Two-state quantum system

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can be seen as a qubit. Two-state systems are the simplest quantum systems that can exist, since the dynamics of a one-state system is trivial; the mathematical framework required for the analysis of two-state systems is that of linear differential equations and linear algebra of two-dimensional spaces. As a result, the dynamics of a two-state system can be solved analytically without any approximation. A well known example of a two-state system is the spin of a spin-1/2 particle such as an electron, whose spin can have values +ħ/2 or −ħ/2, where ħ is the reduced Planck constant. Another example studied in atomic physics, is the transition of an atom to or from an excited state; the state of a two-state quantum system can be described by a two-dimensional complex Hilbert space.

This means every state vector | ψ ⟩ is represented by two complex coordinates: | ψ ⟩ = = c 1 + c 2. If the vectors are normalized, c 1 and c 2 are related by | c 1 | 2 + | c 2 | 2 = 1; the basis vectors are represented as | 0 ⟩ = and | 1 ⟩ = All observable physical quantities associated with this systems are 2 × 2 Hermitian matrices. The Hamiltonian of the system is a 2 × 2 Hermitian matrix; the most general form of the Hamiltonian of a two-state system is given by H = where a 1, a 2, c and d are real numbers. This matrix can be decomposed as, H = a ⋅ σ 0 + c ⋅ σ 1 + d ⋅ σ 2 + b ⋅ σ 3; the matrix σ 0 is the 2 × 2 identity matrix and the matrices σ k are the Pauli matrices. This decomposition simplifies the analysis of the system in the time-independent case where the values of a, b, c and d are constants; the Hamiltonian can be written as: H = a ⋅ σ 0 + r ⋅ σ.

Momentum

In Newtonian mechanics, linear momentum, translational momentum, or momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a direction in three-dimensional space. If m is an object's mass and v is the velocity the momentum is p = m v, In SI units, it is measured in kilogram meters per second. Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, general relativity, it is an expression of one of the fundamental symmetries of time: translational symmetry. Advanced formulations of classical mechanics and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.

In these systems the conserved quantity is generalized momentum, in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function; the momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields and deformable bodies, a momentum density can be defined, a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension; the vector equations are identical to the scalar equations. The momentum of a particle is conventionally represented by the letter p.

It is the product of two quantities, the particle's mass and its velocity: p = m v. The unit of momentum is the product of the units of velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second the momentum is in gram centimeters per second. Being a vector, momentum has direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground; the momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2; the momenta of more than two particles can be added more with the following: p = ∑ i m i v i. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i.

If all the particles are moving, the center of mass will be moving as well. If the center of mass is moving at velocity vcm, the momentum is: p = m v cm; this is known as Euler's first law. If the net force applied to a particle is a constant F, is applied for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t. In differential form, this is Newton's second law. If the net force experienced by a particle changes as a function of time, F, the change in momentum between times t1 and t2 is Δ p = J = ∫ t 1

Wavelength

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is designated by the Greek letter lambda; the term wavelength is sometimes applied to modulated waves, to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths. Wavelength depends on the medium. Examples of wave-like phenomena are sound waves, water waves and periodic electrical signals in a conductor.

A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength; the range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components; the wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, where v is called the phase speed of the wave and f is the wave's frequency.

In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s, thus the wavelength of a 100 MHz electromagnetic wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red 700 nm, to violet 400 nm. For sound waves in air, the speed of sound is 343 m/s; the wavelengths of sound frequencies audible to the human ear are thus between 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light. A standing wave is an undulatory motion. A sinusoidal standing wave includes stationary points of no motion, called nodes, the wavelength is twice the distance between nodes; the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed.

