1.
Quantum mechanics
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Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, early quantum theory was profoundly reconceived in the mid-1920s. The reconceived 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, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. 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 Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, lower energy/frequency means increased time and vice versa, photons of differing frequencies all deliver the same amount of action, but do so in varying time intervals. High frequency waves are damaging to human tissue because they deliver their action packets concentrated in time, the Copenhagen interpretation of Niels Bohr became widely accepted. In the mid-1920s, developments in mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons
2.
Introduction to quantum mechanics
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Quantum mechanics is the science of the very small. It explains the behaviour of matter and its interactions with energy on the scale of atoms, by contrast, classical physics only explains matter and energy on a scale familiar to human experience, including the behaviour of astronomical bodies such as the Moon. Classical physics is still used in much of science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large and the worlds that classical physics could not explain. This article describes how physicists discovered the limitations of classical physics and these concepts are described in roughly the order in which they were first discovered. For a more complete history of the subject, see History of quantum mechanics, Light behaves in some respects like particles and in other respects like waves. Matter—particles such as electrons and atoms—exhibits wavelike behaviour too, some light sources, including neon lights, give off only certain frequencies of light. Quantum mechanics shows that light, along all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its energies, colours. Since one never observes half a photon, a photon is a quantum, or smallest observable amount. More broadly, quantum mechanics shows that many quantities, such as angular momentum, angular momentum is required to take on one of a set of discrete allowable values, and since the gap between these values is so minute, the discontinuity is only apparent at the atomic level. Many aspects of mechanics are counterintuitive and can seem paradoxical. In the words of quantum physicist Richard Feynman, quantum mechanics deals with nature as She is – absurd, thermal radiation is electromagnetic radiation emitted from the surface of an object due to the objects internal energy. If an object is heated sufficiently, it starts to light at the red end of the spectrum. Heating it further causes the colour to change from red to yellow, white, a perfect emitter is also a perfect absorber, when it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, a thermal emitter is known as a black body. In the late 19th century, thermal radiation had been well characterized experimentally. However, classical physics led to the Rayleigh-Jeans law, which, as shown in the figure, agrees with experimental results well at low frequencies, physicists searched for a single theory that explained all the experimental results. The first model that was able to explain the full spectrum of radiation was put forward by Max Planck in 1900
3.
History of quantum mechanics
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The history of quantum mechanics is a fundamental part of the history of modern physics. In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, Ludwig Boltzmann suggested in 1877 that the energy levels of a physical system, such as a molecule, could be discrete. He was a founder of the Austrian Mathematical Society, together with the mathematicians Gustav von Escherich, the earlier Wien approximation may be derived from Plancks law by assuming h ν ≫ k T. This statement has been called the most revolutionary sentence written by a physicist of the twentieth century and these energy quanta later came to be called photons, a term introduced by Gilbert N. Lewis in 1926. In 1913, Bohr explained the lines of the hydrogen atom, again by using quantization, in his paper of July 1913 On the Constitution of Atoms. They are collectively known as the old quantum theory, the phrase quantum physics was first used in Johnstons Plancks Universe in Light of Modern Physics. In 1923, the French physicist Louis de Broglie put forward his theory of waves by stating that particles can exhibit wave characteristics. This theory was for a particle and derived from special relativity theory. Schrödinger subsequently showed that the two approaches were equivalent, heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation started to take shape at about the same time. Starting around 1927, Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron, the Dirac equation achieves the relativistic description of the wavefunction of an electron that Schrödinger failed to obtain. It predicts electron spin and led Dirac to predict the existence of the positron and he also pioneered the use of operator theory, including the influential bra–ket notation, as described in his famous 1930 textbook. These, like other works from the founding period, still stand. The field of chemistry was pioneered by physicists Walter Heitler and Fritz London. Beginning in 1927, researchers made attempts at applying quantum mechanics to fields instead of single particles, early workers in this area include P. A. M. Dirac, W. Pauli, V. Weisskopf, and P. Jordan and this area of research culminated in the formulation of quantum electrodynamics by R. P. Feynman, F. Dyson, J. Schwinger, and S. I. Tomonaga during the 1940s. Quantum electrodynamics describes a quantum theory of electrons, positrons, and the electromagnetic field, the theory of quantum chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated by Politzer, Gross, thomas Youngs double-slit experiment demonstrating the wave nature of light. J. J. Thomsons cathode ray tube experiments, the study of black-body radiation between 1850 and 1900, which could not be explained without quantum concepts
4.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
5.
