Quantum mechanics

Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large scale. Quantum mechanics differs from classical physics in that energy, angular momentum and other quantities of a bound system are restricted to discrete values. Quantum mechanics arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others; the modern theory is formulated in various specially developed mathematical formalisms.

In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor and semiconductors such as the microprocessor and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he described in a paper titled On the nature of light and colours.

This experiment played a major role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays; these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, the 1900 quantum hypothesis of Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. Following Max Planck's solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect.

Around 1900–1910, the atomic theory and the corpuscular theory of light first came to be accepted as scientific fact. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept introduced by Arnold Sommerfeld; this phase is known as old quantum theory. According to Planck, each energy element is proportional to its frequency: E = h ν, where h is Planck's constant. Planck cautiously insisted that this was an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.

In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material, he won the 1921 Nobel Prize in Physics for this work. Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle, with a discrete quantum of energy, dependent on its frequency; the foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wi

Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V, not the zero vector v is an eigenvector of T if T is a scalar multiple of v; this condition can be written as the equation T = λ v, 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 the linear transformation T can be represented as a square matrix A, 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 A v = λ v. There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space to itself, given any basis of the vector space.

For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction, stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations; the prefix eigen- is adopted from the German word eigen for "proper", "characteristic". Utilized to study principal axes of the rotational motion of rigid bodies and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, matrix diagonalization. In essence, an eigenvector v of a linear transformation T is a non-zero 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 T = λ v, 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; the Mona Lisa example pictured at right 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. Notice that points along the horizontal axis do not move at all. 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 take many forms. For example, the linear transformation could be a differential operator like d d x, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as d d x e λ x = λ e λ x. 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 referred to as eigenvectors. If the linear transformation is expressed in the form of an n by n matrix A the eigenvalue equation above for a linear transformation can be rewritten as the matrix multiplication A v = λ v, where the eigenvector v is an n by 1 matrix. For a matrix and eigenvectors can be used to decompose the matrix, for example by diagonalizing it. Eigenvalues and eigenvectors give rise to many related mathematical concepts, 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.

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. If the set of eigenvectors of T form a basis of the domain of T this basis is called an eigenbasis. Eigenvalues are introduced in the context of linear algebra or matrix theory. However, they arose in the study of quadratic forms and differential equations. In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the pri

Physics

Physics is the natural science that studies matter, its motion, behavior through space and time, that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, its main goal is to understand how the universe behaves. Physics is one of the oldest academic disciplines and, through its inclusion of astronomy the oldest. Over much of the past two millennia, chemistry and certain branches of mathematics, were a part of natural philosophy, but during the scientific revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, the boundaries of physics which are not rigidly defined. New ideas in physics explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics enable advances in new technologies.

For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have transformed modern-day society, such as television, domestic appliances, nuclear weapons. Astronomy is one of the oldest natural sciences. Early civilizations dating back to beyond 3000 BCE, such as the Sumerians, ancient Egyptians, the Indus Valley Civilization, had a predictive knowledge and a basic understanding of the motions of the Sun and stars; the stars and planets were worshipped, believed to represent gods. While the explanations for the observed positions of the stars were unscientific and lacking in evidence, these early observations laid the foundation for astronomy, as the stars were found to traverse great circles across the sky, which however did not explain the positions of the planets. According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, all Western efforts in the exact sciences are descended from late Babylonian astronomy.

Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey. Natural philosophy has its origins in Greece during the Archaic period, when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause, they proposed ideas verified by reason and observation, many of their hypotheses proved successful in experiment. The Western Roman Empire fell in the fifth century, this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire resisted the attacks from the barbarians, continued to advance various fields of learning, including physics. In the sixth century Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. In sixth century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noting its flaws.

He introduced the theory of impetus. Aristotle's physics was not scrutinized until John Philoponus appeared, unlike Aristotle who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics John Philoponus wrote: “But this is erroneous, our view may be corroborated by actual observation more than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a small one, and so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other”John Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries during the Scientific Revolution.

Galileo cited Philoponus in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus, it was a step toward the modern ideas of momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further placing emphasis on observation and a priori reasoning, developing early forms of the scientific method; the most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn al-Haytham, in which he conclusively disproved the ancient Greek idea about vision, but came up with a new theory. In the book, he presented a study of the phenomenon of the camera obscura (his thousand-year-old

Leonard Susskind

Leonard Susskind is an American physicist, professor of theoretical physics at Stanford University, founding director of the Stanford Institute for Theoretical Physics. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology, he is a member of the National Academy of Sciences of the US, the American Academy of Arts and Sciences, an associate member of the faculty of Canada's Perimeter Institute for Theoretical Physics, a distinguished professor of the Korea Institute for Advanced Study. Susskind is regarded as one of the fathers of string theory, he was the first to give a precise string-theory interpretation of the holographic principle in 1995 and the first to introduce the idea of the string theory landscape in 2003. Susskind was awarded the 1998 J. J. Sakurai Prize, the 2018 Oskar Klein Medal. Leonard Susskind was born to a Jewish family from the South Bronx section of New York City, he began taking over from his father who had become ill.