For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Wavelength and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. Traveling sinusoidal waves are represented mathematically in terms of their velocity v, frequency f and wavelength λ as: y = A cos = A cos where y is the value of the wave at any position x and time t, A is the amplitude of the wave, they are commonly expressed in terms of wavenumber k and angular frequency ω as: y = A cos = A cos in which wavelength and wavenumber are related to velocity and frequency as: k = 2 π λ = 2 π f v = ω

Wave–particle duality

Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described in terms not only of particles, but of waves. It expresses the inability of the classical concepts "particle" or "wave" to describe the behaviour of quantum-scale objects; as Albert Einstein wrote: It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality. Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, many others, current scientific theory holds that all particles exhibit a wave nature and vice versa; this phenomenon has been verified not only for elementary particles, but for compound particles like atoms and molecules. For macroscopic particles, because of their short wavelengths, wave properties cannot be detected. Although the use of the wave-particle duality has worked well in physics, the meaning or interpretation has not been satisfactorily resolved.

Bohr regarded the "duality paradox" as a metaphysical fact of nature. A given kind of quantum object will exhibit sometimes wave, sometimes particle, character, in different physical settings, he saw such duality as one aspect of the concept of complementarity. Bohr regarded renunciation of the cause-effect relation, or complementarity, of the space-time picture, as essential to the quantum mechanical account. Werner Heisenberg considered the question further, he saw the duality as present for all quantic entities, but not quite in the usual quantum mechanical account considered by Bohr. He saw it in what is called second quantization, which generates an new concept of fields which exist in ordinary space-time, causality still being visualizable. Classical field values are replaced by an new kind of field value, as considered in quantum field theory. Turning the reasoning around, ordinary quantum mechanics can be deduced as a specialized consequence of quantum field theory. Democritus argued that all things in the universe, including light, are composed of indivisible sub-components.

At the beginning of the 11th Century, the Arabic scientist Ibn al-Haytham wrote the first comprehensive Book of optics describing reflection and the operation of a pinhole lens via rays of light traveling from the point of emission to the eye. He asserted. In 1630, René Descartes popularized and accredited the opposing wave description in his treatise on light, The World, showing that the behavior of light could be re-created by modeling wave-like disturbances in a universal medium i.e. luminiferous aether. Beginning in 1670 and progressing over three decades, Isaac Newton developed and championed his corpuscular theory, arguing that the straight lines of reflection demonstrated light's particle nature, only particles could travel in such straight lines, he explained refraction by positing that particles of light accelerated laterally upon entering a denser medium. Around the same time, Newton's contemporaries Robert Hooke and Christiaan Huygens, Augustin-Jean Fresnel, mathematically refined the wave viewpoint, showing that if light traveled at different speeds in different media, refraction could be explained as the medium-dependent propagation of light waves.

The resulting Huygens–Fresnel principle was successful at reproducing light's behavior and was subsequently supported by Thomas Young's discovery of wave interference of light by his double-slit experiment in 1801. The wave view did not displace the ray and particle view, but began to dominate scientific thinking about light in the mid 19th century, since it could explain polarization phenomena that the alternatives could not. James Clerk Maxwell discovered that he could apply his discovered Maxwell's equations, along with a slight modification to describe self-propagating waves of oscillating electric and magnetic fields, it became apparent that visible light, ultraviolet light, infrared light were all electromagnetic waves of differing frequency. In 1901, Max Planck published an analysis that succeeded in reproducing the observed spectrum of light emitted by a glowing object. To accomplish this, Planck had to make a mathematical assumption of quantized energy of the oscillators i.e. atoms of the black body that emit radiation.

Einstein proposed that electromagnetic radiation itself is quantized, not the energy of radiating atoms. Black-body radiation, the emission of electromagnetic energy due to an object's heat, could not be explained from classical arguments alone; the equipartition theorem of classical mechanics, the basis of all classical thermodynamic theories, stated that an object's energy is partitioned among the object's vibrational modes. But applying the same reasoning to the electromagnetic emission of such a thermal object was not so successful; that thermal objects emit light had been long known. Since light was known to be waves of electromagnetism, physicists hoped to describe this emission via classical laws; this became known as the black body problem. Since the equipartition theorem worked so well in describing the vibrational modes of the thermal object itself, it was natural to assume that it would perform well in describing the radiative emission of such objects, but a problem arose if each mode received an equal partition of energy, the short wavelength modes would consume all the energy.

