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, Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle, with a discrete quantum of energy that was dependent on its frequency. The Copenhagen interpretation of Niels Bohr became widely accepted, in the mid-1920s, developments in quantum 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. From Einsteins simple postulation was born a flurry of debating, theorizing, thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927
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.
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
7.
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
8.
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
9.
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 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
11.
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
12.
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
13.
Uncertainty principle
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The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928. Heisenberg offered such an effect at the quantum level as a physical explanation of quantum uncertainty. Thus, the uncertainty principle actually states a fundamental property of quantum systems, since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their research program. These include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems, applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers. The uncertainty principle is not readily apparent on the scales of everyday experience. So it is helpful to demonstrate how it applies to more easily understood physical situations, two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, a nonzero function and its Fourier transform cannot both be sharply localized. In matrix mechanics, the formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value, for example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. According to the de Broglie hypothesis, every object in the universe is a wave, the position of the particle is described by a wave function Ψ. The time-independent wave function of a plane wave of wavenumber k0 or momentum p0 is ψ ∝ e i k 0 x = e i p 0 x / ℏ. In the case of the plane wave, | ψ |2 is a uniform distribution. In other words, the position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. The figures to the right show how with the addition of many plane waves, in mathematical terms, we say that ϕ is the Fourier transform of ψ and that x and p are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, One way to quantify the precision of the position and momentum is the standard deviation σ. Since | ψ |2 is a probability density function for position, the precision of the position is improved, i. e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i. e. increased σp. Another way of stating this is that σx and σp have a relationship or are at least bounded from below
14.
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
15.
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
16.
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
17.
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
18.
Popper's experiment
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Poppers experiment is an experiment proposed by the philosopher Karl Popper to put to the test different interpretations of quantum mechanics. In fact, as early as 1934, Popper started criticising the increasingly more accepted Copenhagen interpretation, nonetheless, Popper gave other and important contributions to the foundations of quantum mechanics in different periods of his long and prolific career. In particular, in the 1980s, he established collaborations and new acquaintances with some illustrious physicists working in the field of foundations of QM, in 1980, Popper proposed his more important, yet overlooked, contribution to QM, a new simplified version of the EPR experiment. The experiment was published only two years later, in the third volume of the Poscript to the Logic of Scientific Discovery. The most widely known interpretation of quantum mechanics is the Copenhagen interpretation put forward by Niels Bohr and it maintains that observations lead to a wavefunction collapse, thereby suggesting the counter-intuitive result that two well separated, non-interacting systems require action-at-a-distance. Popper argued that such non-locality conflicts with common sense, and would lead to a subjectivist interpretation of phenomena, contrarily to the first proposal of 1934, Poppers experiment of 1980 exploits couples of entangled particles, in order to put to the test Heisenbergs uncertainty principle. Poppers proposed experiment consists of a low-intensity source of particles that can generate pairs of particles traveling to the left and to the right along the x-axis. The beams low intensity is so that the probability is high that two particles recorded at the time on the left and on the right are those which have actually interacted before emission. There are two slits, one each in the paths of the two particles, behind the slits are semicircular arrays of counters which can detect the particles after they pass through the slits. These counters are coincident counters that they only detect particles that have passed at the time through A and B. This larger spread in the momentum will show up as particles being detected even at positions that lie outside the regions where particles would normally reach based on their initial momentum spread. Popper suggests that we count the particles in coincidence, i. e. we count only those particles behind slit B, particles which are not able to pass through slit A are ignored. The Heisenberg scatter for both the beams of particles going to the right and to the left, is tested by making the two slits A and B wider or narrower. If the slits are narrower, then counters should come into play which are higher up and lower down, the coming into play of these counters is indicative of the wider scattering angles which go with a narrower slit, according to the Heisenberg relations. Now the slit at A is made very small and the slit at B very wide, Popper wrote that, according to the EPR argument, we have measured position y for both particles with the precision Δ y, and not just for particle passing through slit A. This is because from the initial entangled EPR state we can calculate the position of the particle 2, once the position of particle 1 is known and we can do this, argues Popper, even though slit B is wide open. Therefore, Popper states that fairly precise knowledge about the y position of particle 2 is made, now the scatter can, in principle, be tested with the help of the counters. Popper was inclined to believe that the test would decide against the Copenhagen interpretation, if the test decided in favor of the Copenhagen interpretation, Popper argued, it could be interpreted as indicative of action at a distance
19.
