions of the equalities found in the classical realm.
A set of equalities is obtained from Eq. (43) by first writing them in expanded form as
u^F^G^-G^u^F^=iu^'F^, and F^u^G^-u^G^F^=iu^'G^. E43
Next, we multiply these equalities by the inverse operator to the right or to the left in order to obtain
u^F^G^u^-1F^=G^+iu^'F^u^-1F^, and u^-1G^F^u^G^=F^+iu^-1G^u^'G^.E44
These are a set of generalized shift relationships for the operators G^and F^. The usual shift relationships are obtained when u(x)is the exponential function, i.e.
G^g:=e-igF^/G^eigF^/=G^+g, and F^f:=eifG^/F^e-ifG^/=F^+f .E45
Now, as in Classical Mechanics, the commutator between two operators can be seen as two different derivatives introducing quantum dynamical system as
dP^(f)df=-G^(Q^,P^)Q^=1ihP^(f),G^(Q^,P^), dQ^(f)df=G^(Q^,P^)P^=1iQ^(f),G^(Q^,P^) ,E46
dP^(g)dg=F^(Q^,P^)Q^=1iF^(Q^,P^),P^(g) , and dQ^(g)dg=-F^(Q^,P^)P^=1ihF^(Q^,P^),Q^(g) ,E47
P^f=eifG^/P^e-ifG^/ , Q^f=eifG^/Q^e-ifG^/ , E48
P^g=e-igF^/P^eifF^/ , and Q^g=e-igF^/Q^eigF^/ .E49
These equations can be written in the form of a set of quantum dynamical systems
dz^df=X^G , X^G=G^P^,-G^Q^ , dz^dg=X^F , X^F=-F^P^,F^Q^ ,E50
where z^=Q^,P^.
The inner product between the operator vector fields is
where dl^F2dQ^2+dP^2, evaluated along the gdirection, is the quantum analogue of the square of the line element dlF2=dq2+dp2.
We can define many of the classical quantities but now in the quantum realm. Liouville type operators are
L^^F1iF^, , and L^^G1i,G^ .E52
These operators will move functions of operators along the conjugate directions G^or F^, respectively. This is the case when G^is the Hamiltonian H^of a physical system, a case in which we get the usual time evolution of operator.
There are many equalities that can be obtained as in the classical case. The following diagram shows some of them:
Diagram 2.
Note that the conclusions mentioned at the end of the previous section for classical systems also hold in the quantum realm.
Next, we illustrate the use of these ideas with a simple system.
3. Time evolution using energy and time eigenstates
As a brief application of the abovee ideas, we show how to use the energy-time coordinates and eigenfunctions in the reversible evolution of probability densities.
Earlier, there was an interest on the classical and semi classical analysis of energy transfer in molecules. Those studies were based on the quantum procedure of expanding wave functions in terms of energy eigenstates, after the fact that the evolution of energy eigenstates is quite simple in Quantum Mechanics because the evolution equation for a wave function it|ψ>=H^|ψ>is linear and contains the Hamiltonian operator. In those earlier calculations, an attempt to use the eigenfunctions of a complex classical Liouville operator was made [5-8]. The results in this chapter show that the eigenfunction of the Liouville operator LHis egT(z)and that it do not seems to be a good set of functions in terms of which any other function can be written, as is the case for the eigenfunctions of the Hamiltonian operator in Quantum Mechanics. In this section, we use the time eigenstates instead.
With energy-time eigenstates the propagation of classical densities is quite simple. In order to illustrate our procedure, we will apply it to the harmonic oscillator with Hamiltonian given by (we will use dimensionless units)
Hz=p22+q22 .E53
Given and energy scaling parameter Esand the frequency ωof the harmonic oscillator, the remaining scaling parameters are
ps=mEs , qs=Esmω2 , ts=1ω .E54
We need to define time eigensurfaces for our calculations. The procedure to obtain them is to take the curve q=0as the zero time curve. The forward and backward propagation of the zero time curve generates the time coordinate system in phase space. The trajectory generated with the harmonic oscillator Hamiltonian is
qt=2Ecost+π2 , pt=2Esint+π2 . E55
With the choice of phase we have made, q=0when t=0, which is the requirement for an initial time curve. Then, the equation for the time curve is
p=q tant+π2, or q=p cott+π2 .E56
These are just straight lines passing through the origin, equivalent to the polar coordinates. The value of time on these points is t, precisely. In Fig. 1, we show both coordinate systems, the phase space coordinates (q,p), and the energy time coordinates (E,t)on the plane. This is a periodic system, so we will only consider one period in time.
Figure 1.
Two conjugate coordinate systems for the classical harmonic oscillator in dimensionless units. Blue and black lines correspond to the(q,p)coordinates and the red and green curves to the(E,t)coordinates.
At this point, there are two options for time curves. Both options will cover the plane and we can distinguish between the regions of phase space with negative or positive momentum. One is to use half lines and tin the range from -πto π, with the curve t=0coinciding with the positive paxes. The other option is to use the complete curve including positive and negative momentum values and with t(-π/2,π/2). In the first option, the positive momentum part of a probability density will correspond to the range t(-π/2,π/2), and the negative values will correspond to t-π,-π2(π/2,π). We take this option.
Now, based on the equalities derived in this chapter, we find the following relationship for a marginal density dependent only upon H(z), assuming that the function ρ(H)can be written as a power series of H, ρH=iρiHi,
e-τLHρH=n(-τ)nn!LHniρiHi=iρiHi=ρH , E57
where we have made use of the equality LHH=0. Then, a function of Hdoes not evolve in time, it is a steady state. For a marginal function dependent upon t, we also have that
e-τLHρt=eτd/dtρt=ρt+τ .E58
where we have made use of the result that ddt=-LH. Therefore, a function of tis only shifted in time without changing its shape.
For a function of Hand twe find that
e-τLHρH,t=e-τd/dtρH,t=ρH,t+τ .E59
This means that evolution in energy-time space also is quite simple, it is only a shift of the function along the taxes without a change of shape.
So, let us take a concrete probability density and let us evolve it in time. The probability density, in phase space, that we will consider is
ρz=Hze-q-q02+p-p02/2σ2, E60
with q0,p0=(1,2)and σ=1. A contour plot of this density in phase-space is shown in (a) of Fig. 2. The energy-time components of this density are shown in (b) of the same figure. Time evolution by an amount τcorrespond to a translation along the taxes, from tto t+τ, without changing the energy values. This translation is illustrated in (d) of Fig. 2 in energy-time space and in (c) of the same figure in phase-space.
Recall that the whole function ρ(z)is translated in time with the propagator e-τLH. Then, there are two times involved here, the variable tas a coordinate and the shift in time τ. The latter is the time variable that appears in the Liouville equation of motion dρz;τdτ=-LHρz;τ.
Figure 2.
Contour plots of the time evolution of a probability density on phase-space and on energy-time space. Initial densities (a) in phase space, and (b) in energy-time space. (d) Evolution in energy-time space is accomplished by a shift along thetaxes. (c) In phase space, the density is also translated to the corresponding time eigensurfaces.
This behaviour is also observed in quantum systems. Time eigenfunctions can be defined in a similar way as for classical systems. We start with a coordinate eigenfunction |q>for the eigenvalue q=0and propagate it in time. This will be our time eigenstate
t> =eitH^ q=0> .E61
The projection of a wave function onto this vector is
<tψ>=<q=0e-itH^|ψ>=ψq=0;t ,E62
Which is the time dependent wave function, in the coordinate representation, and evaluated at q=0. This function is the time component of the wave function.
The time component of a propagated wave function for a time τis
<tψ(τ)>=<q=0e-itH^e-iτH^ψ>=<t+τψ> .E63
Then, time evolution is the translation in time representation, without a change in shape. Note that the variable τis the time variable that appears in the Schrödinger equation for the wave function.
Now, assuming a discrete energy spectrum with energy eigenvalue Enand corresponding eigenfunction |n>, in the energy representation we have that
<nψτ>=<ne-iτH^ψ>=e-iτEn<nψ> ,E64
i.e. the wave function in energy space only changes its phase after evolution for a time τ.
4. Concluding remarks
Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both theories.
Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as coordinates for representing dynamical quantities.
Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle. This is the case of the Pauli theorem [9-14] that prevents the existence of a hermitian time operator in Quantum Mechanics. The classical analogue of this puzzle is found in Reference [15].
These were some of the properties and their consequences in which both conjugate variables participate, influencing each other.
1. 1. Woodhouse NMJ. Geometric Quantization. Oxford: Osford University Press; 1991.
2. 2. Wigner E. Phys Rev A 1932; 40 749
3. 3. Husimi K. Proc Phys Math Soc Jpn 1940; 22 264
4. 4. Torres-Vega G, Theoretical concepts of quantum mechanics. Rijeka: InTech; 2012.
5. 5. Jaffé C. Classical Liouville mechanics and intramolecular relaxation dynamics. The Journal of Physical Chemistry 1984; 88 4829.
6. 6. Jaffé C and Brumer C. Classical-quantum correspondence in the distribution dynamics of integrable systems. Journal of Chemical Physics 1985; 82 2330.
7. 7. Jaffé C. Semiclassical quantization of the Liouville formulation of classical mechanics. Journal of Chemical Physics 1988; 88 7603.
8. 8. Jaffé C. Sheldon Kanfer and Paul Brumer, Classical analog of pure-state quantum dynamics. Physical Review Letters 1985; 54 8.
9. 9. Pauli W. Handbuch der Physics. Berlin: Springer-Verlag; 1926
10. 10. Galapon EA, Proc R Soc Lond A 2002; 458 451
11. 11. Galapon EA, Proc R Soc Lond A 2002; 458 2671
12. 12. Galapon EA, quant-ph/0303106
13. 13. Galindo A, Lett Math Phys 1984; 8 495
14. 14. Garrison JC and Wong J, J Math Phys 1970; 11 2242
15. 15. Torres-Vega G, J Phys A: Math Theor 45, 215302 (2012)
Written By
Gabino Torres-Vega
|
8a92de728941d317 | • 10 December 2021
DeepMind AI tackles one of chemistry’s most valuable techniques
Machine-learning algorithm predicts material properties using electron density.
DFT artwork.
The AI predicts the distribution of electrons within a molecule (illustration) and uses it to calculate physical properties.Credit: DeepMind
A team led by scientists at the London-based artificial-intelligence company DeepMind has developed a machine-learning model that suggests a molecule’s characteristics by predicting the distribution of electrons within it. The approach, described in the 10 December issue of Science1, can calculate the properties of some molecules more accurately than existing techniques.
“To make it as accurate as they have done is a feat,” says Anatole von Lilienfeld, a materials scientist at the University of Vienna.
The paper is “a solid piece of work”, says Katarzyna Pernal, a computational chemist at Lodz University of Technology in Poland. But she adds that the machine-learning model has a long way to go before it can be useful for computational chemists.
Predicting properties
In principle, the structure of materials and molecules is entirely determined by quantum mechanics, and specifically by the Schrödinger equation, which governs the behaviour of electron wavefunctions. These are the mathematical gadgets that describe the probability of finding a particular electron at a particular position in space. But because all the electrons interact with one another, calculating the structure or molecular orbitals from such first principles is a computational nightmare, and can be done only for the simplest molecules, such as benzene, says James Kirkpatrick, a physicist at DeepMind.Can artificial intelligence create the next wonder material?
To get around this problem, researchers — from pharmacologists to battery engineers — whose work relies on discovering or developing new molecules have for decades relied on a set of techniques called density functional theory (DFT) to predict molecules’ physical properties. The theory does not attempt to model individual electrons, but instead aims to calculate the overall distribution of the electrons’ negative electric charge across the molecule. “DFT looks at the average charge density, so it doesn’t know what individual electrons are,” says Kirkpatrick. Most properties of matter can then be easily calculated from that density.
Since its beginnings in the 1960s, DFT has become one of the most widely used techniques in the physical sciences: an investigation by Nature’s news team in 2014 found that, of the top 100 most-cited papers, 12 were about DFT. Modern databases of materials’ properties, such as the Materials Project, consist to a large extent of DFT calculations.
But the approach has limitations, and is known to give the wrong results for certain types of molecule, even some as simple as sodium chloride. And although DFT calculations are vastly more efficient than those that start from basic quantum theory, they are still cumbersome and often require supercomputers. So, in the past decade, theoretical chemists have increasingly started to experiment with machine learning, in particular to study properties such as materials’ chemical reactivity or their ability to conduct heat.
Ideal problem
The DeepMind team has made probably the most ambitious attempt yet to deploy AI to calculate electron density, the end result of DFT calculations. “It’s sort of the ideal problem for machine learning: you know the answer, but not the formula you want to apply,” says Aron Cohen, a theoretical chemist who has long worked on DFT and who is now at DeepMind.DeepMind’s AI helps untangle the mathematics of knots
The team trained an artificial neural network on data from 1,161 accurate solutions derived from the Schrödinger equations. To improve accuracy, they also hard-wired some of the known laws of physics into the network. They then tested the trained system on a set of molecules that are often used as a benchmark for DFT, and the results were impressive, says von Lilienfeld. “This is the best the community has managed to come up with, and they beat it by a margin,” he says.
One advantage of machine learning, von Lilienfeld adds, is that although it takes a massive amount of computing power to train the models, that process needs to be done only once. Individual predictions can then be done on a regular laptop, vastly reducing their cost and carbon footprint, compared with having to perform the calculations from scratch every time.
Kirkpatrick and Cohen say that DeepMind is releasing their trained system for anyone to use. For now, the model applies mostly to molecules and not to the crystal structures of materials, but future versions could work for materials, too, the authors say.
1. 1.Kirkpatrick, J. et al. Science 374, 1385–1389 (2021).
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9d17d75e6094f344 | Quantum Foundations Workshop
Workshop: New Topics in Quantum Foundations
November 29 – 30, 2018 Université de Lausanne
Over the past few decades, great work by physicists and philosophers has cleared up much of the early confusion about the foundations of quantum mechanics. Controversies are still ongoing and old prejudices are hard to overcome, but the measurement problem and its possible solutions, the importance of non-locality, the status of probabilities, etc. are now well understood at least by experts. Time to look forward. In this two-day workshop, young researchers from Switzerland and abroad are going to present new research topics in the foundations and metaphysics of quantum physics.
Organizers: Michael Esfeld, Dustin Lazarovici, Andrea Oldofredi
Thursday November 29 UNIL-Anthropole 5060
13h40-13h45 Welcome
Claudio Calosi (Geneva): Quantum Monism
Monism is roughly the view that there is only one fundamental entity. One of the most powerful argument in its favour comes from Quantum Mechanics. Extant discussions of quantum monism are framed independently of any interpretation of the quantum theory. In contrast, this paper argues that matters of interpretation play a crucial role when assessing the viability of monism in the quantum realm. I consider four different interpretations: Modal Interpretations, Bohmian Mechanics, Many Worlds Interpretations, and Wavefunction Realism. In particular, I extensively argue for the following claim: several interpretations of QM do not support monism at a more serious scrutiny, or do so only with further problematic assumptions, or even support different versions of it.
Matthias Egg (Bern): How Scientific Can a Metaphysics of Quantum Mechanics Be?
Ongoing disagreement about the measurement problem in quantum mechanics is a major obstacle for the project of scientific metaphysics, because there do not seem to be any universally accepted scientific standards that would let us decide what the true metaphysics of quantum mechanics is. One response by scientific metaphysicians has been the attempt to dissolve (rather than to solve) the measurement problem. After discussing one particular proposal in that spirit (due to Ladyman and Ross), I will explore the prospects of such an approach in general. The tentative lesson to be drawn seems to be the following: the more narrowly „scientific“ one’s approach to the measurement problem is, the more it is in conflict with the kind of scientific realism that is usually presupposed by the very project of scientific metaphysics.
15h45-16h45 Haktan Akcin (Hong Kong) : What’s really wrong with ontic structural realism: on the possibility of reading off ontology from current fundamental science
16h45-17h15 Coffee break
Davide Romano (Salzburg): The multi-field interpretation of the wave function in Bohm’s theory
In “The wave-function as a multi-field” (EJPS 2018, with Mario Hubert), we have showed that the wave function in the de Broglie-Bohm theory can be interpreted as a (new kind of) physical field, i.e. a multi-field, in three dimensional space. In this talk, I argue that the natural framework for the multi-field view is the original second order Bohm’s theory. In this context, it is possible to construct the multi-field as a real scalar field, to explain which sort of physical interaction is at work between the multi-field and the Bohmian particles and, finally, to clarify some philosophical aspects about the dynamics of the theory.
Antonio Vassallo (Barcelona): A Primitive Ontology for Quantum Spacetime
At the moment, one of the most worked out programs to quantize the general relativistic gravitational field is the so-called “canonical approach”. However, the theories falling in the scope of the canonical program (most notably, loop quantum gravity) have to face at least three huge conceptual issues. The first is that canonical quantum-gravitational states “betray” the spirit of relativity in that they represent purely spatial, as opposed to spatiotemporal, physical degrees of freedom. The second is that the equation that describes the dynamical evolution of these states –the Wheeler-DeWitt equation– does not involve any time-like parameter, thus seemingly cutting off temporal evolution from the physical picture. The third is that quantum-gravitational states are obviously subjected to superpositions and entanglement, which makes very difficult to explain how stable classical spatiotemporal structures can emerge from the quantum regime.
In this talk, I will propose a philosophical framework based on the notion of “self-subsisting” structure, which might help physicists working in the canonical program to move in the direction of ontological clarity. This framework combines (i) a primitive ontology approach to quantum physics, (ii) ontic structural realism, and (iii) a non-standard treatment of dependence relations. Moreover, I will point out how the dynamics of self-subsisting structures can be naturally implemented using shape space physics, which is a theoretical framework for constructing purely relational theories, originally developed by Julian Barbour.
19h30 Drink and snacks in the social room (5099)
Friday November 30 UNIL-Anthropole 4021
Vera Matarese (Centre for Formal Epistemology, Prague): How to understand nomological entities: the case of spin-networks in Loop Quantum Gravity
What are nomological entities? According to Quantum Super-Humeanism (Esfeld 2014, 2018), only matter points and their distance relations exist, while the wave-function and all the dynamical parameters of quantum mechanics should be considered nomological. If this is true, then it is certainly important to achieve a metaphysically rigorous understanding of what the term ‘nomological’ amounts to. In the first part of my talk, after exploring different accounts given to nomological entities within and outside Humeanism, I will raise some metaphysical worries that these accounts have to face. I will dedicate the second part of my talk to what I consider to be a very interesting case of nomological entities: the case of spin-networks in loop quantum gravity.
Karen Crowther (Geneva): As Below, So Before: Synchronic and Diachronic Conceptions of Emergence in Quantum Gravity
The emergence of spacetime from quantum gravity appears to be a striking case-study of emergent phenomena in physics (albeit one that is speculative at present). There are, in fact, two different cases of emergent spacetime in quantum gravity: a “synchronic” conception, applying between different levels of description, and a “diachronic” conception, from the universe “before” and after the “big bang” in quantum cosmology. The purpose of this paper is to explore these two different senses of spacetime emergence; and to see whether, and how, they can be understood in the context of specific extant accounts of emergence in physics.
11h30-12h00 Coffee break
Antoine Tilloy (MPQ Munich): Spontaneous collapse models are Bohmian mechanics applied to a hidden heath bath
Spontaneous collapse models models and Bohmian mechanics are two different solutions to the measurement problem plaguing quantum mechanics. Apart from having a clear primitive ontology, they have a priori nothing in common. At a formal level, collapse models add a non-linear noise term to the Schrödinger equation, and extract the primitive ontology either from the wave function (mass density ontology) or the noise itself (flash ontology). Bohmian mechanics keeps the Schrödinger equation intact but uses the wave function to guide particles, which then make up the primitive ontology. Collapse models modify the predictions, whilst Bohmian mechanics keeps the empirical content intact. However, it turns out that collapse models and their primitive ontology can be exactly recast as Bohmian theories. More precisely, the stochastic wave-function of a collapse model is exactly the Bohmian wave function of the system considered coupled to a carefully tailored bath upon conditioning on the bath Bohmian positions. The noise driving the collapse model is a linear functional of the Bohmian positions. The randomness that seems progressively revealed in the collapse models lies entirely in the initial conditions in the Bohmian theory. The construction of the appropriate bath is not trivial and exploits an old result from the theory of open quantum systems. This reformulation of collapse models as Bohmian theories brings the question of whether there exists realist reconstructions of quantum theory that cannot ultimately be rewritten this way, with some guiding law.
13h00 End of workshop
Practical Information
The workshop takes place at UNIL-Anthropole, Quartier Dorigny, 1015 Lausanne.
Thursday: Room 5060. Friday: Room 4021.
Directions from Lausanne train station:
Take the metro m2, direction “Croisettes”, for one stop to Lausanne-Flon
Then take the metro m1, direction “Renens CFF”, to the metro stop UNIL-Chamberonne (formerly: UNIL-Dorigny)
Directions from the EPFL / SwissTech Hotel
Take the metro m1, direction “Lausanne Flon”, to the metro stop UNIL-Chamberonne (formerly: UNIL-Dorigny)
By car
On the highway, drive in direction “Lausanne-Sud”, take the exit “UNIL-EPFL”.
Follow the direction “UNIL” and then “UNIL-Chamberonne”.
Parking (paid) is available in front of the building (parking Chamberonne)
If you need help booking an accommodation in Lausanne, please contact the organizers. |
06a8ed6b7c194dd6 | Jones Calculus
Jones Calculus
Jones calculus
A quaternion valued wave equation \Psi_{tt} = D^2 \Psi can be solved as usual with a d’Alembert solution \Psi(t) = \cos(D t) \Psi(0) + \sin(D t) D^{-1} \Psi'(0). We can write this more generally as e^{\beta D t} (u(0) - \beta v(0)) where \beta is a unit space quaternion where \psi(0)=u(0) - \beta v(0) is the initial wave. Now, \exp(\beta x) = \cos(x) + \beta \sin(x) holds for any space unit quaternion \beta. Unlike in the complex case, we have now an entire 2-sphere which can be used as a choice for \beta. If u(0) and v(0) are real, then we stay in the plane spanned by 1 and \beta. If u(0) and v(0) are in different plane, then the wave will evolve inside a larger part of the quaternion algebra.
Also as before, the wave equation has not be put in artificially. It appears when letting the system move freely in its symmetry. In the limit of deformation we are given an anti-symmetric matrix B= \beta (b+b^*) and get a unitary evolution \exp(i B t). As we have used Pauli matrices to represent the quaternion algebra on C^2, a wave is now given as a pair (\psi(t),\phi(t)) of complex waves. Using pairs of complex vectors is nothing new in physics. It is the Jones calculus named after Robert Clark Jones (1916-2004) who developed this picture in 1941. Jones was a Harvard graduate who obtained his PhD in 1941 and after some postdoc time at Bell Labs, worked until 1982 at the Polaroid Corporation.
Why would a photography company emply a physisists dealing with quaternion valued waves? The Jones calculus deals with polarization of light. It applies if the electromagnetic waves F =(E,B) have a particular form where E,B are both in a plane and perpendicular to each other. Remember that light is described by a 2-form F=dA which has in 4 dimensions B(4,2)=6 components, three electric and three magnetic components. The Maxwell equations dF=0, d* F=0 are then in a Lorentz gauge d^*A=0 equivalent to a wave equation L A =0, where L is the Laplacian in the Lorentz space. Now, if light has a polarized form, one can describe it with a complex two vector \Psi=(u,v) rather than by giving the 6 components (E,B) of the electromagnetic field. How is this applied? Sun light arrives unpolarized but when scattering at a surface, it catches an amount of polarization. Polarized sunglasses filter out part of this light reducing the glare of reflected light. The effect is also used in LCD technology or for glasses worn in 3D movies. It can not only be used for light, but in radio wave technology, polarization can be used to “double book” frequency channels. And for radar waves, using polarized radar waves can help to avoid seeing rain drops. Even nature has made use of it. Octopi or cuttlefish are able to see polarization patterns. See the encylopedia entry for more. Mathematically the relation with quaternion is no suprise because the linear fibre of a 1-form A(x) at a point is 4-dimensional. Describing the motion of the electromagnetic field potential A (which satisfies the wave equation) is therefore equivalent to a quaternion valued field.
We have to stress however that the connection between a quaternion valued quantum mechanics and wave motion of the electromagnetic field is mostly a mathematical one. First of all, we work in a discrete setup over an arbitrary finite simplicial complex. We don’t even have to take the de Rham complex: any elliptic complex D=d+d* as discribed in a discrete Atiyah-Singer setup will do. The Maxwell equations even don’t need to be 1 forms. If E \oplus F=\oplus E_k + \oplus F_k is the arena of vector spaces on which D:E \to F, F \to E$ acts, then one can see for a given $j \in D_k$ the equations dF=0,d^*F=j as the Maxwell equation in that space. For F=dA and gauge d^*A=0, the Maxwell equations reduce to the Poisson equation D^2 A=j which in the case of an absense of “current” j gives the wave equation D^2 A=0 meaning that A is a harmonic k-form. Now, in a classical de Rham setup on a simplicial complex G, A is just an anti-symmetric function on k-dimensional simplices of the complex. Still, in this setup, when describing light on a space of k-forms, it is given by real valued functions. If we Lax deform the elliptic complex, then the exterior derivatives become complex but still, the harmonic forms do not change because the Laplacian does not change. Also note that we don’t incorporate time into the simplicial complex (yet). Time evolution is given by an external real quantity leading to a differential equation. The wave equation u_{tt}=Lu can be described as a Schrödinger equation u_t = i Du. We have seen that when placing three complex evolutions together that we can get a quaternion valued evolution. But the waves in that evolution have little to do with the just described Maxwell equations in vacuum, which just describes harmonic functions in the elliptic complex.
We will deal with the problematic of time elsewhere. Just to state now that describing a space time with a finite simplicial complex does not seem to work. It migth be beautiful and interesting to describe finite discrete space times but one can hardly solve the Kepler problem with it. Mathematically close to the Einstein equations is to describe simplicial complexes with a fixed number of simplices which have maximal or minimal Euler characteristic among all complexes. Anyway, describing physics with waves evolving on finite geometries is appealing because the mathematics of its quantum mechanics is identical to the mathematics of the quantum mechanics in the continuum, just that everything is finite dimensional. Yes there are certain parts of quantum mechanics which appear needing infinite dimensions but if one is interested in the PDE’s, the Schroedinger respectivly the wave equation on such a space there are many interesting problems already in finite dimensions. The question how fast waves travel is also iteresting in the nonlinear Lax set-up. See This HCRP project from 2016 of Annie Rak. In principle the mathematics of PDE’s on simplicial complexes (which are actually ordinary differential equations) has more resemblence with the real thing because if one numerically computes any PDE using a finite element method, one essentially does this.
Here is a photograph showing Robert Clark Jones:
Robert Jones Robert Jones
Source: Emilio Segrè Visual Archives.
There are other places in physics where complex vector-valued fields appear. In quantum mechanics it appears from SU(2) symmetries, two level systems, isospin or weak isospin. Essentially everywhere, where two quantities can be exchanged, the SU(2) symmetry appears. A quaternion valued field is also an example of a non-abelian gauge field. In that case, one is interested (without matter) in the Lagrangian |F|^2/2 with F=dA+A \wedge A, where A is the connection 1-form. Summing the Lagranging over space gives the functional. One is interested then in critical points. The satisfy d_A^* F=0, d_A F=0 meaning that they are “harmonic” similarly as in the abelian case, where harmonic functions are critical points of the quadratic Lagrangian. There are differences however. In the Yang-Mills case, one looks at SU(2) meaning that the fields are quaternions of length 1. When we look at the Lax (or asymptotically for large t, the Schrödinger evolution) of quaternion valued fields \psi(t), then for exach fixed simplex x, the field value \psi(t,x) is a quaternion, not necessarily a unit quaternion.
[Remark. A naive idea put forward in the “particle and primes allegory” is to see a particle realized if it has an integer value. The particles and prime allegory draws a striking similarity between structures in the standard model and combinatorics of primes in associative complete division algebras. The later is pure mathematics. As there are symmetry groups acting on the primes, it is natural to look at the equivalence classes. The symmetry groups in the division algebras are U(1) and SU(2) but there is also a natural SU(3) action due to the exchange of the space generators i,j,k in the quaternion algebra. This symmetry does not act linearly on the apace, but it produces an other (naturally called strong) equivalence relation. The weak (SU(2)) and strong equivalence relations combined lead to pictures of Mesons and Baryons among the Hadrons while the U(1) symmetry naturally leads to pictures of Electron-Positron pairs and Neutrini in the Lepton case. The nomenclature essentially pairs the particle structure seen in the standard model with the prime structure in the division algebras. As expected, the analogy does not go very far. The fundamental theorem of algebra for quaternions leads to some particle processes like pair creation and annihilation and recombination but not all. It does not explain for example a transition from a Hadron to a Lepton. The set-up also leads naturally to charges with values 1/3 or 2/3 but not all. Also, number theory has entered physics in many places, it is not clear why “integers” should appear at all in a quantum field theory. What was mentioned in the particles and primes allegory is the possibility to see particles only realized at a simplex x, if the field value is an integer there. As in a non-linear integrable Hamiltonian system like the Lax evolution soliton solutions are likely to appear and so, if the wave takes some integer value p at some time t and position x, it will at a later time have that value p at a different position. The particle has traveled. But as during the time it has jumped from one vertex to an other, it can have changed to a gauge equivalent particle. If the integer value is not prime, it decomposes as a product of primes. Taking a situation where space is a product of other spaces allows to model particle interactions. One can then ask why a particle like an electron modeled by some non-real prime is so stable and why if we model an electron-positron pair by a 4k+1 prime, the position of the electron and positron are different. A Fock space analogy is to view space as an element in the strong ring, where every part is a particle. Still the mathematics is the same, we have a geometric space G with a Dirac operator D. Time evolution is obtained by letting D go in its symmetry group.] |
84cc86942c0bbf75 | The Two Operators
The Two Operators
The strong ring
The strong ring generated by simplicial complexes produces a category of geometric objects which carries a ring structure. Each element in the strong ring is a “geometric space” carrying cohomology (simplicial, and more general interaction cohomologies) and has nice spectral properties (like McKean Singer) and a “counting calculus” in which Euler characteristic is the most natural functional. Unlike the class of simplicial complexes or the class of discrete CW complexes or Stanley-Raiser ring elements, this ring combines the property that it is a “Cartesian closed category” and that the arithmetic is compatible with cohomology and spectra of connection Laplacians L of G. The strong ring is isomorphic to a subring of the strong Sabidussi ring via the ring homomorphism G \to G' attaching to a complex its signed connection graph. Like the Stanley-Reisner ring, also the full Sabidussi ring on the category of all finite simple graphs is too large. The strong ring solves the problem to define a category of finite geometries which is Cartesian closed, has a ring arithmetic, compatibility with cohomology (Kuenneth) and a finite potential theory (energy theorem) and spectral compatibility in the sense that multiplication leads to products of spectra and tensor products of connection Laplacians. Each of the two Laplacians also carries nonlinear discrete partial differential equations (PDE’s). The first one is a Lax pair and integrable, the eigenvalues of the Dirac operator being the integrals. For the second, the Helmhotz system, we suspect integrability as it looks like a nonlinear Schrödinger equation. In the first case, the description is given in the Heisenberg picture, deforming the operators. In the second case, the discrete PDE will deform states of the Hilbert space. Of course both can be seen in the Schroedinger or Heisenberg picture. We have implemented both dynamical systems on the computer, but still require to do more experiments. Interesting would be to study how both play together. We hope of course to see soliton like solutions (nonlinear systems are nicer in explaining particle structures and featuring particles with different speeds).