He enrolled in the City College of New York as an engineering student, graduating with a B. S. in physics in 1962. In an interview in the Los Angeles Times, Susskind recalls the moment he discussed with his father this change in career path: "When I told my father I wanted to be a physicist, he said: ‘Hell no, you ain’t going to work in a drug store’. I said, "No, not a pharmacist." I said, ‘Like Einstein.’ He poked me in the chest with a piece of plumbing pipe. ‘You ain’t going to be no engineer’, he said. ‘You’re going to be Einstein.’" Susskind studied at Cornell University under Peter A. Carruthers where he earned his Ph. D. in 1965. Susskind was an assistant professor of physics an associate professor at Yeshiva University, after which he went for a year to the Tel Aviv University, returning to Yeshiva to become a professor of physics. Since 1979 he has been professor of physics at Stanford University, since 2000 has held the Felix Bloch professorship of physics. Susskind was awarded the 1998 J. J. Sakurai Prize for his "pioneering contributions to hadronic string models, lattice gauge theories, quantum chromodynamics, dynamical symmetry breaking".

Susskind's hallmark, according to colleagues, has been the application of "brilliant imagination and originality to the theoretical study of the nature of the elementary particles and forces that make up the physical world". In 2007, Susskind joined the faculty of Perimeter Institute for Theoretical Physics in Waterloo, Canada, as an associate member, he has been elected to the National Academy of Sciences and the American Academy of Arts and Sciences. He is a distinguished professor at Korea Institute for Advanced Study. Susskind was one of at least three physicists, alongside Yoichiro Nambu and Holger Bech Nielsen, who independently discovered during or around 1970 that the Veneziano dual resonance model of strong interactions could be described by a quantum mechanical model of oscillating strings, was the first to propose the idea of the string theory landscape. Susskind has made important contributions in the following areas of physics: The independent discovery of the string theory model of particle physics The theory of quark confinement The development of Hamiltonian lattice gauge theory known as Kogut-Susskind fermions The theory of scaling violations in deep inelastic electroproduction The theory of symmetry breaking sometimes known as "technicolor theory" The second, yet independent, theory of cosmological baryogenesis String theory of black hole entropy The principle of black hole complementarity The causal patch hypothesis The holographic principle M-theory, including development of the BFSS matrix model Introduction of holographic entropy bounds in physical cosmology The idea of an anthropic string theory landscape The Census Taker's Hat Most application of ideas from information and computation theory, such as quantum complexity, to the physics and thermodynamics of black holes, holographic theories in general.

Susskind is the author of several popular science books. The Cosmic Landscape: String Theory and the Illusion of Intelligent Design is Susskind's first popular science book, published by Little and Company on December 12, 2005, it is Susskind's attempt to bring his idea of the anthropic landscape of string theory to the general public. In the book, Susskind describes how the string theory landscape was an inevitable consequence of several factors, one of, Steven Weinberg's prediction of the cosmological constant in 1987; the question addressed here is. Susskind explains that Weinberg calculated that if the cosmological constant was just a little different, our universe would cease to exist; the Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics is Susskind's second popular science book, published by Little and Company on July 7, 2008. The book is his most famous work and explains what he thinks would happen to the information and matter stored in a black hole when it evaporates.

The book sparked from a debate that started in 1981, when there was a meeting of physicists to try to decode some of the mysteries about how particles of particular elemental compounds function. During this discussion Stephen Hawking stated that the information inside a black hole is lost forever as the black hole evaporates, it took 28 years for Leonard Susskind to formulate his theory. He published his theory in his book, The Black Hole War. Like The Cosmic Landscape, The Black Hole War is aimed at the

Electric field

An electric field surrounds an electric charge, exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. Mathematically the electric field is a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point; the SI unit for electric field strength is volt per meter. Newtons per coulomb is used as a unit of electric field strengh. Electric fields are created by time-varying magnetic fields. Electric fields are important in many areas of physics, are exploited electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature. From Coulomb's law a particle with electric charge q 1 at position x 1 exerts a force on a particle with charge q 0 at position x 0 of F = 1 4 π ε 0 q 1 q 0 2 r ^ 1, 0 where r 1, 0 is the unit vector in the direction from point x 1 to point x 0, ε0 is the electric constant in C2 m−2 N−1When the charges q 0 and q 1 have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other.

When the charges have unlike signs the force is negative, indicating the particles attract. To make it easy to calculate the Coulomb force on any charge at position x 0 this expression can be divided by q 0, leaving an expression that only depends on the other charge E = F q 0 = 1 4 π ε 0 q 1 2 r ^ 1, 0 This is the electric field at point x 0 due to the point charge q 1. Since this formula gives the electric field magnitude and direction at any point x 0 in space it defines a vector field. From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, toward the charge if it is negative, its magnitude decreases with the inverse square of the distance from the charge. If there are multiple charges, the resultant Coulomb force on a charge can be found by summing the vectors of the forces due to each charge; this shows the electric field obeys the superposition principle: the total electric field at a point due to a collection of charges is just equal to the vector sum of the electric fields at that point due to the individual charges.

E = E 1 + E 2 + E 3 + ⋯ = 1 4 π ε 0 q 1 2 r ^ 1 + 1 4 π ε 0 q 2 ( x 2 −