Quantum eraser experiment
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Next, the experimenter marks through which slit each photon went and demonstrates that thereafter the interference pattern is destroyed. This stage indicates that it is the existence of the information that causes the destruction of the interference pattern. Third, the information is erased, whereupon the interference pattern is recovered. A key result is that it does not matter whether the procedure is done before or after the photons arrive at the detection screen. Quantum erasure technology can be used to increase the resolution of advanced microscopes, the quantum eraser experiment described in this article is a variation of Thomas Youngs classic double-slit experiment. It establishes that when action is taken to determine which slit a photon has passed through, when a stream of photons is marked in this way, then the interference fringes characteristic of the Young experiment will not be seen. The experiment described in this article is capable of creating situations in which a photon that has been marked to reveal through which slit it has passed can later be unmarked and this experiment involves an apparatus with two main sections. After two entangled photons are created, each is directed into its own section of the apparatus, anything done to learn the path of the entangled partner of the photon being examined in the double-slit part of the apparatus will influence the second photon, and vice versa. In doing so, the experimenter restores interference without altering the part of the experimental apparatus. In delayed-choice experiments quantum effects can mimic an influence of future actions on past events, however, the temporal order of measurement actions is not relevant. First, a photon is shot through a nonlinear optical device. This crystal converts the photon into two entangled photons of lower frequency, a process known as spontaneous parametric down-conversion. These entangled photons follow separate paths, one photon goes directly to a detector, while the second photon passes through the double-slit mask to a second detector. Both detectors are connected to a circuit, ensuring that only entangled photon pairs are counted. A stepper motor moves the second detector to scan across the target area and this configuration yields the familiar interference pattern. This polarization is measured at the detector, thus marking the photons, finally, a linear polarizer is introduced in the path of the first photon of the entangled pair, giving this photon a diagonal polarization. Entanglement ensures a complementary diagonal polarization in its partner, which passes through the double-slit mask and this alters the effect of the circular polarizers, each will produce a mix of clockwise and counter-clockwise polarized light. Thus the second detector can no longer determine which path was taken, a double slit with rotating polarizers can also be accounted for by considering the light to be a classical wave
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Delayed choice quantum eraser
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The experiment was designed to investigate peculiar consequences of the well-known double-slit experiment in quantum mechanics, as well as the consequences of quantum entanglement. The delayed choice quantum eraser experiment investigates a paradox, If a photon manifests itself as though it had come by a single path to the detector, then common sense says it must have entered the double-slit device as a particle. If a photon manifests itself as though it had come by two paths, then it must have entered the double-slit device as a wave. If the experimental apparatus is changed while the photon is in mid‑flight and this is the standard view, and recent experiments have supported it. In the basic double slit experiment, a beam of light is directed perpendicularly towards a wall pierced by two parallel slit apertures. If a detection screen is put on the side of the double slit wall, a pattern of light and dark fringes will be observed. Other atomic-scale entities such as electrons are found to exhibit the same behavior when fired toward a double slit, by decreasing the brightness of the source sufficiently, individual particles that form the interference pattern are detectable. This is an idea that contradicts our everyday experience of discrete objects and this which-way experiment illustrates the complementarity principle that photons can behave as either particles or waves, but not both at the same time. However, technically feasible realizations of this experiment were not proposed until the 1970s, which-path information and the visibility of interference fringes are hence complementary quantities. However, in 1982, Scully and Drühl found a loophole around this interpretation and they proposed a quantum eraser to obtain which-path information without scattering the particles or otherwise introducing uncontrolled phase factors to them. Lest there be any misunderstanding, the pattern does disappear when the photons are so marked. However, the interference pattern if the which-path information is further manipulated after the marked photons have passed through the double slits to obscure the which-path markings. Since 1982, multiple experiments have demonstrated the validity of the quantum eraser. A simple version of the eraser can be described as follows. In the two diagrams in Fig.1, photons are emitted one at a time from a laser symbolized by a yellow star and they pass through a 50% beam splitter that reflects or transmits 1/2 of the photons. The reflected or transmitted photons travel along two possible paths depicted by the red or blue lines, in the bottom diagram, a second beam splitter is introduced at the top right. It can direct either beam toward either exit port, thus, photons emerging from each exit port may have come by way of either path. By introducing the second beam splitter, the information has been erased
21.