Remarks and reiterations of points made elsewhere:
• The full Stanley-Reisner ring is a ring, but it is too large. Its elements somehow behave like measurable sets and do not carry cohomology nor an Euler caracteristic compatible with cohomology and homotopy. It contains for example elements like f=xy which is an “edge without vertices”, we want elements like f=xy+x+y which is the graph K2. The Euler characteristic of f=xy is -1. There is no Euler-Poincare formula which links this to cohomology. Discrete CW complexes are natural, carry cohomology but forming products is problematic. Its the strong ring generated by products of simplicial complexes which works nicely. It is a finite structure resembling the concpt of compactly generated Hausdorff space in classical topology. Having algebraic structures associated to geometric objects is the heart of algebraic topology. But the point of view here is not to attach an algebraic object like a ring to a geometric object but to see a geometric object as an element in an algebraic one, here a ring. We “calculate WITH spaces”. The category of geometries is an algebraic object. This arithmetic generalizes the usual arithmetic, which is the special case if the geometric objects are zero dimensional (like pebbles used by the earlierst mathematicians in pre-Babylonian time). In the strong ring, primes are either zero dimensional classical rational primes or then elements in the ring which are simplicial complexes. Some of them might decay in the full Sabidussi ring of all graphs, which is a natural “ring extension” of the strong ring.
• While we have compatibility with cohomology on the entire space of graphs (which as usual are seen as simplicial Whitney complexes and not just one-dimensional skeletons to which classical graph theory reduces them), the corresponding Cartesian product is not associative, as multiplying with 1=K_1 is the Barycentric refinement.
• Graphs with the weak Cartesian product form a Cartesian closed category too, but this assumes to see graphs as one-dimensional simplicial complexes, which is a point of view from the last century. Graphs have much more structure as they can be equipped with more interesting simplicial complex structures, in particularly the canonical Whitney complex for which the discrete topological results are identical to the continuum. As pointed out at various places, we like graphs for their intuitive appeal, because they are hard wired into computer algebra systems and because after one Barycentric refinement, one always deals with Whitney complexes. The language of abstract simplicial complexes is equally suited but much less intuitive as one can not visualize them nicely unless they are Whitney complexes of graphs.
• The connection graph G’ is in general not homotopic to G, but only homotopic to G1, the Barycentric refinement of G. Also the Barycentric refinement G1 of a ring element G, is the Whitney complex of a graph. The dimension of the connection graph G’ is larger in general but G_1 and G are homotopic.
• In some sense the strong ring produces a purely geometric quantum field theory where the individual simplicial complexes are elementary particles as they are algebraic primes in the strong ring. Taking the product of two positive dimensional spaces produces a “particle state” in which the particles are correlated. The energy spectra multiply.
• We can make use of the strong ring and define a discrete Euclidean space Zd which when equipped with the connection Laplacian has a mass gap also in the infinite volume limit. This is really exciting as perturbation theory becomes trivial. No strong implicit function theorems are needed, the classical implicit function theorem works as the operators remain bounded and invertible even in the infinite volume limit. We perturb around hyperbolic systems. This is very un-usual as we usually perturb around a trivial integrable case, which leads to small divisor subtleties.
• An obvious task is to implement Euclidean rotation and translation symmetries on such a geometry. Together with scaling operations we can approximate also more complex rotations: just zoom into the space first, then implement a transformation in which the matrices are integers, then zoom back. Its what computer programmers always do when implementing Lie group symmetries in computer games. They never, ever would dream about using floating point arithmetic to do that. Rotating using integer matrices is good enough and saves valuable computation time for the GPU.
Two towering operators
Two operators
Both the Hodge and the connection Laplacians define nonlinear integrable Hamiltonian systems [P.S. we currently only suspect the Helmholtz system to be integrable. There is no proof yet]. The first one defines an evolution of space (as the exterior derivative d by Connes distance defines a metric d(x,y) = sup {f | |df|<1} |f(x)-f(y)|). The second is an evolution of waves which looks like a non-linear Schrödinger evolution which at least in the zero temperature and large time version is the classical linear Schrödinger evolution of a combination of the two Laplacians. For the Hodge Laplacian H it is the kernel, the Harmonic forms which is topologically interesting and Künneth which shows the compatibility with the arithmetic. For the connection Laplacian L, it is the inverse g=L-1, the Green function values which produces a finite (!) potential theory and an other relation with Euler characteristic. For a simplicial complex G or more generally for any element in the strong ring, the two Hamiltonians H(G) and L(G) live on the same Hilbert space. Both operators show some kind of compatibility. The map G \to p(G) is a ring homomorphism. The spectrum G \to \sigma(G) is multiplicative.
The Hodge operator H is affiliated to calculus given by an exterior derivative d appearing in the Stokes equation X(δ G) = d X(G) for signed valuations, where δis the boundary operator for simplicial complexes. The connection operator L is oblivious to the initial orientation of simplices and like the Dirac operator D=d+d* encodes incidences. But while D does not count intersections of simplices for which the difference of dimension is larger than 1, the operator L does count such intersections.
Just because of these analogies, we like to see L more and more as an object parallel to the Dirac operator and not to the Laplacian. Also the spectral picture supports this. Both D and L have both positive and negative spectrum. It is D2 and L2 which have non-negative spectrum. Still, for here, we continue to look at L as a Laplacian. We like L also because the entries of the inverse g have topological interpretations. We don’t have that (yet) for the inverse of the square L2.
The Hodge Laplacian H=D^2= (d+d^*)^2 decomposes into blocks H_k of form Laplacians. It is important because the kernels of H_k are isomorphic to the cohomology groups. The Nullety of H_k is the k’th Betti number b_k(G). The Super nullity \sum_k (-1)^k b_k(G) is the Euler characteristic. It is also the super trace of the identity or by McKean-Singer, the super trace of the heat kernel e^{-t H} as well as the super trace of L (by definition) and the super trace of its inverse (a Gauss-Bonnet formula). The Dirac operator D compares more closely to the connection operator and D^2=H is probably more in spirit to L^2 as both have non-negative spectrum. The connection operator L encodes the incidence of simplices in G as it is L(x,y)=1 if x,y intersect and 0 else. Both the boundary operation \delta and the exterior derivative (incidence matrices) d depend on an orientation of the simplices but this “gauge choice” does not matter for any cohomologically or spectrally relevant quantities. It is a choice of basis in the Hilbert space [which always exist, we don’t insist on compatibility of the orientations which for non-orientable graphs like the projective plane or Moebius strip do not exist]. The Hodge Laplacian H for example is independent of the gauge. We usually are not aware of this choice of orientation. In the case of an orientable manifold with boundary, the orientations are naturally linked in the sense that the orientation of a subsimplex matches the orientation of the larger one. For the operator L, the kernel is irrelevant as it is trivial by the unimodularity theorem.
Two nonlinear Hamiltonian systems
We have a natural dynamical isospectral deformation of the Dirac operator D in the form of a Lax pair D'=[D,B] where B=d-d^*. It is natural as it is a symmetry of the geometry but it dramatically effects geometry. Space expands with an initial inflation bump. This is the case for any simplicial complex and extends to all strong ring elements. The deformation is a nonlinear integrable dynamical system. As stated it is a scattering case. There is a complex version which renders the deformed d complex and which is asymptotically the Schrödinger evolution. Also the operator L features a natural dynamics. As the Green function g=L^{-1} satisfies the energy theorem \chi(G) = E(\psi) = \sum_{x,y} g(x,y) = <\psi,g \psi>, where \psi is the constant wave being \psi(x)=1, one can take E as a Hamiltonian and look at i \psi' = E_{\overline{\psi}} a dynamics which preserves both |\psi|^2 and energy E(\psi). As Euler characteristic shares with entropy compatibility with the algebraic structure in the ring, we can look at the Helmhotz system
Is it relevant in physics?
The strong ring and its integrable dynamical systems belongs to pure mathematics: it is a ring of geometric objects with nice mathematical properties in which each element has operators which can be deformed using nonlinear partial difference equations. The first deformation can be seen in the continuum also by deforming exterior derivatives on compact Riemannian manifolds (leading to Pseudo partial differential operators but featuring the same kind of expansion featuring an initial inflation bump). For the second deformation, there is no continuum analogue yet, at least not classically. [The reasons are various. We don’t have a classical analogue of the connection graph and we also don’t have a good definition of Shannon entropy for general measures. Staying within finite structures really makes entropy mathematically solid. Notions of entropy as used in physics are often vague, especially when used in cosmology.] The Barycentric limit produces not the usual Euclidean manifolds but leads to dyadic analogue structures. In the limiting case we actually deform almost periodic operators.
But one can ask if there is physics involved. The development of calculus always almost by definition ran parallel to questions in physics, in particularly in fluid dynamics and later in electromagnetism. Obviously, H is associated to fundamental particle interactions. The Maxwell equations for example are equivalent to the Poisson equation H_1 A =j. The reason is that in a special gauge d^* A=0, the electromagnetic field F=dA satisfies dF=0, d^* F = j. Also the gravitational field equation H_0 V = \rho (a Poisson equation too but for scalar fields) gives the gravitational potential V so that the field F=dV satisfies the Gauss equation {\rm div}(F) = d^* F = \rho. Similarly as gravitational potentials V and electromagnetic potentials A are described by 0-forms and 1-forms, solving H_k U_k = \rho_k gives k-forms U_k which lead to “fields” F_k = d U_k. Allowing the Dirac operator to move freely in the isospectral set produces a dynamics which naturally leads to a complex Schrödinger equation. Its really interesting that the evolution produces a diagonal part in the Dirac operator which is independent of the deformed exterior derivative parts. The effect of the flow is that parts of the dynamical kinetic energy of space given by the exterior derivatives is fueled into a potential theoretical part which is not visible. [The Dirac operator which is initially D=d+d* develops a diagonal part D=d+d* + b.] The diagonal part b is kind of like “dark matter”: it is present but not visible on the Laplacian side. The Hodge Laplacian H=D2 does not move under the isospectral deformation of the Dirac operator. Classical geometry does not see that system! But under the hood, on the level of the Dirac operator, the geometric effect is dramatic and pretty universal. We can run that system for any simplicial complex or now on any element in the strong ring and always get the qualitative same behavior: expansion with an inflationary single bumpy start.
[ For the connection operator L, which does not appear to have such an isospectral deformation the Euler characteristic or internal energy \chi(G) of the complex is formally related to the Hilbert action. In physics, gravity has various descriptions. It has been geometrically trivialized in relativity theory by the assumption that particles just move on geodesics in a pseudo Riemannian manifold, where the metric is defined by matter. In the Standard model there is the Higgs mechanism for assigning mass to certain particles. Also this does not seem to complete the enigma of defining “mass” as the mass of some particles like neutrini is believed not to come from the Higgs mechanism. Then there are still undiscovered gravitons. In any case, there are both geometric and quantum field aspects for gravity. ]
So, when we look at any individual element in the ring of geometric structures generated by simplicial complexes, then we have a finite dimensional Hilbert space and two dynamical systems: the first is a nonlinear isospectral deformation which asymptotically produces the Schrödinger equation and so quantum mechanics, the second features an internal finite potential theory with bounded Green functions. Both operators are related to Euler characteristic, the most important functional in geometry, in some sense the only functional compatible with the algebraic ring structure. Similarly, also the functional on waves, the entropy is unique in the class of functional which are compatible with arithmetic. We don’t have to justify the naturality of isospectral deformation, it is like explaining the rotation of a rock thrown into empty space (a rigid body motion free of external forces is a Lax pair too in any dimension leading to the integrable evolution, a geodesic flow on a Lie group. This integrable system is discribed in an appendix of Arnold who also looks that infinite dimensional case, where it is the Euler equations of fluid dynamics.) It just happens without any input (it would be very strange if a rock would NOT rotate, as the identity has zero measure in the group of all rotations). The choice of the energy functional in the L case however needed to be justified and we see that the Helmholtz functional has two aspects, internal potential energy and entropy. They were both uniquely selected by compatibility with arithmetic (a theorem of Meyer telling that Euler characteristic is the only valuation (counting function) compatible with addition and multiplication, and then a theorem of Shannon which renders Entropy unique. Of course, both functionals require a normalization to justify their uniqueness). Besides that there is the success of the “Gibbs and Helmholtz approach” to free energy who saw it as a more fundamental quantity than energy. Free energy features a healthy competition between boring static energy equilibria sinks in the zero temperature limit and equally boring chaos in the infinite temperature limit. Chemistry in particular shows that interesting processes happen when these two principles compete. The minimal energy principle and maximal entropy principle combined make life possible.
Now, lets look at both systems together. What happens is that the Helmholtz energy changes if we deform the operator alone. Obviously, when looking at the isospectral deformation of the operator D (in the Heisenberg picture) then we simultaneously have to deform the waves (in the Schrödinger picture). While the potential energy does not change if we both deform L and \psi, the entropy changes. This is not a big surprise as entropy is more like a mathematical trick to incorporate any other aspects of the system which are not part of the Hamiltonian under consideration. The geometric space is “thrown into a heat bath” so to speak and the Gibbs measures appear as critical points.
The Lax and Helmholtz systems can be run together: if U(t) D(t) U(t)^* = D(0), and \psi(t) is a solution of the Helmholtz system, just look at U(t) \psi(t). This evolution could even be extended to quaternion valued fields. There is one caveat: the isospectral deformation given by the symmetry of the Dirac operator does not preserve the entropy part of the Helmholtz Hamiltonian. Intuition given by the fact that the deformation of the exterior derivative produces an expansion suggests that the entropy increases, leading to an arrow of time but this is not a surprise as already the expansive nature of the evolution leads to an arrow of time. By the way, it might appear paradox that we have a Hamiltonian system with that feature as running the system backwards is also a solution. But what happens is that if would run the system backwards, then we would eventually reach the case where D=d+d* has no diagonal term and then get the usual expansion as D(t) = d+d* + b(t) will produce a diagonal term and reduce the size of the exterior derivatives and thus the Connes scale used to measure distance in space. |
c0b839cc90c86ca7 | Linear Combinations of f Orbitals
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It is less common to find the atomic orbitals illustrated in chemistry textbooks than the , , and orbitals. Boundary surface pictures of any of these atomic orbitals typically only show the real part of these complex functions and often leave out the sign information as well. The one-electron wavefunctions resulting from the solution of the Schrödinger equation for the hydrogen atom are complex functions except when . The real forms of atomic orbitals can be constructed by taking appropriate linear combinations of the complex forms. Here, boundary surfaces of the orbitals are colored to indicate the real and imaginary components as well as the positive and negative signs.
Contributed by: Lisa M. Goss (March 2011)
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1bd9d44bb8c13bab | Being You: Are we ready to understand consciousness?
Artificial consciousness seems to be a hot topic these days. This week’s Economist includes two invited articles about the matter (one arguing that we are progressing fast towards artificial consciousness by Google engineer Blaise Agüera y Arcas and another arguing that we are very far from such a thing by renowned scholar and author Douglas Hofstadter). Several media report of an ongoing discussion within Google of whether LaMDA, Google’s latest Large Language Model (LLM) has some sort of consciousness.
Amid all this hoopla, Anil Seth book brings a welcome breath of fresh air on a very old and difficult problem. Understanding consciousness, in general, and artificial consciousness, in particular, remains an open and obscure problem. As I recently wrote on a short PNAS commentary,
Although it has been the subject of human thought for many centuries, consciousness remains a mysterious and controversial topic. Every one of us is overly familiar with the phenomenon, but there is little agreement on what it is, what it entails, and how it is created. Certainly, no other phenomenon is simultaneously so familiar and so hard to explain.
Still, we are not doomed to remain in the dark forever. Seth’s book does an excellent job at clarifying what consciousness is all about and makes some serious contributions to our understanding of the phenomenon. Seth describes very clearly what are the different aspects of consciousness that need an explanation, and argues convincingly that consciousness is the result of our brain ability to model the world and predict the future including our own role in the unfolding sequence of events. Seth’s argument fits well with several other existing theories, including Baar’s Global Workspace Theory, probably the most popular and widely accepted proposal.
However, Seth makes a clear argument for the relevance of several factors that are not that explicit in other models and also makes important connections with other theories, including Damásio’s focus on the importance of emotions. In the process, Seth takes a stab at the idea that we will never be able to understand consciousness and, in particular, that the hard problem of consciousness will remain forever outside of our reach.
Overall, one is left with the idea that sometime, in the not too distant future, we will have a clear theory of what consciousness is, and how it is produced. Will that lead to artificial consciousness, to the creation of machines that are, at least in some ways, conscious? Here, Seth and I diverge, because Seth seems to shy away from the most natural conclusion that his book lead us into: consciousness is a phenomenon that results from a very specific way to process information about the world and systems that work in that way will, undoubtedly, be conscious, in one way or the other.
Extraterrestrial: The First Sign of Alien Life?
Avi Loeb is not exactly someone who one may call an outsider to the scientific community. As a reputed scholar and the longest serving chair of Harvard’s Department of Astronomy, he is a well-known and reputed physicist, with many years of experience in astrophysics and cosmology. It is therefore somewhat surprising that in this book he strongly supports an hypothesis that is anything but widely accepted in the scientific community: ʻOumuamua, the first interstellar object ever detected in our solar system may be an artifact created by an alien civilization.
We are not talking here about alien conspiracies, UFOs or little green men from Mars. Loeb’s idea, admirably explained, is that there are enough strange things about ʻOumuamua to raise the real possibility that it is not simply a strange rock and that it may be an artificial construct, maybe a lightsail or a beacon.
There are, indeed, several strange things about this object, discovered by a telescope in Hawaii, in October 2017. It was the first object ever discovered near the Sun that did not orbit our star; its luminosity changed radically, by a factor of about 10; it is very bright for its size; and, perhaps more strangely, it exhibited non‑gravitational acceleration as its orbit did not exactly match the orbit of a normal rock with no external forces applied other than the gravity of the Sun.
None of these abnormalities, per se, would be enough to raise eyebrows. But, all combined, they do indeed make for a strange object. And Loeb’s point is, exactly, that the possibility that ‘Oumuamua is an artifact of alien origin should be taken seriously by the scientific community. And yet, he argues, anything that has to do with extraterrestrial life is not considered serious science, leading to a negative bias and to a lack of investment in what should be one of the most important scientific questions: are we alone in the Universe? As such, SETI, the Search for Extra-Terrestrial Life, does not get the recognition and the funding it deserves. Paradoxically, other fields whose theories may never be confirmed by experiment nor have any real impact on us, such as multiverse based explanations of quantum mechanics or string theory, are considered serious fields, attract much more funding, and are more favorably viewed by young researchers.
The book makes for very interesting reading, both for the author’s positions about ‘Oumuamua itself and for his opinions about today’s scientific establishment.
Possible minds
John Brockman’s project of bringing together 25 pioneers in Artificial Intelligence to discuss the promises and perils of the field makes for some interesting reading. This collection of short essays lets you peer inside the minds of such luminaries as Judea Pearl, Stuart Russell, Daniel Dennett, Frank Wilczek, Max Tegmark, Steven Pinter or David Deutsch, to name only a few. The fact that each one of them contributed with an essay that is only a dozen pages long does not hinder the transmission of the messages and ideas they support. On the contrary, it is nice to read about Pearl’s ideas about causality or Tegmark’s thoughts on the future of intelligence in a short essay. Although the essays do not replace longer and more elaborate texts, they certainly give the reader the gist of the central arguments that, in many cases, made the authors well-known. Although the organization of the essay varies from author to author, all contributions are relevant and entertaining, whether they come from lesser-known artists or from famous scientists such as George Church, Seth Loyd, or Rodney Brooks.
The texts in this book did not appear out of thin air. In fact, the invited contributors were given the same starting point: Norbert Wiener’s influential book “The Human Use of Human Beings”, a prescient text authored more than 70 years ago by one of the most influential researchers in the field that, ultimately, originally coined as cybernetics ultimately led to digital computers and Artificial Intelligence. First published in 1950, Wiener’s book serves as the starting point for 25 interesting takes on the future of computation, artificial intelligence, and humanity. Whether you believe that the future of humanity will be digital or are concerned that we are losing our humanity, there will be something in this book for you.
Is the Universe a mathematical structure?
In his latest book, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality, Max Tegmark tries to answer what is maybe the most fundamental question in science and philosophy: what is the nature of reality?
Our understanding of reality has certainly undergone deep change in the last few centuries. From Galileo and Newton, to Maxwell, Einstein, Bohr and Heisenberg, Physics has evolved by leaps and bounds, as well as our understanding of the place of humans in the Universe. And yet, in some respects, we know little more than the ancient Greeks. Is the visible Universe all that exists? Could other universes, with different laws of physics, exist? Does the universe split into several universes every time a quantum observation takes place? Why is mathematics such a good model for physics (an old question) and could there exist other universes which obey different mathematical structures? These questions are not arbitrary ones, as their answers take us into the four levels of the multiverse proposed by Tegmark.
As you dive into it, the book takes us into an ever-expanding model of reality. Tegmark defines four level of multiverses: the first one consisting of all the (possibly infinite) spacetime of which we see only a ball with a radius of 14 billion light-years, since the rest is too far for light to have reached us; the second one which possibly holds other parts of spacetime which obey different laws of physics; a third one, implied by the many-worlds interpretation of quantum physics; and a fourth one, where other mathematical structures, different from the spacetime we know and love, define the rules of the game.
It is certainly a lot to take in, in a book that has less than 400 pages, and the reader may feel dizzy at times. But, in the process, Tegmark does his best at explaining what inflation is and why it plays such an important role in cosmology, how the laws of quantum physics can be viewed simply as an equation (the Schrödinger equation) describing the evolution of a point in Hilbert space, doing away with all the difficult-to-explain consequences of the Copenhagen interpretation, the difficulties caused by the measure problem, why is the space so flat, and many, many other fascinating topics in modern physics.
Since the main point of the book is to help is understand our place in this not only enormous Universe but unthinkably enormous multiverse, he brings us back to Earth (literally) with a few disturbing questions, such as:
• What is the role of intelligence and consciousness in this humongous multiverse?
• Why is this Universe we see amenable to life, in some places, and why have we been so lucky to be born exactly here?
• Shall one view oneself as a random sample of an intelligent being existing in the universe (the SSA, or Self-Sampling Assumption proposed by Bostrom in his book Anthropic Bias)
• If the SSA is valid, does it imply the Doomsday Argument, that it is very unlikely that humans will last for a long time because such a fact that would make it highly unlikely that I would have been born so soon?
All in all, a fascinating read, if at times is reads more like sci-fi than science!
Chinese translation of The Digital Mind
The Chinese translation of my book, The Digital Mind, is now available. For those who want to dust off their (simplified) Chinese, it can be found in the usual physical and online bookstores, including Amazon and Regrettably, I cannot directly assess the quality of the translation, you will have to decide for yourself. Or maybe you’d rather go for the more mundane English version, published by MIT Press, or the Portuguese one, published by IST Press.
You’re not the customer, you’re the product!
The attention that each one of us pays to an item and the time we spend on a site, article, or application is the most valuable commodity in the world, as witnessed by the fact that the companies that sell it, wholesale, are the largest in the world. Attracting and selling our attention is, indeed, the business of Google and Facebook but also, to a larger extent, of Amazon, Apple, Microsoft, Tencent, or Alibaba. We may believe we are the customers of these companies but, in fact, many of the services provided serve, only, to attract our attention and sell it to the highest bidder, in the form of publicity of personal information. In the words of Richard Serra and Carlota Fay Schoolman, later reused by a number of people including Tom Johnson, if you are not paying “You’re not the customer; you’re the product.
Attracting and selling attention is an old business, well described in Tim Wu’s book The Attention Merchants. First created by newspapers, then by radios and television, the market of attention came to maturity with the Internet. Although newspapers, radio programs, and television shows have all been designed to attract our attention and use it to sell publicity, none of them had the potential of the Internet, which can attract and retain our attention by tailoring the contents to each and everyone’s content.
The problem is that, with excessive customization, comes a significant and very prevalent problem. As sites, social networks, and content providers fight to attract our attention, they show us exactly the things we want to see, and not the things as they are. Each person lives, nowadays, in a reality that is different from anyone else’s reality. The creation of a separate and different reality, for each person, has a number of negative side effects, that include the creation of paranoia-inducing rabbit holes, the radicalization of opinions, the inability to establish democratic dialogue, and the diffiulty to distinguish reality from fabricated fiction.
Wu’s book addresses, in no light terms, this issue, but the Netflix documentary The Social Dilemma makes an even stronger point that customized content, as shown to us by social networks and other content providers is unraveling society and creating a host of new and serious problems. Social networks are even more worrying than other content providers because they create pressure in children and young adults to conform to a reality that is fabricated and presented to them in order to retain (and resell) their attention.
Decoding the code of life
We have known, since 1953, that the DNA molecule encodes the genetic information that transmits characteristics from ancestors to descendants, in all types of lifeforms on Earth. Genes, in the DNA sequences, specify the primary structure of proteins, the sequence of amino acids that are the components of the proteins, the cellular machines that do the jobs required to keep a cell alive. The secondary structure of proteins specifies some of the ways a protein folds locally, in structures like alpha helices and beta sheets. Methods that can determine reliably the secondary structure of proteins have existed for some time. However, determining the way a protein folds globally in space (its tertiary structure, the shape it assumes) has remained, mostly, an open problem, outside the reach of most algorithms, in the general case.
The Critical Assessment of protein Structure Prediction (CASP) competition, started in 1994, took place every two years since then and made it possible for hundreds of competing teams to test their algorithms and approaches in this difficult problem. Thousands of approaches have been tried, to some success, but the precision of the predictions was still rather low, especially for proteins that were not similar to other known proteins.
A number of different challenges have taken place over the years in CASP, ranging from ab-initio prediction to the prediction of structure using homology information and the field has seen steady improvements, over time. However, the entrance of DeepMind into the competition upped the stakes and revolutionized the field. As DeepMind itself reports in a blog post, the program AlphaFold 2, a successor of AlphaFold, entered the 2020 edition of CASP and managed to obtain a score of 92.4%, measured in the Global Distance Test (GDT) scale, which ranges from 0 to 100. This value should be compared with the value 58.9% obtained by AlphaFold (the previous version of this year’s winner) in 2018, and the 40% score obtained by the winner of the 2016 competition.
Structure of insulin
Even though details of the algorithm have still not been published, the information provided in the DeepMind post provides enough information to realize that this result is a very significant one. Although the whole approach is complex and the system integrates information from a number of sources, it relies on an attention-based neural network, which is trained end-to-end to learn which amino acids are close to each other, and at which distance.
Given the importance of the problem on areas like biology, medical science and pharmaceutics, it is to be expected that this computational approach to the problem of protein structure determination will have a significant impact in the future. Once more, rather general machine learning techniques, which have been developed over the last decades, have shown great potential in real world problems.
Novacene: the future of humanity is digital?
As it says on the cover of the book, James Lovelock may well be “the great scientific visionary of our age“. He is probably best known for the Gaia Hypothesis, but he made several other major contributions. While working for NASA, he was the first to propose looking for chemical biomarkers in the atmosphere of other planets as a sign of extraterrestrial life, a method that has been extensively used and led to a number of interesting results, some of them very recent. He has argued for climate engineering methods, to fight global warming, and a strong supporter of nuclear energy, by far the safest and less polluting form of energy currently available.
Lovelock has been an outspoken environmentalist, a strong voice against global warming, and the creator of the Gaia Hypothesis, the idea that all organisms on Earth are part of a synergistic and self-regulating system that seeks to maintain the conditions for life on Earth. The ideas he puts forward in this book are, therefore, surprising. To him, we are leaving the Anthropocene (a geological epoch, characterized by the profound effect of men on the Earth environment, still not recognized as a separate epoch by mainstream science) and entering the Novacene, an epoch where digital intelligence will become the most important form of life on Earth and near space.
Although it may seem like a position inconsistent with his previous arguments about the nature of life on Earth, I find the argument for the Novacene era convincing and coherent. Again, Lovelock appears as a visionary, extrapolating to its ultimate conclusion the trend of technological development that started with the industrial revolution.
As he says, “The intelligence that launches the age that follows the Anthropocene will not be human; it will be something wholly different from anything we can now conceive.”
To me, his argument that artificial intelligence, digital intelligence, will be our future, our offspring, is convincing. It will be as different from us as we are from the first animals that appeared hundreds of millions ago, which were also very different from the cells that started life on Earth. Four billion years after the first lifeforms appeared on Earth, life will finally create a new physical support, that does not depend on DNA, water, or an Earth-like environment and is adequate for space.
Could Venus possibly harbor life?
Two recently published papers, including one in Nature Astronomy (about the discovery itself) and this one in Astrobiology (describing a possible life cycle), report the existence of phosphine in the upper atmosphere of Venus, a gas that cannot be easily generated by non-biological processes in the conditions believed to exist in that planet. Phosphine may, indeed, turn out to be a biosignature, an indicator of the possible existence of micro-organisms in a planet that was considered, up to now, barren. Search for life in our solar system has been concentrated in other bodies, more likely to host micro-organisms, like Mars of the icy moons of outer planets.
The findings have been reported in many media outlets, including the NY Times and The Economist, raising interesting questions about the prevalence of life in the universe and the possible existence of life in one of our nearest neighbor planets. If the biological origin of phosphine were to be confirmed, it would qualify as the discovery of the century, maybe the most important discovery in the history of science! We are, however, far from that point. A number of things may make this finding another false alarm. Still, it is quite exciting that what has been considered a possible sign of life has been found so close to us and even a negative result would increase our knowledge about the chemical processes that generate this compound until now believed to be a reliable biomarker.
This turns out to be a first step, not a final result. Quoting from the Nature Astronomy paper:
Even if confirmed, we emphasize that the detection of PH3 is not robust evidence for life, only for anomalous and unexplained chemistry. There are substantial conceptual problems for the idea of life in Venus’s clouds—the environment is extremely dehydrating as well as hyperacidic. However, we have ruled out many chemical routes to PH3, with the most likely ones falling short by four to eight orders of magnitude (Extended Data Fig. 10). To further discriminate between unknown photochemical and/or geological processes as the source of Venusian PH3, or to determine whether there is life in the clouds of Venus, substantial modelling and experimentation will be important. Ultimately, a solution could come from revisiting Venus for in situ measurements or aerosol return.
The Book of Why
Correlation is not causation is a mantra that you may have heard many times, calling attention to the fact that no matter how strong the relations one may find between variables, they are not conclusive evidence for the existence of a cause and effect relationship. In fact, most modern AI and Machine Learning techniques look for relations between variables to infer useful classifiers, regressors, and decision mechanisms. Statistical studies, with either big or small data, have also generally abstained from explicitly inferring causality between phenomena, except when randomized control trials are used, virtually the unique case where causality can be inferred with little or no risk of confounding.
In The Book of Why, Judea Pearl, in collaboration with Dana Mackenzie, ups the ante and argues not only that one should not stay away from reasoning about causes and effects, but also that the decades-old practice of avoiding causal reasoning has been one of the reasons for our limited success in many fields, including Artificial Intelligence.