Interference (wave propagation)
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In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves or matter waves. If a crest of a wave meets a crest of wave of the same frequency at the same point. If a crest of one wave meets a trough of another wave, constructive interference occurs when the phase difference between the waves is an even multiple of π, whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values. Consider, for example, what happens when two identical stones are dropped into a pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped, when the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement, in other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above, prime examples of light interference are the famous Double-slit experiment, laser speckle, optical thin layers and films and interferometers. Dark areas in the slit are not available to the photons. Thin films also behave in a quantum manner, the above can be demonstrated in one dimension by deriving the formula for the sum of two waves. Suppose a second wave of the frequency and amplitude but with a different phase is also traveling to the right W2 = A cos where ϕ is the phase difference between the waves in radians. Constructive interference, If the phase difference is a multiple of pi. Interference is essentially a redistribution process. The energy which is lost at the interference is regained at the constructive interference. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave, assuming that the two waves are in phase at the point B, then the relative phase changes along the x-axis. Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, a fringe pattern is produced, where the separation of the maxima is d f = λ sin θ
6.
Casimir effect
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In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. They are named after the Dutch physicist Hendrik Casimir who predicted them in 1948, the typical example is of the two uncharged conductive plates in a vacuum, placed a few nanometers apart. In a classical description, the lack of an external field means that there is no field between the plates, and no force would be measured between them and this force has been measured and is a striking example of an effect captured formally by second quantization. However, the treatment of conditions in these calculations has led to some controversy. In fact, Casimirs original goal was to compute the van der Waals force between molecules of the conductive plates. Thus it can be interpreted without any reference to the zero-point energy of quantum fields, predictions of the force were later extended to finite-conductivity metals and dielectrics, and recent calculations have considered more general geometries. It was not until 1997, however, that a direct experiment, subsequent experiments approach an accuracy of a few percent. Because the strength of the force falls off rapidly with distance, on a submicron scale, this force becomes so strong that it becomes the dominant force between uncharged conductors. In fact, at separations of 10 nm—about 100 times the size of an atom—the Casimir effect produces the equivalent of about 1 atmosphere of pressure. Any medium supporting oscillations has an analogue of the Casimir effect, for example, beads on a string as well as plates submerged in noisy water or gas illustrate the Casimir force. Since the value of this depends on the shapes and positions of the conductors and dielectrics. Vibrations in this field propagate and are governed by the wave equation for the particular field in question. The second quantization of field theory requires that each such ball-spring combination be quantized, that is. At the most basic level, the field at point in space is a simple harmonic oscillator. Excitations of the field correspond to the particles of particle physics. However, even the vacuum has a complex structure, so all calculations of quantum field theory must be made in relation to this model of the vacuum. The vacuum has, implicitly, all of the properties that a particle may have, spin, or polarization in the case of light, energy, on average, most of these properties cancel out, the vacuum is, after all, empty in this sense. One important exception is the energy or the vacuum expectation value of the energy
7.
Coherence (physics)
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In physics, two wave sources are perfectly coherent if they have a constant phase difference and the same frequency. Coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave, or between several waves or wave packets. Interference is nothing more than the addition, in the mathematical sense, a single wave can interfere with itself, but this is still an addition of two waves. Constructive or destructive interferences are limit cases, and two waves interfere, even if the result of the addition is complicated or not remarkable. Two waves are said to be coherent if they have a constant relative phase, spatial coherence describes the correlation between waves at different points in space, either lateral or longitudinal. Temporal coherence describes the correlation between waves observed at different moments in time, both are observed in the Michelson–Morley experiment and Youngs interference experiment. Similarly, if in an experiment, the space between the two slits is increased, the coherence dies gradually and finally the fringes disappear, showing spatial coherence. In both cases, the fringe amplitude slowly disappears, as the difference increases past the coherence length. The property of coherence is the basis for commercial applications such as holography, a precise definition is given at degree of coherence. The cross-spectral density and the spectral density are defined as the Fourier transforms of the cross-correlation. In this case the coherence is a function of frequency, in that case, coherence is a function of wavenumber. The coherence varies in the interval 0 ⩽ γ x y 2 ⩽1, if γ x y 2 =1 it means that the signals are perfectly correlated or linearly related and if γ x y 2 =0 they are totally uncorrelated. If a linear system is being measured, x being the input and y the output, however, if non-linearities are present in the system the coherence will vary in the limit given above. The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function, the cross-correlation quantifies the ability to predict the phase of the second wave by knowing the phase of the first. As an example, consider two waves perfectly correlated for all times, at any time, phase difference will be constant. As will be discussed below, the second wave need not be a separate entity and it could be the first wave at a different time or position. In this case, the measure of correlation is the autocorrelation function, degree of correlation involves correlation functions. 545-550 These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof
8.