Wheeler's delayed choice experiment
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Wheelers delayed choice experiment is actually several thought experiments in quantum physics, proposed by John Archibald Wheeler, with the most prominent among them appearing in 1978 and 1984. Some interpreters of these experiments contend that a photon either is a wave or is a particle, Wheelers intent was to investigate the time-related conditions under which a photon makes this transition between alleged states of being. His work has been productive of many revealing experiments, however, he himself seems to be very clear on this point. Either it was a wave or a particle, either it went both ways around the galaxy or only one way, actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago to be is to be perceived and this line of experimentation proved very difficult to carry out when it was first conceived. Nevertheless, it has proven very valuable over the years since it has led researchers to provide increasingly sophisticated demonstrations of the duality of single quanta. As one experimenter explains, Wave and particle behavior can coexist simultaneously, Wheelers delayed choice experiment refers to a series of thought experiments in quantum physics, the first being proposed by him in 1978. Another prominent version was proposed in 1983, all of these experiments try to get at the same fundamental issues in quantum physics. According to the complementarity principle, a photon can manifest properties of a particle or of a wave, what characteristic is manifested depends on whether experimenters use a device intended to observe particles or to observe waves. When this statement is applied very strictly, one could argue that by determining the type one could force the photon to become manifest only as a particle or only as a wave. Detection of a photon is a process because a photon can never be seen in flight. A photon always appears at some highly localized point in space, suppose that a traditional double-slit experiment is prepared so that either of the slits can be blocked. If both slits are open and a series of photons are emitted by the then a interference pattern will quickly emerge on the detection screen. The interference pattern can only be explained as a consequence of wave phenomena, if only one slit is available then there will be no interference pattern, so experimenters may conclude that each photon decides to travel as a particle as soon as it is emitted. One way to investigate the question of when a photon decides whether to act as a wave or a particle in an experiment is to use the interferometer method. If the apparatus is changed so that a beam splitter is placed in the upper-right corner. Experimenters must explain these phenomena as consequences of the nature of light. They may affirm that each photon must have traveled by both paths as a wave else that photon could not have interfered with itself
22.
Mathematical formulation of quantum mechanics
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The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Many of these structures are drawn from functional analysis, an area within pure mathematics that was influenced in part by the needs of quantum mechanics. These formulations of quantum mechanics continue to be used today, at the heart of the description are ideas of quantum state and quantum observable which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, probability theory was used in statistical mechanics. Geometric intuition played a role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule, planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Plancks constant in his honor, in 1905, Einstein explained certain features of the photoelectric effect by assuming that Plancks energy quanta were actual particles, which were later dubbed photons. All of these developments were phenomenological and challenged the theoretical physics of the time, Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. The most sophisticated version of formalism was the so-called Sommerfeld–Wilson–Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the mathematical status of quantum theory remained uncertain for some time. In 1923 de Broglie proposed that wave–particle duality applied not only to photons but to electrons, the physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity. Werner Heisenbergs matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra, later in the same year, Schrödinger created his wave mechanics. Schrödingers formalism was considered easier to understand, visualize and calculate as it led to differential equations, within a year, it was shown that the two theories were equivalent. It was Max Born who introduced the interpretation of the square of the wave function as the probability distribution of the position of a pointlike object. Borns idea was taken over by Niels Bohr in Copenhagen who then became the father of the Copenhagen interpretation of quantum mechanics. Schrödingers wave function can be seen to be related to the classical Hamilton–Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in fact, in these early years, linear algebra was not generally popular with physicists in its present form. He is the third, and possibly most important, pillar of that field and his work was particularly fruitful in all kinds of generalizations of the field
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Phase space formulation