Pearl’s main point is that causal reasoning is not only essential for higher-level intelligence but is also the natural way we, humans, think about the world. Pearl, a world-renowned researcher for his work in probabilistic reasoning, has made many contributions to AI and statistics, including the well known Bayesian networks, an approach that exposes regularities in joint probability distributions. Still, he thinks that all those contributions pale in comparison with the revolution he speared on the effective use of causal reasoning in statistics.
Pearl argues that statistical-based AI systems are restricted to finding associations between variables, stuck in what he calls rung 1 of the Ladder of Causation: Association. Seeing associations leads to a very superficial understanding of the world since it restricts the actor to the observation of variables and the analysis of relations between them. In rung 2 of the Ladder, Intervention, actors can intervene and change the world, which leads to an understanding of cause and effect. In rung 3, Counterfactuals, actors can imagine different worlds, namely what would have happened if the actor did this instead of that.
This may seem a bit abstract, but that is where the book becomes a very pleasant surprise. Although it is a book written for the general public, the authors go deeply into the questions, getting to the point where they explain the do-calculus, a methodology Pearl and his students developed to calculate, under a set of dependence/independence assumptions, what would happen if a specific variable is changed in a possibly complex network of interconnected variables. In fact, graphic representations of these networks, causal diagrams, are at the root of the methods presented and are extensively used in the book to illustrate many challenges, problems, and paradoxes.
In fact, the chapter on paradoxes is particularly entertaining, covering the Monty Hall, Berkson, and Simpson’s paradoxes, all of them quite puzzling. My favorite instance of Simpson’s paradox is the Berkeley admissions puzzle, the subject of a famous 1975 Science article. The paradox comes from the fact that, at the time, Berkeley admitted 44% of male candidates to graduate studies, but only 35% of female applicants. However, each particular department (departments decide the admissions in Berkeley, as in many other places) made decisions that were more favorable to women than men. As it turns out, this strange state of affairs has a perfectly reasonable explanation, but you will have to read the book to find out.
The book contains many fascinating stories and includes a surprising amount of personal accounts, making for a very entertaining and instructive reading.
Note: the ladder of causation figure is from the book itself. |
e8c02d9f2e813fd2 | The first Josephson phase-battery device
Image: Andrea Iorio.
In science we try to be very precise with the wording of our texts. One of the consequences of this is that we are not afraid of repeating some words in a paragraph, or even in a sentence, if that contributes to avoid possible misunderstandings, even though it may be not recommended stylistically. Another feature of scientific communication is its abhorrence of polysemy, that is why definitions abound in any scientific text. And, as we are going to talk about a ‘phase battery’, definitions are badly needed.
If I say battery, you may think about the battery in your car, and that would be fine as an example. The battery in your car is able to produce a voltage of 12 V so that the electric devices in the car can work. But, actually, an electric battery is a number of electric cells joined together; in the case of the car battery, it usually consists of 6 cells connected in series. A battery is just a number of things of the same type joined together.
A phase could be a homogeneous part of a heterogenous system, but this is not the one we are interested now. The phase we are talking about is the stage that a periodic motion has reached, usually by comparison with another such motion of the same frequency. If we represent those periodic motions as waves, they will be in phase if their maximum and minimum values occur at the same instants; otherwise, there is said to be a phase difference.
Putting both words together, phase battery, even though we have given their definitions, still makes no sense. We need something more.
We know that quantum systems are described by sets of wave functions, mathematical expressions involving the coordinates of a particle in space, that are solutions of the Schrödinger equation. There we have our waves. Then, if a classical battery converts chemical energy into a persistent voltage bias that can power electronic circuits, a phase battery is a quantum device that provides a persistent phase bias to the wave function of a quantum circuit. Phase batteries represent a key element for quantum technologies based on phase coherence. Now, a team of researchers demonstrate 1 a phase battery in a hybrid superconducting circuit.
Josephson phase-battery device. a) Conceptual scheme of a Josephson phase battery composed of an InAs nanowire (red) embedded between two superconducting poles (blue) converting the spin polarization of surface unpaired spins (yellow) into a phase bias. b) Schematic illustration of the hybrid InAs-nanowire–aluminium-SQUID interferometer used to quantify the phase bias φ0 provided by the two Josephson junctions (in red).
At the base of phase-coherent superconducting circuits is the Josephson effect: a quantum phenomenon describing the flow of a supercurrent in weak links between two superconductors. When we approximate two superconducting materials at low temperature, so that they are only separated by a very thin layer (less than 10 nanometres thick) of an insulating material, some new and very interesting electrical effects can be observed. First, a supercurrent, a current with zero resistance, can flow through the barrier. If this current exceeds a critical value, this conductivity is lost; the barrier then only allows the small current flow due to the tunnel effect and a voltage develops across the junction. If we apply a magnetic field below the critical current value we will find that the net current through the barrier depends on the magnetic field that we are applying. If the field increases beyond a critical value the supercurrent vanishes. A junction like the one we have just described is called a Josephson junction.
But, what if instead of an insulating material we use a conducting metal in a Josephson junction? In this case superconducting correlations are induced in the normal metal due to the proximity effect. At the interface between a normal conductor and a superconductor, charge is transferred from the normal conductor to the superconductor.
The point is that the Josephson current is intimately connected to the macroscopic phase difference between the two superconductors via the so-called current–phase relationship. The implementation of a phase battery is prevented by symmetry constraints, either time-reversal or inversion, which impose a rigidity on the superconducting phase, a universal constraint valid for any quantum phase. It follows that, if both time-reversal and inversion symmetries could be broken, a finite phase shift can be induced. A junction like this, a φ0-junction, will generate a constant phase bias in an open circuit configuration, while when inserted into a closed superconducting loop it will induce a so-called anomalous Josephson current.
Lateral hybrid junctions made of materials with a strong spin–orbit interaction or topological insulators are ideal candidates to engineer Josephson φ0-junctions. The new phase battery (see figure) consists of a Josephson junction made of an indium arsenide (InAs) nanowire (in red) embedded (proximitized) between two aluminium (Al) superconducting poles (in blue).
The researchers find that the ferromagnetic polarization of the unpaired-spin states is efficiently converted into a persistent phase bias across the wire, leading to the anomalous Josephson effect. The application of an external in-plane magnetic field and, thereby, achieve a continuous tuning of the phase. Or, in other words, the quantum phase battery can be charged and discharged.
This quantum element, providing a controllable and localized phase bias, can find key applications in different quantum circuits such as an energy tuner for superconducting flux and hybrid qubits, or a persistent multi-valued phase-shifter for superconducting quantum memories as well as superconducting rectifiers.
1. Elia Strambini, Andrea Iorio, Ofelia Durante, Roberta Citro, Cristina Sanz-Fernández, Claudio Guarcello, Ilya V. Tokatly, Alessandro Braggio, Mirko Rocci, Nadia Ligato, Valentina Zannier, Lucia Sorba, F. Sebastián Bergeret and Francesco Giazotto (2020) A Josephson phase battery Nature Nanotechnology doi: 10.1038/s41565-020-0712-7
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Quantum Physics I
PHYS 3111
First of a two-quarter sequence. The Schrödinger equation: interpretation of wave functions; the uncertainty principle; stationary states; the free particle and wave packets; the harmonic oscillator; square well potentials. Hilbert space: observables, commutator algebra, eigenfunctions of a Hermitian operator; the hydrogen atom and hydrogenic atoms. Prerequisites: PHYS 2252, PHYS 2260, PHYS 2556, PHYS 3612 and MATH 2070. |
ccfe4b27708228df | Wednesday, September 7, 2016
Quantum God (short story)
(link to pdf version
(link to Italian version, translation by Erica Mannoni)
2033 AD. The entire population of the planet was watching, most of them through the eyes of the media, waiting for Lord Q to perform a miracle and save the world. Thousands of people gathered around his tent, meditating, praying, praising him, and hoping for the miracle. The asteroid was heading toward the Earth. All previous attempts to destroy the asteroid failed, because it was a black hole. It was detected only by the way it bent the light and the trajectories of other asteroids in the Solar System. So the asteroid continued undisturbed to threaten the Earth, and Quentin, named by his followers Lord Q, was the only hope.
* * *
Roxanne joined Quentin’s group one year before, not as a believer, but as one of the last skeptics alive, by now a dying species. In the previous decades, scientific and technological progress continued to hunt God into the farthest and most obscure explanatory gaps, into oblivion. Until the emergence of Lord Q three years earlier, when everything was turned upside-down. Since then, he performed the most scientifically incredible miracles, normally attributed to a deity. Roxanne, a reputed physicist with a hobby of debunking pseudoscience, received a grant from a philanthropist who asked her to either find a scientific explanation for Quentin’s miracles, or prove that they are authentic.
When she joined the group, the believers disliked her for her skepticism, which remained unchanged even now, a year later. The only reason they tolerated her was because Quentin seemed to have a strange affection for her. She was allowed to be in his proximity all the time, and this made them dislike her even more, but they had to accept her. Quentin liked her, and was continuously amused by the suspicious look in her eyes, which was visible even when she was surprised by his miracles. For a year she followed him everywhere, witnessed him healing people, stopping natural catastrophes, wars, crime, and bringing back faith. She even saw him bringing back to life the president of the United States, killed by a rare form of cancer. But she continued to say that there must be a scientific explanation for everything.
Quentin never ceased to be intrigued by her disbelief and continued to watch her reaction as he performed his miracles. Once, he started to make flowers grow up out of nowhere and blossom in seconds, covering every piece of ground where Roxanne stepped. She was surprised, she blushed, and she told him that this is harassment. He didn’t know whether she was joking, so he stopped. It would have been the easiest thing for him to make her fall in love with him or even become a believer, but he didn’t want it to be like this. He loved her for being independent, and he would have never traded the chance to see her surprised and at the same time unimpressed by his powers for having an obedient Roxanne in place, a fervent adorer like the rest of his followers.
* * *
As Roxanne was waiting, worried to see what Quentin will do about the asteroid, Tom approached her.
– He will make it, don’t worry, he said.
– I know, she replied.
Tom was the first and only friend Roxanne made among Quentin’s followers. He used to be the leader of the third group of scientists sent by the James Randi Educational Foundation. The group tried to debunk Quentin’s miracles and find mistakes in the reports of the previous two groups, which declared that this was indeed the first authentic miracle recorded by the foundation. He saw no other choice but to accept Quentin’s miracles as true. The Foundation had to award the Psychic Prize for the first time in the history. They offered Quentin one billion dollars, which was the value of the prize at that time. Quentin donated the money to charity, of course. Tom became a believer, and in fact the most ardent one, given that he really looked for the smoke and mirrors and didn’t find any.
During one of their first conversations, Tom told Roxanne that he believed that all these miracles can be explained if Quentin had the ability to control quantum probabilities.
The behavior of particles is governed by quantum mechanics, which is very different from what we observe in our day-to-day life where things behave more like in classical mechanics. According to quantum mechanics, there is always an infinitesimally small probability for something apparently impossible for us to happen even in the real life. In the quantum world, if you observe that a particle – for example an atom – is in a small region of space, at a later time there is a small, but non-zero chance to find it in any other region of space. The reason is that quantum uncertainty makes the particle you know as being in a certain place to have an undetermined velocity, and therefore be able to move anywhere. Hence, at a later time, the particle will potentially be in all places simultaneously. When you observe it again, you will find it in one of these potential places. You can never know where it will be, only know the probability to find it in a given place. This probability is given by a formula called the Born rule. The probability to find the particle in a given place is small, nearly infinitesimally small, but not zero. The same works for more particles or atoms. It’s true that the probabilities become smaller and smaller as the number of involved atoms increases, but it never truly vanishes.
So Tom told Roxanne that he thinks Quentin performs his miracles by selecting which of the potential positions of a particle becomes true. If he can make a particle be where he wants, he can move objects. He can rearrange matter at will. Roxanne remembered that the neuroscientist Adam Hobson used to entertain similar ideas some years ago, but she always considered him a crackpot. She said:
– This happens only for quantum measurements, while Quentin, through his senses, makes classical observations, just like any of us. But, given that any observation we make is eventually a quantum one, maybe your hypothesis is true. However, the probability to control like this even a grain of sand is almost zero.
Tom said:
– Well, the chances are one in a billion of billions of billions… whatever, let’s just say a chance in a gazillion – but Quentin seems to bring those odds into existence.
– But even if he could do this, how can he influence larger objects, which behave classically due to decoherence?
Tom said:
– Decoherence suppresses the probability that the object behaves in a quantum way, but that probability never becomes truly zero. So there is always a very small chance that even larger objects behave in a quantum way, and apparently Quentin has a way to make this chance happen.
Roxanne replied that quantum probabilities are just those the Born rule says they are, and she doesn’t believe anyone can really break this law even a bit. So she can’t accept Tom’s suggestion, especially since it would mean Quentin breaking them. Tom said that for him the alternative is even worse because otherwise Quentin would break other laws, which are exact.
– Forced to choose between an exact physical law and a probabilistic law, Tom said, I would choose to sacrifice the probabilistic one.
– What about the many-worlds interpretation? she asked.
According to the many-worlds interpretation, every possible alternative result of a quantum observation is realized in an alternative world. This way, all possibilities already existing before the observation continue to exist in independent worlds, as if the world splits in many alternative histories.
Roxanne continued:
– Assuming the many-worlds interpretation is true, if Quentin’s miracles are explained because he controls the probabilities, this means that in the vast majority of the alternative worlds he doesn’t control them, just like any of us. And even for us, there is a very tiny chance that the possibility we wish becomes true, but that chance is so small, that it practically never happens. And even if miracles are just very improbable but still possible events, the Born rule has to remain valid in each of the alternative worlds. So the chances that Quentin remains in a world in which he always gets to make his miracles are the same – one in a gazillion.
Tom said he wants to think about this. Next day he said:
– What if he suppresses the possibility of the other worlds when he controls the probabilities?
Roxanne said she has to think about it.
Together, Roxanne and Tom analyzed every miracle made by Quentin, and indeed found that these could be achieved if Quentin had the ability to control quantum probabilities. Healing people, levitating, moving objects, all these seemed to them plausibly explainable if he would really control the quantum probabilities for the particles constituting the objects. But she was still not satisfied, she wanted to know how he does all these, and whether he really breaks the Born rule. She was sure that there must be a better explanation.
* * *
In the last minutes before the asteroid was about to hit Earth, Quentin got out of his tent holding a TV set. He put the TV on a table as the TV anchor was reporting the most recent news about the asteroid. Quentin sat on the grass and started to meditate. After several minutes, his peaceful face, his entire body started to glow. He began to levitate. He raised his eyes to the sky, then his hand, and smiled warmly. Shortly after, the anchor reported in an explosion of joy that the black hole changed its place, and it was no longer a threat to Earth. After a worldwide celebration, life on Earth continued to exist as before.
* * *
A few weeks later, one night, Quentin was walking with Roxanne. After a full day of prodding him with all kinds of devices, she kept asking him all sorts of questions. He told her with a comforting voice to relax and just enjoy the night. Suddenly, she saw the stars moving in the sky, until they formed the image of her face. Really terrified, she yelled:
– Why did you do that?
Confused, he asked her what the problem was. She said:
– You just moved thousands of stars to impress me by drawing my portrait!
– So what? he said.
– You probably just killed dozens of civilizations!
She looked again in the sky, and the stars were back to their usual positions. He laughed:
– It was an illusion. I didn’t rearrange the stars in the sky, apparently I just bent the light rays coming from them. Or maybe I moved them back, I’m not sure…
They laughed, but she was still frightened.
* * *
For days, Roxanne kept studying Quentin with various high-tech devices. The money was not a problem for her sponsor. She scanned him, monitored his brain activity, recorded everything, and sent the data to specialized laboratories, for more thorough analysis. She found that he had a device implanted in his brain, following a car crash that happened three years earlier. Quentin refused to talk about the implant. But the implant seemed to do nothing relevant that would explain his abilities. It was simply a device that monitored his brain activity, collecting data from a number of places on his cortex. The implant was also stimulating some regions of his brain from time to time, but nothing relevant.
She also found something that surprised her even more: everywhere in Quentin’s body, there were billions of tiny spheres. She didn’t see them initially, they were too small, but she eventually found them after a more detailed analysis of Quentin’s tissues. She collected several of them and sent them to a laboratory. Just like the implant in his brain, the spheres seemed to be useless, or at least they didn’t exchange energy or information with the body at all, so she was very curious about their role.
When the result came back, she was perplexed. The tiny spheres were nanobombs, this was their only functionality. Remote controlled nanobombs, programmed to blow if they receive a certain signal, but obviously never detonated because that signal was never sent. Why on Earth would he have an implant that collects brain activity and never does anything with it, and why have billions of nanobombs also doing nothing at all? Anyway, the presence of nanobombs prevented her from trying to disable Quentin’s brain implant to see if it was the source of his powers.
* * *
Roxanne told Quentin what she found, and insisted that he should give her more explanations. He said that he will tell her more, if she promised to keep the secret.
– I have something to confess, he said. I am not the first one with these powers. Professor Adam Hobson, my uncle, who saved me after the car crash, he used to have them as well.
– You mean Adam Hobson, the guy with that crazy theory about the quantum brain, who disappeared a few years ago and was never found? Roxanne said.
Quentin told her how he had the car accident, and Adam saved his life with his highly advanced surgical robots. He said he had to implant a device in Quentin’s brain. Then Adam personally conducted the recovery therapy, teaching Quentin how to gain control of his body again and how to control his thoughts. After the recovery, Adam revealed more to Quentin. He said that both of them had implants, and that the implants allowed their minds to control matter.
Quentin continued:
– Uncle Adam taught me how to make miracles, by wishing them and by thinking at the changes I should be observing in the world after every miracle. He told me that he can already do this, and I will soon be able to do it too, and that both of us are godlike beings. He said there is no room for two gods, and that he will soon leave this world for a better one. He also said that I will leave this world soon too, and to tell everyone who cares about me that I will go to a better world. Then Uncle Adam activated my implant, and then he vanished in a bright explosion. It was the last time I saw him. I don’t know how this device functions. I just wish for things to happen, visualize what to expect once they happen until I have that feeling that they will do, and then they happen exactly as I visualized them. I learned that this way I can do pretty much anything.
Roxanne said:
– But this doesn’t make sense. The implant doesn’t do anything, it just collects your brain activity and stimulates it to make you feel happy. I just don’t understand…
* * *
Quentin was doing his morning meditation, when Roxanne came with a desperate look on her face, yelling from afar:
– You have to stop doing any miracle right now! This is gonna kill you!
– What?! said Quentin. What do you mean? What happened?
– You know Tom’s hypothesis that the way you do your miracles is by controlling quantum probabilities? Well, you can’t control them, nobody can!
– OK… so what? Quentin said.
– I know what’s in your head, what that implant does to you, she said.
– Well, good to know you finally got it. I’m all ears. Sit down here with me…
Roxanne caught her breath, but didn’t sit on the grass near Quentin. Instead she kept circling him, explaining:
– Whenever there’s a choice between more quantum alternatives, new worlds are created, in which each of these possibilities become reality. You can’t control which world is ours, you exist in all of them. Including in those in which your miracle doesn’t work. But when you make a miracle, you visualize the desired result, and your brain implant collects this information from your brain. Then, it compares it with what you observe afterwards. If your wish becomes real, then nothing happens.
– I don’t get it, Quentin said. Nothing happens, so the device does nothing. Then how can this explain anything?
Roxanne grabbed his shoulders and looked at him frantically:
– But if your wish doesn’t come true, which is almost always the case, then the implant detonates the billions of bombs in your body. You explode into a bright light, and you die.
– I never died…
– You will always find yourself, of course, in a world in which you are not killed, hence where your wish came true. Your implant is a quantum suicide device, inspired by the quantum suicide thought experiment proposed in the eighties to test the many-worlds interpretation. Gazillions of worlds are created whenever you make a miracle, and gazillions of copies of you are killed in all of these worlds! Gazillions of copies of us are left in tears… In all worlds, except in those very, very, very rare in which your wish comes true!
Seeing her crying, Quentin dispersed the clouds in the sky and made out of thin air a rain of flower petals.
– See, nothing happened, dummy…
Cristi Stoica, May 17, 2016
Tuesday, May 3, 2016
Are Single-World Interpretations of Quantum Theory Inconsistent?
A recent eprint caught my atention: Single-world interpretations of quantum theory cannot be self-consistent by Daniela Frauchiger and Renato Renner. In the abstract we read
The article contains an experiment based on Wigner's friend thought experiment, from which is deduced in a Theorem that there cannot exist a theory T that satisfies the following conditions:
(QT) Compliance with quantum theory: T forbids all measurement results that are forbidden by standard [non-relativistic] quantum theory (and this condition holds even if the measured system is large enough to contain itself an experimenter).
(SW) Single-world: T rules out the occurrence of more than one single outcome if an
experimenter measures a system once.
(SC) Self-consistency: T's statements about measurement outcomes are logically consistent (even if they are obtained by considering the perspectives of different experimenters).
A proof of the inconsistency of Bohmian mechanics (discovered by de Broglie and rediscovered and further developed by David Bohm) would already be a big deal, because despite being rejected with enthusiasm by many quantum theorists, it was never actually refuted, neither by reasoning, nor by experiment. Bohmian mechanics is based on two objects: the pilot-wave, which is very similar to the standard wavefunction and evolves according to the Schrödinger equation, and the Bohmian trajectory, which is an integral curve of the current associated to the Schrödinger equation. While one would expect the Bohmian trajectory to be the trajectory of a physical particle, all observables and physical properties, including mass, charge, spin, properties like non-locality and contextuality, are attributes of the wave, and not of the Bohmian particle. This explains in part why BM is able to satisfy (QT). The pilot-wave itself evolves unitarily, not being subject to the collapse. Decoherence (first discovered by Bohm when developing this theory) plays a major role. The only role played by the Bohmian trajectory seems (to me at least) to be to point which outcome was obtained during an experiment. In other words, the pilot-wave behaves just like in the Many-Worlds Interpretation, and the Bohmian trajectory is used only to select a single-world. But the other single-worlds are equally justified, once we accepted all branches of the pilot-wave to be equally real, and the Bohmian trajectory really plays no role. I will come back later with a more detailed argumentation of what I said here about Bohmian mechanics, but I repeat, this is not a refutation of BM, rather some arguments coming from my personal taste and expectations of what a theory of QT should do. Anyway, if the result of the Frauchiger-Renner paper is correct, this will show not only that the Bohmian trajectory is not necessary, but also that it is impossible in the proposed experiment. This would be really strange, given that the Bohmian trajectory is just an integral curve of a vector field in the configuration space, and it is perfectly well defined for almost all initial configurations. This would be a counterexample given by Bohmian mechanics itself to the Frauchiger-Renner theorem. Or is the opposite true?
But when you read their paper you realize that any theory compatible with standard quantum theory (which satisfies QT and SW) has to be inconsistent, including therefore standard QT itself. Despite the fact that the paper analyzes all three options obtained by negating each of the three conditions, it is pretty transparent that the only alternative has to be Many-Worlds. In fact, even MW, where each world is interpreted as a single-world, seems to be ruled out. If correct, this may be the most important result in the foundations of QT in decades.
Recall that the Many-Worlds Interpretation is considered by most of its supporters as being the logical consequence of the Schrödinger equation, without needing to assume the wavefunction collapse. The reason is that the unitary evolution prescribed by the Schrödinger equation contains in it all possible results of the measurement of a quantum system, in superposition. And since each possible result lies in a branch of the wavefunction that can no longer interfere with the other branches, there will be independent branches behaving as separate worlds. Although there are some important open questions in the MWI, the official point of view is that the most important ones are already solved without assuming more than the Schrödinger equation. So perhaps for them this result would add nothing. But for the rest of us, it would really be important.
My first impulse was that there is a circularity in the proof of the Frauchiger-Renner theorem: they consider that it is possible to perform an experiment resulting in the superposition of two different classical states of a system. Here by "classical state" I understand of course still a quantum state, but one which effectively looks classical, as a measurement device is expected to be before and after the measurement. In other words, their experiment is designed so that an observer sees a superposition of a dead cat and an alive one. Their experiment is cleverly designed so that two such observations of "Schrödinger cats" lead to inconsistencies, if (SW) is assumed to be true. So my first thought was that this means they already assume MWI, by allowing an observer to observe a superposition between a classical state that "happened" and one that "didn't happen".
But the things are not that simple, because even if a quantum state looks classical, it is still quantum. And there seem to be no absolute rule to forbid the superposition of two classical states. Einselection (environment-induced superselection) is a potential answer, but so far it is still an open problem, and at any rate, unlike the usual superselection rules, it is not an exact rule, but again an effective one (even if it would be proven to resolve the problem). So the standard formulation of QT doesn't actually forbid superpositions of classical states. Well, in Bohr's interpretation there are quantum and there are classical objects, and the distinction is unbreakable, so for him the extended Wigner's friend experiment proposed by Frauchiger and Renner would not make sense. But if we want to include the classical level in the quantum description, it seems that there is nothing to prevent the possibility, in principle, of this experiment.
Reading the Frauchiger-Renner paper made me think that there is an important open problem in QT, because it doesn't seem to prescribe how to deal with classical states:
Does QT allow quantum measurements of classical (macroscopic) systems, so that the resulting states are non-classical superpositions of their classical states?
I am not convinced that we are allowed to do this even in principle (in practice seems pretty clear it is impossible), but also I am not convinced why we are forbidden. To me, this is a big open problem. Can the answer to this question be derived logically from the principles of standard QT, or should it be added as an independent, new principle?
My guess is that we don't have a definitive solution yet. It is therefore a matter of choice: those accepting that we are allowed to perform any quantum measurements on classical states, perhaps already accept MWI, and consider that it is a logical consequence of the Schrödinger equation. Those who think that one can't perform on classical states quantum measurements that result in Schrödinger cats, will of course object to the result of the paper of Frauchiger and Renner and consider its proof circular.
I will not rush with the verdict about the Frauchiger-Renner paper. But I think at least the open problem I mentioned deserves more attention. Nevertheless, if their result is true, it will pose a big problem not only to Bohmian mechanics, but also to standard QT. And also to my own proposed interpretation, which is based on the possibility of a single-world unitary solution of the Schrödinger equation (see my recent paper On the Wavefunction Collapse and the references therein).
Monday, May 2, 2016
An attempt to refute my Big-Bang singularity solution
Sunday, March 27, 2016
Faster than light signaling leads to paradoxes
You may have encountered statements like this one made by Sabine:
This is correct. Special relativity implies that, if faster than light signaling would be true, you would be able to signal to your own past, and this can lead to paradoxes. Here I will explain how exactly this can happen. This is rather elementary special relativity stuff, but I realized there is much confusion around it. First, I never saw a precise scenario in which faster than light (FTL) signaling can be used to signal back to your own past, so I will give one. Second, I have the feeling that when people make statements like this,
• they either refer to the fact that, if an observer A sends FTL signals in her own future, for another observer B it may look like sending in back in time, in B's reference frame, as in this figure:
Orange lines represent light cones, blue represent timelike curves (observers), red represents the proper space of an observer, and green represents FTL signals. While the picture represents the proper space of A as a horizontal red line, the proper space of B is oblique, due to the Lorentz transformation (relativity of simultaneity).
The first scenario is not that paradoxical, because observer B can always reinterpret the signal from A to B as a signal going in his own future, from B to A. But even in this case, we will have the problem of who actually created the message in the first place.
• or they refer to examples where the observer sends an FTL signal toward her own past, as in this figure:
The second scenario is the usual example of causality violation due to FTL you will find, but is refutable on the grounds that you are not allowed to send signals directly to your own past, or to receive signals directly coming from your own future.
Here is how FTL signaling would imply that one can signal back in time, using only signals sent in the future and received from the past, with respect to the proper reference frame:
The inertial observer A accelerates away from B, then sends an FTL signal at t₀. Observer B receives it at t'₀ in his proper time, then accelerates away from observer A, then sends it back, at t'₁. Observer A receives the signal at t₋₁, where t₋₁< t₀.
So indeed FTL implies signaling back in your own past, even if FTL signals are sent only to the proper future and received only from the proper past.
Let us see how this allows paradoxes. Suppose that earlier A and B agreed on the following: if A receives the message "Yes", she sends the message "No", and if she receives "No", she sends "Yes". If B receives a signal, he just resends it without changing. Then, we have a paradox: does A send the message "Yes", or "No"? It is similar to the liar paradox, since if she sends "Yes", then she receives "Yes", so she sends "No", and so on. But it is also like grandfather's paradox, because B can send instead of a message, a killing FTL ray, to kill A or her grandfather before she was born.
So far there is no evidence of FTL signaling, except for some misunderstandings of the EPR "paradox". I don't know either of a fundamental physical law which prevents it, given that tachyonic solutions are mathematically consistent, both in special relativity, and in quantum field theory. But as we have seen, FTL would lead to time travel paradoxes.
Saturday, February 13, 2016
Gravitational waves, evidence of the fourth dimension of spacetime
Most of the headlines are right: gravitational waves are a long-known prediction of General Relativity, and their detection show that the theory is correct. I waited a bit to see if an important consequence of this fact will be uncovered, but it seems it did not, so let me tell you: This experiment refutes a great deal of alternatives to General Relativity proposed in the last decades. You perhaps already noticed that many physicists brag on social networks or even in online articles that the detection of gravitational waves confirmed not necessarily GR, but also the alternatives to GR they endorse. But in fact this experimental result refutes those alternative theories in which the background metric of spacetime is fixed, as well as those in which space is a three-dimensional thing that is not part of a four-dimensional spacetime, as in GR is. I will discuss first the latter. Many relativists would say that such theories were already refuted, but if you talk with a supporter of such a theory, you will hear that it is not necessarily so. The idea of a 3-dimensional space still could be defended, with the price of complicating the things. But in my opinion, LIGO just put the last nail in the coffin of such theories. Because gravitational waves are waves of spacetime, and not of space. They are waves of the Weyl curvature tensor, which simply vanishes in a space with less than four dimensions!
The number of those trying to replace GR with other theories increased very much lately. The main reasons may be that they don't know how to handle singularities, or that they don't know how to enforce to gravity the few methods we know to quantize fields, so they come up with alternative theories. While I don't think it is easy to replace GR with something that explain as much starting from as little as GR does, I agree that these alternative should be explored (by others, of course). Related to whether there is a 3-dimensional space or a 4-dimensional spacetime, you can find reasons to doubt the fourth dimension too. First, even Galilean space and time can be joined in a four-dimensional spacetime, but not as tight as in Relativity. In Relativity, indeed, Lorentz transformations mix the time and space directions, leading to length contraction and time dilation, but some think that these are sort of due to the perspective of the observer, without needing a fourth dimension. In addition, many quantities become unified in the four-dimensional spacetime, such as energy and momentum, electric and magnetic fields etc. But maybe these are all just circumstantial evidence of the fourth dimension. You can take any theory and make it satisfy some four-dimensional transformations. Especially since the evolution equations are hyperbolic, you can do this. Also, you can express any equation in Physics in curvilinear coordinates, and this doesn't mean four dimensions, neither that the invariance to diffeomorphisms means something physical. So people cooked up or even revived various alternatives to GR, in which three-dimensional space is not part of a spacetime. If such a theory does not include curvature, it will not predict gravitational waves. Also, if it admits curvature, but only of the three-dimensional space, nothing in four dimensions, it still doesn't predict gravitational waves out of this curvature. So now the proponents of alternative to GR will have to adjust their theories. Maybe some predict naturally some sort of gravitational waves, but most don't, so they will put the waves by hand. The Cotton tensor, which is somewhat analogous to the Weyl tensor in three dimensions, because its vanishing means conformal flatness, is believed sometimes to give the gravitational waves. But the Cotton tensor vanishes in vacuum, where the Ricci tensor vanishes too. So this can't give gravitational waves in three-dimensional spacetime.