Quantum decoherence
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Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons behave like waves and are described by a wavefunction and these waves can interfere, leading to the peculiar behaviour of quantum particles. As long as there exists a definite relation between different states, the system is said to be coherent. This coherence is a property of quantum mechanics, and is necessary for the function of quantum computers. However, when a system is not perfectly isolated, but in contact with its surroundings, the coherence decays with time. As a result of this process, the behaviour is lost. Decoherence was first introduced in 1970 by the German physicist H. Dieter Zeh and has been a subject of research since the 1980s. Decoherence can be viewed as the loss of information from a system into the environment, viewed in isolation, the systems dynamics are non-unitary. Thus the dynamics of the system alone are irreversible, as with any coupling, entanglements are generated between the system and environment. These have the effect of sharing quantum information with—or transferring it to—the surroundings, Decoherence has been used to understand the collapse of the wavefunction in quantum mechanics. Decoherence does not generate actual wave function collapse and it only provides an explanation for the observation of wave function collapse, as the quantum nature of the system leaks into the environment. That is, components of the wavefunction are decoupled from a coherent system, a total superposition of the global or universal wavefunction still exists, but its ultimate fate remains an interpretational issue. Specifically, decoherence does not attempt to explain the measurement problem, rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive. Decoherence represents a challenge for the realization of quantum computers. Simply put, they require that coherent states be preserved and that decoherence is managed, to examine how decoherence operates, an intuitive model is presented. The model requires some familiarity with quantum theory basics, analogies are made between visualisable classical phase spaces and Hilbert spaces. A more rigorous derivation in Dirac notation shows how decoherence destroys interference effects, next, the density matrix approach is presented for perspective. An N-particle system can be represented in non-relativistic quantum mechanics by a wavefunction, ψ and this has analogies with the classical phase space
9.
Energy level
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A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy and these discrete values are called energy levels. The energy spectrum of a system with discrete energy levels is said to be quantized. In chemistry and atomic physics, a shell, or a principal energy level. The closest shell to the nucleus is called the 1 shell, followed by the 2 shell, then the 3 shell, the shells correspond with the principal quantum numbers or are labeled alphabetically with letters used in the X-ray notation. Each shell can contain only a number of electrons, The first shell can hold up to two electrons, the second shell can hold up to eight electrons, the third shell can hold up to 18. The general formula is that the nth shell can in principle hold up to 2 electrons, since electrons are electrically attracted to the nucleus, an atoms electrons will generally occupy outer shells only if the more inner shells have already been completely filled by other electrons. However, this is not a requirement, atoms may have two or even three incomplete outer shells. For an explanation of why electrons exist in these shells see electron configuration, if the potential energy is set to zero at infinite distance from the atomic nucleus or molecule, the usual convention, then bound electron states have negative potential energy. If an atom, ion, or molecule is at the lowest possible level, it. If it is at an energy level, it is said to be excited. If more than one quantum state is at the same energy. They are then called degenerate energy levels, quantized energy levels result from the relation between a particles energy and its wavelength. For a confined particle such as an electron in an atom, only stationary states with energies corresponding to integral numbers of wavelengths can exist, for other states the waves interfere destructively, resulting in zero probability density. Elementary examples that show mathematically how energy levels come about are the particle in a box, the first evidence of quantization in atoms was the observation of spectral lines in light from the sun in the early 1800s by Joseph von Fraunhofer and William Hyde Wollaston. The notion of levels was proposed in 1913 by Danish physicist Niels Bohr in the Bohr theory of the atom. The modern quantum mechanical theory giving an explanation of these levels in terms of the Schrödinger equation was advanced by Erwin Schrödinger and Werner Heisenberg in 1926. When the electron is bound to the atom in any closer value of n, assume there is one electron in a given atomic orbital in a hydrogen-like atom
10.