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The phase space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations, the two key features of the phase space formulation are that the quantum state is described by a quasiprobability distribution and operator multiplication is replaced by a star product. The theory was developed by Hilbrand Groenewold in 1946 in his PhD thesis. This formulation is statistical in nature and offers connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two. The conceptual ideas underlying the development of mechanics in phase space have branched into mathematical offshoots such as algebraic deformation theory. The phase space distribution f of a state is a quasiprobability distribution. There are several different ways to represent the distribution, all interrelated, the most noteworthy is the Wigner representation, W, discovered first. Other representations include the Glauber-Sudarshan P, Husimi Q, Kirkwood-Rihaczek, Mehta, Rivier and these alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, the phase space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within an interval, for example, by integrating the Wigner function over all momenta and over the position interval. If Â is an operator representing an observable, it may be mapped to phase space as A through the Wigner transform, conversely, this operator may be recovered via the Weyl transform. The expectation value of the observable with respect to the phase space distribution is ⟨ A ^ ⟩ = ∫ A W d p d x. Moreover, it can, in general, take negative values even for states, with the unique exception of coherent states. Regions of such negative value are provable to be small, they extend to compact regions larger than a few ħ. They are shielded by the uncertainty principle, which does not allow precise localization within phase-space regions smaller than ħ, the fundamental noncommutative binary operator in the phase space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★. Each representation of the distribution has a different characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner-Weyl representation, for notational convenience, we introduce the notion of left and right derivatives
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Path integral formulation
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The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. Unlike previous methods, the path integral allows a physicist to easily change coordinates between very different canonical descriptions of the quantum system. Another advantage is that it is in easier to guess the correct form of the Lagrangian of a theory. Possible downsides of the approach include that unitarity of the S-matrix is obscure in the formulation, the path-integral approach has been proved to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach go away. The Schrödinger equation is an equation with an imaginary diffusion constant. The basic idea of the integral formulation can be traced back to Norbert Wiener. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 article, the complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian as a starting point, in quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a later time differs from the state at the current time by the result of acting with the Hamiltonian operator. For states with an energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle. The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames, so the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics. The Hamiltonian is a function of the position and momentum at one time, the Lagrangian is a function of the position now and the position a little later. The relation between the two is by a Legendre transform, and the condition that determines the classical equations of motion is that the action has an extremum, in quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with discretization in time, the Legendre transform becomes ϵ H = p − ϵ L and p = ∂ L ∂ q ˙, where the partial derivative with respect to q ˙ holds q fixed. The inverse Legendre transform is ϵ L = ϵ p q ˙ − ϵ H, where q ˙ = ∂ H ∂ p, and the partial derivative now is with respect to p at fixed q
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Dirac equation
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In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and it was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved, moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, in the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1/2 particles. The Dirac equation in the originally proposed by Dirac is. The p1, p2, p3 are the components of the momentum, also, c is the speed of light, and ħ is the Planck constant divided by 2π. These fundamental physical constants reflect special relativity and quantum mechanics, respectively, Diracs purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have a bearing on the problem of atomic spectra, the new elements in this equation are the 4 ×4 matrices αk and β, and the four-component wave function ψ. There are four components in ψ because the evaluation of it at any point in configuration space is a bispinor. It is interpreted as a superposition of an electron, a spin-down electron, a spin-up positron. These matrices and the form of the function have a deep mathematical significance. The algebraic structure represented by the matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Cliffords ideas had emerged from the work of the German mathematician Hermann Grassmann in his Lineale Ausdehnungslehre. The latter had been regarded as well-nigh incomprehensible by most of his contemporaries, the appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics. The Dirac equation is similar to the Schrödinger equation for a massive free particle. The left side represents the square of the momentum operator divided by twice the mass, space and time derivatives both enter to second order. This has a consequence for the interpretation of the equation. Because the equation is second order in the derivative, one must specify initial values both of the wave function itself and of its first-time derivative in order to solve definite problems
26.