What about theories with more dimensions? For instance, Kaluza theory is an extension of GR to 5 dimensions, which is able to obtain the sourceless electromagnetic field from the extra dimension. You can also obtain other gauge theories as Kaluza-type theory. Such modification predict gravitational waves too.
What about String Theory? It is said that String Theory includes GR, so it must include gravitational waves too, isn't it? But the reason why is said to include GR is because it contains closed strings, which have spin 2, and they are identified with the still hypothetical gravitons (not even predicted by GR alone) just because they have spin 2. But if your theory has spin-2 particles, even if you call them gravitons, it doesn't mean you have included GR. String Theory usually works on fixed background, which usually is flat, or with constant curvature as in the anti-de Sitter spacetime. I am not aware of a successful way to include GR in String Theory such that gravity is an effect of spacetime curvature. If this can be done, can it predict gravitational waves in a natural way? Can it even include GR in a natural way?
To my surprise, the advocates of theories which don't have dynamical background, or are based on three-dimensional space, didn't take the chance to predict that there are no gravitational waves, as their theories imply. They should have done this, and they should have waited for the confirmation of their prediction by LIGO. My guess is that maybe they doubted that GR will be refuted - nobody wants to make predictions which contradict GR in regimes that can be experimentally verified. Whenever we could test the predictions of GR, they were always confirmed, so I think not even those supporting alternative theories actually believed that it will be refuted this time. So I guess that's why they didn't say that their theory predicts no gravitational waves, and that they really think that LIGO will show there are. Instead, now you can see that some claim that gravitational waves confirm their theories too. Like for examples they are waves of space alone, and not of spacetime, which is not true, unless you put them in your theory by hand (while in GR they are just there, not a mobile or replaceable part). So I expect to see a lot of papers in which it is explain that their theory was there too, along with GR, when gravitational waves were predicted.
Since the model was based on calculations made using GR applied to two colliding black holes, LIGO confirmed GR (again): it confirmed gravitational waves, and black holes (again). This does not exclude though the possibility that other modifications, alternatives or extensions of GR can work out similar predictions. So further experiments may be needed. But what I can say is that the theories that remained are modifications of GR that still explain gravity as spacetime curvature, and still make use of the four-dimensional spacetime. Theories that at purpose mimic most of GR.
Space is dead, long live spacetime!
Tuesday, January 12, 2016
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8e1998dd51baebdb | Approximability of Optimization Problems through Adiabatic Quantum Computation
Book description
The adiabatic quantum computation (AQC) is based on the adiabatic theorem to approximate solutions of the Schrödinger equation. The design of an AQC algorithm involves the construction of a Hamiltonian that describes the behavior of the quantum system. This Hamiltonian is expressed as a linear interpolation of an initial Hamiltonian whose ground state is easy to compute, and a final Hamiltonian whose ground state corresponds to the solution of a given combinatorial optimization problem. The adiabatic theorem asserts that if the time evolution of a quantum system described by a Hamiltonian is large enough, then the system remains close to its ground state. An AQC algorithm uses the adiabatic theorem to approximate the ground state of the final Hamiltonian that corresponds to the solution of the given optimization problem. In this book, we investigate the computational simulation of AQC algorithms applied to the MAX-SAT problem. A symbolic analysis of the AQC solution is given in order to understand the involved computational complexity of AQC algorithms. This approach can be extended to other combinatorial optimization problems and can be used for the classical simulation of an AQC algorithm where a Hamiltonian problem is constructed. This construction requires the computation of a sparse matrix of dimension 2ⁿ × 2ⁿ, by means of tensor products, where n is the dimension of the quantum system. Also, a general scheme to design AQC algorithms is proposed, based on a natural correspondence between optimization Boolean variables and quantum bits. Combinatorial graph problems are in correspondence with pseudo-Boolean maps that are reduced in polynomial time to quadratic maps. Finally, the relation among NP-hard problems is investigated, as well as its logical representability, and is applied to the design of AQC algorithms. It is shown that every monadic second-order logic (MSOL) expression has associated pseudo-Boolean maps that can be obtained by expanding the given expression, and also can be reduced to quadratic forms.
Table of Contents: Preface / Acknowledgments / Introduction / Approximability of NP-hard Problems / Adiabatic Quantum Computing / Efficient Hamiltonian Construction / AQC for Pseudo-Boolean Optimization / A General Strategy to Solve NP-Hard Problems / Conclusions / Bibliography / Authors' Biographies
Table of contents
1. Preface
2. Acknowledgments
3. Introduction
4. Approximability of NP-hard Problems
1. Basic Definitions
2. Probabilistic Proof Systems
3. Optimization Problems
1. Approximation Algorithms
4. Randomized Complexity Classes
1. The Complexity Class BPP
2. The Complexity Class RP
3. The Complexity Class ZPP
4. Quantum Complexity
5. Randomness and Determinism
1. Derandomization of the Class BPP
2. Derandomization Techniques
5. Adiabatic Quantum Computing
1. Basic Definitions
1. Linear Operators
2. Quantum States and Evolution
3. The Adiabatic Theorem
1. Adiabatic Evolution
2. Quantum Computation by Adiabatic Evolution
4. Adiabatic Paths
1. Geometric Berry Phases
2. Geometric Quantum Computation
6. Efficient Hamiltonian Construction
1. AQC Applied to the MAX-SAT Problem
1. Satisfiability Problem
2. AQC Formulation of SAT
2. Procedural Hamiltonian Construction
1. Hyperplanes in the Hypercube
2. The Hamiltonian Operator H_E
3. The Hamiltonian Operator H_Z_
7. AQC for Pseudo-Boolean Optimization
1. Basic Transformations
2. AQC for Quadratic Pseudo-Boolean Maps
1. Hadamard Transform
2. _x Transform
3. k-Local Hamiltonian Problems
1. Reduction of Graph Problems to the 2-Local Hamiltonian Problem
4. Graph Structures and Optimization Problems
1. Relational Signatures
2. First-Order Logic
3. Second-Order Logic
4. Monadic Second-Order Logic Decision and Optimization Problems
5. MSOL Optimization Problems and Pseudo-Boolean Maps
8. A General Strategy to Solve NP-Hard Problems
1. Background
1. Basic Notions
2. Tree Decompositions
2. Procedural Modification of Tree Decompositions
1. Modification by the Addition of an Edge
2. Iterative Modification
3. Branch Decompositions
4. Comparison of Time Complexities
3. A Strategy to Solve NP-Hard Problems
1. Dynamic Programming Approach
2. The Courcelle Theorem
3. Examples of Second-Order Formulae
4. Dynamic Programming Applied to NP-Hard Problems
5. The Classical Ising Model
6. Quantum Ising Model
9. Conclusions
10. Bibliography (1/2)
11. Bibliography (2/2)
12. Authors' Biographies
Product information
• Title: Approximability of Optimization Problems through Adiabatic Quantum Computation
• Author(s): William Cruz-Santos, Guillermo Morales-Luna
• Release date: September 2014
• Publisher(s): Morgan & Claypool Publishers
• ISBN: 9781627055574 |
f85e549386150dac | June 1, 2016
The scientific self-elimination of Heterodoxy
Comment on Jamie Morgan on ‘Economists confuse Greek method with science’
You say: “One underlying question is if economics is to be a science — what kind of ‘science’ can it be?” False, economics has only the choice between complying with well-defined standards or to be thrown out of science. Science was there before economics was there and the Greeks defined it as episteme = knowledge in contradistinction to doxa = opinion. This demarcation line is often hard to draw in the concrete case but it nevertheless exists.
This is the meaning of SCIENTIFIC truth (= formal and material consistency) which is different from religious or philosophical truth. A heterodox economist who says that ‘there is no truth’ shoots himself in the foot because if anything goes and nothing matters Orthodoxy cannot be criticized/falsified/rejected. To deny the true/false criterion means to kick oneself out of science. Asad Zaman is a prominent example.
Another widespread error of Heterodoxy is to maintain that rejection/debunking of Orthodoxy is sufficient. It is not: “The problem is not just to say that something might be wrong, but to replace it by something — and that is not so easy.” (Feynman, 1992, p. 161)
Replacement means in concrete terms to replace the methodologically unacceptable microfoundations, which are nothing but the explicit formal specification of methodological individualism, by correct macrofoundations. This means in even more concrete terms that one needs a total replacement for this axiomatic hard core: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.” (Weintraub, 1985, p. 147)
The green cheese behavioral assumptions HC1, HC2, HC4 define economics as a social science. The crux of the so-called social sciences, though, is this: “By having a vague theory it is possible to get either result. ... It is usually said when this is pointed out, ‘When you are dealing with psychological matters things can’t be defined so precisely’. Yes, but then you cannot claim to know anything about it.” (Feynman, 1992, p. 159). Because of this, behavioral assumptions cannot be taken into the axiomatic foundations of a theory. Remember Aristotle: “When the premises are certain, true, and primary, and the conclusion formally follows from them, this is demonstration, and produces scientific knowledge of a thing.”
As a matter of principle, behavioral assumptions/propositions are NOT certain enough and therefore they cannot be used as axiomatic foundation of economics. This is the fatal methodological blunder of Orthodoxy. Economics has to be redefined as system science and put on objective (= non-behavioral) foundations.*
The mistake of Heterodoxy is to define itself in opposition to Orthodoxy. This negative identity has to be turned into a positive identity by spelling out the foundational propositions that define Heterodoxy. The fault of Orthodoxy has never been to apply the axiomatic-deductive method but to choose shaky behavioral assumptions as axioms. The fault of heterodox economics is that it cannot define itself with a handfull of objective and certain foundational propositions. What is built upon shaky foundations eventually falls apart. This is what happened to Orthodoxy and this is why Heterodoxy never got off the ground.
Does the world expect from economists to find out how people behave? No, this is the proper job of psychology, sociology, anthropology, history, political science, evolution theory, criminology, etcetera. Does the world expect from economists to figure out what profit is? Yes, of course, no philosopher, psychologist, biologist, or sociologist will ever try to figure this out.
Have orthodox or heterodox economists figured out what profit is? No: “A satisfactory theory of profits is still elusive.” (Desai, 2008, p. 10). So, economists can be defined as scientific write-offs who give economic policy advice without ever having understood the pivotal concept of their subject matter.** If there ever was a scientific failure worse than the flat earth theory then it is economics defined as social science. And exactly this is what Orthodoxy and Heterodoxy have in common.
Egmont Kakarot-Handtke
* See ‘Economics is NOT a science of behavior
** See ‘How the intelligent non-economist can refute every economist hands down
For additional aspects see cross-references Heterodoxy.
REPLY to Jamie Morgan on Jun 4
My main points in a nutshell:
(i) Methodology is important — but only if it helps to promote the real thing.
(ii) The real thing is to answer the question how the actual (world-) economy works.
(iii) So, compared to physics the subject matter is the universe and not the learning-disabled fruit-fly called homo oeconomicus.
(iv) Because of this, economics has to get out of folk psychology, folk sociology, folk history, and folk politics. Economics is a system science.
(v) One cannot perceive the economy with the two natural eyes but only with the third eye of theory. So economics is abstract and not accessible by way of storytelling and misplaced concreteness.
(vi) Theory is composed of elementary premises (= axioms) and the superstructure of derived propositions. A theory is true if it satisfies the criteria of formal and material consistency. True theory is the precondition of any policy advice. Policy advice without true theory is an abuse of science for agenda pushing. This is what economics is today. Both Orthodoxy and Heterodoxy is stuck at the proto-scientific level.
(vii) Orthodoxy is microfounded (see the axioms HC1 to HC5 in the post above). This is the methodological root error/mistake.
(viii) The task of Heterodoxy is to replace microfoundations by macrofoundations.
REPLY to Ken Zimmerman on Jun 5
You say: “if you mean by ‘real thing’ the historical processes through which economic theories, actions, and ways of life are constructed, then we’re on the same page.”
The real thing is theoretical economics, that is, the formally and materially correct explanation of how the (actual-monetary-world) economy works. The true theory is the precondition for the understanding of present and past economic reality.
It is not only that Orthodoxy has failed but Heterodoxy, too. The representative economist — including historians since the German Historical School and the American Institutionialists since Veblen — has until this day NO idea of what profit is. This is comparable to a medieval physicist who has no proper understanding of the fundamental concept of energy. The contribution of historians to economic theory has been zero.
So, we are certainly NOT on the same page. As I see it there is no chance that you will ever get out from behind the curve. For details see ‘The future of economics: why you will probably not be admitted to it, and why this is a good thing’.
REPLY to Jamie Morgan on Jun 9
Imagine somebody throwing three golf balls amidst a cyclone over their shoulder into a very large sandbox. Clearly, the three balls form a triangle but no one can predict its form and size. Yet, the mathematician can tell with certainty that the sum of angles is 180 degrees (if the sandbox is Euclidean).
Science is about invariants, that is, properties or relationships which remain unchanged over time. A famous example is E=mc2 which describes not a single historical event but something that is the case always and everywhere. Non-scientists and historians are glued to the ever changing surface, so they produce stories while scientists produce laws.
Here, for example, is the First Economic Law, which shows the relationship between the firm, the market, and the income distribution for the pure consumption economy.
Needless to say that all variables are measurable, hence, the First Economic Law is a testable proposition. This is the way how economics gets out of the proto-scientific stage. Or, as Popper put it: “It is the optimistic theory that science, that is real knowledge about the hidden real world, though certainly very difficult, is nevertheless attainable, at least for some of us.”
At the moment neither orthodox nor heterodox economists are among the “some of us”.
REPLY to Asad Zaman on Jun 9
The pure consumption economy is, of course, the most elementary case. The problem is that BOTH orthodox AND heterodox economists do not even understand the simple things. Because of this, they have NO chance to understand anything: “There can be no doubt whatsoever that a problem which has not yet been solved in all its aspects under its simplest conditions will be still more difficult to tackle if other, ‘more realistic’ assumptions are being made.” (Morgenstern, 1941, p. 373)
From the pure consumption economy follows by successive differentiation the complete employment equation which contains profit/loss, profit distribution, saving/dissaving, investment/disinvestment (2012), public deficit spending, and import/export.
This equation then describes the actual monetary economy exhaustively and — that is decisive — it is testable. So, everybody who thinks that the axiomatically founded structural employment equation is false can try to refute it. This is how matters are settled since the ancient Greeks invented the scientific method.
If you have a better methodology it would be appropriate to present a testable proposition about how the actual economy works. At the end of the day methodological discussions must result in an improved understanding of the economy or else they are vacuous. In particular, it would be interesting to learn something about Zaman’s Profit Law. After all, profit is the pivotal phenomenon of the capitalist economy. Who does not understand profit understands nothing (2014).
1–23. URL
REPLY to Jamie Morgan on Jun 10
(i) You write: “... and top be clear Euclid is not science it is maths-geometry.” This is not the only methodological point where you are way behind the curve. Note that science is ultimately the perfect SYNTHESIS of logic and experience: “Hilbert and Einstein again agree that geometry is a natural science based on real experiments and measurements. Thus, similarly to Einstein, Hilbert can assert: Geometry is nothing but a branch of physics; in no way whatsoever do geometrical truths differ essentially from physical truths nor are they of a different nature.” (Majer, 1995, p. 280)
(ii) You ask: “How would you respond to: Axiom 1: people sometimes follow rules; Axiom 2: rules change.” My answer: there is NO such thing as a behavioral axiom.* To accept behavioral assumptions as axioms is the cardinal methodological error/mistake of Orthodoxy and Heterodoxy.**
Logical consistency is secured by applying the axiomatic-deductive method and empirical consistency is secured by applying state-of-the-art testing. Isn’t it curious that genuine scientists have no problem at all with this methodology since the ancient Greeks but that so-called social scientists cannot get their head around it?
(iv) A good rule for your methodological thoughts is: Whenever you meet with approval from Asad Zaman, Ken Zimmerman, Robert Locke, or other would-be scientists you can be sure that you have lost your way.
Majer, U. (1995). Geometry, Intuition and Experience: From Kant to Husserl. Erkenntnis, 42(2): 261–285. URL
* See ‘Austrian blather
** See ‘Economics is NOT a science of behavior
REPLY to Jamie Morgan and Ken Zimmerman on Jun 14
(i) Science does not explain everything, but non-science explains nothing. Scientific explanation comes in the communicative format of theory. Non-science comes in the format of storytelling.
(ii) Science is well-defined by material and formal consistency. These criteria are demanding and it is often not clear in the concrete case how to apply them. So the need for specification arises. This is where methodology comes in.
(iii) Nobody is forced to do science. But if one decides to do science one has to stick to the rules. As in all walks of life, some people either do not understand the rules or misapply them. Here again, methodology can be helpful. The proper role of methodology is NOT to soften scientific standards but to enforce them.*
(iv) Economics is a science as clearly communicated in the title “Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel”.
(v) Orthodox economics, though, does NOT satisfy scientific standards. More specifically: Walrasianism, Keynesianism, Marxianism, Austrianism is PROVABLE false.
(vi) Economics does not live up to the claim as stated in (iv). And this is the exact point where Heterodoxy comes it. Either Heterodoxy fully replaces Orthodoxy or both are eventually thrown out of science. There cannot be a pluralism of false theories.
(vii) To replace a theory means in very practical terms to replace the foundational hard core, a.k.a. axioms, that is the Walrasian propositions HC1 to HC5 as enumerated above.
(viii) Due to form, at this critical juncture Ken Zimmerman again asks a silly question from behind the curve: “Egmont, ever ask yourself how and why were the axioms made? From God? From some great human law givers? From the universe?” And this brings us directly back to the key issue of this thread, viz. to Asad Zaman’s cognitive dissonance with the ancient Greeks.
“To Plato’s question, ‘Granted that there are means of reasoning from premises to conclusions, who has the privilege of choosing the premises?’ the correct answer, I presume, is that anyone has this privilege who wishes to exercise it, but that everyone else has the privilege of deciding for himself what significance to attach to the conclusions, and that somewhere there lies the responsibility, through the choice of the appropriate premises, to see to it that judgment, information, and perhaps even faith, hope and charity, wield their due influence on the nature of economic thought.” (Viner, 1963, p.12)
(ix) So, nobody hinders Jamie Morgan, Asad Zaman, Ken Zimmerman and the rest of Heterodoxy to employ ‘the privilege of choosing the premises’. All there is to do is to take care that “the premises are certain, true, and primary” (Aristotle). Heterodoxy defines itself by its axioms or it is scientifically non-existent.
Viner, J. (1963). The Economist in History. American Economic Review, 53(2): pp. 1–22. URL
* See also ‘The insignificance of Gödel’s theorem for economics’.
REPLY to Asad Zaman on Jun 14
Yes, I can, see Wikipedia (picture at the right-hand side)
“But it was a second and more important quality that struck readers of the Principia. At the head of Book I stand the famous Axioms, or the Laws of motion: … For readers of that day, it was this deductive, mathematical aspect that was the great achievement.” (Truesdell, quoted in Schmiechen, 2009, p. 213)
In Newton's own words: “Could all the phaenomena of nature be deduced from only thre [sic] or four general suppositions there might be great reason to allow those suppositions to be true.” (Westfall, 2008, p. 642)
Westfall, R. S. (2008). Never at Rest. A Biography of Isaac Newton. Cambridge: Cambridge University Press, 17th edition.
REPLY to blockethe on Jun 14
I appreciate your work about management. But this is an entirely different matter. Your example shows that you do not understand what the subject matter of economics is. Strictly speaking, management is the subject matter of psychology/sociology and NOT of economics. The subject matter of economics is the (world-) economy as a whole.
Imagine a physicist is asked to figure out how the universe works and after some time he comes back and says: “The universe is much too large, not of direct relevance to our daily lives, and ultimately incomprehensible, so I have analyzed the molehills in my front garden — with surprising results.”
One can say without contradiction that this physicist has done valuable empirical work but failed at the original task.
The error/mistake of the microfoundations approach is to take green cheese behavioral assumptions as axioms. And this, indeed, is what Poincaré has told Walras in no uncertain terms: “Walras approached Poincaré for his approval. ... But Poincaré was devoutly committed to applied mathematics and did not fail to notice that utility is a nonmeasurable magnitude. ... He also wondered about the premises of Walras’s mathematics: It might be reasonable, as a first approximation, to regard men as completely self-interested, but the assumption of perfect foreknowledge ‘perhaps requires a certain reserve’.” (Porter, 1994, p. 154)
No genuine scientist ever accepted or will ever accept the Walrasian axioms. Walrasians are de facto out of science since Walras.
Most economists have not realized that economics is NOT a science of human nature/behavior/action — not of individual behavior, not of social behavior, not of rational behavior, not of irrational behavior, not of sincerity, not of corruption. All these issues belong entirely to the realms of psychology, sociology, anthropology, political science, history, criminology, social philosophy, etcetera.
As you have certainly noticed,* I do not propose to change the Walrasian behavioral axioms but to completely REPLACE them by objective structural axioms. Economics is NOT a social science but a system science.
That’s the absolute minimum for a start. This set is obviously superior to Walrasian and Keynesian axioms and leads to testable propositions. Empirical tests decide whether A1 to A3 are acceptable and NOT vacuous methodological filibuster.
Porter, T. M. (1994). Rigor and Practicality: Rival Ideals of Quantification in Nineteenth-Century Economics. In P. Mirowski (Ed.), Natural Images in Economic Thought, pages 128–170. Cambridge: Cambridge University Press.
* If not see the documentation on this blogspot
REPLY to Jamie Morgan on Jun 15
You say: “What concerns me is that you consistently respond to all inquiry with assertion ...”. True, but I give you the reference to the comprehensive argument. What concerns me is that you consistently overlook that your questions and arguments have already been thoroughly answered.*
You say: “the failure to agree is not itself a failure — since sometimes it reminds us of the limits of knowledge and the boundaries of ignorance — which is basic also to Socratic dialogue…”
Trivially true, we cannot know everything, but from this does not follow that we cannot know something. It is this limited but certain Something that science and theoretical economics is all about.
Nobody needs a reminder that there are limits of knowledge and nobody needs the false modesty of ‘I know that I know nothing’. For a philosopher this is fine but for a scientist this is self-disqualifying.
In economics, we are FAR away from the limits of knowledge. In fact, the problem is the exact OPPOSITE: “we know little more now about ‘how the economy works,’ or about the modus operandi of the invisible hand than we knew in 1790, after Adam Smith completed the last revision of The Wealth of Nations.” (Clower, 1999, p. 401)
What we actually have are multiple approaches that are PROVABLE false. There are (at least) four heterodox profit theories and you can tell nobody that they are all true.** This lack of consistency has NOTHING to do with the limits of knowledge or the failure to agree but with incompetence and intellectual sloppiness and poor methodology and the persistent ignorance/violation of scientific standards.***
Economics is a proto-scientific swamp: “We are lost in a swamp, the morass of our ignorance. ... We have to find the roots and get ourselves out! ... Braids or bootstraps are necessary for two purposes: to pull ourselves out of the swamp and, afterwards, to keep our bits and pieces together in an orderly fashion.” (Schmiechen, 2009, p. 11)
This is in simple words was axiomatization is all about. What concerns me is that you and Asad Zaman and many others on this blog do not grasp what the ancient Greeks grasped more than 2000 years ago.
As long as economists do not have a consistent defininition of the pivotal concepts profit and income it is absurd to philosophize about the limits of knowledge. Economists are over the ears in the swamp of ignorance and have NO idea of how to pull themselves out.
This is the concrete historical situation: Heterodoxy either REPLACES the vacuous Walrasian axioms HC1 to HC5 and comes forward with TESTABLE propositions about how the economy works or it goes down the scientific drain together with Orthodoxy.
Clower, R. W. (1999). Post-Keynes Monetary and Financial Theory. Journal of Post Keynesian Economics, 21(3): 399–414. URL
* For a test go occasionally to this blogspot and enter for example Gödel or Duhem-Quine or Zaman in the search field.
** See ‘Heterodoxy, too, is scientific junk
*** See also ‘The prophets of wish-wash, ignoramus et ignorabimus, and preemptive vanitization
REPLY to Asad Zaman
You say: “Even though Newton calls his four laws axioms, what he means by axioms is very different from what you mean by axioms.”
What I mean is the SAME what Newton meant. And this is the SAME what Popper meant:
“The attempt is made to collect all the assumptions, which are needed, but no more, to form the apex of the system. They are usually called the ‘axioms’ (or ‘postulates’, or ‘primitive propositions’; no claim of truth is implied in the term ‘axiom’ as here used). The axioms are chosen in such a way that all the other statements belonging to the theoretical system can be derived from the axioms by purely logical or mathematical transformations.” (1980, p. 71)
And this is the SAME what Einstein meant:
“But the axioms Science is the attempt to make the chaotic diversity of our sense-experience correspond to a logically uniform system of thought ...” (quoted in Clower, 1998, p. 409)
Newton, Popper, Einstein referred to the context of JUSTIFICATION. Peirce’s abduction refers to the context of DISCOVERY and it is the SAME as Popper’s Conjectures and Refutations:
“It is a great mistake to suppose that the mind of the active scientist is filled with propositions which, if not proved beyond all reasonable cavil, are at least extremely probable. On the contrary, he entertains hypotheses which are almost wildly incredible, and treats them with respect for the time being. Why does he do this? Simply because any scientific proposition whatever is always liable to be refuted and dropped at short notice. A hypothesis is something which looks as if it might be true and were true, and which is capable of verification or refutation by comparison with facts. The best hypothesis, in the sense of the one most recommending itself to the inquirer, is the one which can be the most readily refuted if it is false. This far outweighs the trifling merit of being likely. For after all, what is a likely hypothesis? It is one which falls in with our preconceived ideas. But these may be wrong. Their errors are just what the scientific man is out gunning for more particularly. But if a hypothesis can quickly and easily be cleared away so as to go toward leaving the field free for the main struggle, this is an immense advantage.” (Peirce, 1931, 1.120)
You constantly CONFUSE the context of discovery with the context of justification. What Peirce said about the axiomatic-deductive method is the SAME what Aristotle, Newton, Einstein, Popper said and what I mean:
“Inference, which is the machinery of logic, is the process by which one belief determines another belief, habit or action. A successful inference is one that leads from true premises to true conclusions.” (quoted in Hoover, 1994, p. 300)
The clearly stated premises, a.k.a axioms/postulates/principles/primitive propositions, of Orthodoxy (HC1 to HC5 above) are provably false. What I mean with true macrofoundations I have clearly stated (A1 to A3 above). Now it is YOUR TURN to clearly state your economic axioms. Subsequently, the truth of the respective premises is indirectly established by testing the conclusions. This settles the matter.
As Peirce said: “That the settlement of opinion is the sole end of inquiry is a very important proposition.” (1992, p. 115)
Clower, R. W. (1998). New Microfoundations for the Theory of Economic Growth? In G. Eliasson, C. Green, and C. R. McCann (Eds.), Microfoundations of Economic Growth., pages 409–423. Ann Arbour, MI: University of Michigan Press.
Hoover, K. D. (1994). Pragmatism, Pragmaticism and Economic Method. In R. E. Backhouse (Ed.), New Directions in Economic Methodology, pages 286–315. London, New York, NY: Routledge.
Peirce, C. S. (1992). The Fixation of Belief. In N. Houser, and C. Kloesel (Eds.), The Essential Peirce. Selected Philosophical Writings., volume 1, pages 109–123. Bloomington, IN: Indiana University Press.
REPLY to Jamie Morgan on Jun 17
You say: “Perhaps there is a different way of thinking about these problems…”.
This exactly is the scientific problem: there are always many ways and opinions and questions and interests, and it can be a tedious task to figure out what is true and what is false.
The difference between scientists and the rest is that scientists attempt to get a clear-cut either/or answer with the highest possible degree of certainty. Without this certainty, science cannot be cumulative. In other words, if one does not make sure that the elementary Law of the Lever or the Law of Falling Bodies is certain beyond reasonable doubt one will never arrive at more complex relationships like the Law of Gravitation or the Schrödinger equation. Physicists are so high up the ladder because each rung from the first one onward is certain and stable and reliable and can carry heavy weight. This is what logical and empirical consistency is all about.
Economics is not cumulative but circles since Adam Smith at a rather low level around the same issues. Take value or capital theory as an example or take Wicksell’s pertinent characterization.
Such is the pseudo-scientist's tautologically true answer. However, many are perfectly satisfied with this inconclusive sitcom stuff and euphemize it as Socratic. But, clearly, confused wish-wash is not that highly appreciated among genuine scientists.
Economics is still at the proto-scientific level because of scientific incompetence and because many simply prefer the swampy lowlands where “nothing is clear and everything is possible” (Keynes) over the hard rocks of true/false.
Those, who have ― for whatever reason ― established themselves in the swampy lowlands of plausible myths defend it with phrases like: there is no truth, nobody knows anything, uncertainty is ontological, the effect is sometimes in one direction, sometimes in another and sometimes nil, everybody has their own truths, quantum physics says whether the cat is dead or alive depends on the observer, knowledge is arrogance, ignorance is humility, truth is relative and culture-specific, Gödel has proved that logic has limits and ― the emotional solidarity of the incompetent ― we are ALL fallible humans. Yes, indeed, but we are NOT ALL imbeciles who accept utility maximization and equilibrium as scientific explanation of how markets work or who maintain that a dozen false profit theories are better than one correct theory.
The fact of the matter is: scientists and swampies can never be friends.
REPLY to Asad Zaman on Jun 17
(i) You and I and everybody who uses the zero in a calculation care whether the calculation is correct and does NOT care whether the Arabs, Indians, Egyptians, or Greeks invented the zero. It is the same with fire making or methodology. The actual use and the history of concepts or tools are different things. It is the very task of the economist to figure out how the economy works and NOT to clarify who invented what. The history of scientific thought is valuable in its own right but it is a DISTRACTION in the context of economics.
(ii) It is misleading and counterproductive to play the ‘experimental method of the Arabs’ against the ‘axiomatic-deductive method of the Greeks’. Science is defined by material AND formal consistency. It is the sophisticated COMBINATION of empirics and logic, or experiment and axiomatics, which delivers the winning formula of science. Incompetent scientists fall either on the side of crude observational empiricism or vacuous formalism.
(iii) Your account of the history of science is in almost every respect false or confused. Two examples suffice here, your characterization of Bacon* and your misplaced debunking of axiomatics: “Here is what the Greeks grasped more than 2000 years ago, based on an axiomatic-deductive approach to the natural sciences: 1. The Earth is the center of the universe.” It is common knowledge that: “Aristarchus of Samos was an ancient Greek astronomer and mathematician who presented the first known model that placed the Sun at the center of the known universe with the Earth revolving around it.” (Wikipedia)
(iv) I wonder whether anybody checks the content of the WEA Pedagogy Blog. Just for the records: I do NOT accept this easy to disprove garbage as Heterodoxy economics or methodology. The Pedagogy Block has to be clearly labeled as Asad Zaman’s idiosyncratic contribution which does NOT represent any official consensus of the WEA. The Pedagogy Blog is sufficient to expel RWER-Heterodoxy from the sciences.