Quantum entanglement
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Measurements of physical properties such as position, momentum, spin, and polarization, performed on entangled particles are found to be appropriately correlated. Later, however, the predictions of quantum mechanics were verified experimentally. Recent experiments have measured entangled particles within less than one hundredth of a percent of the time of light between them. According to the formalism of theory, the effect of measurement happens instantly. It is not possible, however, to use this effect to transmit information at faster-than-light speeds. Research is also focused on the utilization of entanglement effects in communication and computation, the counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen. In this study, they formulated the EPR paradox, an experiment that attempted to show that quantum mechanical theory was incomplete. They wrote, We are thus forced to conclude that the description of physical reality given by wave functions is not complete. However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered and he shortly thereafter published a seminal paper defining and discussing the notion, and terming it entanglement. Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, Einstein later famously derided entanglement as spukhafte Fernwirkung or spooky action at a distance. The EPR paper generated significant interest among physicists and inspired much discussion about the foundations of quantum mechanics, until recently each had left open at least one loophole by which it was possible to question the validity of the results. However, in 2015 the first loophole-free experiment was performed, which ruled out a class of local realism theories with certainty. The work of Bell raised the possibility of using these super-strong correlations as a resource for communication and it led to the discovery of quantum key distribution protocols, most famously BB84 by Charles H. Bennett and Gilles Brassard and E91 by Artur Ekert. Although BB84 does not use entanglement, Ekerts protocol uses the violation of a Bells inequality as a proof of security, in entanglement, one constituent cannot be fully described without considering the other. Quantum systems can become entangled through various types of interactions, for some ways in which entanglement may be achieved for experimental purposes, see the section below on methods. Entanglement is broken when the entangled particles decohere through interaction with the environment, for example, as an example of entanglement, a subatomic particle decays into an entangled pair of other particles. For instance, a particle could decay into a pair of spin-½ particles. The special property of entanglement can be observed if we separate the said two particles
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Ground state
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The ground state of a quantum mechanical system is its lowest-energy state, the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with greater than the ground state. In the quantum theory, the ground state is usually called the vacuum state or the vacuum. If more than one ground state exists, they are said to be degenerate, many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state, according to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state, thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a crystal lattice, have a unique ground state. It is also possible for the highest excited state to have zero temperature for systems that exhibit negative temperature. In 1D, the state of the Schrödinger equation has no nodes. This can be proved considering the energy of a state with a node at x =0, i. e. ψ =0. Consider the average energy in this state ⟨ ψ | H | ψ ⟩ = ∫ d x where V is the potential, now consider a small interval around x =0, i. e. x ∈. Take a new wave function ψ ′ to be defined as ψ ′ = ψ, x < − ϵ and ψ ′ = − ψ, x > ϵ, if ϵ is small enough then this is always possible to do so that ψ ′ is continuous. Note that the energy density | d ψ ′ d x |2 < | d ψ d x |2 everywhere because of the normalization. For definiteness let us choose V ≥0, then it is clear that outside the interval x ∈ the potential energy density is smaller for the ψ ′ because | ψ ′ | < | ψ | there. Therefore, the energy is unchanged up to order ϵ2 if we deform the state with a node ψ into a state without a node ψ ′. We can therefore remove all nodes and reduce the energy, which implies that the wave function cannot have a node. The wave function of the state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The wave function of the state of a hydrogen atom is a spherically-symmetric distribution centred on the nucleus. The electron is most likely to be found at a distance from the equal to the Bohr radius
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Quantum state
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In quantum physics, quantum state refers to the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, knowledge of the quantum state together with the rules for the systems evolution in time exhausts all that can be predicted about the systems behavior. A mixture of states is again a quantum state. Quantum states that cannot be written as a mixture of states are called pure quantum states. Mathematically, a quantum state can be represented by a ray in a Hilbert space over the complex numbers. The ray is a set of nonzero vectors differing by just a scalar factor, any of them can be chosen as a state vector to represent the ray. A unit vector is usually picked, but its phase factor can be chosen freely anyway, nevertheless, such factors are important when state vectors are added together to form a superposition. Hilbert space is a generalization of the ordinary Euclidean space and it all possible pure quantum states of the given system. If this Hilbert space, by choice of representation, is exhibited as a function space, a more complicated case is given by the spin part of a state vector | ψ ⟩ =12, which involves superposition of joint spin states for two particles with spin 1⁄2. A mixed quantum state corresponds to a mixture of pure states, however. Mixed states are described by so-called density matrices, a pure state can also be recast as a density matrix, in this way, pure states can be represented as a subset of the more general mixed states. For example, if the spin of an electron is measured in any direction, e. g. with a Stern–Gerlach experiment, the Hilbert space for the electrons spin is therefore two-dimensional. A mixed state, in case, is a 2 ×2 matrix that is Hermitian, positive-definite. These probability distributions arise for both mixed states and pure states, it is impossible in quantum mechanics to prepare a state in all properties of the system are fixed. This is exemplified by the uncertainty principle, and reflects a difference between classical and quantum physics. Even in quantum theory, however, for every observable there are states that have an exact. In the mathematical formulation of mechanics, pure quantum states correspond to vectors in a Hilbert space. The operator serves as a function which acts on the states of the system
