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A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is ''one'' wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters or (lower-case and capital psi). For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete. Such a set of observables, by a postulate of quantum mechanics, are Hermitian linear operators on the space of states representing a set of physical observables, like position, momentum and spin that can, in principle, be simultaneously measured with arbitrary precision. Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This is the superposition principle of quantum mechanics. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. The inner product is a measure of the overlap between physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The actual space depends on the system's degrees of freedom (hence on the chosen representation and coordinate system) and the exact form of the Hamiltonian entering the equation governing the dynamical behavior. In the non-relativistic case, disregarding spin, this is the Schrödinger equation. The Schrödinger equation determines the allowed wave functions for the system and how they 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. The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical "space", and in this respect it differs fundamentally from water waves or waves on a string.〔Heisenberg, W. (1927/1985/2009). Heisenberg is translated by , (from ).〕 For a given system, the choice of which relevant degrees of freedom to use are not unique, and correspondingly the domain of the wave function is not unique. It may be taken to be a function of all the position coordinates of the particles over ''position space'', or the momenta of all the particles over ''momentum space'', the two are related by a Fourier transform. These descriptions are the most important, but they are not the only possibilities. Just like in classical mechanics, canonical transformations may be used in the description of a quantum system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic discrete degree of freedom. In general, for a particle with ''half-integer'' spin the wave function is a spinor, for a particle with ''integer'' spin the wave function is a tensor. Particles with spin zero are called scalar particles, those with spin 1 vector particles, and more generally for higher integer spin, tensor particles. The terminology derives from how the wave functions transform under a rotation of the coordinate system. No ''elementary'' particle with spin or higher is known, except for the hypothesized spin graviton. Other discrete variables can be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g. a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g. z-component of spin). These values are often displayed in a column matrix (e.g. a column vector for a non-relativistic electron with spin ). In the Copenhagen interpretation, an interpretation of quantum mechanics, the squared modulus of the wave function, , is a real number interpreted as the probability density of measuring a particle as being at a given place at a given time or having a definite momentum, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, 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. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function and calculate the statistical distributions for measurable quantities. The unit of measurement for depends on the system, and can be found by dimensional analysis of the normalization condition for the system. For one particle in three dimensions, its units are ()−3/2, because an integral of over a region of three-dimensional space is a dimensionless probability. ==Historical background== In 1905 Einstein postulated the proportionality between the frequency of a photon and its energy, ,〔 (in German), (in English)〕 and in 1916 the corresponding relation between photon momentum and wavelength, .〔 and a nearly identical version translated in .〕 In 1923, De Broglie was the first to suggest that the relation , now called the De Broglie relation, holds for ''massive'' particles, the chief clue being Lorentz invariance, and this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent wave–particle duality for both massless and massive particles. In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. 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, and others, developing "matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent. In 1926, Schrödinger published the famous wave equation now named after him, indeed the Schrödinger equation, based on classical Conservation of energy using quantum operators and the de Broglie relations such that the solutions of the equation are the wave functions for the quantum system. 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 how elastic scattering of a wave packet representing a particle off a target appears; it spreads out in all directions.〔, translated in at pages 52–55.〕 While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of probability amplitude.〔〔, translated in . Also (here ).〕 This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics. In 1927, Hartree and Fock made the first step in an attempt to solve the ''N''-body wave function, and developed the ''self-consistency cycle'': an iterative algorithm to approximate the solution. Now it is also known as the Hartree–Fock method. The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater. Schrödinger did encounter an equation for the wave function that satisfied relativistic energy conservation ''before'' he published the non-relativistic one, but discarded it as it predicted negative probabilities and negative energies. In 1927, Klein, Gordon and Fock also found it, but incorporated the electromagnetic interaction and proved that it was Lorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the Klein–Gordon equation. In 1927, Pauli phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the Pauli equation. Pauli found the wave function was not described by a single complex function of space and time, but needed two complex numbers, which respectively correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, Dirac found an equation from the first successful unification of special relativity and quantum mechanics applied to the electron, now called the Dirac equation. In this, the wave function is a ''spinor'' represented by four complex-valued components:〔 two for the electron and two for the electron's antiparticle, the positron. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other relativistic wave equations were found. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wave function」の詳細全文を読む スポンサード リンク
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