(v) You say: “My critique of axiomatics is based on axiomatics as defined by Lionel Robbins, who is the founder of current economic methodology. ‘The propositions of economic theory, like all scientific theory, are obviously deductions from a series of postulates. And the chief of these postulates are all assumptions involving in some way simple and indisputable facts of experience….’ With this methodology, axioms are certain, and logical deductions are certain so there is no room for conflict with experience. This is exactly the same as axiomatic Greek geometry, which can never be wrong, because it is based on certainties and driven by logic.”
This is as confused as it can get. First of all, Robbins explicitly claims that his postulates are based on “simple and indisputable facts of experience”. This is what you praise as the superior method of the Arabs. Robbins is the very prophet of what Popper called observationism: “These are not postulates the existence of whose counterpart in reality admits of extensive dispute once their nature is fully realized. We do not need controlled experiments to establish their validity: they are so much the stuff of our everyday experience that they only have to be stated to be recognized as obvious.” (Robbins, 1935, p. 79)
(vi) Robbins’s methodological blunder was that he based economics on BEHAVIORAL axioms (e.g. constrained optimization). Yet, there is NO such thing as a ‘certain, true, and primary’ (Aristotle) behavioral axiom.
To make a long argument short: Robbins’s definition of economics has to be changed from: “Economics is the science which studies human behavior as a relationship between ends and scarce means which have alternative uses.” (Robbins, 1935, p. 16) to “Economics is the science which studies how the economic system works.” (2013, p. 20)
Objective structural axioms lead to relationships that are readily testable, e.g. the Profit Law or the employment equation.**
(vii) The lethal error/mistake of Orthodoxy does NOT consist in the application of the axiomatic-deductive method but in taking green cheese behavioral assumptions as axioms. From this follows that economics has to move from behavioral microfoundations to structural macrofoundations.
(viii) Science is about true/false and NOTHING else. We agree that Robbins based economics upon provable false axioms.
* Go occasionally to this blogspot and enter Bacon in the search field
** See ‘Unemployment ― the fatal consequence of economists’ scientific incompetence’, |
2e09f566581a269f | Mathematical model
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Not to be confused with the same term used in model theory, a branch of mathematical logic. An artifact that is used to illustrate a mathematical idea may also be called a mathematical model, the usage of which is the reverse of the sense explained in this article.
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.
Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.
Model classifications in mathematics[edit]
Mathematical models are usually composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters of interest, that can be quantified. Operators can act with or without variables.[1] Models can be classified in the following ways:
• Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.
Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by differential equations.
• Explicit vs. implicit: If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations (known as linear programming, not to be confused with linearity as described above), the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newton's method (if the model is linear) or Broyden's method (if non-linear). For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model; while a continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic): A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of initial conditions. Conversely, in a stochastic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions.
• Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models.[2] Application of catastrophe theory in science has been characterized as a floating model.[3]
Significance in the natural sciences[edit]
Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models.
Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits relativity theory and quantum mechanics must be used, even these do not apply to all situations and need further refinement. It is possible to obtain the less accurate models in appropriate limits, for example relativistic mechanics reduces to Newtonian mechanics at speeds much less than the speed of light. Quantum mechanics reduces to classical physics when the quantum numbers are high. For example the de Broglie wavelength of a tennis ball is insignificantly small, so classical physics is a good approximation to use in this case.
It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are such as a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis.
Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean.
Some applications[edit]
Since prehistorical times simple models such as maps and diagrams have been used.
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.
Building blocks[edit]
In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables.
Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).
Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases.
For example, in economics students often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.
A priori information[edit]
Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.
Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.
In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification [4] can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.
Subjective information[edit]
Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: one specifies a prior probability distribution (which can be subjective) and then updates this distribution based on empirical data. An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown, so the experimenter would need to make an arbitrary decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of the subjective information is necessary in this case to get an accurate prediction of the probability, since otherwise one would guess 1 or 0 as the probability of the next flip being heads, which would be almost certainly wrong.[5]
In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling; the essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a Paradigm shift offers radical simplification.
For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.
Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe. If the modeling is done by a neural network, the optimization of parameters is called training. In more conventional modeling through explicitly given mathematical functions, parameters are determined by curve fitting.
Model evaluation[edit]
A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation.
Fit to empirical data[edit]
Usually the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics.
Defining a metric to measure distances between observed and predicted data is a useful tool of assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role.
While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from non-parametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form.
Scope of the model[edit]
Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data.
The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points outside the observed data is called extrapolation.
As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.
Philosophical considerations[edit]
Many types of modeling implicitly involve claims about causality. This is usually (but not always) true of models involving differential equations. As the purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied.
An example of such criticism is the argument that the mathematical models of Optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.[6]
• One of the popular examples in computer science is the mathematical models of various machines, an example is the Deterministic finite automaton which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s.
The state diagram for M
M = (Q, Σ, δ, q0, F) where
S1 S2 S1
S2 S1 S2
The state S1 represents that there has been an even number of 0s in the input so far, while S2 signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M will finish in state S1, an accepting state, so the input string will be accepted.
The language recognized by M is the regular language given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleene star, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".
• Many everyday activities carried out without a thought are uses of mathematical models. A geographical map projection of a region of the earth onto a small, plane surface is a model[7] which can be used for many purposes such as planning travel.
• Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as dead reckoning when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.[8][9]
• Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
• Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function V\!:\mathbb{R}^3\!\rightarrow\mathbb{R} and the trajectory, that is a function \mathbf{r}\!:\mathbb{R}\rightarrow\mathbb{R}^3, is the solution of the differential equation:
-\frac{\mathrm{d}^2\mathbf{r}(t)}{\mathrm{d}t^2}m=\frac{\partial V[\mathbf{r}(t)]}{\partial x}\mathbf{\hat{x}}+\frac{\partial V[\mathbf{r}(t)]}{\partial y}\mathbf{\hat{y}}+\frac{\partial V[\mathbf{r}(t)]}{\partial z}\mathbf{\hat{z}},
that can be written also as:
m\frac{\mathrm{d}^2\mathbf{r}(t)}{\mathrm{d}t^2}=-\nabla V[\mathbf{r}(t)].
subject to:
Modeling requires selecting and identifying relevant aspects of a situation in the real world.
See also[edit]
1. ^
2. ^ Andreski, Stanislav (1972). Social Sciences as Sorcery. St. Martin’s Press. ISBN 0-14-021816-5.
3. ^ Truesdell, Clifford (1984). An Idiot’s Fugitive Essays on Science. Springer. pp. 121–7. ISBN 3-540-90703-3.
5. ^ MacKay, D.J. Information Theory, Inference, and Learning Algorithms, Cambridge, (2003-2004). ISBN 0-521-64298-1
6. ^ Pyke, G. H. (1984). "Optimal Foraging Theory: A Critical Review". Annual Review of Ecology and Systematics 15: 523–575. doi:10.1146/ edit
7. ^, definition of map projection
8. ^ Gallistel (1990). The Organization of Learning. Cambridge: The MIT Press. ISBN 0-262-07113-4.
9. ^ Whishaw, I. Q.; Hines, D. J.; Wallace, D. G. (2001). "Dead reckoning (path integration) requires the hippocampal formation: Evidence from spontaneous exploration and spatial learning tasks in light (allothetic) and dark (idiothetic) tests". Behavioural Brain Research 127 (1–2): 49–69. doi:10.1016/S0166-4328(01)00359-X. PMID 11718884. edit
Further reading[edit]
Specific applications
External links[edit]
General reference material
Philosophical background |
d0d47563a962e858 | SVDPACK comprises four numerical (iterative) methods for computing the singular value decomposition (SVD) of large sparse matrices using double precision ANSI Fortran-77. A compatible ANSI-C version (SVDPACKC) is also available. This software package implements Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left- and right-singular vectors) for large sparse matrices. The package has been ported to a variety of machines ranging from supercomputers to workstations: CRAY Y-MP, CRAY-2S, Alliant FX/80, SPARCstation 10, IBM RS/6000-550, DEC 5000-100, and HP 9000-750. The development of SVDPACK wa motivated by the need to compute large rank approximations to sparse term-document matrices from information retrieval applications. Future updates to SVDPACK(C), will include out-of-core updating strategies, which can be used, for example, to handle extremely large sparse matrices (on the order of a million rows or columns) associated with extremely large databases in query-based information retrieval applications.
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15. Aswani Kumar, Cherukuri; Srinivas, Suripeddi: Latent semantic indexing using eigenvalue analysis for efficient information retrieval (2006)
16. Bruns, T.E.: Zero density lower bounds in topology optimization (2006)
17. Doescher, Erwin; De Campos Velho, Haroldo F.; Ramos, Fernando M.: Criteria for mixed grids in computational fluid dynamics (2006)
18. Kontoghiorghes, Erricos John: Handbook of parallel computing and statistics. (2006)
19. Xu, Shuting; Zhang, Jun; Han, Dianwei; Wang, Jie: Singular value decomposition based data distortion strategy for privacy protection (2006) ioport
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f740f66dfda8a954 | Schrödinger equation
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A fundamental equation in quantum mechanics that determines, together with corresponding additional conditions, a wave function $ \psi(t,\mathbf{q}) $ characterizing the state of a quantum system. For a non-relativistic system of spin-less particles, it was formulated by E. Schrödinger in 1926. It has the form $$ i \hbar \frac{\partial}{\partial t} [\psi(t,\mathbf{q})] = \hat{H} \psi(t,\mathbf{q}), $$ where $ \hat{H} = H(\hat{\mathbf{p}},\hat{\mathbf{r}}) $ is the Hamiltonian operator constructed by the following general rule: In the classical Hamiltonian function $ H(\mathbf{p},\mathbf{r}) $, the particle momenta $ \mathbf{p} $ and their coordinates $ \mathbf{r} $ are replaced by operators that have, respectively, the following form in the coordinate representation $ \mathbf{q} = (r_{1},\ldots,r_{N}) $ and in the momentum representation $ \mathbf{p} = (p_{1},\ldots,p_{N}) $: $$ \hat{p}_{i} = \frac{\hbar}{i} \frac{\partial}{\partial r_{i}} \quad \text{and} \quad \hat{r}_{i} = r_{i}; \qquad \hat{p}_{i} = p_{i} \quad \text{and} \quad \hat{r}_{i} = - \frac{\hbar}{i} \frac{\partial}{\partial p_{i}}; \qquad i \in \{ 1,\ldots,N \}. $$ For charged particles in an electromagnetic field, characterized by a vector potential $ \mathbf{A}(t,\mathbf{r}) $, the quantity $ \mathbf{p} $ is replaced by $ \mathbf{p} + \dfrac{e}{c} \mathbf{A}(t,\mathbf{r}) $. In these representations, the Schrödinger equation is a partial differential equation. For example, for particles in the potential field $ U(\mathbf{r}) $, the equation becomes $$ i \hbar \frac{\partial}{\partial t} [\psi(t,\mathbf{r})] = - \frac{\hbar^{2}}{2 m} {\Delta \psi}(t,\mathbf{r}) + U(\mathbf{r}) \psi(t,\mathbf{r}). $$
Discrete representations are possible, in which the function $ \psi $ is a multi-component function and the operator $ \hat{H} $ has the form of a matrix. If a wave function is defined in the space of occupation numbers, then the operator $ \hat{H} $ is represented by some combinations of creation and annihilation operators (i.e., the second quantization representation).
The generalization of the Schrödinger equation to the case of a non-relativistic particle with spin $ \dfrac{1}{2} $ (a two-component wave-function $ \psi(t,\mathbf{r}) $) is called the Pauli equation (1927); to the case of a relativistic particle with spin $ \dfrac{1}{2} $ (a four-component wave-function $ \psi $) — the Dirac equation (1928); to the case of a relativistic particle with spin $ 0 $ — the Klein–Gordon equation (1926); to the case of a relativistic particle with spin $ 1 $ (the wave-function $ \psi $ is a vector) — the Proca equation (1936); etc.
The solution of the Schrödinger equation is defined in the class of functions that satisfy the normalization condition $ \langle \psi(t,\mathbf{q}),\psi(t,\mathbf{q}) \rangle = 1 $ for all $ t $ (the angled brackets mean an integration or a summation over all values of $ \mathbf{q} $). To find the solution, it is necessary to formulate initial and boundary conditions, corresponding to the character of the problem under consideration. The most characteristic among such problems are:
1. The stationary Schrödinger equation and the determination of admissible values of the energy of the system. Assuming that $ \psi(t,\mathbf{q}) = \phi(\mathbf{q}) e^{- i E t / \hbar} $, and requiring in conformity with the normalization condition and the condition of absence of flows at infinity that the wave function and its gradients vanish when $ \| \mathbf{r} \| \to \infty $, one obtains an equation for the eigenvalues $ E_{n} $ and eigenfunctions $ \phi_{n} $ of the Hamiltonian operator: $$ \hat{H} {\phi_{n}}(\mathbf{q}) = E_{n} {\phi_{n}}(\mathbf{q}). $$ Characteristic examples of the exact solution to this problem are: The eigenfunctions and energy levels for a harmonic oscillator, a hydrogen atom, etc.
2. The quantum-mechanical scattering problem. The Schrödinger equation is solved under boundary conditions that correspond at a large distance from the scattering center (described by a potential $ U(\mathbf{r}) $) to the plane waves falling on it and the spherical waves arising from it. Taking into consideration this boundary condition, the Schrödinger equation can be written as an integral equation, the first iteration of which with respect to the term containing $ U(\mathbf{r}) $ corresponds to the so-called Born approximation. This equation is also called the Lippman–Schwinger equation.
3. The case where the Hamiltonian of the system depends on time, $ H = {H_{0}}(\mathbf{p},\mathbf{r}) + U(t,\mathbf{p},\mathbf{r}) $, is usually considered in the framework of time-dependent perturbation theory. This is a theory of quantum transition, the determination of the system’s reaction to an external perturbation (dynamic susceptibility) and characteristics of relaxation processes.
To solve the Schrödinger equation, one usually applies approximate methods, regular methods (different types of perturbation theories), variational methods, etc.
[1] A. Messiah, “Quantum mechanics”, 1, North-Holland (1961).
[2] L.D. Landau, E.M. Lifshitz, “Quantum mechanics”, Pergamon (1965). (Translated from Russian)
[3] L.I. Schiff, “Quantum mechanics”, McGraw-Hill (1955).
A comprehensive treatise on the mathematics of the Schrödinger equation is [a4].
[a1] R.P. Feynman, R.B. Leighton, M. Sands, “The Feynman lectures on physics”, III, Addison-Wesley (1965).
[a2] S. Gasiorowicz, “Quantum physics”, Wiley (1974).
[a3] J.M. Lévy-Lehlond, “Quantics-rudiments of quantum physics”, North-Holland (1990). (Translated from French)
[a4] F.A. Berezin, M.A. Shubin, “The Schrödinger equation”, Kluwer (1991). (Translated from Russian)
How to Cite This Entry:
Schroedinger equation. Encyclopedia of Mathematics. URL: |
2affdc80e391fab9 | Louche Part 2: Quantum Tunneling
(Continued from Louche Part 1: Feed Stock for Energy Beings)
I started Louche Part 1 with a definition of the word louche, so I want to also define the term Quantum Tunneling, since it’s not a word that anyone would use in casual conversation…unless that someone is talking to me over coffee and croissants, in which case, there would be other strange words thrown into the conversation. But I digress…
Quantum Tunneling
Tunneling is often explained using the Heisenberg uncertainty principle and the wave–particle duality of matter. [1]
effettunnelIn Taobabe-speak, quantum tunneling is a process where particles are fired at a barrier repeatedly so that it bounces back from the barrier.
Once in a random while, the wave portion of the particle continues forward and jumps through the barrier, which creates a measurable voltage on the other side.
As the waves randomly jump across the barrier, the voltage on the other side also randomly fluctuates. A sampling of that voltage can then be use to generate random data.
Aggregate this sampling over a lengthy period of time and a truly random noise source can be obtained via this equation, appropriately named, Schrödinger equation.
The Schrödinger equation
where \hbar is the reduced Planck’s constant, m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ is the Schrödinger wave function, V is the potential energy of the particle (measured relative to any convenient reference level), E is the energy of the particle that is associated with motion in the x-axis (measured relative to V), and M(x) is a quantity defined by V(x) – E which has no accepted name in physics. [1]
In this graph, you can see that the wave being measured is less defined, and certainly weaker than the wave being bounced back from the barrier, but it, nevertheless, continues to exist, and is actually measurable.
This measurable wave on the other side of reality is what encryption scientists use to create passwords that are almost impossible to crack. It is, in fact, the basis for quantum encryption and what will be used in the future to keep networks secure.
This all sounds great and cool, but you might be wondering why I am calling attention to random data generation, and what, if anything, it has to do with louche.
Well wouldn’t you like to know?
As luck would have it, a bunch of scientists have been on this quest for finding louche for a very long time. From the very start, these really smart folks were already on the forefront of trying to find measurable louche.
Since louche is described as energies created by human emotions, scientists from Princeton University’s PEAR lab, at the Institute of Noetic Sciences, began looking around the spacetime of our energy systems experiments to look at how deeply our minds are connected to the fabric of physical reality.
Based upon results from the data they have published, there is an interesting connection to our minds and how we can have an unexplained ordering affect on chaotic systems. The Global Consciousness Project has been showing that a few dozen such systems, called random number generators spread around the world can produce anomalies when global events happen that polarize human attention.
Here is a graph that was generated during the attacks of 9/11. As you can see, the energies of the world, at the point of the attack, spiked in between the first crash and the second crash, and continued to spike as the towers began to collapse.
According to scientists at the Noetic Sciences, the odds that the combined Global Consciousness Project data are due to chance is less than one in one hundred billion. The implication is that there’s some deep connection between the mind and physical reality, which we don’t yet fully understand. [2]
Jumping on the band wagon of the work done by the Institute of Noetic Sciences, a company called Entangled is in the process of creating an app that you can download onto your phone.
When downloaded, the app converts hardware functions on your phone into a physical random number generator, which then uses your personal emotions and thoughts to transform into your very own mind meter. The information is then uploaded and fed into a database which is supposed to be able to keep track of what they call a consciousness technology. [3]
smirkPersonally, I’m not all that convinced that an app on my phone would do anything all that useful other than broadcast to some private entity my exact location at all times.
Think about it. Using the phone’s hardware does not allow for the quantum tunneling aspect of randomization. That makes this random number generator NOT a true random generator.
Now, all this is interesting, but it still does not answer my question of HOW generating random numbers translate into louche.
To get at the answer to the connection between these two entities will require that I dig into some geometry of spacetime. It is a big subject, and one that took me awhile to grasp, as I am rather lazy and tend to put off thinking about mathematical constructs until such time as I can no longer put it off and have to think about it.
Since this post is getting rather lengthy, I’ll address the HOW in my next posting.
(Continue to Louche Part 3: Third Density Barrier)
[1] Wikipedia Quantum Tunneling
[2] Institute of Noetic Sciences
[3] Entangle Consciousness App |
44abbe85a8655cd9 | (\376\377\000P\000i\000n\000g\000b\000a\000c\000k\000s) Assume that the variational wave function is a Gaussian of the form Ne (r ) 2; where Nis the normalization constant and is a variational parameter. ISBN 9780122405501, 9780323157476 0000000838 00000 n Note that the best value was obtained for Z=27/16 instead of Z= 2. Chapter 14 illustrates the use of variational methods in quantum mechanics. Introduction. Print Book & E-Book. How does this variational energy compare with the exact ground state energy? The helium atom has two electrons bound to a nucleus with charge Z = 2. trailer << /Size 105 /Info 84 0 R /Encrypt 88 0 R /Root 87 0 R /Prev 185419 /ID[<8c7b44dfda6e475ded266644b4c1926c>] >> startxref 0 %%EOF 87 0 obj << /Type /Catalog /Pages 82 0 R /Metadata 85 0 R /PageLabels 80 0 R >> endobj 88 0 obj << /Filter /Standard /R 3 /O (�'%�d��T%�\).����) /U (�9r�'P�*����m ) /P -1340 /V 2 /Length 128 >> endobj 103 0 obj << /S 738 /L 843 /Filter /FlateDecode /Length 104 0 R >> stream 8.3 Analytic example of variational method - Binding of the deuteron Say we want to solve the problem of a particle in a potential V(r) = −Ae−r/a. endobj ]3 e r=na 2r na l L2l+1 n l l1 2r na Ym( ;˚) (3) and the form of the Bohr radius a: a= 4ˇ 0h¯2 me2 (4) where the e2 in the denominator is the product of the two charges, so it goes over to Ze2 for a hyrdogen-like atom, we can see that the ground state of a hydrogen-like atom (nlm=100) is Variational and perturbative approaches to the confined hydrogen atom with a moving nucleus Item Preview remove-circle Share or Embed This Item. Exercise 2.2: Hydrogen atom Up: Examples of linear variational Previous: Exercise 2.1: Infinite potential Hydrogen atom. Variational calculations for Hydrogen and Helium Recall the variational principle. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary … A. Amer2) 1) Mathematics Department, Faculty of Science, Alexandria University, Alexandria, Egypt E-mail address: sbdoma@yahoo.com 2) Mathematics Department, Faculty of … Stark effect, the Zeeman effect, fine structure, and hyperfine structure, in the hydrogen atom. Abstract: Variational perturbation theory was used to solve the Schrödinger equation for a hydrogen atom confined at the center of an impenetrable cavity. Variational approach to a hydrogen atom in a uniform magnetic field of arbitrary strength M. Bachmann, H. Kleinert, and A. Pelster Institut fu ¨r Theoretische Physik, Freie Univ The experimental data are presented for comparison. Applying the method of Lagrange multipliers to the RR variational principle, we must ex-tremize h jHj i (h j i 1) or Z H d3r Z d3r 1: (1) Taking the variational derivative with respect to we get H = 0. eigenfuctions of the 2D confined hydrogen atom. Trial wave functions depending on the variational parameters are constructed for this purpose. Finally, in Sec. ... the ground-state energy of the hydrogen atom-like system made up of particles 1 and 3, can One example of the variational method would be using the Gaussian function as a trial function for the hydrogen atom ground state. Ground State Energy of the Helium Atom by the Variational Method. 0000002058 00000 n The free complement method for solving the Schrodinger and Dirac equations has been applied to the hydrogen¨ atom in extremely strong magnetic fields. In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states.This allows calculating approximate wavefunctions such as molecular orbitals. Lecture notes Numerical Methods in Quantum Mechanics Corso di Laurea Magistrale in Fisica Interateneo Trieste { Udine Anno accademico 2019/2020 Paolo Giannozzi University of Udine Contains software and material written by Furio Ercolessi1 and Stefano de Gironcoli2 1Formerly at University of Udine 2SISSA - Trieste Last modi ed April 7, 2020 >> One example of the variational method would be using the Gaussian function as a trial function for the hydrogen atom ground state. ... Download PDF . We use neither perturbation nor variational methods for the excited states. 0000034431 00000 n Variational method – The method is based on the variational principle, which says that, if for a system with Hamiltonian H ˆ we calculate the number ε = Φ ∣ H ˆ Φ Φ ∣ Φ, where Φ stands for an arbitrary function, then the number ε ≥ E 0, with E 0 being the ground-state eigenvalue of H ˆ. Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. The basis for this method is the variational principle.. Variational Methods ... and the ψ100(r) hydrogen ground state is often a good choice for radially symmetric, 3-d problems. For very strong fields such as those observed on the surfaces of white dwarf and neutron stars, we calculate the highly accurate non-relativistic and relativistic energies of the hydrogen atom. 1. Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. 4, we give c. Stochastic variational method 80 3. Improved variational method that solves the energy eigenvalue problem of the hydrogen atom. 0000001895 00000 n Variational Methods Michael Fowler 2/28/07 Introduction So far, we have concentrated on problems that were analytically solvable, such as the simple harmonic oscillator, the hydrogen atom, and square well type potentials. /Length 2707 %���� Its polarizability was already calculated by using a simple version of the perturbation theory (p. 743). We know the ground state energy of the hydrogen atom is -1 Ryd, or -13.6 ev. This is a model for the binding energy of a deuteron due to the strong nuclear force, with A=32MeV and a=2.2fm. Keywords: Variational methods, Monte Carlo methods, Atomic structure. 0000034304 00000 n %PDF-1.5 Given a Hamiltonian the method consists This problem could be solved by the variational method by obtaining the energy of as a function of the variational parameter , and then minimizing to … 0000000745 00000 n previous home next PDF. 0000003312 00000 n 7.3 Hydrogen molecule ion A second classic application of the variational principle to quantum mechanics is to the singly-ionized hydrogen molecule ion, H+ 2: Helectron = ~2 2m r2 e2 4ˇ 0 1 r1 + 1 r2! PHY 491: Atomic, Molecular, and Condensed Matter Physics Michigan State University, Fall Semester 2012 Solve by: Wednesday, September 12, 2012 Homework 2 { Solution 2.1. AND B. L. MOISEIWITSCH University College, London (Received 4 August 1950) The variational methods proposed by … The Variational Monte Carlo method 83 7. Variational method – The method is based on the variational principle, which says that, if for a system with Hamiltonian H ˆ we calculate the number ε = Φ ∣ H ˆ Φ Φ ∣ Φ, where Φ stands for an arbitrary function, then the number ε ≥ E 0, with E 0 being the ground-state eigenvalue of H ˆ. Variational Perturbation Theory of the Confined Hydrogen Atom H. E. Montgomery, Jr. Chemistry Department, Centre College, 600 West Walnut Street, Danville, KY 40422-1394, USA. Ground state and excited state energies and expectation values calculated from the perturbation wavefunction are comparable in accuracy to results from direct numerical solution. Application of Variational method,Hydrogen,Helium atom,Comparison with perturbation theory NPTEL IIT Guwahati. 1 0 obj The Helium Atom and Variational Principle: Approximation Methods for Complex Atomic Systems The hydrogen atom wavefunctions and energies, we have seen, are deter-mined as a combination of the various quantum "dynamical" analogues of Variational Methods of Approximation The concept behind the Variational method of approximating solutions to the Schrodinger Equation is based on: a) An educated guess as to the functional form of the wave function. Calculate the ground state energy of a hydrogen atom using the variational principle. DOI: 10.1021/ed2003675. We used the linear variational method with the basis set of a free particle in a circle. Application of variational method for three-color three-photon transitions in hydrogen atom implanted in Debye plasmas November 2009 Physics of Plasmas 16(11):113301-113301-10
variational method hydrogen atom pdf
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ad50bbf08c19f01f | Euclidean quantum gravity
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In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics.
Introduction in layperson's terms[edit]
The Wick rotation[edit]
In physics, a Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to dynamics problems in n dimensions, by transposing their descriptions in n+1 dimensions, by trading one dimension of space for one dimension of time. More precisely, it substitutes a mathematical problem in Minkowski space into a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.
It is called a rotation because when complex numbers are represented as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of \pi/2 about the origin.
For example, a Wick rotation could be used to relate a macroscopic event temperature diffusion (like in a bath) to the underlying thermal movements of molecules. If we attempt to model the bath volume with the different gradients of temperature we would have to subdivide this volume into infinitesimal volumes and see how they interact. We know such infinitesimal volumes are in fact water molecules. If we represent all molecules in the bath by only one molecule in an attempt to simplify the problem, this unique molecule should walk along all possible paths that the real molecules might follow. Path integral formulation is the conceptual tool used to describe the movements of this unique molecule, and Wick rotation is one of the mathematical tools that are very useful to analyse an integral path problem.
Application in quantum mechanics[edit]
In a somewhat similar manner, the motion of a quantum object as described by quantum mechanics implies that it can exist simultaneously in different positions and have different speeds. It differs clearly to the movement of a classical object (e.g. a billiard ball), since in this case a single path with precise position and speed can be described. A quantum object does not move from A to B with a single path, but moves from A to B by all ways possible at the same time. According to the principle of superposition (Richard Feynman's integral of path in 1963), the path of the quantum object is described mathematically as a weighted average of all those possible paths. In 1966 an explicitly gauge invariant functional-integral algorithm was found by DeWitt, which extended Feynman's new rules to all orders. What is appealing in this new approach is its lack of singularities when they are unavoidable in general relativity.
Another operational problem with general relativity is the difficulty to do calculations, because of the complexity of the mathematical tools used. Integral of path in contrast has been used in mechanics since the end of the 19th century and is well known. In addition Path integral is a formalism used both in mechanics and quantum theories so it might be a good starting point for unifying general relativity and quantum theories. Some quantum features like the Schrödinger equation and the heat equation are also related by Wick rotation. So the Wick relation is a good tool to relate a classical phenomena to a quantum phenomena. The ambition of Euclidean quantum gravity is to use the Wick rotation to find connections between a macroscopic phenomenon, gravity, and something more microscopic.
More rigorous treatment[edit]
\int \mathcal{D}\bold{g}\, \mathcal{D}\phi\, \exp\left(\int d^4x \sqrt{|\bold{g}|}(R+\mathcal{L}_\mathrm{matter})\right)
where φ denotes all the matter fields. See Einstein–Hilbert action.
Relation to ADM Formalism[edit]
• Bryce S. DeWitt, Quantum Theory of Gravity - The Manifestly Covariant Theory, Phys. Rev. D 162, 1195 (1967).
• Richard P. Feynman, Lectures on Gravitation, Notes by F.B. Morinigo and W.G. Wagner, CalTech 1963 (Addison Wesley 1995).
• Gary W. Gibbons and Stephen W. Hawking (eds.), Euclidean quantum gravity, World Scientific (1993)
• Herbert W. Hamber, Quantum Gravitation - The Feynman Path Integral Approach, Springer Publishing 2009, ISBN 978-3-540-85293-3.
• Stephen W. Hawking, The Path Integral Approach to Quantum Gravity, in General Relativity - An Einstein Centenary Survey, Cambridge U. Press, 1977.
• James B. Hartle and Stephen W. Hawking, "Wave function of the Universe." Phys. Rev. D 28 (1983) 2960–2975, eprint. Formally relates Euclidean quantum gravity to ADM formalism.
• Martin J.G. Veltman, Quantum Theory of Gravitation, in Methods in Field Theory, Les Houches Session XXVIII, North Holland 1976. |
ecb2cdd4fe955765 | A Reinterpretation of the Copenhagen Interpretation of Quantum Theory
San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley
A Reinterpretation of the Copenhagen
Interpretation of Quantum Theory
Historical Background
In the early 1920's Werner Heisenberg in Copenhagen under the guidance of the venerable Niels Bohr and Max Born and Pascual Jordan of Göttingen University were developing the New Quantum Theory of physics. Heisenberg, Born and Jordan were in their early 20's, the wunderkinder of physics. By 1925 Heisenberg had developed Matrix Mechanics, a marvelous intellectual achievement based upon infinite square matrices. Then in 1926 the Austrian physicist, Erwin Schrödinger, in six journal articles established Wave Mechanics based upon partial differential equations. The wunderkinder of quantum theory were not impressed by Schrödinger, an old man in his late thirties without any previous work in quantum theory and Heisenberg made some disparaging remarks about Wave Mechanics. But Schrödinger produced an article establishing that Wave Mechanics and Matrix Mechanics were equivalent. Wave Mechanics was easier to use and became the dominant approach to quantum theory.