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Quantum teleportation
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Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for faster-than-light transport or communication of classical bits. While it has proven possible to one or more qubits of information between two atoms, this has not yet been achieved between molecules or anything larger. Although the name is inspired by the commonly used in fiction, there is no relationship outside the name. The seminal paper first expounding the idea of quantum teleportation was published by C. H. Bennett, G. Brassard, Crépeau, R. Jozsa, A. Peres and W. K. Wootters in 1993. Since then, quantum teleportation was first realized with single photons and later demonstrated with various systems such as atoms, ions, electrons. The record distance for quantum teleportation is 143 km and these results were confirmed by two studies with statistical significance over 5 standard deviations which were published in December 2015. In matters relating to quantum or classical information theory, it is convenient to work with the simplest possible unit of information, in classical information this is a bit, commonly represented using one or zero. The quantum analog of a bit is a bit, or qubit. Qubits encode a type of information, called quantum information, which differs sharply from classical information, for example, quantum information can be neither copied nor destroyed. Quantum teleportation provides a mechanism of moving a qubit from one location to another, however, as of 2013, only photons and single atoms have been employed as information bearers. In essence, a kind of quantum channel between two sites must be established first, before a qubit can be moved. Teleportation also requires a classical information link to be established, as two classical bits must be transmitted to each qubit. The reason for this is that the results of the measurements must be communicated, the need for such links may, at first, seem disappointing, however, this is not unlike ordinary communications, which requires wires, radios or lasers. Whats more, Bell states are most easily shared using photons from lasers, the quantum states of single atoms have been teleported. Physicists have teleported the qubits encoded in the state of atoms, they have not teleported the nuclear state. It is therefore false to say an atom has been teleported, the quantum state of an atom has. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, an important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems. These correlations hold even when measurements are chosen and performed independently, out of contact from one another
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Qubit
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In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit. A qubit is a two-state quantum-mechanical system, such as the polarization of a single photon, in a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a superposition of states at the same time, a property that is fundamental to quantum computing. The concept of the qubit was unknowingly introduced by Stephen Wiesner in 1983, in his proposal for quantum money, the coining of the term qubit is attributed to Benjamin Schumacher. The paper describes a way of compressing states emitted by a source of information so that they require fewer physical resources to store. This procedure is now known as Schumacher compression, the bit is the basic unit of information. It is used to represent information by computers, an analogy to this is a light switch—its off position can be thought of as 0 and its on position as 1. A qubit has a few similarities to a bit, but is overall very different. There are two possible outcomes for the measurement of a qubit—usually 0 and 1, like a bit, the difference is that whereas the state of a bit is either 0 or 1, the state of a qubit can also be a superposition of both. It is possible to encode one bit in one qubit. However, a qubit can hold more information, e. g. up to two bits using superdense coding. For a system of n components, a description of its state in classical physics requires only n bits. The two states in which a qubit may be measured are known as basis states, as is the tradition with any sort of quantum states, they are represented by Dirac—or bra–ket—notation. This means that the two basis states are conventionally written as |0 ⟩ and |1 ⟩. A pure qubit state is a superposition of the basis states. When we measure this qubit in the basis, the probability of outcome |0 ⟩ is | α |2. Because the absolute squares of the amplitudes equate to probabilities, it follows that α and β must be constrained by the equation | α |2 + | β |2 =1. It might at first sight seem that there should be four degrees of freedom, as α and β are complex numbers with two degrees of freedom each
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Spin (physics)
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In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. Spin is one of two types of angular momentum in mechanics, the other being orbital angular momentum. In some ways, spin is like a vector quantity, it has a definite magnitude, all elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number. The SI unit of spin is the or, just as with classical angular momentum, very often, the spin quantum number is simply called spin leaving its meaning as the unitless spin quantum number to be inferred from context. When combined with the theorem, the spin of electrons results in the Pauli exclusion principle. Wolfgang Pauli was the first to propose the concept of spin, in 1925, Ralph Kronig, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested an physical interpretation of particles spinning around their own axis. The mathematical theory was worked out in depth by Pauli in 1927, when Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it. As the name suggests, spin was originally conceived as the rotation of a particle around some axis and this picture is correct so far as spin obeys the same mathematical laws as quantized angular momenta do. On the other hand, spin has some properties that distinguish it from orbital angular momenta. Although the direction of its spin can be changed, a particle cannot be made to spin faster or slower. The spin of a particle is associated with a magnetic dipole moment with a g-factor differing from 1. This could only occur if the internal charge of the particle were distributed differently from its mass. The conventional definition of the quantum number, s, is s = n/2. Hence the allowed values of s are 0, 1/2,1, 3/2,2, the value of s for an elementary particle depends only on the type of particle, and cannot be altered in any known way. The spin angular momentum, S, of any system is quantized. The allowed values of S are S = ℏ s = h 4 π n, in contrast, orbital angular momentum can only take on integer values of s, i. e. even-numbered values of n. Those particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while particles with integer spins. The two families of particles obey different rules and broadly have different roles in the world around us, a key distinction between the two families is that fermions obey the Pauli exclusion principle, that is, there cannot be two identical fermions simultaneously having the same quantum numbers