Rydberg formula
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The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on 5 November 1888, in 1880, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he found that he could simplify his calculations by using the wavenumber as his unit of measurement and he plotted the wavenumbers of successive lines in each series against consecutive integers which represented the order of the lines in that particular series. Finding that the curves were similarly shaped, he sought a single function which could generate all of them. This did not work very well, Rydberg therefore rewrote Balmers formula in terms of wavenumbers, as n = n 0 −4 n 0 m 2. This suggested that the Balmer formula for hydrogen might be a case with m ′ =0 and C0 =4 n 0, where n 0 =1 h. The term Co was found to be a universal constant common to all elements and this constant is now known as the Rydberg constant, and m is known as the quantum defect. As stressed by Niels Bohr, expressing results in terms of wavenumber, the fundamental role of wavenumbers was also emphasized by the Rydberg-Ritz combination principle of 1908. The fundamental reason for this lies in quantum mechanics, lights wavenumber is proportional to frequency 1 λ = f c, and therefore also proportional to lights quantum energy E. Thus,1 λ = E h c. Rydbergs 1888 classical expression for the form of the series was not accompanied by a physical explanation. In Bohrs conception of the atom, the integer Rydberg n numbers represent electron orbitals at different integral distances from the atom. A frequency emitted in a transition from n1 to n2 therefore represents the energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2. Later models found that the values for n1 and n2 corresponded to the quantum numbers of the two orbitals. The number of protons in the nucleus of this element, n 1 and n 2 are integers such that n 1 < n 2, corresponding to the principal quantum numbers of the orbitals occupied before. Examples would include He+, Li2+, Be3+ etc. where no other electrons exist in the atom and this is analogous to the Lyman-alpha line transition for hydrogen, and has the same frequency factor. Its frequency is thus the Lyman-alpha hydrogen frequency, increased by a factor of 2. This formula of f = c/λ = ⋅2 is historically known as Moseleys law, see the biography of Henry Moseley for the historical importance of this law, which was derived empirically at about the same time it was explained by the Bohr model of the atom. Rydberg–Ritz combination principle Balmer series Hydrogen line Sutton, Mike, getting the numbers right, The lonely struggle of the 19th century physicist/chemist Johannes Rydberg
27.
Interpretations of quantum mechanics
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An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing and this question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. The definition of quantum theorists terms, such as functions and matrix mechanics. Although the Copenhagen interpretation was originally most popular, quantum decoherence has gained popularity, thus the many-worlds interpretation has been gaining acceptance. The authors reference a similarly informal poll carried out by Max Tegmark at the Fundamental Problems in Quantum Theory conference in August 1997, in Tegmarks poll, the Everett interpretation received 17% of the vote, which is similar to the number of votes in our poll. A general law is a regularity of outcomes, whereas a causal mechanism may regulate the outcomes, a phenomenon can receive interpretation either ontic or epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception, in a broad sense, scientific theory can be viewed as offering scientific realism—approximately true description or explanation of the natural world—or might be perceived with antirealism. A realist stance seeks the epistemic and the ontic, whereas an antirealist stance seeks epistemic, in the 20th centurys first half, antirealism was mainly logical positivism, which sought to exclude unobservable aspects of reality from scientific theory. The instrumentalist view is carried by the quote of David Mermin, Shut up and calculate. Other approaches to conceptual problems introduce new mathematical formalism. In classical field theory, a property at a given location in the field is readily derived. In Heisenbergs formalism, on the hand, to derive physical information about a location in the field, one must apply a quantum operation to a quantum state. Schrödingers formalism describes a waveform governing probability of outcomes across a field, yet how do we find in a specific location a particle whose wavefunction of mere probability distribution of existence spans a vast region of space. The act of measurement can interact with the state in peculiar ways. Yet quantum decoherence grants that all the possibilities can be real, Quantum entanglement, as illustrated in the EPR paradox, seemingly violates principles of local causality. Complementarity holds that no set of physical concepts can simultaneously refer to all properties of a quantum system. For instance, wave description A and particulate description B can each describe quantum system S, as now well known, the origin of complementarity lies in the non-commutativity of operators that describe quantum objects. Since the intricacy of a system is exponential, it is difficult to derive classical approximations
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Many-worlds interpretation