Schrödinger's field had been optics and he had been prompted to start to work in quantum theory by the work of Louis de Broglie which asserted that particles have a wave aspect just as radiation phenomena have a particle aspect. Schrödinger's equations involved an unspecified variable which was called the wave function. He thought that it would have an interpretation similar to such variables involved in optics. However Niels Bohr and the wunderkinder had a different interpretation. Max Born at Göttingen University wrote to Bohr suggesting that the squared magnitude of the wave function in Schrödinger's equation was a probability density function. Bohr replied that he and the other physicists with him in Copenhagen had never considered any other interpretation of the wave function. This interpretation of the wave function became part of what was known as the Copenhagen Interpretation. Erwin Schrödinger did not agree with this interpretation. Bohr had a predelection to emphasize the puzzling aspects of quantum theory. He said something to the effect of:
If you are not shocked by the nature of quantum theory then you do not understand it.
The Copenhagen Interpretation came to mean, among other things, that
Some Simple Terminology
The static appearance of an object is its appearance when it is not moving. The dynamic appearance is that of an object moving so fast that it appears as a blur over its trajectory. This is because any observation involves a time-averaging. The homey example is that of a rapidly rotating fan that appears as a blurred disk.
There is a simple, yet profound, theorem that the expected value of the effect of a charged particle executing a periodic path is the same as that of an object in which the density of the charge is proportional to the time spent in various locations of the path. Charged here could be gravitational mass, electric charge, magnetic charge or charge with respect to the nuclear strong force.
It is very simple to compute the rate of revolution of subatomic particle. For an electron in a hydrogen atom it is about 7 quadrillion times per second. At this frequency any time-averaged observation is then equal to its expected value. Thus an electron revolving about the proton in a hydrogen atom dynamically appears to be a tubular ring. The Copenhagen Interpretation treats this tubular ring as a concatenation of nodules of probability density. But the probability density is the classical time-spent probability from the electron's motion. Equally well the tubular ring could be considered a static object with the properties of the electron smeared throughout its extent in proportion to the time spent by the electron in various parts of the path.
The Correspondence Principle
The Copenhagen Interpretation is largely due to Niels Bohr and Werner Heisenberg. But Bohr also articulated the Correspondence Principle. He said that the validity of classical physics was well established so for a piece of quantum theoretic analysis to be valid its limit when scaled up to the macro level had to be compatible with the classical analysis. It is very important to note that the observable world at the macro level involved averaging over time and space. Physical systems are not observed at instants because no energy can be transferred at an instant. Likewise there can be no observations made at a point in space. Therefore for a quantum analysis to be compatible with the classical analysis at the macro level it must not only be scaled up but also averaged over time or space.
For an example, consider a harmonic oscillator; i.e., a physical system in which the restoring force on a particle is proportional to its deviation from equilibrium. The graph below shows the probability density function for a harmonic oscillator with a principal quantum number of 60.
The heavy line is the probability density function for a classical harmonic oscillator. That probability density is proportional to the reciprocal of the speed of the particle. As can be seen that heavy line is roughly the spatial average of the probability density function derived from the solution of Schrödinger's equation for a harmonic oscillator.
As the energy of the quantum harmonic oscillator increases fluctuations in probability density become more dense and hence no matter how short the interval over which they are averaged there will be some energy level at which the average is equal to the classical time-spent probability density function.
A classical oscillator executing a closed path periodically is a deterministic system but there is still a legitimate probability density function for it which is the probability of finding the particle in some interval ds of its path at a randomly chosen time. The time interval dt spent in a path interval ds about the point s in the path is ds/v(s) where v(s) is the speed of the particle at point s of the path. The probability density function PTS is then given by
PTS(s) = 1/(Tv(s))
where T is the time period for the path; i.e., T=∫ds/v(s).
If the solution to the Schrödinger equation for a physical system gives a probability density function then the limit as the energy increases without bound is also a probability density function. The spatial averaged limit has to also be a probability density function. For compatibility according to the Correspondence Principle that spatially average limit of the quantum system has to be the time-spent probability density function. That indicates that the quantum probability density function from Schrödinger's equation also is in the nature of a time-spent probability density function. This means that the quantum probability density can be translated into the motion of quantum system. This involves sequences of relatively slow movement and then relatively fast movement. The positions of relatively slow movement correspond to what the Copenhagen Interpretation. designates as allowable states and the places of relatively fast movement are what the Copenhagen Interpretation designates as quantum jumps or leaps. When the periodic motion of quantum system is being executed at quadrillions of times per second it may seem like the particle exists simultaneously at multiple locations but that is not the physical reality. It is only the dynamic appearance. A rapidly rotating fan seems to have the fan smeared over a blurred disk.
For one dimensional systems there is no question but that the above is the proper interpretation of wave mechanics. For two and three dimensional systems the situation is murky. The Schrödinger equations for such systems cannot be solved analytically except through resort to the separation-of-variables assumption. But the separation-of-variables assumption is not compatible with a particle having a trajectory.
The Copenhagen Interpretation accepts such solutions and asserts that generally a particle does not exist in the physical world unless it is subjected to a measurement that forces its probability density function to collapse to a point value.
The alternate interpretation is that the solutions developed through the use of the separation-of-variables assumption are not valid quantum analysis.
1. The solutions to Schrödinger's equations correspond to probability density functions. Their spatially-averaged asymptotic limits also correspond to probability density functions. According to the Correspondence Principle these spatially-averaged asymptotic limits must be equal to the classical solution. The only relevant probability density distribution for a deterministic classical situation is the time-spent probability density distribution.
2. At any scale the solutions to Schrödinger's equations correspond to the time-spent probability density distributions and, in effect, correspond to the dynamic appearance of particles in motion. For one dimensional sitations the Copenhagen Interpretation is valid in terms of the dynamic appearance of particles.
3. The conventional solutions for the two and three dimensional cases are derived from the Separation-of-Variables assumption. This assumption is incompatible with particleness and solutions derived from it do not satisfy the Correspondence Principle and therefore are not valid quantum analysis.
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Section 8.2: The Quantum-mechanical Free-particle Solution
Please wait for the animation to completely load.
In order to tackle the free-particle problem, we begin with the Schrödinger equation in one dimension with V(x) = 0,
[−(ħ2/2m)∂2Ψ(x,t)/∂x2 = (∂/∂t)Ψ(x,t) . (8.2)
We can simplify the analysis somewhat by performing a separation of variables and therefore considering the time-independent Schrödinger equation:
−(ħ2/2m)(d2/dx2) ψ(x) = E ψ(x) , (8.3)
which we can rewrite as:
[(d2/dx2) + k2] ψ(x) = 0 , (8.4)
where k2 ≡ 2mE/ħ2. We find the solutions to the equation above are of the form ψ(x) = Aexp(ikx) where we allow k to take both positive and negative values.2 Unlike a bound-state problem, such as the infinite square well, there are no boundary conditions to restrict the k and therefore E values. However, each plane wave has a definite k value and therefore a definite momentum (and also a definite energy) since pψ(x) = −(d/dx) ψ(x) = ħkψ(x), again with k taking on both positive and negative values (so that pψ(x) = ± ħ|k|ψ(x)). The time dependence is now straightforward from the Schrödinger equation:
(∂/∂t) Ψ(x,t) = EΨ(x,t) , (8.6)
or by acting the time-evolution operator, UT(t) = eiHt/ħ, on ψ(x, t). Both procedures yield ψ(x,t) = AeikxiEt/ħ (again k can take positive and negative values) and since E = p2/2m = ħ2k2/2m, we also have that
ψ(k > 0)(x,t) = Aexp(ikxiħk2t/2m) or ψ(k < 0)(x,t) = Aexp(-i|k|xiħk2t/2m), (8.7)
where ħk2/2m ≡ ω. These solutions describe both right-moving (k > 0) and left-moving (k < 0) plane waves. Recall that solutions to the classical wave equation are in the form of f(kx −/+ ωt) for a wave moving to the right (−) or left (+). These quantum-mechanical plane waves, however, are complex functions and can be written in the form f(±|k|x − ωt).
RestartIn the animation, ħ = 2m = 1. A right-moving plane wave is represented in terms of its amplitude and phase (as color) and also its real, cos(kx - ħk2t/2m), and imaginary, sin(kx ħk2t/2m), parts.
What is the velocity of this wave? If this were a classical free particle with non-relativistic velocity, E = mv2/2 = p2/2m and vclassical = p/m as expected. But what about our solution? The velocity of our wave is ω/k which gives: ħk/2m = p/2m, half of the expected (classical) velocity! This velocity is the phase velocity. If instead we consider the group velocity,
vg = ∂ω/∂k, (8.8)
we find that
vg = ∂(ħk2/2m)/∂k = ħk/m, (8.9)
the expected (classical) velocity.
Consider the right-moving wave,
ψ(x,t) = Aexp(ikxiħk2t/2m) ,
which has a definite momentum, p = ħk. We notice that the amplitude of the wave function, A, is a finite constant over all space. However, we also find that ∫ ψ*ψ dx = ∞ [integral from −∞ to +∞] even though ψ*ψ = |A|2 is finite. While the plane wave is a definite-momentum solution to the Schrödinger equation, it is not a localized solution. In this case then, we must discard, or somehow modify, these solutions if we wish a localized and normalized free-particle description.
2The most general solution to the differential equation is
ψ(x) = Aeikx + Beikx (8.5)
with k values positive.
3We can however, box normalize the wave function. In box normalization, we normalize the wave function such that over a finite region of space the wave function is normalized.
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e26265fa3a9903aa | Many-worlds interpretation
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• MWI is not some crazy speculative idea that runs afoul of Occam’s razor. On the contrary, MWI really is just the “obvious, straightforward” reading of quantum mechanics itself, if you take quantum mechanics literally as a description of the whole universe, and assume nothing new will ever be discovered that changes the picture.
• The “many worlds interpretation” seems to me an extravagant, and above all an extravagantly vague, hypothesis. I could almost dismiss it as silly. And yet…It may have something distinctive to say in connection with the “Einstein Podolsky Rosen puzzle,” and it would be worthwhile, I think, to formulate some precise version of it to see if this is really so. And the existence of all possible worlds may make us more comfortable about the existence of our own world…which seems to be in some ways a highly improbable one.
• John S. Bell, "Six possible worlds of quantum mechanics", Proceedings of the Nobel Symposium 65: Possible Worlds in Arts and Sciences. (1986)
• The conclusion, therefore, is that multiple worlds automatically occur in quantum mechanics. They are an inevitable part of the formalism. The only remaining question is: what are you going to do about it? There are three popular strategies on the market: anger, denial, and acceptance.
The “anger” strategy says “I hate the idea of multiple worlds with such a white-hot passion that I will change the rules of quantum mechanics in order to avoid them.” And people do this! In the four options listed here, both dynamical-collapse theories and hidden-variable theories are straightforward alterations of the conventional picture of quantum mechanics. In dynamical collapse, we change the evolution equation, by adding some explicitly stochastic probability of collapse. In hidden variables, we keep the Schrödinger equation intact, but add new variables — hidden ones, which we know must be explicitly non-local. Of course there is currently zero empirical evidence for these rather ad hoc modifications of the formalism, but hey, you never know.
The “denial” strategy says “The idea of multiple worlds is so profoundly upsetting to me that I will deny the existence of reality in order to escape having to think about it.” Advocates of this approach don’t actually put it that way, but I’m being polemical rather than conciliatory in this particular post. And I don’t think it’s an unfair characterization. This is the quantum Bayesianism approach, or more generally “psi-epistemic” approaches. The idea is to simply deny that the quantum state represents anything about reality; it is merely a way of keeping track of the probability of future measurement outcomes. Is the particle spin-up, or spin-down, or both? Neither! There is no particle, there is no spoon, nor is there the state of the particle’s spin; there is only the probability of seeing the spin in different conditions once one performs a measurement. I advocate listening to David Albert’s take at our WSF panel.
The final strategy is acceptance. That is the Everettian approach. The formalism of quantum mechanics, in this view, consists of quantum states as described above and nothing more, which evolve according to the usual Schrödinger equation and nothing more. The formalism predicts that there are many worlds, so we choose to accept that. This means that the part of reality we experience is an indescribably thin slice of the entire picture, but so be it. Our job as scientists is to formulate the best possible description of the world as it is, not to force the world to bend to our pre-conceptions.
So, while most accounts say that Bohr won the debate, my view is that Einstein, as usual, was seeking an explanation of reality, while his rivals were advocating nonsense. Everett’s interpretation doesn’t make Einstein a demigod. But it does make him right.
• In fact the physicists have no good point of view. Somebody mumbled something about a many-world picture, and that many-world picture says that the wave function ψ is what's real, and damn the torpedos if there are so many variables, NR. All these different worlds and every arrangement of configurations are all there just like our arrangement of configurations, we just happen to be sitting in this one. It's possible, but I'm not very happy with it.
• Richard Feynman, "Simulating Physics with Computers", International Journal of Theoretical Physics, volume 21, 1982, p. 467-488
• There is, I think, no sense at all to be made of the splitting of worlds-plus-agents in many worlds. Of course, one can repeat the words over and over until one becomes deaf to the nonsense, but it remains nonsense nevertheless. Curiously, those who favor this interpretation concentrate their defense on dealing with some obvious technical issues: preferred basis, getting the right probabilities via “measures of existence” (or the like), questions of identity and individuation across worlds, and so on. But the fundamental question is just to explain what it means to talk of splitting worlds, and why we should not just write it off, à la Wittgenstein, as language on holiday. (Einstein once described the writings of Hegel as “word-music.” Perhaps that would be a gentler way of dismissing many worlds.)
• Arthur Fine, in M. Schlosshauer (ed.), Elegance and Enigma, The Quantum Interviews (2011)
• The greatest danger I see in the many-worlds/one-Hilbert-space point of view (beside the ridiculous silliness of it all) is the degree to which it is a dead end. The degree to which it is morally bankrupt. Charlie, by thinking that he has taken some of the anthropocentrism out of the picture, has actually emptied the world of all content.
Beyond that though, I think, many-worlds empties the world of content in a way that’s even worse than classical determinism. Let me explain. In my mind, both completely deterministic ontologies and completely indeterministic ones are equally unpalatable. This is because, in both, all our consciousnesses, all our great works of literature, everything that we know, even the coffee maker in my kitchen, are but dangling appendages, illusions. In the first case, the only truth is the Great Initial Condition. In the second, it is the great “I Am That I Am.” But many-worlds compounds that trouble in a far worse fashion by stripping away even those small corners of mystery. It is a world in which anything goes, and everything does. What could be more empty than that?
• Christopher A. Fuchs, Letters to Herb Bernstein, “Epiphenomena Chez Dyer”, 02 August 1999
• It is true that the MWI, in this realist form, avoids some of the paradoxes of QM. The so-called “measurement problem,” for example, is no longer a problem because whenever a measurement occurs, there is no “collapse of the wave function” (or rotation of the state vector in a different terminology). All possible outcomes take place. Schrödinger’s notorious cat is never in a mixed state of alive and dead. It lives in one universe, dies in another. But what a fantastic price is paid for these seeming simplicities! It is hard to imagine a more radical violation of Occam’s razor, the law of parsimony which urges scientists to keep entities to a minimum.
• Martin Gardner, "Multiverses and Blackberries", Skeptical Inquirer (2001)
• The many-worlds theory is incoherent for reasons which have been often pointed out: since there are no frequencies in the theory there is nothing for the numerical predictions of quantum theory to mean. This fact is often disguised by the choice of fortuitous examples. A typical Schrödinger-cat apparatus is designed to yield a 50 percent probability for each of two results, so the “splitting” of the universe in two seems to correspond to the probabilities. But the device could equally be designed to yield a 99 percent probability of one result and 1 percent probability of the other. Again the world “splits” in two; wherein lies the difference between this case and the last?
Defenders of the theory sometimes try to alleviate this difficulty by demonstrating that in the long run (in the limit as one repeats experiments an infinite number of times) the quantum probability assigned to branches in which the observed frequencies match the quantum predictions approaches unity. But this is a manifest petitio principii. If the connection between frequency and quantum “probability” has not already been made, the fact that the assigned “probability” approaches unity cannot be interpreted as approach to certainty of an outcome. All of the branches in which the observed frequency diverges from the quantum predictions still exist, indeed they are certain to exist. It is not highly likely that I will experience one of the frequencies rather than another, it is rather certain that for each possible frequency some descendants of me (descendants through world-splitting) will see it. And in no sense will “more” of my descendants see the right frequency rather than the wrong one: just the opposite is true. So approach of some number to unity cannot help unless the number already has the right interpretation. It is also hard to see how such limiting cases help us: we never get to one since we always live in the short run. If the short-run case can be solved, the theorems about limits are unnecessary; if they can’t be then the theorems are irrelevant.
• Tim Maudlin, Quantum Non-Locality and Relativity (3rd ed., 2011), Introduction
• I regard this last issue as a problem in the interpretation of quantum mechanics, even though I do not believe that consciousness (as a physical phenomenon) collapses (as a physical process) the wave packet (as an objective physical entity). But because I do believe that physics is a tool to help us ŀnd powerful and concise expressions of correlations among features of our experience, it makes no sense to apply quantum mechanics (or any other form of physics) to our very awareness of that experience. Adherents of the many-worlds interpretation make this mistake. So do those who believe that conscious awareness can ultimately be reduced to physics, unless they believe that the reduction will be to a novel form of physics that transcends our current understanding, in which case, as Rudolf Peierls remarked, whether such an explanation should count as “physical” is just a matter of terminology.
• N. David Mermin, in M. Schlosshauer (ed.), Elegance and Enigma, The Quantum Interviews (2011)
• The philosophical moral behind my question is this: once you give up the distinction between actuality and possibility—as the Many Worlds interpretation in effect does, by postulating that all the quantum mechanical possibilities are actualized, each in its own physical universe—once you say that all possible outcomes are, ontologically speaking, equally actual— the notion of ‘probability’ loses all meaning. ‘No collapse and no hidden variables’ is incoherent.
• Hilary Putnam, "A Philosopher Looks at Quantum Mechanics (Again)", The British journal for the philosophy of science 56.4 (2005): 615-634.
• Some very good theorists seem to be happy with an interpretation of quantum mechanics in which the wavefunction only serves to allow us to calculate the results of measurements. But the measuring apparatus and the physicist are presumably also governed by quantum mechanics, so ultimately we need interpretive postulates that do not distinguish apparatus or physicists from the rest of the world, and from which the usual postulates like the Born rule can be deduced. This effort seems to lead to something like a "many worlds" interpretation, which I find repellent. Alternatively, one can try to modify quantum mechanics so that the wavefunction does describe reality, and collapses stochastically and nonlinearly, but this seems to open up the possibility of instantaneous communication. I work on the interpretation of quantum mechanics from time to time, but have gotten nowhere.
• Steven Weinberg, in "Questions and answers with Steven Weinberg", Physics Today (2013)
The debate should already be over. It should have been over fifty years ago. The state of evidence is too lopsided to justify further argument. There is no balance in this issue. There is no rational controversy to teach. The laws of probability theory are laws, not suggestions; there is no flexibility in the best guess given this evidence. Our children will look back at the fact that we were STILL ARGUING about this in the early 21st-century, and correctly deduce that we were nuts.
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1d044d43619b496e | There are plenty of free particles — particles outside any square well —in the universe, and quantum physics has something to say about them. The discussion starts with the Schrödinger equation:
Say you’re dealing with a free particle whose general potential, V(x) = 0. In that case, you’d have the following equation:
And you can rewrite this as
where the wave number, k, is
You can write the general solution to this Schrödinger equation as
If you add time-dependence to the equation, you get this time-dependent wave function:
That’s a solution to the Schrödinger equation, but it turns out to be unphysical. To see this, note that for either term in the equation, you can’t normalize the probability density,
as long as A and B aren’t both equal to zero.
What’s going on here? The probability density for the position of the particle is uniform throughout all x! In other words, you can’t pin down the particle at all.
This is a result of the form of the time-dependent wave function, which uses an exact value for the wave number,
So what that equation says is that you know E and p exactly. And if you know p and E exactly, that causes a large uncertainty in x and t — in fact, x and t are completely uncertain. That doesn’t correspond to physical reality.
For that matter, the wave function
Marylouise, can you format the EQ above as a gif? Thanks, Alexa.
as it stands, isn’t something you can normalize. Trying to normalize the first term, for example, gives you this integral:
EQ needs to be a gif.
Remember that the asterisk symbol (*) means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.
And for the first term of
EQ needs to be a gif.
And the same is true of the second term in |
366f25e685c9e307 | Relational Quantum Mechanics
First published Mon Feb 4, 2002; substantive revision Wed Jan 2, 2008
Relational quantum mechanics is an interpretation of quantum theory which discards the notions of absolute state of a system, absolute value of its physical quantities, or absolute event. The theory describes the way systems affect one another in the course of physical interactions. State and physical quantities refer always to the interaction, or the relation, between two systems. Nevertheless, the theory is assumed to be complete. The physical content of quantum theory is understood as expressing the net of relations connecting all different physical systems.
1. Introduction
Quantum theory is our current general theory of physical motion. The theory is the core component of the momentous change that our understanding of the physical world has undergone during the first decades of the 20th century. It is one of the most successful scientific theories ever: it is supported by vast and unquestionable empirical and technological effectiveness and is today virtually unchallenged. But the interpretation of what the theory actually tells us about the physical world raises a lively debate, which has continued with alternating fortunes, from the early days of the theory in the late twenties, to nowadays. The relational interpretation is an attempt to take the theory at its face value and take seriously the picture of reality it provides. The core idea is to read the theory as a theoretical account of the way distinct physical systems affect one another when they interact (and not of the way physical systems “are”), and the idea that this account exhausts all that can be said about the physical world. The physical world is thus seen as a net of interacting components, where there is no meaning to the state of an isolated system. A physical system (or, more precisely, its contingent state) is reduced to the net of relations it entertains with the surrounding systems, and the physical structure of the world is identified as this net of relationships.
The possibility that the physical content of an empirically successful physical theory could be debated should not surprise: examples abound in the history of science. For instance, the great scientific revolution was fueled by the grand debate on whether the effectiveness of the Copernican system could be taken as an indication that the Earth was not in fact at the center of the universe. In more recent times, Einstein's celebrated first major theoretical success, special relativity, consisted to a large extent just in understanding the physical meaning (simultaneity is relative) of an already existing effective mathematical formalism (the Lorentz transformations). In these cases, as in the case of quantum mechanics, a very strictly empiricist position could have circumvented the problem altogether, by reducing the content of the theory to a list of predicted numbers. But perhaps science can offer us more than such a list; and certainly science needs more than such a list to find its ways.
The difficulty in the interpretation of quantum mechanics derives from the fact that the theory was first constructed for describing microscopic systems (atoms, electrons, photons) and the way these interact with macroscopic apparatuses built to measure their properties. Such interactions are denoted as “measurements”. The theory consists in a mathematical formalism, which allows probabilities of alternative outcomes of such measurements to be calculated. If used just for this purpose, the theory raises no difficulty. But we expect the macroscopic apparatuses themselves—in fact, any physical system in the world—to obey quantum theory, and this seems to raise contradictions in the theory.
1.1 The Problem
In classical mechanics, a system S is described by a certain number of physical variables. For instance, an electron is described by its position and its spin (intrinsic angular momentum). These variables change with time and represent the contingent properties of the system. We say that their values determine, at every moment, the “state” of the system. A measurement of a system's variable is an interaction between the system S and an external system O, whose effect on O, depends on the actual value q of the variable (of S) which is measured. The characteristic feature of quantum mechanics is that it does not allow us to assume that all variables of the system have determined values at every moment (this irrespectively of whether or not we know such values). It was Werner Heisenberg who first realized the need to free ourselves from the belief that, say, an electron has a well determined position at every time. When it is not interacting with an external system that can detect its position, the electron can be “spread out” over different positions. In the jargon of the theory, one says that the electron is in a “quantum superposition” of two (or many) different positions. It follows that the state of the system cannot be captured by giving the value of its variables. Instead, quantum theory introduces a new notion of “state” of a system, which is different from a list of values of its variables. Such a new notion of state was developed in the work of Erwin Schrödinger in the form of the “wave function” of the system, usually denoted by Ψ. Paul Adrien Maurice Dirac gave a general abstract formulation of the notion of quantum state, in terms of a vector Ψ moving in an abstract vector space. The time evolution of the state Ψ is deterministic and is governed by the Schrödinger equation. From the knowledge of the state Ψ, one can compute the probability of the different measurement outcomes q. That is, the probability of the different ways in which the system S can affect a system O in an interaction with it. The theory then prescribes that at every such ‘measurement’, one must update the value of Ψ, to take into account which of the different outcomes has happened. This sudden change of the state Ψ depends on the specific outcome of the measurement and is therefore probabilistic. It is called the “collapse of the wave function”.
The problem of the interpretation of quantum mechanics takes then different forms, depending on the relative ontological weight we choose to assign to the wave function Ψ or, respectively, to the sequence of the measurement outcomes q, q′, q″, …. If we take Ψ as the “real” entity which fully represents the actual state of affairs of the world, we encounter a number of difficulties. First, we have to understand how Ψ can change suddenly in the course of a measurement: if we describe the evolution of two interacting quantum systems in terms of the Schrödinger equation, no collapse happens. Furthermore, the collapse, seen as a physical process, seems to depend on arbitrary choices in our description and shows a disturbing amount of nonlocality. But even if we can circumvent the collapse problem, the more serious difficulty of this point of view is that it appears to be impossible to understand how specific observed values q, q′, q″, … can emerge from the same Ψ. A better alternative is to take the observed values q, q′, q″, … as the actual elements of reality, and view Ψ just as a bookkeeping device, determined by the actual values q, q′, q″, … that happened in past. From this perspective, the real events of the world are the “realization” (the “coming to reality”, the “actualization”) of the values q, q′, q″, … in the course of the interaction between physical systems. This actualization of a variable q in the course of an interaction can be denoted as the quantum event q. An exemple of a quantum event is the detection of an electron in a certain position. The position variable of the electron assumes a determined value in the course of the interaction between the electron and an external system and the quantum event is the “manifestation” of the electron in a certain position. Quantum events have an intrinsically discrete (“quantized”) granular structure.
The difficulty of this second option is that if we take the quantum nature of all physical systems into account, the statement that a certain specific event q “has happened” (or, equivalently that a certain variable has or has not taken the value q) can be true and not-true at the same time. To clarify this key point, consider the case in which a system S interacts with another system (an apparatus) O, and exhibits a value q of one of its variables. Assume that the system O obeys the laws of quantum theory as well, and use the quantum theory of the combined system formed by O and S in order to predict the way this combined system can later interact with a third system O′. Then quantum mechanics forbids us to assume that q has happened. Indeed, as far as its later behavior is concerned, the combined system S+O may very well be in a quantum superposition of alternative possible values q, q′, q″, …. This “second observer” situation captures the core conceptual difficulty of the interpretation of quantum mechanics: reconciling the possibility of quantum superposition with the fact that the observed world is characterized by uniquely determined events q, q′, q″, …. More precisely, it shows that we cannot disentangle the two: according to the theory an observed quantity (q) can be at the same time determined and not determined. An event may have happened and at the same time may not have happened.
2. Relational view of quantum states
The way out from this dilemma suggested by the relational interpretations is that the quantum events, and thus the values of the variables of a physical system S, namely the q's, are relational. That is, they do not express properties of the system S alone, but rather refer to the relation between two systems.
The best developed of these interpretations is relational quantum mechanics (Rovelli 1996, 1997). For evaluations and critical account of this view of quantum theory, see for instance van Fraassen (2010) and Bitbol (2007). The central tenet of relational quantum mechanics is that there is no meaning in saying that a certain quantum event has happened or that a variable of the system S has taken the value q: rather, there is meaning in saying that the event q has happened or the variable has taken the value q for O, or with respect to O. The apparent contradiction between the two statements that a variable has or hasn't a value is resolved by indexing the statements with the different systems with which the system in question interacts. If I observe an electron at a certain position, I cannot conclude that the electron is there: I can only conclude that the electron as seen by me is there. Quantum events only happen in interactions between systems, and the fact that a quantum event has happened is only true with respect to the systems involved in the interaction. The unique account of the state of the world of the classical theory is thus fractured into a multiplicity of accounts, one for each possible “observing” physical system. In the words of Rovelli (1996): “Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world”.
This relativisation of actuality is viable thanks to a remarkable property of the formalism of quantum mechanics. John von Neumann was the first to notice that the formalism of the theory treats the measured system (S ) and the measuring system (O) differently, but the theory is surprisingly flexible on the choice of where to put the boundary between the two. Different choices give different accounts of the state of the world (for instance, the collapse of the wave function happens at different times); but this does not affect the predictions on the final observations. Von Neumann only described a rather special situation, but this flexibility reflects a general structural property of quantum theory, which guarantees the consistency among all the distinct “accounts of the world” of the different observing systems. The manner in which this consistency is realized, however, is subtle.
What appears with respect to O as a measurement of the variable q (with a specific outcome), appears with respect to O′ simply as the establishing of a correlation between S and O (without any specific outcome). As far as the observer O is concerned, a quantum event has happened and a property q of a system S has taken a certain value. As far as the second observer O′ is concerned, the only relevant element of reality is that a correlation is established between S and O. This correlation will manifest itself only in any further observation that O′ would perform on the S+O system. Up to the time in which it physically interacts with S+O, the system O′ has no access to the actual outcomes of the measurements performed by O on S . This actual outcome is real only with respect to O (Rovelli 1996, pp. 1650–52). Consider for instance a two-state system O (say, a light-emitting diode, or l.e.d., which can be on or off) interacting with a two-state system S (say, the spin of an electron, which can be up or down). Assume the interaction is such that if the spin is up (down) the l.e.d. goes on (off). To start with, the electron can be in a superposition of its two states. In the account of the state of the electron that we can associate with the l.e.d., a quantum event happens in the interaction, the wave function of the electron collapses to one of two states, and the l.e.d. is then either on or off. But we can also consider the l.e.d./electron composite system as a quantum system and study the interactions of this composite system with another system O′. In the account associated to O′, there is no event and no collapse at the time of the interaction, and the composite system is still in the superposition of the two states [spin up/l.e.d. on] and [spin down/l.e.d. off] after the interaction. It is necessary to assume this superposition because it accounts for measurable interference effects between the two states: if quantum mechanics is correct, these interference effects are truly observable by O′. So, we have two discordant accounts of the same events. Can the two discord accounts be compared and does the comparison lead to contradiction? They can be compared, because the information on the first account is stored in the state of the l.e.d. and O′ has access to this information. Therefore O and O′ can compare their accounts of the state of the world.
However, the comparison does not lead to contradiction because the comparison is itself a physical process that must be understood in the context of quantum mechanics. Indeed, O′ can physically interact with the electron and then with the l.e.d. (or, equivalently, the other way around). If, for instance, he finds the spin of the electron up, quantum mechanics predicts that he will then consistently find the l.e.d. on (because in the first measurement the state of the composite system collapses on its [spin up/l.e.d. on] component). That is, the multiplicity of accounts leads to no contradiction precisely because the comparison between different accounts can only be a physical quantum interaction. This internal self-consistency of the quantum formalism is general, and it is perhaps its most remarkable aspect. This self consistency is taken in relational quantum mechanics as a strong indication of the relational nature of the world.