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Quantum superposition
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Quantum superposition is a fundamental principle of quantum mechanics. Mathematically, it refers to a property of solutions to the Schrödinger equation, since the Schrödinger equation is linear, an example of a physically observable manifestation of superposition is interference peaks from an electron wave in a double-slit experiment. Another example is a logical qubit state, as used in quantum information processing. Here |0 ⟩ is the Dirac notation for the state that will always give the result 0 when converted to classical logic by a measurement. Likewise |1 ⟩ is the state that will convert to 1. The numbers that describe the amplitudes for different possibilities define the kinematics, the dynamics describes how these numbers change with time. This list is called the vector, and formally it is an element of a Hilbert space. The quantities that describe how they change in time are the transition probabilities K x → y, which gives the probability that, starting at x, the particle ends up at y time t later. When no time passes, nothing changes, for 0 elapsed time K x → y = δ x y, the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the change in the probability. Quantum amplitudes give the rate at which amplitudes change in time, the reason it is multiplied by i is that the condition that U is unitary translates to the condition, = I H † − H =0 which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities, which have an interpretation as energy levels. For a particle that has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. By redefining the phase of the wavefunction in time, ψ → ψ e i 2 c t, but this phase rotation introduces a linear term. I d ψ n d t = c ψ n +1 −2 c ψ n + c ψ n −1, the analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. The analogous expression in quantum mechanics is the path integral, a generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions
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Symmetry in quantum mechanics
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In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints, the notational conventions used in this article are as follows. Boldface indicates vectors, four vectors, matrices, and vectorial operators, wide hats are for operators, narrow hats are for unit vectors. The summation convention on the repeated indices is used, unless stated otherwise. Generally, the correspondence between continuous symmetries and conservation laws is given by Noethers theorem and this can be done for displacements, durations, and angles. Additionally, the invariance of certain quantities can be seen by making changes in lengths and angles. In what follows, transformations on only one-particle wavefunctions in the form, Ω ^ ψ = ψ are considered, unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state must be invariant under these transformations. The inverse is the Hermitian conjugate Ω ^ −1 = Ω ^ †, the results can be extended to many-particle wavefunctions. Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers, i. e. the operator equals its Hermitian conjugate, following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall Let G be a Lie group, ξN. the dimension of the group, N, is the number of parameters it has. The generators satisfy the commutator, = i f a b c X c where fabc are the constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra, due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices. The representations of the group are denoted using a capital D and defined by, representations are linear operators that take in group elements and preserve the composition rule, D D = D. A representation which cannot be decomposed into a sum of other representations, is called irreducible. It is conventional to label irreducible representations by a number n in brackets, as in D, or if there is more than one number. Representations also exist for the generators and the notation of a capital D is used in this context. An example of abuse is to be found in the defining equation above
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Quantum tunnelling
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Quantum tunnelling or tunneling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun. It has important applications to modern devices such as the diode, quantum computing. The effect was predicted in the early 20th century and its acceptance as a physical phenomenon came mid-century. Tunnelling is often explained using the Heisenberg uncertainty principle and the duality of matter. Pure quantum mechanical concepts are central to the phenomenon, so quantum tunnelling is one of the implications of quantum mechanics. Quantum tunnelling was developed from the study of radioactivity, which was discovered in 1896 by Henri Becquerel, radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903. 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. J. J. Thomson commented the finding warranted further investigation, 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. Friedrich Hund was the first to notice of tunnelling in 1927 when he was calculating the ground state of the double-well potential. Its first application was an explanation for alpha decay, which was done in 1928 by George Gamow and independently by Ronald Gurney. After attending a seminar by Gamow, Max Born recognised the generality of tunnelling and he realised that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Shortly thereafter, both 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, in 2016, the quantum tunneling of water was discovered. Quantum tunnelling falls under the domain of quantum mechanics, the study of what happens at the quantum scale and this process cannot be directly perceived, but much of its understanding is shaped by the microscopic world, which classical mechanics cannot adequately explain. Classical mechanics predicts that particles that do not have enough energy to surmount a barrier 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 or in the extreme case, bury itself inside the wall
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Vacuum state