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The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternate histories and futures are real, the theory is also referred to as MWI, the relative state formulation, the Everett interpretation, the theory of the universal wavefunction, many-universes interpretation, or just many-worlds. The original relative state formulation is due to Hugh Everett in 1957, later, this formulation was popularized and renamed many-worlds by Bryce Seligman DeWitt in the 1960s and 1970s. The decoherence approaches to interpreting quantum theory have been explored and developed. MWI is one of many multiverse hypotheses in physics and philosophy and it is currently considered a mainstream interpretation along with the other decoherence interpretations, collapse theories, and hidden variable theories such as the Bohmian mechanics. Before many-worlds, reality had always viewed as a single unfolding history. Many-worlds, however, views reality as a tree, wherein every possible quantum outcome is realised. Many-worlds reconciles the observation of events, such as random radioactive decay. In Dublin in 1952 Erwin Schrödinger gave a lecture in which at one point he jocularly warned his audience that what he was about to say might seem lunatic. He went on to assert that when his Nobel equations seem to be describing several different histories, they are not alternatives and this is the earliest known reference to the many-worlds. The idea of MWI originated in Everetts Princeton Ph. D, the phrase many-worlds is due to Bryce DeWitt, who was responsible for the wider popularisation of Everetts theory, which had been largely ignored for the first decade after publication. Under scrutiny of the environment, only pointer states remain unchanged, other states decohere into mixtures of stable pointer states that can persist, and, in this sense, exist, They are einselected. These ideas complement MWI and bring the interpretation in line with our perception of reality, Deutsch is dismissive that many-worlds is an interpretation, saying that calling it an interpretation is like talking about dinosaurs as an interpretation of fossil records. As with the interpretations of quantum mechanics, the many-worlds interpretation is motivated by behavior that can be illustrated by the double-slit experiment. When particles of light are passed through the slit, a calculation assuming wave-like behavior of light can be used to identify where the particles are likely to be observed. Yet when the particles are observed in experiment, they appear as particles. Everetts Ph. D. work provided such an alternative interpretation. e, the state of the observer and the observed are correlated after the observation is made. This led Everett to derive from the unitary, deterministic dynamics alone the notion of a relativity of states, thus the appearance of the objects wavefunctions collapse has emerged from the unitary, deterministic theory itself
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Relational quantum mechanics
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This article is intended for those already familiar with quantum mechanics and its attendant interpretational difficulties. Readers who are new to the subject may first want to read the introduction to quantum mechanics and this interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been expanded upon by a number of theorists. It is inspired by the key idea behind Special Relativity, that the details of an observation depend on the frame of the observer. The physical content of the theory is not to do with objects themselves, as Rovelli puts it, Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world. The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, however, it is held by RQM that this applies to all physical objects, whether or not they are conscious or macroscopic. Any measurement event is simply as an ordinary physical interaction. This incorrect assumption, he said, was that of an observer-independent state of a system, david Mermin has contributed to the relational approach in his Ithaca interpretation. The moniker Zero Worlds has been popularized by Garret to contrast with the Many Worlds Interpretation and this problem was initially discussed in detail in Everetts thesis, The Theory of the Universal Wavefunction. Consider observer O, measuring the state of the quantum system S and we assume that O has complete information on the system, and that O can write down the wavefunction | ψ ⟩ describing it. At the same time, there is another observer O ′, who is interested in the state of the entire O - S system, for our purposes here, we can assume that in a single experiment, the outcome is the eigenstate | ↑ ⟩. So, we may represent the sequence of events in this experiment, with observer O doing the observing, as follows and this is observer O s description of the measurement event. Now, any measurement is also an interaction between two or more systems. According to O, at t 2, the system S is in a determinate state, and, if quantum mechanics is complete, then so is his description. But, for O ′, S is not uniquely determinate, but, if quantum mechanics is complete, then the description that O ′ gives is also complete. Thus the standard formulation of quantum mechanics allows different observers to give different accounts of the same sequence of events. There are many ways to overcome this perceived difficulty, what makes O s description better than that of O ′, or vice versa. Alternatively, we could claim that quantum mechanics is not a theory. Yet another option is to give a preferred status to an observer or type of observer