In fact, one may conjecture that this peculiar consistency between the observations of different observers is the missing ingredient for a reconstruction theorem of the Hilbert space formalism of quantum theory. Such a reconstruction theorem is still unavailable: On the basis of reasonable physical assumptions, one is able to derive the structure of an orthomodular lattice containing subsets that form Boolean algebras, which “almost”, but not quite, implies the existence of a Hilbert space and its projectors' algebra (see the entry quantum logic and probability theory.) Perhaps an appropriate algebraic formulation of the condition of consistency between subsystems could provide the missing hypothesis to complete the reconstruction theorem.
Bas van Fraassen has given an extensive critical discussion of this interpretation; he has also suggested an improvement, in the form of an additional postulate weakly relating the description of the same system given by different observers (van Fraassen 2010). Michel Bitbol has analyzed the relational interpretation of quantum mechanics from a Kantian perspective, substituting functional reference frames for physical (or naturalized) observers (Bitbol 2007).
3. Correlations
The conceptual relevance of correlations in quantum mechanics,—a central aspect of relational quantum mechanics—is emphasized by David Mermin, who analyses the statistical features of correlation (Mermin 1998), and arrives at views close to the relational ones. Mermin points out that a theorem on correlations in Hilbert space quantum mechanics is relevant to the problem of what exactly quantum theory tells us about the physical world. Consider a quantum system S with internal parts s, s′,…, that may be considered as subsystems of S , and define the correlations among subsystems as the expectation values of products of subsystems' observables. It can be proved that, for any resolution of S into subsystems, the subsystems' correlations determine uniquely the state of S. According to Mermin, this theorem highlights two major lessons that quantum mechanics teaches us: first, the relevant physics of S is entirely contained in the correlations both among the s, s′,…, themselves (internal correlations) and among the s′,…, and other systems (external correlations); second, correlations may be ascribed physical reality whereas, according to well-known ‘no-go’ theorems, the quantities that are the terms of the correlations cannot (Mermin 1998).
4. Self-reference and self-measurement
From a relational point of view, the properties of a system exists only in reference to another system. What about the properties of a system with respect to itself? Can a system measure itself? Is there any meaning of the correlations of a system with itself? Implicit in the relational point of view is the intuition that a complete self-measurement is impossible. It is this impossibility that forces all properties to be referred to another system. The issue of the self-measurement has been analyzed in details in two remarkable works, from very different perspectives, but with similar conclusions, by Marisa Dalla Chiara and by Thomas Breuer.
4.1 Logical aspect of the measurement problem
Marisa Dalla Chiara (1977) has addressed the logical aspect of the measurement problem. She observes that the problem of self-measurement in quantum mechanics is strictly related to the self-reference problem, which has an old tradition in logic. From a logical point of view the measurement problem of quantum mechanics can be described as a characteristic question of “semantical closure” of a theory. To what extent can quantum mechanics apply consistently to the objects and the concepts in terms of which its metatheory is expressed? Dalla Chiara shows that the duality in the description of state evolution, encoded in the ordinary (i.e. von Neumann's) approach to the measurement problem, can be given a purely logical interpretation: “If the apparatus observer O is an object of the theory, then O cannot realize the reduction of the wave function. This is possible only to another O′, which is ‘external’ with respect to the universe of the theory. In other words, any apparatus, as a particular physical system, can be an object of the theory. Nevertheless, any apparatus which realizes the reduction of the wave function is necessarily only a metatheoretical object ” (Dalla Chiara 1977, p. 340). This observation is remarkably consistent with the way in which the state vector reduction is justified within the relational interpretation of quantum mechanics. When the system S+O is considered from the point of view of O′, the measurement can be seen as an interaction whose dynamics is fully unitary, whereas by the point of view of O the measurement breaks the unitarity of the evolution of S. The unitary evolution does not break down through mysterious physical jumps, due to unknown effects, but simply because O is not giving a full dynamical description of the interaction. O cannot have a full description of the interaction of S with himself (O), because his information is correlation information and there is no meaning in being correlated with oneself. If we include the observer into the system, then the evolution is still unitary, but we are now dealing with the description of a different observer.
4.2 Impossibility of complete self-measurement
As is well known, from a purely logical point of view self-reference properties in formal systems impose limitations on the descriptive power of the systems themselves. Thomas Breuer has shown that, from a physical point of view, this feature is expressed by the existence of limitations in the universal validity of physical theories, no matter whether classical or quantum (Breuer 1995). Breuer studies the possibility for an apparatus O to measure its own state. More precisely, of measuring the state of a system containing an apparatus O. He defines a map from the space of all sets of states of the apparatus to the space of all sets of states of the system. Such a map assigns to every set of apparatus states the set of system states that is compatible with the information that—after the measurement interaction—the apparatus is in one of these states. Under reasonable assumptions on this map, Breuer is able to prove a theorem stating that no such map can exist that can distinguish all the states of the system. An apparatus O cannot distinguish all the states of a system S containing O. This conclusion holds irrespective of the classical or quantum nature of the systems involved, but in the quantum context it implies that no quantum mechanical apparatus can measure all the quantum correlations between itself and an external system. These correlations are only measurable by a second external apparatus, observing both the system and the first apparatus.
5. Other relational views
5.1 Quantum reference systems
A relational view of quantum mechanics has been proposed also by Gyula Bene (1997). Bene argues that quantum states are relative in the sense that they express a relation between a system to be described and a different system, containing the former as a subsystem and acting for it as a quantum reference system (here the system is contained in the reference system, while in Breuer's work the system contains the apparatus). Consider again a measuring system (O) that has become entangled with a measured system (S ) during a measurement. Once again, the difficulty of quantum theory is that there is an apparent contradiction between the fact that the quantity q of the system assumes an observed value in the measurement, while the composite S+O system still has to be considered in a superposition state, if we want to properly predict the outcome of measurements on the S+O system. This apparent contradiction is resolved by Bene by relativizing the state not to an observer, as in the relational quantum mechanics sketched in Section 2, but rather to a relevant composite system. That is: there is a state of the system S relative to S alone, and a state of the system S relative to the S+O composite system. (Similarly, there is a state of the system O relative to itself alone, and a state of the system O relative to the S+O ensemble.) The ensemble with respect to which the state is defined is called by Bene the quantum reference system . The state of a system with respect to a given quantum reference system correctly predicts the probability distributions of any measurement on the entire reference system. This dependence of the states of quantum systems from different quantum systems that act as reference systems is viewed as a fundamental property that holds no matter whether a system is observed or not.
5.2 Sigma algebra of interactive properties
Similar views have been expressed by Simon Kochen in unpublished but rather well-known notes (Kochen 1979, preprint). In Kochen's words: “The basic change in the classical framework which we advocate lies in dropping the assumption of the absoluteness of physical properties of interacting systems… Thus quantum mechanical properties acquire an interactive or relational character.” Kochen uses a σ-algebra formalism. Each quantum system has an associated Hilbert space. The properties of the system are established by its interaction with other quantum systems, and these properties are represented by the corresponding projection operators on the Hilbert space. These projectors are elements of a Boolean σ-algebra, determined by the physics of the interaction between the two systems. Suppose a quantum system S can interact with quantum systems Q, Q′,…. In each case, S will acquire an interaction σ-algebra of properties σ(Q), σ(Q′) since the interaction between S and Q may be finer grained than the interaction between S and Q′. Thus, interaction σ-algebras may have non-trivial intersections. The family of all Boolean σ-algebras forms a category, with the sets of the projectors of each σ-algebra as objects. In Kochen's words: “Just as the state of a composite system does not determine states of its components, conversely, the states of the… correlated systems do not determine the state of the composite system […] We thus resolve the measurement problem by cutting the Gordian knot tying the states of component systems uniquely to the state of the combined system.” This is very similar in spirit to the Bene approach and to Rovelli's relational quantum mechanics, but the precise technical relation between the formalisms utilized in these approaches has not yet been analysed in full detail.
Further approaches at least formally related to Kochen's have been proposed by Healey (1989), who also emphasises an interactive aspect of his approach, and by Dieks (1989). See also the entry on modal interpretations of quantum mechanics.
5.3 Quantum theory of the universe
Relational views on quantum theory have been defended also by Lee Smolin (1995) and by Louis Crane (1995) in a cosmological context. If one is interested in the quantum theory of the entire universe, then, by definition, an external observer is not available. Breuer's theorem shows then that a quantum state of the universe, containing all correlations between all subsystems, expresses information that is not available, not even in principle, to any observer. In order to write a meaningful quantum state, argue Crane and Smolin, we have to divide the universe in two components and consider the relative quantum state predicting the outcomes of the observations that one component can make on the other.
5.4 Relation with Everett's relative-state interpretation
Relational ideas underlie also the interpretations of quantum theory inspired by the work of Everett. Everett’ original work (Everett 1975) relies on the notion of “relative state” and has a marked relational tone (see the entry on Everett's relative-state formulation of quantum). In the context of Everettian accounts, a state may be taken as relative either (more commonly) to a “world”, or “branch”, or (sometimes) to the state of another system (see for instance Saunders 1996, 1998). While the first variant (relationalism with respect to branches) is far from the relational views described here, the second variant (relationalism with respect to the state of a system) is closer.
However, it is different to say that something is relative to a system or that something is relative to a state of a system. Consider for instance the situation described in the example of Section 5: According to the relational interpretation, after the first measurement the quantity q has a given value and only one for O, while in Everettian terms the quantity q has a value for one state of O and a different value for another state of O, and the two are equally real. In Everett, there is an ontological multiplicity of realities, which is absent in the relational point of view, where physisical quantities are uniquely determined, once two systems are given.
The difference derives from a very general interpretational difference between Everettian accounts and the relational point of view. Everett (at least in its widespread version) takes the state Ψ as the basis of the ontology of quantum theory. The overall state Ψ includes different possible branches and different possible outcomes. On the other hand, the relational interpretation takes the quantum events q, that is, the actualizations of values of physical quantities, as the basic elements of reality (see Section 1.1 above) and such q's are assumed to be univocal. The relational view avoids the traditional difficulties in taking the q's as univocal simply by noticing that a q does not refer to a system, but rather to a pair of systems.
For a comparison between the relational interpretation and other current interpretations of quantum mechanics, see Rovelli 1996.
6. Some consequences of the relational point of view
A number of open conceptual issues in quantum mechanics appear in a different light when seen in the context of a relational interpretation of the theory. In particular, the Einstein-Podolsky-Rosen (EPR) correlations have a substantially different interpretation within the perspective of the relational interpretation of quantum mechanics. Laudisa (2001) has argued that the non-locality implied by the conventional EPR argument turns out to be frame-dependent, and this result supports the “peaceful coexistence” of quantum mechanics and special relativity. More radically, Rovelli and Smerlak (2006) argue that these correlations do not entail any form of “non-locality”, when viewed in the context of this interpretation, essentially because there is a quantum event relative to an observer that happens at a spacelike separation from this observer. The abandonment of strict Einstein realism implied by the relational stance permits to reconcile quantum mechanics, completeness and locality.
Also, the relational interpretation allows one to give a precise definition of the time (or, better, the probability distribution of the time) at which a measurement happens, in terms of the probability distribution of the correlation between system and apparatus, as measurable by a third observer (Rovelli 1998).
Finally, it has been suggested in Rovelli (1997) that the relationalism at the core of quantum theory pointed out by the relational interpretations might be connected with the spatiotemporal relationalism that characterizes general relativity. Quantum mechanical relationalism is the observation that there are no absolute properties: properties of a system S are relative to another system O with which S is interacting. General relativistic relationalism is the well known observation that there is no absolute localization in spacetime: localization of an object S in spacetime is only relative to the gravitational field, or to any other object O, to which S is contiguous. There is a connection between the two, since interaction between S and O implies contiguity and contiguity between S and O can only be checked via some quantum interaction. However, because of the difficulty of developing a consistent and conceptually transparent theory of quantum gravity, so far this suggestion has not been developed beyond the stage of a simple intuition.
7. Conclusion
Relational interpretations of quantum mechanics propose a solution to the interpretational difficulties of quantum theory based on the idea of weakening the notions of the state of a system, event, and the idea that a system, at a certain time, may just have a certain property. The world is described as an ensemble of events (“the electron is the point x”) which happen only relatively to a given observer. Accordingly, the state and the properties of a system are relative to another system only. There is a wide diversity in style, emphasis, and language in the authors that we have mentioned. Indeed, most of the works mentioned have developed independently from each other. But it is rather clear that there is a common idea underlying all these approaches, and the convergence is remarkable.
Werner Heisenberg first recognized that the electron does not have a well defined position when it is not interacting. Relational interpretations push this intuition further, by stating that, even when interacting, the position of the electron is only determined in relation to a certain observer, or to a certain quantum reference system, or similar.
In physics, the move of deepening our insight into the physical world by relativizing notions previously used as absolute has been applied repeatedly and very successfully. Here are a few examples. The notion of the velocity of an object has been recognized as meaningless, unless it is indexed with a reference body with respect to which the object is moving. With special relativity, simultaneity of two distant events has been recognized as meaningless, unless referred to a specific state of motion of something. (This something is usually denoted as “the observer” without, of course, any implication that the observer is human or has any other peculiar property besides having a state of motion. Similarly, the “observer system” O in quantum mechanics need not to be human or have any other property beside the possibility of interacting with the “observed” system S.) With general relativity, the position in space and time of an object has been recognized as meaningless, unless it is referred to the gravitational field, or to some other dynamical physical entity. The move proposed by the relational interpretations of quantum mechanics has strong analogies with these, but is, in a sense, a longer jump, since all physical events and the entirety of the contingent properties of any physical system are taken to be meaningful only as relative to a second physical system. The claim of the relational interpretations is that this is not an arbitrary move. Rather, it is a conclusion which is difficult to escape, following from the observation—explained above in the example of the “second observer”—that a variable (of a system S) can have a well determined value q for one observer (O) and at the same time fail to have a determined value for another observer (O′).
This way of thinking the world has certainly heavy philosophical implications. The claim of the relational interpretations is that it is nature itself that is forcing us to this way of thinking. If we want to understand nature, our task is not to frame nature into our philosophical prejudices, but rather to learn how to adjust our philosophical prejudices to what we learn from nature.
• Bene, G., 1997, “Quantum reference systems: a new framework for quantum mechanics”, Physica, A242: 529–560.
• Breuer, T., 1993, “The impossibility of accurate state self-measurements”, Philosophy of Science, 62: 197-214.
• Crane, L., 1993, “Clock and Category: Is Quantum Gravity Algebraic?”, Journal of Mathematical Physics, 36: 6180–6193.
• Dalla Chiara, M.L., 1977, “Logical self-reference, set theoretical paradoxes and the measurement problem in quantum mechanics”, Journal of Philosophical Logic, 6: 331–347.
• Everett H., 1957, “‘Relative State’ Formulation of Quantum Mechanics,” Reviews of Modern Physics, 29: 454–462.
• Dieks, D., 1989, “Resolution of the Measurement Problem through Decoherence of the Quantum State”, Physics letters, A142: 439–446.
• Kochen, S., 1979, “The interpretation of quantum mechanics”, Princeton: Princeton University Preprint.
• Laudisa, F., 2001, “The EPR Argument in a Relational Interpretation of Quantum Mechanics”, Foundations of Physics Letters, 14: 119–132.
• Mermin, N.D., 1998, “What is quantum mechanics trying to tell us?”, American Journal of Physics, 66: 753–767.
• Rovelli, C., 1996, “Relational quantum mechanics”, International Journal of Theoretical Physics, 35: 1637–1678.
• –––, 1997, “Half way through the woods”, in J. Earman and J.D. Norton (eds.), The Cosmos of Science, Pittsburgh: University of Pittsburgh Press.
• –––, 1998, “‘Incerto tempore, incertisque loci’: Can we compute the exact time at which a quantum measurement happens?”, Foundations of Physics, 28: 1031–1043.
• Rovelli, C., and Smerlak, M., 2007, “Relational EPR”, Foundations of Physics, 37: 427–445.
• Saunders, S., 1996, “Relativism”, in R. Clifton (ed.), Perspectives on Quantum Reality, Dordrecht: Kluwer.
• –––, 1998, “Time, quantum mechanics and probability”, Synthese, 114: 373–404.
• van Fraassen, B., 2010, “Rovelli's World”, Foundations of Physics 40(4): 390–417.
Other Internet Resources
• Bitbol, M., “Physical Relations or Functional Relations? A non-metaphysical construal of Rovelli's Relational Quantum Mechanics”, Philosophy of Science Archive: (2007).
• Smolin, L., “The Bekenstein bound, topological quantum field theory and pluralistic quantum field theory”, Penn State preprint CGPG-95/8-7, 1995, Los Alamos Archive gr-qc/9508064.
[Please contact the authors with further suggestions.]
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300cf102a2788f9e | Physics, Bernoulli, Daniel: model of gas pressure [Credit: Encyclopædia Britannica, Inc.; based on Daniel Bernoulli, Hydrodynamica (1738)]Bernoulli, Daniel: model of gas pressureEncyclopædia Britannica, Inc.; based on Daniel Bernoulli, Hydrodynamica (1738)science that deals with the structure of matter and the interactions between the fundamental constituents of the observable universe. In the broadest sense, physics (from the Greek physikos) is concerned with all aspects of nature on both the macroscopic and submicroscopic levels. Its scope of study encompasses not only the behaviour of objects under the action of given forces but also the nature and origin of gravitational, electromagnetic, and nuclear force fields. Its ultimate objective is the formulation of a few comprehensive principles that bring together and explain all such disparate phenomena.
The scope of physics
Hooke, Robert: Hooke’s law of elasticity of materials [Credit:]Hooke, Robert: Hooke’s law of elasticity of is generally taken to mean the study of the motion of objects (or their lack of motion) under the action of given forces. Classical mechanics is sometimes considered a branch of applied mathematics. It consists of kinematics, the description of motion, and dynamics, the study of the action of forces in producing either motion or static equilibrium (the latter constituting the science of statics). The 20th-century subjects of quantum mechanics, crucial to treating the structure of matter, subatomic particles, superfluidity, superconductivity, neutron stars, and other major phenomena, and relativistic mechanics, important when speeds approach that of light, are forms of mechanics that will be discussed later in this section.
In classical mechanics the laws are initially formulated for point particles in which the dimensions, shapes, and other intrinsic properties of bodies are ignored. Thus in the first approximation even objects as large as the Earth and the Sun are treated as pointlike—e.g., in calculating planetary orbital motion. In rigid-body dynamics, the extension of bodies and their mass distributions are considered as well, but they are imagined to be incapable of deformation. The mechanics of deformable solids is elasticity; hydrostatics and hydrodynamics treat, respectively, fluids at rest and in motion.
The three laws of motion set forth by Isaac Newton form the foundation of classical mechanics, together with the recognition that forces are directed quantities (vectors) and combine accordingly. The first law, also called the law of inertia, states that, unless acted upon by an external force, an object at rest remains at rest, or if in motion, it continues to move in a straight line with constant speed. Uniform motion therefore does not require a cause. Accordingly, mechanics concentrates not on motion as such but on the change in the state of motion of an object that results from the net force acting upon it. Newton’s second law equates the net force on an object to the rate of change of its momentum, the latter being the product of the mass of a body and its velocity. Newton’s third law, that of action and reaction, states that when two particles interact, the forces each exerts on the other are equal in magnitude and opposite in direction. Taken together, these mechanical laws in principle permit the determination of the future motions of a set of particles, providing their state of motion is known at some instant, as well as the forces that act between them and upon them from the outside. From this deterministic character of the laws of classical mechanics, profound (and probably incorrect) philosophical conclusions have been drawn in the past and even applied to human history.
Lying at the most basic level of physics, the laws of mechanics are characterized by certain symmetry properties, as exemplified in the aforementioned symmetry between action and reaction forces. Other symmetries, such as the invariance (i.e., unchanging form) of the laws under reflections and rotations carried out in space, reversal of time, or transformation to a different part of space or to a different epoch of time, are present both in classical mechanics and in relativistic mechanics, and with certain restrictions, also in quantum mechanics. The symmetry properties of the theory can be shown to have as mathematical consequences basic principles known as conservation laws, which assert the constancy in time of the values of certain physical quantities under prescribed conditions. The conserved quantities are the most important ones in physics; included among them are mass and energy (in relativity theory, mass and energy are equivalent and are conserved together), momentum, angular momentum, and electric charge.
The study of gravitation
Laser Interferometer Space Antenna [Credit: Encyclopædia Britannica, Inc.]Laser Interferometer Space AntennaEncyclopædia Britannica, Inc.This field of inquiry has in the past been placed within classical mechanics for historical reasons, because both fields were brought to a high state of perfection by Newton and also because of its universal character. Newton’s gravitational law states that every material particle in the universe attracts every other one with a force that acts along the line joining them and whose strength is directly proportional to the product of their masses and inversely proportional to the square of their separation. Newton’s detailed accounting for the orbits of the planets and the Moon, as well as for such subtle gravitational effects as the tides and the precession of the equinoxes (a slow cyclical change in direction of the Earth’s axis of rotation) through this fundamental force was the first triumph of classical mechanics. No further principles are required to understand the principal aspects of rocketry and space flight (although, of course, a formidable technology is needed to carry them out).
curvature: curved space-time [Credit: Encyclopædia Britannica, Inc.]curvature: curved space-timeEncyclopædia Britannica, Inc.The modern theory of gravitation was formulated by Albert Einstein and is called the general theory of relativity. From the long-known equality of the quantity “mass” in Newton’s second law of motion and that in his gravitational law, Einstein was struck by the fact that acceleration can locally annul a gravitational force (as occurs in the so-called weightlessness of astronauts in an Earth-orbiting spacecraft) and was led thereby to the concept of curved space-time. Completed in 1915, the theory was valued for many years mainly for its mathematical beauty and for correctly predicting a small number of phenomena, such as the gravitational bending of light around a massive object. Only in recent years, however, has it become a vital subject for both theoretical and experimental research. (Relativistic mechanics refers to Einstein’s special theory of relativity, which is not a theory of gravitation.)
The study of heat, thermodynamics, and statistical mechanics
Heat is a form of internal energy associated with the random motion of the molecular constituents of matter or with radiation. Temperature is an average of a part of the internal energy present in a body (it does not include the energy of molecular binding or of molecular rotation). The lowest possible energy state of a substance is defined as the absolute zero (−273.15 °C, or −459.67 °F) of temperature. An isolated body eventually reaches uniform temperature, a state known as thermal equilibrium, as do two or more bodies placed in contact. The formal study of states of matter at (or near) thermal equilibrium is called thermodynamics; it is capable of analyzing a large variety of thermal systems without considering their detailed microstructures.
First law
The first law of thermodynamics is the energy conservation principle of mechanics (i.e., for all changes in an isolated system, the energy remains constant) generalized to include heat.
Second law
The second law of thermodynamics asserts that heat will not flow from a place of lower temperature to one where it is higher without the intervention of an external device (e.g., a refrigerator). The concept of entropy involves the measurement of the state of disorder of the particles making up a system. For example, if tossing a coin many times results in a random-appearing sequence of heads and tails, the result has a higher entropy than if heads and tails tend to appear in clusters. Another formulation of the second law is that the entropy of an isolated system never decreases with time.
Third law
The third law of thermodynamics states that the entropy at the absolute zero of temperature is zero, corresponding to the most ordered possible state.
Statistical mechanics
Brownian motion [Credit: ]Brownian motionThe science of statistical mechanics derives bulk properties of systems from the mechanical properties of their molecular constituents, assuming molecular chaos and applying the laws of probability. Regarding each possible configuration of the particles as equally likely, the chaotic state (the state of maximum entropy) is so enormously more likely than ordered states that an isolated system will evolve to it, as stated in the second law of thermodynamics. Such reasoning, placed in mathematically precise form, is typical of statistical mechanics, which is capable of deriving the laws of thermodynamics but goes beyond them in describing fluctuations (i.e., temporary departures) from the thermodynamic laws that describe only average behaviour. An example of a fluctuation phenomenon is the random motion of small particles suspended in a fluid, known as Brownian motion.
Quantum statistical mechanics plays a major role in many other modern fields of science, as, for example, in plasma physics (the study of fully ionized gases), in solid-state physics, and in the study of stellar structure. From a microscopic point of view the laws of thermodynamics imply that, whereas the total quantity of energy of any isolated system is constant, what might be called the quality of this energy is degraded as the system moves inexorably, through the operation of the laws of chance, to states of increasing disorder until it finally reaches the state of maximum disorder (maximum entropy), in which all parts of the system are at the same temperature, and none of the state’s energy may be usefully employed. When applied to the universe as a whole, considered as an isolated system, this ultimate chaotic condition has been called the “heat death.”
The study of electricity and magnetism
Although conceived of as distinct phenomena until the 19th century, electricity and magnetism are now known to be components of the unified field of electromagnetism. Particles with electric charge interact by an electric force, while charged particles in motion produce and respond to magnetic forces as well. Many subatomic particles, including the electrically charged electron and proton and the electrically neutral neutron, behave like elementary magnets. On the other hand, in spite of systematic searches undertaken, no magnetic monopoles, which would be the magnetic analogues of electric charges, have ever been found.
The field concept plays a central role in the classical formulation of electromagnetism, as well as in many other areas of classical and contemporary physics. Einstein’s gravitational field, for example, replaces Newton’s concept of gravitational action at a distance. The field describing the electric force between a pair of charged particles works in the following manner: each particle creates an electric field in the space surrounding it, and so also at the position occupied by the other particle; each particle responds to the force exerted upon it by the electric field at its own position.
ultraviolet radiation: types of electromagnetic radiation [Credit: Encyclopædia Britannica, Inc.]ultraviolet radiation: types of electromagnetic radiationEncyclopædia Britannica, Inc.Classical electromagnetism is summarized by the laws of action of electric and magnetic fields upon electric charges and upon magnets and by four remarkable equations formulated in the latter part of the 19th century by the Scottish physicist James Clerk Maxwell. The latter equations describe the manner in which electric charges and currents produce electric and magnetic fields, as well as the manner in which changing magnetic fields produce electric fields, and vice versa. From these relations Maxwell inferred the existence of electromagnetic waves—associated electric and magnetic fields in space, detached from the charges that created them, traveling at the speed of light, and endowed with such “mechanical” properties as energy, momentum, and angular momentum. The light to which the human eye is sensitive is but one small segment of an electromagnetic spectrum that extends from long-wavelength radio waves to short-wavelength gamma rays and includes X-rays, microwaves, and infrared (or heat) radiation.
diffraction grating [Credit: Courtesy of Bausch & Lomb, Rochester, N.Y.]diffraction gratingCourtesy of Bausch & Lomb, Rochester, N.Y.Because light consists of electromagnetic waves, the propagation of light can be regarded as merely a branch of electromagnetism. However, it is usually dealt with as a separate subject called optics: the part that deals with the tracing of light rays is known as geometrical optics, while the part that treats the distinctive wave phenomena of light is called physical optics. More recently, there has developed a new and vital branch, quantum optics, which is concerned with the theory and application of the laser, a device that produces an intense coherent beam of unidirectional radiation useful for many applications.
The formation of images by lenses, microscopes, telescopes, and other optical devices is described by ray optics, which assumes that the passage of light can be represented by straight lines, that is, rays. The subtler effects attributable to the wave property of visible light, however, require the explanations of physical optics. One basic wave effect is interference, whereby two waves present in a region of space combine at certain points to yield an enhanced resultant effect (e.g., the crests of the component waves adding together); at the other extreme, the two waves can annul each other, the crests of one wave filling in the troughs of the other. Another wave effect is diffraction, which causes light to spread into regions of the geometric shadow and causes the image produced by any optical device to be fuzzy to a degree dependent on the wavelength of the light. Optical instruments such as the interferometer and the diffraction grating can be used for measuring the wavelength of light precisely (about 500 micrometres) and for measuring distances to a small fraction of that length.
Atomic and chemical physics
Millikan oil-drop experiment [Credit: Encyclopædia Britannica, Inc.]Millikan oil-drop experimentEncyclopædia Britannica, Inc.One of the great achievements of the 20th century was the establishment of the validity of the atomic hypothesis, first proposed in ancient times, that matter is made up of relatively few kinds of small, identical parts—namely, atoms. However, unlike the indivisible atom of Democritus and other ancients, the atom, as it is conceived today, can be separated into constituent electrons and nucleus. Atoms combine to form molecules, whose structure is studied by chemistry and physical chemistry; they also form other types of compounds, such as crystals, studied in the field of condensed-matter physics. Such disciplines study the most important attributes of matter (not excluding biologic matter) that are encountered in normal experience—namely, those that depend almost entirely on the outer parts of the electronic structure of atoms. Only the mass of the atomic nucleus and its charge, which is equal to the total charge of the electrons in the neutral atom, affect the chemical and physical properties of matter.
Although there are some analogies between the solar system and the atom due to the fact that the strengths of gravitational and electrostatic forces both fall off as the inverse square of the distance, the classical forms of electromagnetism and mechanics fail when applied to tiny, rapidly moving atomic constituents. Atomic structure is comprehensible only on the basis of quantum mechanics, and its finer details require as well the use of quantum electrodynamics (QED).
Atomic properties are inferred mostly by the use of indirect experiments. Of greatest importance has been spectroscopy, which is concerned with the measurement and interpretation of the electromagnetic radiations either emitted or absorbed by materials. These radiations have a distinctive character, which quantum mechanics relates quantitatively to the structures that produce and absorb them. It is truly remarkable that these structures are in principle, and often in practice, amenable to precise calculation in terms of a few basic physical constants: the mass and charge of the electron, the speed of light, and Planck’s constant (approximately 6.62606957 × 10−34 joule∙second), the fundamental constant of the quantum theory named for the German physicist Max Planck.
Condensed-matter physics
transistor [Credit: AT&T Bell Labs/Science Photo Library/Photo Researchers, Inc.]transistorAT&T Bell Labs/Science Photo Library/Photo Researchers, Inc.This field, which treats the thermal, elastic, electrical, magnetic, and optical properties of solid and liquid substances, grew at an explosive rate in the second half of the 20th century and scored numerous important scientific and technical achievements, including the transistor. Among solid materials, the greatest theoretical advances have been in the study of crystalline materials whose simple repetitive geometric arrays of atoms are multiple-particle systems that allow treatment by quantum mechanics. Because the atoms in a solid are coordinated with each other over large distances, the theory must go beyond that appropriate for atoms and molecules. Thus conductors, such as metals, contain some so-called free electrons, or valence electrons, which are responsible for the electrical and most of the thermal conductivity of the material and which belong collectively to the whole solid rather than to individual atoms. Semiconductors and insulators, either crystalline or amorphous, are other materials studied in this field of physics.
Nuclear physics
particle physics: particle tracks from collision of niobium nuclei [Credit: Courtesy of the Department of Physics and Astronomy, Michigan State University]particle physics: particle tracks from collision of niobium nucleiCourtesy of the Department of Physics and Astronomy, Michigan State UniversityThis branch of physics deals with the structure of the atomic nucleus and the radiation from unstable nuclei. About 10,000 times smaller than the atom, the constituent particles of the nucleus, protons and neutrons, attract one another so strongly by the nuclear forces that nuclear energies are approximately 1,000,000 times larger than typical atomic energies. Quantum theory is needed for understanding nuclear structure.