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In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles, zero-point field is sometimes used as a synonym for the vacuum state of an individual quantized field. According to present-day understanding of what is called the state or the quantum vacuum. According to quantum mechanics, the state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into. The QED vacuum of quantum electrodynamics was the first vacuum of quantum theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga and Schwinger, today the electromagnetic interactions and the weak interactions are unified in the theory of the electroweak interaction. The Standard Model is a generalization of the QED work to all the known elementary particles. Quantum chromodynamics is the portion of the Standard Model that deals with strong interactions and it is the object of study in the Large Hadron Collider and the Relativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions. In this case the vacuum value of any field operator vanishes. For quantum field theories in which perturbation theory breaks down at low energies field operators may have non-vanishing vacuum expectation values called condensates. In the Standard Model, the vacuum expectation value of the Higgs field. In many situations, the state can be defined to have zero energy. The vacuum state is associated with a zero-point energy, and this zero-point energy has measurable effects, in the laboratory, it may be detected as the Casimir effect. In physical cosmology, the energy of the cosmological vacuum appears as the cosmological constant, in fact, the energy of a cubic centimeter of empty space has been calculated figuratively to be one trillionth of an erg. An outstanding requirement imposed on a potential Theory of Everything is that the energy of the vacuum state must explain the physically observed cosmological constant. For a relativistic theory, the vacuum is Poincaré invariant. Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEVs, the VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case the vacuum has less symmetry than the theory allows, in principle, quantum corrections to Maxwells equations can cause the experimental electrical permittivity ε of the vacuum state to deviate from the defined scalar value ε0 of the electric constant
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Wave function
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A wave function in quantum physics is a description of the quantum state of a system. The wave function is a probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a function are the Greek letters ψ or Ψ. The wave function is a function of the degrees of freedom corresponding to some set of commuting observables. Once such a representation is chosen, the function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom. Other discrete variables can also be included, such as isospin and these values are often displayed in a column matrix. According to the principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions. The Schrödinger equation determines how wave functions evolve over time, a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name wave function, and gives rise to wave–particle duality, however, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. The integral of this quantity, over all the degrees of freedom. This general requirement a wave function must satisfy is called the normalization condition, since the wave function is complex valued, only its relative phase and relative magnitude can be measured. In 1905 Einstein postulated the proportionality between the frequency of a photon and its energy, E = hf, and in 1916 the corresponding relation between photon momentum and wavelength, λ = h/p, the equations represent wave–particle duality for both massless and massive particles. In the 1920s and 1930s, quantum mechanics was developed using calculus and those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing wave mechanics. Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, Schrödinger subsequently showed that the two approaches were equivalent. However, no one was clear on how to interpret it, at first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with the scattering of a wave packet representing a particle off a target. While a scattered particle may scatter in any direction, it not break up
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Matter wave
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Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter can exhibit wave-like behavior, for example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave is referred to as the de Broglie hypothesis due to having been proposed by Louis de Broglie in 1924. Matter waves are referred to as de Broglie waves, the de Broglie wavelength is the wavelength, λ, associated with a massive particle and is related to its momentum, p, through the Planck constant, h, λ = h p. The wave-like behavior of matter is crucial to the theory of atomic structure. In 1900, this division was exposed to doubt, when, investigating the theory of black body thermal radiation and it was thoroughly challenged in 1905. Extending Plancks investigation in several ways, including its connection with the photoelectric effect, light quanta are now called photons. In the modern convention, frequency is symbolized by f as is done in the rest of this article, einstein’s postulate was confirmed experimentally by Robert Millikan and Arthur Compton over the next two decades. De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties, the relationship is now known to hold for all types of matter, all matter exhibits properties of both particles and waves. In 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve—the matter wave analogue of Maxwell’s equations—and used it to derive the energy spectrum of hydrogen, furthermore, neutral atoms and even molecules have been shown to be wave-like. In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a nickel target. The angular dependence of the electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for x-rays. At the same time George Paget Thomson at the University of Aberdeen was independently firing electrons at very thin metal foils to demonstrate the same effect, before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be exhibited only by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter, when the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons. This was a result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, the Davisson–Germer experiment showed the wave-nature of matter, advances in laser cooling have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal de Broglie wavelengths come into the micrometre range and this effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of phenomena is based on the wave properties of neutral atoms
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Afshar experiment