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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

Universe

The Universe is all of space and time and their contents, including planets, stars and all other forms of matter and energy. While the spatial size of the entire Universe is unknown, it is possible to measure the size of the observable universe, estimated to be 93 billion light years in diameter. In various multiverse hypotheses, a universe is one of many causally disconnected constituent parts of a larger multiverse, which itself comprises all of space and time and its contents; the earliest scientific models of the Universe were developed by ancient Greek and Indian philosophers and were geocentric, placing Earth at the center of the Universe. Over the centuries, more precise astronomical observations led Nicolaus Copernicus to develop the heliocentric model with the Sun at the center of the Solar System. In developing the law of universal gravitation, Isaac Newton built upon Copernicus' work as well as observations by Tycho Brahe and Johannes Kepler's laws of planetary motion. Further observational improvements led to the realization that the Sun is one of hundreds of billions of stars in the Milky Way, one of at least hundreds of billions of galaxies in the Universe.

Many of the stars in our galaxy have planets. At the largest scale galaxies are distributed uniformly and the same in all directions, meaning that the Universe has neither an edge nor a center. At smaller scales, galaxies are distributed in clusters and superclusters which form immense filaments and voids in space, creating a vast foam-like structure. Discoveries in the early 20th century have suggested that the Universe had a beginning and that space has been expanding since and is still expanding at an increasing rate; the Big Bang theory is the prevailing cosmological description of the development of the Universe. Under this theory and time emerged together 13.799±0.021 billion years ago and the energy and matter present have become less dense as the Universe expanded. After an initial accelerated expansion called the inflationary epoch at around 10−32 seconds, the separation of the four known fundamental forces, the Universe cooled and continued to expand, allowing the first subatomic particles and simple atoms to form.

Dark matter gathered forming a foam-like structure of filaments and voids under the influence of gravity. Giant clouds of hydrogen and helium were drawn to the places where dark matter was most dense, forming the first galaxies and everything else seen today, it is possible to see objects that are now further away than 13.799 billion light-years because space itself has expanded, it is still expanding today. This means that objects which are now up to 46.5 billion light-years away can still be seen in their distant past, because in the past when their light was emitted, they were much closer to the Earth. From studying the movement of galaxies, it has been discovered that the universe contains much more matter than is accounted for by visible objects; this unseen matter is known as dark matter. The ΛCDM model is the most accepted model of our universe, it suggests that about 69.2%±1.2% of the mass and energy in the universe is a cosmological constant, responsible for the current expansion of space, about 25.8%±1.1% is dark matter.

Ordinary matter is therefore only 4.9% of the physical universe. Stars and visible gas clouds only form about 6% of ordinary matter, or about 0.3% of the entire universe. There are many competing hypotheses about the ultimate fate of the universe and about what, if anything, preceded the Big Bang, while other physicists and philosophers refuse to speculate, doubting that information about prior states will be accessible; some physicists have suggested various multiverse hypotheses, in which our universe might be one among many universes that exist. The physical Universe is defined as all of their contents; such contents comprise all of energy in its various forms, including electromagnetic radiation and matter, therefore planets, stars and the contents of intergalactic space. The Universe includes the physical laws that influence energy and matter, such as conservation laws, classical mechanics, relativity; the Universe is defined as "the totality of existence", or everything that exists, everything that has existed, everything that will exist.

In fact, some philosophers and scientists support the inclusion of ideas and abstract concepts – such as mathematics and logic – in the definition of the Universe. The word universe may refer to concepts such as the cosmos, the world, nature; the word universe derives from the Old French word univers, which in turn derives from the Latin word universum. The Latin word was used by Cicero and Latin authors in many of the same senses as the modern English word is used. A term for "universe" among the ancient Greek philosophers from Pythagoras onwards was τὸ πᾶν, tò pân, defined as all matter and all space, τὸ ὅλον, tò hólon, which did not include the void. Another synonym was ho kósmos. Synonyms are found in Latin authors and survive in modern languages, e.g. the German words Das All and Natur for Universe. The same synonyms are found in English, such as everything, the cosmos, the world (as in the many-worlds interpr