Like excited atoms, unstable radioactive nuclei (either naturally occurring or artificially produced) can emit electromagnetic radiation. The energetic nuclear photons are called gamma rays. Radioactive nuclei also emit other particles: negative and positive electrons (beta rays), accompanied by neutrinos, and helium nuclei (alpha rays).
A principal research tool of nuclear physics involves the use of beams of particles (e.g., protons or electrons) directed as projectiles against nuclear targets. Recoiling particles and any resultant nuclear fragments are detected, and their directions and energies are analyzed to reveal details of nuclear structure and to learn more about the strong force. A much weaker nuclear force, the so-called weak interaction, is responsible for the emission of beta rays. Nuclear collision experiments use beams of higher-energy particles, including those of unstable particles called mesons produced by primary nuclear collisions in accelerators dubbed meson factories. Exchange of mesons between protons and neutrons is directly responsible for the strong force. (For the mechanism underlying mesons, see below Fundamental forces and fields.)
In radioactivity and in collisions leading to nuclear breakup, the chemical identity of the nuclear target is altered whenever there is a change in the nuclear charge. In fission and fusion nuclear reactions in which unstable nuclei are, respectively, split into smaller nuclei or amalgamated into larger ones, the energy release far exceeds that of any chemical reaction.
Particle physics
neutron: depiction of protons, neutrons, pions, and other hadrons [Credit: Encyclopædia Britannica, Inc.]neutron: depiction of protons, neutrons, pions, and other hadronsEncyclopædia Britannica, Inc.One of the most significant branches of contemporary physics is the study of the fundamental subatomic constituents of matter, the elementary particles. This field, also called high-energy physics, emerged in the 1930s out of the developing experimental areas of nuclear and cosmic-ray physics. Initially investigators studied cosmic rays, the very-high-energy extraterrestrial radiations that fall upon the Earth and interact in the atmosphere (see below The methodology of physics). However, after World War II, scientists gradually began using high-energy particle accelerators to provide subatomic particles for study. Quantum field theory, a generalization of QED to other types of force fields, is essential for the analysis of high-energy physics. Subatomic particles cannot be visualized as tiny analogues of ordinary material objects such as billiard balls, for they have properties that appear contradictory from the classical viewpoint. That is to say, while they possess charge, spin, mass, magnetism, and other complex characteristics, they are nonetheless regarded as pointlike.
During the latter half of the 20th century, a coherent picture evolved of the underlying strata of matter involving two types of subatomic particles: fermions (baryons and leptons), which have odd half-integral angular momentum (spin 1/2, 3/2) and make up ordinary matter; and bosons (gluons, mesons, and photons), which have integral spins and mediate the fundamental forces of physics. Leptons (e.g., electrons, muons, taus), gluons, and photons are believed to be truly fundamental particles. Baryons (e.g., neutrons, protons) and mesons (e.g., pions, kaons), collectively known as hadrons, are believed to be formed from indivisible elements known as quarks, which have never been isolated.
Quarks come in six types, or “flavours,” and have matching antiparticles, known as antiquarks. Quarks have charges that are either positive two-thirds or negative one-third of the electron’s charge, while antiquarks have the opposite charges. Like quarks, each lepton has an antiparticle with properties that mirror those of its partner (the antiparticle of the negatively charged electron is the positive electron, or positron; that of the neutrino is the antineutrino). In addition to their electric and magnetic properties, quarks participate in both the strong force (which binds them together) and the weak force (which underlies certain forms of radioactivity), while leptons take part in only the weak force.
Baryons, such as neutrons and protons, are formed by combining three quarks—thus baryons have a charge of −1, 0, or 1. Mesons, which are the particles that mediate the strong force inside the atomic nucleus, are composed of one quark and one antiquark; all known mesons have a charge of −2, −1, 0, 1, or 2. Most of the possible quark combinations, or hadrons, have very short lifetimes, and many of them have never been seen, though additional ones have been observed with each new generation of more powerful particle accelerators.
The quantum fields through which quarks and leptons interact with each other and with themselves consist of particle-like objects called quanta (from which quantum mechanics derives its name). The first known quanta were those of the electromagnetic field; they are also called photons because light consists of them. A modern unified theory of weak and electromagnetic interactions, known as the electroweak theory, proposes that the weak force involves the exchange of particles about 100 times as massive as protons. These massive quanta have been observed—namely, two charged particles, W+ and W, and a neutral one, W0.
In the theory of the strong force known as quantum chromodynamics (QCD), eight quanta, called gluons, bind quarks to form baryons and also bind quarks to antiquarks to form mesons, the force itself being dubbed the “colour force.” (This unusual use of the term colour is a somewhat forced analogue of ordinary colour mixing.) Quarks are said to come in three colours—red, blue, and green. (The opposites of these imaginary colours, minus-red, minus-blue, and minus-green, are ascribed to antiquarks.) Only certain colour combinations, namely colour-neutral, or “white” (i.e., equal mixtures of the above colours cancel out one another, resulting in no net colour), are conjectured to exist in nature in an observable form. The gluons and quarks themselves, being coloured, are permanently confined (deeply bound within the particles of which they are a part), while the colour-neutral composites such as protons can be directly observed. One consequence of colour confinement is that the observable particles are either electrically neutral or have charges that are integral multiples of the charge of the electron. A number of specific predictions of QCD have been experimentally tested and found correct.
Quantum mechanics
Although the various branches of physics differ in their experimental methods and theoretical approaches, certain general principles apply to all of them. The forefront of contemporary advances in physics lies in the submicroscopic regime, whether it be in atomic, nuclear, condensed-matter, plasma, or particle physics, or in quantum optics, or even in the study of stellar structure. All are based upon quantum theory (i.e., quantum mechanics and quantum field theory) and relativity, which together form the theoretical foundations of modern physics. Many physical quantities whose classical counterparts vary continuously over a range of possible values are in quantum theory constrained to have discontinuous, or discrete, values. Furthermore, the intrinsically deterministic character of values in classical physics is replaced in quantum theory by intrinsic uncertainty.
According to quantum theory, electromagnetic radiation does not always consist of continuous waves; instead it must be viewed under some circumstances as a collection of particle-like photons, the energy and momentum of each being directly proportional to its frequency (or inversely proportional to its wavelength, the photons still possessing some wavelike characteristics). Conversely, electrons and other objects that appear as particles in classical physics are endowed by quantum theory with wavelike properties as well, such a particle’s quantum wavelength being inversely proportional to its momentum. In both instances, the proportionality constant is the characteristic quantum of action (action being defined as energy × time)—that is to say, Planck’s constant divided by 2π, or ℏ.
atom: Bohr model [Credit: Encyclopædia Britannica, Inc.]atom: Bohr modelEncyclopædia Britannica, Inc.In principle, all of atomic and molecular physics, including the structure of atoms and their dynamics, the periodic table of elements and their chemical behaviour, as well as the spectroscopic, electrical, and other physical properties of atoms, molecules, and condensed matter, can be accounted for by quantum mechanics. Roughly speaking, the electrons in the atom must fit around the nucleus as some sort of standing wave (as given by the Schrödinger equation) analogous to the waves on a plucked violin or guitar string. As the fit determines the wavelength of the quantum wave, it necessarily determines its energy state. Consequently, atomic systems are restricted to certain discrete, or quantized, energies. When an atom undergoes a discontinuous transition, or quantum jump, its energy changes abruptly by a sharply defined amount, and a photon of that energy is emitted when the energy of the atom decreases, or is absorbed in the opposite case.
Although atomic energies can be sharply defined, the positions of the electrons within the atom cannot be, quantum mechanics giving only the probability for the electrons to have certain locations. This is a consequence of the feature that distinguishes quantum theory from all other approaches to physics, the uncertainty principle of the German physicist Werner Heisenberg. This principle holds that measuring a particle’s position with increasing precision necessarily increases the uncertainty as to the particle’s momentum, and conversely. The ultimate degree of uncertainty is controlled by the magnitude of Planck’s constant, which is so small as to have no apparent effects except in the world of microstructures. In the latter case, however, because both a particle’s position and its velocity or momentum must be known precisely at some instant in order to predict its future history, quantum theory precludes such certain prediction and thus escapes determinism.
Compton effect [Credit: Encyclopædia Britannica, Inc.]Compton effectEncyclopædia Britannica, Inc.The complementary wave and particle aspects, or wave–particle duality, of electromagnetic radiation and of material particles furnish another illustration of the uncertainty principle. When an electron exhibits wavelike behaviour, as in the phenomenon of electron diffraction, this excludes its exhibiting particle-like behaviour in the same observation. Similarly, when electromagnetic radiation in the form of photons interacts with matter, as in the Compton effect in which X-ray photons collide with electrons, the result resembles a particle-like collision and the wave nature of electromagnetic radiation is precluded. The principle of complementarity, asserted by the Danish physicist Niels Bohr, who pioneered the theory of atomic structure, states that the physical world presents itself in the form of various complementary pictures, no one of which is by itself complete, all of these pictures being essential for our total understanding. Thus both wave and particle pictures are needed for understanding either the electron or the photon.
Although it deals with probabilities and uncertainties, the quantum theory has been spectacularly successful in explaining otherwise inaccessible atomic phenomena and in thus far meeting every experimental test. Its predictions, especially those of QED, are the most precise and the best checked of any in physics; some of them have been tested and found accurate to better than one part per billion.
Relativistic mechanics
In classical physics, space is conceived as having the absolute character of an empty stage in which events in nature unfold as time flows onward independently; events occurring simultaneously for one observer are presumed to be simultaneous for any other; mass is taken as impossible to create or destroy; and a particle given sufficient energy acquires a velocity that can increase without limit. The special theory of relativity, developed principally by Albert Einstein in 1905 and now so adequately confirmed by experiment as to have the status of physical law, shows that all these, as well as other apparently obvious assumptions, are false.
Specific and unusual relativistic effects flow directly from Einstein’s two basic postulates, which are formulated in terms of so-called inertial reference frames. These are reference systems that move in such a way that in them Isaac Newton’s first law, the law of inertia, is valid. The set of inertial frames consists of all those that move with constant velocity with respect to each other (accelerating frames therefore being excluded). Einstein’s postulates are: (1) All observers, whatever their state of motion relative to a light source, measure the same speed for light; and (2) The laws of physics are the same in all inertial frames.
time dilation [Credit: Encyclopædia Britannica, Inc.]time dilationEncyclopædia Britannica, Inc.The first postulate, the constancy of the speed of light, is an experimental fact from which follow the distinctive relativistic phenomena of space contraction (or Lorentz-FitzGerald contraction), time dilation, and the relativity of simultaneity: as measured by an observer assumed to be at rest, an object in motion is contracted along the direction of its motion, and moving clocks run slow; two spatially separated events that are simultaneous for a stationary observer occur sequentially for a moving observer. As a consequence, space intervals in three-dimensional space are related to time intervals, thus forming so-called four-dimensional space-time.
The second postulate is called the principle of relativity. It is equally valid in classical mechanics (but not in classical electrodynamics until Einstein reinterpreted it). This postulate implies, for example, that table tennis played on a train moving with constant velocity is just like table tennis played with the train at rest, the states of rest and motion being physically indistinguishable. In relativity theory, mechanical quantities such as momentum and energy have forms that are different from their classical counterparts but give the same values for speeds that are small compared to the speed of light, the maximum permissible speed in nature (about 300,000 kilometres per second, or 186,000 miles per second). According to relativity, mass and energy are equivalent and interchangeable quantities, the equivalence being expressed by Einstein’s famous mass-energy equation E = mc2, where m is an object’s mass and c is the speed of light.
The general theory of relativity is Einstein’s theory of gravitation, which uses the principle of the equivalence of gravitation and locally accelerating frames of reference. Einstein’s theory has special mathematical beauty; it generalizes the “flat” space-time concept of special relativity to one of curvature. It forms the background of all modern cosmological theories. In contrast to some vulgarized popular notions of it, which confuse it with moral and other forms of relativism, Einstein’s theory does not argue that “all is relative.” On the contrary, it is largely a theory based upon those physical attributes that do not change, or, in the language of the theory, that are invariant.
Conservation laws and symmetry
Since the early period of modern physics, there have been conservation laws, which state that certain physical quantities, such as the total electric charge of an isolated system of bodies, do not change in the course of time. In the 20th century it has been proved mathematically that such laws follow from the symmetry properties of nature, as expressed in the laws of physics. The conservation of mass-energy of an isolated system, for example, follows from the assumption that the laws of physics may depend upon time intervals but not upon the specific time at which the laws are applied. The symmetries and the conservation laws that follow from them are regarded by modern physicists as being even more fundamental than the laws themselves, since they are able to limit the possible forms of laws that may be proposed in the future.
Conservation laws are valid in classical, relativistic, and quantum theory for mass-energy, momentum, angular momentum, and electric charge. (In nonrelativistic physics, mass and energy are separately conserved.) Momentum, a directed quantity equal to the mass of a body multiplied by its velocity or to the total mass of two or more bodies multiplied by the velocity of their centre of mass, is conserved when, and only when, no external force acts. Similarly angular momentum, which is related to spinning motions, is conserved in a system upon which no net turning force, called torque, acts. External forces and torques break the symmetry conditions from which the respective conservation laws follow.
In quantum theory, and especially in the theory of elementary particles, there are additional symmetries and conservation laws, some exact and others only approximately valid, which play no significant role in classical physics. Among these are the conservation of so-called quantum numbers related to left-right reflection symmetry of space (called parity) and to the reversal symmetry of motion (called time reversal). These quantum numbers are conserved in all processes other than the weak force.
Other symmetry properties not obviously related to space and time (and referred to as internal symmetries) characterize the different families of elementary particles and, by extension, their composites. Quarks, for example, have a property called baryon number, as do protons, neutrons, nuclei, and unstable quark composites. All of these except the quarks are known as baryons. A failure of baryon-number conservation would exhibit itself, for instance, by a proton decaying into lighter non-baryonic particles. Indeed, intensive search for such proton decay has been conducted, but so far it has been fruitless. Similar symmetries and conservation laws hold for an analogously defined lepton number, and they also appear, as does the law of baryon conservation, to hold absolutely.
Fundamental forces and fields
beta particle: fission of uranium nucleus [Credit: Encyclopædia Britannica, Inc.]beta particle: fission of uranium nucleusEncyclopædia Britannica, Inc.The four basic forces of nature, in order of increasing strength, are thought to be: (1) the gravitational force between particles with mass; (2) the electromagnetic force between particles with charge or magnetism or both; (3) the colour force, or strong force, between quarks; and (4) the weak force by which, for example, quarks can change their type, so that a neutron decays into a proton, an electron, and an antineutrino. The strong force that binds protons and neutrons into nuclei and is responsible for fission, fusion, and other nuclear reactions is in principle derived from the colour force. Nuclear physics is thus related to QCD as chemistry is to atomic physics.
According to quantum field theory, each of the four fundamental interactions is mediated by the exchange of quanta, called vector gauge bosons, which share certain common characteristics. All have an intrinsic spin of one unit, measured in terms of Planck’s constant ℏ. (Leptons and quarks each have one-half unit of spin.) Gauge theory studies the group of transformations, or Lie group, that leaves the basic physics of a quantum field invariant. Lie groups, which are named for the 19th-century Norwegian mathematician Sophus Lie, possess a special type of symmetry and continuity that made them first useful in the study of differential equations on smooth manifolds (an abstract mathematical space for modeling physical processes). This symmetry was first seen in the equations for electromagnetic potentials, quantities from which electromagnetic fields can be derived. It is possessed in pure form by the eight massless gluons of QCD, but in the electroweak theory—the unified theory of electromagnetic and weak force interactions—gauge symmetry is partially broken, so that only the photon remains massless, with the other gauge bosons (W+, W, and Z) acquiring large masses. Theoretical physicists continue to seek a further unification of QCD with the electroweak theory and, more ambitiously still, to unify them with a quantum version of gravity in which the force would be transmitted by massless quanta of two units of spin called gravitons.
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ce931dd378daefdf | Molecular Modeling and Electronic Structure Calculations with QC-Lab
by Marcelo Carignano
Molecular Modeling and Electronic Structure Calculations
George Schatz, Baudilio Tejerina, Shelby Hatch and Jennifer Roden
Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113
QC_Lab_module.pdf (275 KB, uploaded by 1 month 2 weeks ago)
This laboratory is designed to use the program GAMESS (General Atomic Molecular Electronic Structure System, developed in Gordon research group at Iowa State) through a website called nanoHUB to determine the geometric and electronic properties of numerous small molecules. GAMESS uses ab initio and semi-empirical calculations to determine these properties. Ab initio (“from first principles”) calculations solve the Schrödinger equation using the exact computational expression for the energy of the electrons.1 The particular ab initio method that we will use for this lab is called Hartree-Fock (HF). HF uses an approximate wavefunction to solve Schrödinger, so the resulting molecular properties are approximate, but for many applications the accuracy is adequate for interpreting experiments. Semi-empirical calculations use an approximate energy expression for the electrons, but solve for the exact wavefunction associated with this expression. Usually the energy expression uses empirical parameters (found experimentally) to match molecular properties, but the resulting properties are still approximations to the correct values. The semi-empirical method that we used in this lab is called PM3. This stands for “parameterized model 3,” which was the third (and best) method that the original authors of the method developed.
The underlying theory for GAMESS will be described in the lecture. In brief, GAMESS self-consistently solves the Schrödinger equation. The self-consistent method is an iterative approach that minimizes the energy by adjusting the wave functions of the molecules. Further information can be obtained from the GAMESS user guide (GAMESS_Manual).
Pre-Lab Information
GAMESS is located in the QC-Lab tool on the nanoHUB website. To access this tool you must first create a user account (see Appendix A for directions).
Make sure to include the results that you obtain (bond distances, bond angles, energy values, and charge per atom) in your lab notebooks.
The units on these results should be the same as the units used in GAMESS (distances in Angstroms (Å), angles in degrees, energy in Hartrees).
The results should be presented neatly, preferably in a tabular form with some experimental data obtained from literature also recorded. Literature values can be located through the National Institute of Standards and Technology (NIST) Computational Chemistry Comparison and Benchmark Database (CCCBD) at
It is important to note that calculated energy values cannot be directly compared to experimental energy values.2
PROBLEM 1: Warm-up and Practice
In this problem, the properties of three small linear molecules (CO, H2, and N2) will be calculated using a semi-empirical and an ab initio method. The calculation for CO will be shown in full detail and the remaining two molecules (H2 and N2) will be left for the reader to perform individually.
Carbon Monoxide Walk through:
1. (In lab notebook) Determine the input coordinates.
—A Draw the Lewis Dot structure to determine basic bonding and lone pairs, :C__=__O:
—B Redraw the molecule using the appropriate Valence Shell Electron Pair Repulsion (VSEPR) model to determine basic structure. Linear
—C Place molecule on Cartesian coordinates using the average bond lengths given in the textbook.
——i Make x-axis the bonding axis:
C 0.0 0.0 0.0
O 1.12 0.0 0.0
——ii Reassign the origin of the coordinates in order to achieve the highest symmetry possible. This will reduce number of calculations and speed up the process (trivial for small linear molecules but helpful for larger non-linear molecules).
C -0.56 0.0 0.0
O 0.56 0.0 0.0
—D Be sure to show all of your work in your notebook.
2. (In Nanohub) Perform semi-empirical calculations.
—A After setting up a nanoHUB account and launching the QC-lab tool, select new from the ‘QC task’ pull-down menu.
—B Delete the text in the atomic coordinates box and enter the input coordinates (X, Y, Z) that you determined, being sure to first put the atomic symbol and atomic number. Be sure to note the syntax (spacing, using the decimal, etc.) See screenshot on next page.
—C Leave the ‘Molecular Point Group,’ ‘Symmetry Order,’ and ‘Coordinate Style’ at the default setting (Cn, 1, unique).3
—D Click on the ‘Theoretical Model’ tab (circled in previous image).
—E Leave the default ‘Job Control Parameters’ settings for now (Run, Geometry Optimization, Restricted Hartree-Fock Calculation, 0 and 1).4
—F Under the ‘Basis Set’ tab, set the ‘Basis Set for’ pull down menu to ‘Semi-empirical calculations.’ This action will refresh your screen and bring you back to the ‘Job Control Parameters’ tab so click back into the ‘Basis Set’ tab (you might notice other tabs have vanished; this is okay).
—G In the ‘Hamiltonian’ pull-down, select PM3.
—H Click the ‘Simulate’ button in the lower right.
—I After the job has finished running, an image of the molecule will appear in the window.
—J In order to obtain the necessary information, select the ‘Output Log’ from the ‘Results’ pull-down. The output contains all the results and information about the calculation.
—KThe following key words can be found using the ‘Find’ feature to locate the desired information:
——i. LOCATED = the location in the output with the optimized coordinates and bond distances are printed
——ii. Slightly above the word LOCATED will be a value for TOTAL WALL CLOCK TIME, NSERCH and ENERGY, which gives the time the calculation took, the number of steps taken by the computer to obtain an optimized geometry and the energy for the optimized structure.
——iii.MOPAC CHARGES = this is located below the LOCATED and gives the charge for each atom.
—L. In addition to searching the output file, results can also be obtained through MacMolPlot. Clicking on the launchmolviewer tab at the bottom of the screen and using the pull down menu to select MacMolPlot will access this software.
——i. Select ‘Open’ from the ‘File’ menu to display all the jobs that were run in this session of QC-lab with the largest numbered file corresponding to the most recent calculation. Select the proper file and click open.
——ii.’ The molecule should appear in the window with the energy written in the bottom left corner.
——iii. The bond length (and angle) can be found by using the ‘Z-Matrix Calculator’ from the ‘Subwindow’ menu. The atoms will be numbered based on the order their coordinates were input (in this case C will be 1 and O will be 2). The assigned atom numbers can be displayed on the molecule image by selecting ‘Atom Number’ under the ‘Atom Labels’ menu in the ‘View’ pull-down.
——iv. Close out of the ‘Z-Matrix Calculator’ using the hyphen bar but DO NOT close the MacMolPlot window. Click back into the QC-Lab v2.0 window.
——v. Note: MacMolPlot cannot give you atomic charge information.
3. Perform Ab Initio Calculations
—A. In the QC-Lab v2.0 window, click on the ‘INPUT’ button in the bottom left corner.
—B. Click on the ‘Molecular Geometry’ tab and verify that your input geometry is the same as you entered for the semi-empirical calculation. Alternately, you can replace this input geometry with your optimized geometry obtained using the semi-empirical method. That may reduce the number of steps (nserch) the computer will have to use. This is a common technique for more complex structures; however, is not necessary for simple structures.
—C. Click on the ‘Theoretical Model’ tab and select the ‘Basis Set’ tab. In the ‘Basis set for’ pull-down select ‘All-electron calculation.’ This will refresh your session and switch the screen back to the ‘Job Control Parameters’ tab; in addition it will add more tabs.
—D. Click on the ‘Basis Set’ tab again and select 6-31G from the ‘Basis Set’ pull down.
—E. Press the Simulate button.
—F. When your job has finished running, an image of the molecule will be present on the screen. The output log can be accessed in the same way it was in the semi-empirical calculation. However, you need to search for slightly different words using the ‘Find’ function.
——i. LOCATED = will take you to the region in the output where you can easily find the optimized coordinates, bond lengths, nserch, and total wall clock time.
——ii. TOTAL MULLIKEN = will give you the charge on the atoms, but be careful because the TOTAL MULLIKEN is printed off for each nserch and the only charge that matters is the one corresponding to the optimized geometry at the bottom of the output.
—G. You could also use the MacMolPlot GUI to process the results, keeping in mind that you cannot get atomic charge data from it.
H2 and N2 Practice:
Perform the same calculations (semi-empirical PM3 and ab initio 6-31G) except replace CO with H2 and N2. Report the same results.
Problem 1 Questions:
1.) What method of calculation compared the best with experimental results?
2.) Which method of calculation took a longer amount of time?
3.) Keeping 1 and 2 in mind, why would it be more beneficial to take the geometry generated from the semi-empirical calculation and set it as an input for the ab initio calculation?
1) The time-independent Schrödinger equation is Hψ=Eψ where ψ is the wavefunction representing atomic/ electronic positions, E is the energy and H is the Hamiltonian, an operator which takes the wavefunction and finds the energy.
2) The calculated energy values are absolute energy values while the experimental values obtained are typically energy differences between states. Comparison between experimental and computational energy values can be made. How?
3) These three selections have to do with taking advantage of symmetry to determine the input geometry with the different point groups dictating what symmetry operations (mirror of plane, rotation, inversion, etc.) need to be performed on a minimal number of input atoms to obtain the whole molecule. For water you would need to enter Cnv, 2, and unique with the input of one O and one H. The second H would be found through the performed symmetry operations.
4) These setting are very important in telling the software what calculations to run.
Execution Type Tells the software to run the calculation or check the input
Run Type Tells the software what type of calculation to run
(geometry optimization = find ‘lowest’ energy geometry,
Hessian = determines more information about the potential energy surface and can be used to find vibration information,
single energy = finds energy of input geometry)
SCF Type Provides computational details, to be discussed later
Molecular Charge Indicates if the molecules in an ion (positively or negatively charged)
Spin Multiplicity Gives the software information on how many electrons are spin up or spin down
(which has to do with the number of unpaired electrons)
Created on , Last modified on |
b9d6657e3d599fad | Take the 2-minute tour ×
I'm using the Crank-Nicholson method to solve the time-dependent Schrödinger equation with the split-operator method. I'm getting some weird results that are probably the result of a bug somewhere in my code.
Just in case, I thought I should probably check to see if the method I'm using is unstable.
Does the split-operator method change the stability properties of the Crank-Nicolson method?
If so, how?
share|improve this question
I haven't analyzed or experimented with this specific case, but certainly the use of ADI can in general affect the stable timestep. However, in the case of the implicit trapezoidal rule, I wouldn't expect any change as it is A-stable. – David Ketcheson Feb 13 '12 at 6:40
@David Ketcheson: You may want to write that up as an answer. – Dan Feb 16 '12 at 17:49
1 Answer 1
up vote 1 down vote accepted
The time-dependent Schrödinger equation is not really a heat equation. Still, the Crank–Nicolson method is well suited for its solution. However, the Crank-Nicolson method is fully implicit, so the statement "doing the implicit part with ADI" sounds a bit suspicious. It probably means that the diffusion like part is done with ADI. I wonder a bit whether that means the potential part is treated by an analytical solution together with another application of an operator-split scheme. (But why would we call this a Crank-Nicolson method?)
For the heat equation, normally the ADI methods Peaceman–Rachford, Douglas–Rachford and Douglas–Gunn get discussed. I'm not so sure how much this analysis carries over to fake/formal heat equations, but at least Douglas–Rachford is certainly unsuitable for the Schrödinger equation. There certainly are stable ADI schemes that can be used for the Schrödinger equations (probably Douglas–Gunn works), but an arbitrary ADI scheme that works well for the heat equation is not guaranteed to also work well for the Schrödinger equation. But even if it would be unstable, it would probably be only weakly unstable, so that you should still be able to get "some" results. So really "weird" results probably have a different origin than the stability of the ADI scheme.
share|improve this answer
"Implicit part" was just poor wording. I was using Crank-Nicholson steps on dimension. I was using the Baker-Campbell-Hausdorf expansion to get the particular form that I used. – Dan Feb 1 '13 at 21:51
@Dan Because you reference the Baker-Campbell-Hausdorf expansion, I guess you do exactly what I called "application of an operator-split scheme". I guess it's a Strang splitting. When I google for "Strang splitting", the first hit contains "time-dependent Schrödinger", "Baker-Campbell-Hausdorff formula" and also introduces the Strang splitting... – Thomas Klimpel Feb 1 '13 at 22:28
Your Answer
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cf8208994f9b8910 | 45: Schrodinger
Explain xkcd: It's 'cause you're dumb.
Jump to: navigation, search
There was no alt-text until you moused over
[edit] Explanation
This comic is a joke creating a humorously false synthesis, combining the principals of quantum superposition and the effects of reading a comic one panel at a time.
Schrödinger's cat is a thought experiment that illuminates the notion that a particle only resolves itself to its state upon observation, and until this observation it is in all of its possible states simultaneously. In the thought experiment a cat is both dead and alive until observation, likewise in this comic the comic is both funny and unfunny until the comic is observed (or read).
Black Hat and Cueball are likening the last panel to the box with the cat: until you read it, it is in a mixed state (a superposition) of both funny and unfunny. In the last panel Black Hat says "Shit."
The joke is that after reading the last panel the comic is both funny (as it is unexpected) and not funny (as the last line was a non sequitur and therefore there is no climax) at the same time, thus proving Black Hat and Cueball wrong, hence them expressing discontent with the word shit.
The title text, which Randall here calls the alt-text, suggests that the alt text did not exist until the mouse over action occurred.
[edit] Schrödinger's cat
Schrödinger thought the Copenhagen interpretation was absurd, and devised the below thought experiment to show this. The experiment goes as follows: Put a cat in a box, he said, with a device triggered by the decay of an atom with a half-life of one hour that would release a poisonous gas if triggered. Then, after waiting an hour, the Copenhagen interpretation would say that the atom is in a superposition of decayed and undecayed states, and thus, by extension, the cat would be in a superposition of alive and dead states. Only when the box is opened would the wave-function for the cat collapse into either alive or dead states. This thought experiment is not meant to be taken literally as every interaction of a particle with another constitutes an observation, and many particles must interact for a cat to die, but still his argument was that since it is absurd for a cat to be both alive and dead, it is absurd for an atom to be both decayed and undecayed.
If this experiment were to be performed the cat would not be both dead and alive.
[edit] Transcript
[Black Hat and Cueball are standing next to each other. Above them the text is written in a box with shades around it.]
Schrödinger's Comic
[Black Hat and Cueball are still standing next to each other, but Cueball has lifted his arms above his head. The text is again written in a box with shades around it.]
[Black Hat and Cueball are still standing next to each other, Cueball arms are down again. The text is again written in a box with shades around it.]
[Black Hat and Cueball are still standing next to each other. Cueball has become smaller and smaller through the three frames after the first. Quite clearly here in the last panel. The text is again written in a box with shades around it.]
[edit] Trivia
• This was the 42nd comic originally posted to LiveJournal.
• There had been a break of almost a month between this and the previous comic.
• This time was probably used to prepare the launch of the new xkcd site.
• Original title: "Drawing: Schrodinger"
• For the first time in eight comics, and only the second time since after the first day on LiveJournal, is the weekday not part of the title on LiveJournal.
• But apart from in the very next comic, the extra word "Drawing" was still added to the title for this and the four comics after the next, in spite of the simultaneous release on xkcd.
• There were no original Randall quote for this comic.
• This was the first comic to be posted simultaneous (i.e. on the same day) on both LiveJournal and the new xkcd site.
• This comic was thus one of the last 11 comics posted on LiveJournal.
• The Schrödinger equation was enhanced by Paul Dirac only three years later in 1928: Dirac equation. It did combine the Schrödinger world with Einstein, e.g. relativity.
• Black Hat's hat is beginning to shorten from its top-hat look, although its height varies between panels. (As does Cueballs height compared to Black Hat.)
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