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The Afshar experiment is an optics experiment, devised and carried out by Shahriar Afshar at Harvard University in 2004, which is a variation of the double slit experiment in quantum mechanics. Afshars experiment uses a variant of Thomas Youngs classic double-slit experiment to create patterns to investigate complementarity. Such interferometer experiments typically have two arms or paths a photon may take, one of Afshars assertions is that, in his experiment, it is possible to check for interference fringes of a photon stream while at the same time observing each photons path. The results were presented at a Harvard seminar in March 2004, the experiment was featured as the cover story in the July 24,2004 edition of New Scientist. Afshar presented his work also at the American Physical Society meeting in Los Angeles and his peer-reviewed paper was published in Foundations of Physics in January 2007. Afshar claims that his experiment invalidates the complementarity principle and has far-reaching implications for the understanding of quantum mechanics, according to Cramer, Afshars results support Cramers own transactional interpretation of quantum mechanics and challenge the many-worlds interpretation of quantum mechanics. This claim has not been published in a reviewed journal. The experiment uses a similar to that for the double-slit experiment. In Afshars variant, light generated by a laser passes through two closely spaced circular pinholes, after the dual pinholes, a lens refocuses the light so that the image of each pinhole falls on separate photon-detectors. When the light acts as a wave, because of quantum interference one can observe that there are regions that the photons avoid, called dark fringes. A grid of wires is placed just before the lens so that the wires lie in the dark fringes of an interference pattern which is produced by the dual pinhole setup. If one of the pinholes is blocked, the pattern will no longer be formed. Consequently, the quality is reduced. When one pinhole is closed, the grid of wires causes appreciable diffraction in the light, the effect is not dependent on the light intensity. Afshar argues that this contradicts the principle of complementarity, since it shows both complementary wave and particle characteristics in the same experiment for the same photons. Afshar has responded to critics in his academic talks, his blog. She proposes that Afshars experiment is equivalent to preparing an electron in a spin-up state and this does not imply that one has found out the up-down spin state and the sideways spin state of any electron simultaneously. In addition she underscores her conclusion with an analysis of the Afshar setup within the framework of the interpretation of quantum mechanics
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Bell test experiments
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Under local realism, correlations between outcomes of different measurements performed on separated physical systems have to satisfy certain constraints, called Bell inequalities. John Bell derived the first inequality of this kind in his paper On the Einstein-Podolsky-Rosen Paradox, Bells Theorem states that the predictions of quantum mechanics, concerning correlations, being inconsistent with Bells inequality, cannot be reproduced by any local hidden variable theory. However, this doesnt disprove hidden variable theories that are such as Bohmian mechanics. A Bell test experiment is one designed to test whether or not the real world satisfies local realism, in practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons, rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments, such experiments fall into two classes, depending on whether the analysers used have one or two output channels. The diagram shows an optical experiment of the two-channel kind for which Alain Aspect set a precedent in 1982. Coincidences are recorded, the results being categorised as ++, +−, −+ or −−, four separate subexperiments are conducted, corresponding to the four terms E in the test statistic S. For each selected value of a and b, the numbers of coincidences in each category are recorded, the experimental estimate for E is then calculated as, E = /. Once all four E’s have been estimated, an estimate of the test statistic S = E − E + E + E can be found. If S is numerically greater than 2 it has infringed the CHSH inequality, the experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories. A strong assumption has had to be made, however, to use of expression. It has been assumed that the sample of detected pairs is representative of the emitted by the source. That this assumption may not be true comprises the fair sampling loophole, the derivation of the inequality is given in the CHSH Bell test page. Prior to 1982 all actual Bell tests used single-channel polarisers and variations on an inequality designed for this setup, the latter is described in Clauser, Horne, Shimony and Holts much-cited 1969 article as being the one suitable for practical use. Counts are taken as before and used to estimate the test statistic, S = / N, where the symbol ∞ indicates absence of a polariser. If S exceeds 0 then the experiment is declared to have infringed Bells inequality, in order to derive, CHSH in their 1969 paper had to make an extra assumption, the so-called fair sampling assumption. This means that the probability of detection of a given photon, if this assumption were violated, then in principle a local hidden variable model could violate the CHSH inequality. In a later 1974 article, Clauser and Horne replaced this assumption by a weaker, no enhancement assumption, deriving a modified inequality, see the page on Clauser
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Double-slit experiment
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A simpler form of the double-slit experiment was performed originally by Thomas Young in 1801. He believed it demonstrated that the theory of light was correct. The experiment belongs to a class of double path experiments. Changes in the lengths of both waves result in a phase shift, creating an interference pattern. Another version is the Mach–Zehnder interferometer, which splits the beam with a mirror, furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit, and not through both slits. However, such experiments demonstrate that particles do not form the pattern if one detects which slit they pass through. These results demonstrate the principle of wave–particle duality, other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit. Additionally, the detection of individual impacts is observed to be inherently probabilistic. The experiment can be done with much larger than electrons and photons. The largest entities for which the experiment has been performed were molecules that each comprised 810 atoms. However, when this experiment is actually performed, the pattern on the screen is a diffraction pattern in which the light is spread out. The smaller the slit, the greater the angle of spread, the top portion of the image shows the central portion of the pattern formed when a red laser illuminates a slit and, if one looks carefully, two faint side bands. More bands can be seen with a highly refined apparatus. Diffraction explains the pattern as being the result of the interference of waves from the slit. If one illuminates two parallel slits, the light from the two slits again interferes, here the interference is a more-pronounced pattern with a series of light and dark bands. The width of the bands is a property of the frequency of the illuminating light, however, the later discovery of the photoelectric effect demonstrated that under different circumstances, light can behave as if it is composed of discrete particles. These seemingly contradictory discoveries made it necessary to go beyond classical physics, the double-slit experiment has become a classic thought experiment, for its clarity in expressing the central puzzles of quantum mechanics. In reality, it contains the only mystery, Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment