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Quantum Physics

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[B]Quantum mechanics (QM)[/B] is a branch of [URL="http://en.wikipedia.org/wiki/Physics"]physics[/URL] describing the behavior of [URL="http://en.wikipedia.org/wiki/Energy"]energy[/URL] and [URL="http://en.wikipedia.org/wiki/Matter"]matter[/URL] at the atomic and subatomic scales. The name derives from the observation that some physical quantities—such as the energy of an electron bound into an atom or molecule—can be changed only by discrete amounts, or [URL="http://en.wikipedia.org/wiki/Quanta"]quanta[/URL], rather than being capable of varying by any amount. The [URL="http://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality"]wave–particle duality[/URL] of energy and matter at the atomic scale provides a unified view of the behavior of particles such as [URL="http://en.wikipedia.org/wiki/Photon"]photons[/URL] and [URL="http://en.wikipedia.org/wiki/Electron"]electrons[/URL]. [URL="http://en.wikipedia.org/wiki/Photons"]Photons[/URL] are the quanta of light, and have energy values proportional to their frequency via the [URL="http://en.wikipedia.org/wiki/Planck_constant"]Planck constant[/URL]. An electron bound in an [URL="http://en.wikipedia.org/wiki/Atomic_orbital"]atomic orbital[/URL] has quantized values of [URL="http://en.wikipedia.org/wiki/Angular_momentum#Angular_momentum_in_quantum_mechanics"]angular momentum[/URL] and energy. The unbound electron does not exhibit quantized energy levels, but is associated with a [URL="http://en.wikipedia.org/wiki/Compton_wavelength"]quantum mechanical wavelength[/URL], as are all massive particles. The full significance of the Planck constant is expressed in physics through the abstract mathematical notion of [URL="http://en.wikipedia.org/wiki/Action_%28physics%29"]action[/URL]. The [URL="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics"]mathematical formulation of quantum mechanics[/URL] is abstract and its implications are often non-intuitive. The centerpiece of this mathematical system is the [URL="http://en.wikipedia.org/wiki/Wavefunction"]wavefunction[/URL]. The wavefunction is a mathematical function of time and space that can provide information about the position and momentum of a particle, but only as probabilities, as dictated by the constraints imposed by the [URL="http://en.wikipedia.org/wiki/Uncertainty_principle"]uncertainty principle[/URL]. Mathematical manipulations of the wavefunction usually involve the [URL="http://en.wikipedia.org/wiki/Bra-ket_notation"]bra-ket notation[/URL], which requires an understanding of [URL="http://en.wikipedia.org/wiki/Complex_number"]complex numbers[/URL] and [URL="http://en.wikipedia.org/wiki/Linear_functional"]linear functionals[/URL]. Many of the results of QM can only be expressed mathematically and do not have models that are as easy to visualize as those of [URL="http://en.wikipedia.org/wiki/Classical_mechanics"]classical mechanics[/URL]. For instance, the [URL="http://en.wikipedia.org/wiki/Ground_state"]ground state[/URL] in quantum mechanical model is a non-zero energy state that is the lowest permitted energy state of a system, rather than a more traditional system that is thought of as simply being at rest with zero kinetic energy. [B]Contents[/B] [[URL="http://javascript%3Cb%3E%3C/b%3E:toggleToc%28%29"]hide[/URL]] [LIST] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Overview"]1 Overview[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#History"]2 History[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Quantum_mechanics_and_classical_physics"]3 Quantum mechanics and classical physics[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Theory"]4 Theory[/URL] [LIST] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulation"]4.1 Mathematical formulation[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Interactions_with_other_scientific_theories"]4.2 Interactions with other scientific theories[/URL] [/LIST] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Example"]5 Example[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Attempts_at_a_unified_field_theory"]6 Attempts at a unified field theory[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Relativity_and_quantum_mechanics"]7 Relativity and quantum mechanics[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Applications"]8 Applications[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Philosophical_consequences"]9 Philosophical consequences[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#See_also"]10 See also[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Notes"]11 Notes[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#References"]12 References[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#Further_reading"]13 Further reading[/URL] [*][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#External_links"]14 External links[/URL] [/LIST] [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=1"]edit[/URL]] Overview[/B] Main article: [URL="http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics"]Introduction to quantum mechanics[/URL] The word [I]quantum[/I] derives from [URL="http://en.wikipedia.org/wiki/Latin_language"]Latin[/URL] meaning "how great" or "how much".[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-0"][1][/URL] In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain [URL="http://en.wikipedia.org/wiki/Physical_quantity"]physical quantities[/URL], such as the [URL="http://en.wikipedia.org/wiki/Energy"]energy[/URL] of an [URL="http://en.wikipedia.org/wiki/Atom"]atom[/URL] at rest (see Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics that deals with atomic and subatomic systems which is today called quantum mechanics. It is the underlying [URL="http://en.wikipedia.org/wiki/Mathematical"]mathematical[/URL] framework of many fields of [URL="http://en.wikipedia.org/wiki/Physics"]physics[/URL] and [URL="http://en.wikipedia.org/wiki/Chemistry"]chemistry[/URL], including [URL="http://en.wikipedia.org/wiki/Condensed_matter_physics"]condensed matter physics[/URL], [URL="http://en.wikipedia.org/wiki/Solid-state_physics"]solid-state physics[/URL], [URL="http://en.wikipedia.org/wiki/Atomic_physics"]atomic physics[/URL], [URL="http://en.wikipedia.org/wiki/Molecular_physics"]molecular physics[/URL], [URL="http://en.wikipedia.org/wiki/Computational_physics"]computational physics[/URL], [URL="http://en.wikipedia.org/wiki/Computational_chemistry"]computational chemistry[/URL], [URL="http://en.wikipedia.org/wiki/Quantum_chemistry"]quantum chemistry[/URL], [URL="http://en.wikipedia.org/wiki/Particle_physics"]particle physics[/URL], [URL="http://en.wikipedia.org/wiki/Nuclear_chemistry"]nuclear chemistry[/URL], and [URL="http://en.wikipedia.org/wiki/Nuclear_physics"]nuclear physics[/URL]. The foundations of quantum mechanics were established during the first half of the twentieth century by [URL="http://en.wikipedia.org/wiki/Werner_Heisenberg"]Werner Heisenberg[/URL], [URL="http://en.wikipedia.org/wiki/Max_Planck"]Max Planck[/URL], [URL="http://en.wikipedia.org/wiki/Louis_de_Broglie"]Louis de Broglie[/URL], [URL="http://en.wikipedia.org/wiki/Albert_Einstein"]Albert Einstein[/URL], [URL="http://en.wikipedia.org/wiki/Niels_Bohr"]Niels Bohr[/URL], [URL="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger"]Erwin Schrödinger[/URL], [URL="http://en.wikipedia.org/wiki/Max_Born"]Max Born[/URL], [URL="http://en.wikipedia.org/wiki/Von_Neumann"]John von Neumann[/URL], [URL="http://en.wikipedia.org/wiki/Paul_Dirac"]Paul Dirac[/URL], [URL="http://en.wikipedia.org/wiki/Wolfgang_Pauli"]Wolfgang Pauli[/URL], [URL="http://en.wikipedia.org/wiki/David_Hilbert"]David Hilbert[/URL], and [URL="http://en.wikipedia.org/wiki/Category:Quantum_physicists"]others[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-1"][2][/URL] Some fundamental aspects of the theory are still actively studied.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-2"][3][/URL] Quantum mechanics is essential to understand the behavior of systems at [URL="http://en.wikipedia.org/wiki/Atom"]atomic[/URL] length scales and smaller. For example, if [URL="http://en.wikipedia.org/wiki/Classical_mechanics"]classical mechanics[/URL] governed the workings of an atom, [URL="http://en.wikipedia.org/wiki/Electron"]electrons[/URL] would rapidly travel towards and collide with the [URL="http://en.wikipedia.org/wiki/Atomic_nucleus"]nucleus[/URL], making stable atoms impossible. However, in the natural world the electrons normally remain in an uncertain, non-deterministic "smeared" (wave–particle wave function) orbital path around or through the nucleus, defying [URL="http://en.wikipedia.org/wiki/Classical_electromagnetism"]classical electromagnetism[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-3"][4][/URL] Quantum mechanics was initially developed to provide a better explanation of the atom, especially the [URL="http://en.wikipedia.org/wiki/Spectrum"]spectra[/URL] of [URL="http://en.wikipedia.org/wiki/Light"]light[/URL] emitted by different [URL="http://en.wikipedia.org/wiki/Isotope"]atomic species[/URL]. The quantum theory of the atom was developed as an explanation for the electron's staying in its [URL="http://en.wikipedia.org/wiki/Atomic_orbital"]orbital[/URL], which could not be explained by [URL="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion"]Newton's laws of motion[/URL] and by [URL="http://en.wikipedia.org/wiki/Maxwell%27s_equations"]Maxwell's laws[/URL] of classical electromagnetism. In the formalism of quantum mechanics, the state of a system at a given time is described by a [URL="http://en.wikipedia.org/wiki/Complex_number"]complex[/URL] [URL="http://en.wikipedia.org/wiki/Wave_function"]wave function[/URL] (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex [URL="http://en.wikipedia.org/wiki/Vector_space"]vector space[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-4"][5][/URL] This abstract mathematical object allows for the calculation of [URL="http://en.wikipedia.org/wiki/Probability"]probabilities[/URL] of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of [URL="http://en.wikipedia.org/wiki/Conjugate_variables"]conjugate variables[/URL], such as position and momentum, with accuracy. For instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Contours of constant probability, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's [URL="http://en.wikipedia.org/wiki/Uncertainty_principle"]uncertainty principle[/URL] quantifies the inability to precisely locate the particle given its conjugate.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-5"][6][/URL] The other [URL="http://en.wikipedia.org/wiki/Exemplar"]exemplar[/URL] that led to quantum mechanics was the study of [URL="http://en.wikipedia.org/wiki/Electromagnetic_wave"]electromagnetic waves[/URL] such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, [URL="http://en.wikipedia.org/wiki/Albert_Einstein"]Albert Einstein[/URL] further developed this idea to show that an electromagnetic wave such as light could be described by a particle called the [URL="http://en.wikipedia.org/wiki/Photon"]photon[/URL] with a discrete energy dependent on its frequency. This led to a [URL="http://en.wikipedia.org/wiki/Photon_polarization"]theory of unity[/URL] between subatomic particles and electromagnetic waves called [URL="http://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality"]wave–particle duality[/URL] in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics describes the world of the very small, it also is needed to explain certain [URL="http://en.wikipedia.org/wiki/Macroscopic"]macroscopic[/URL] quantum systems such as [URL="http://en.wikipedia.org/wiki/Superconductivity"]superconductors[/URL] and [URL="http://en.wikipedia.org/wiki/Superfluid"]superfluids[/URL]. Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account: (I) the [URL="http://en.wikipedia.org/wiki/Quantization_%28physics%29"]quantization[/URL] (discretization) of [URL="http://en.wikipedia.org/wiki/Canonical_conjugate_variables"]certain physical quantities[/URL], (II) [URL="http://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality"]wave–particle duality[/URL], (III) the [URL="http://en.wikipedia.org/wiki/Uncertainty_principle"]uncertainty principle[/URL], and (IV) [URL="http://en.wikipedia.org/wiki/Quantum_entanglement"]quantum entanglement[/URL]. Each of these phenomena is described in detail in subsequent sections. [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=2"]edit[/URL]] History[/B] Main article: [URL="http://en.wikipedia.org/wiki/History_of_quantum_mechanics"]History of quantum mechanics[/URL] The history of quantum mechanics began with the 1838 discovery of [URL="http://en.wikipedia.org/wiki/Cathode_rays"]cathode rays[/URL] by [URL="http://en.wikipedia.org/wiki/Michael_Faraday"]Michael Faraday[/URL], the 1859 statement of the [URL="http://en.wikipedia.org/wiki/Black_body_radiation"]black body radiation[/URL] problem by [URL="http://en.wikipedia.org/wiki/Gustav_Kirchhoff"]Gustav Kirchhoff[/URL], the 1877 suggestion by [URL="http://en.wikipedia.org/wiki/Ludwig_Boltzmann"]Ludwig Boltzmann[/URL] that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by [URL="http://en.wikipedia.org/wiki/Max_Planck"]Max Planck[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-6"][7][/URL] Planck's hypothesis stated that any energy is radiated and absorbed in quantities divisible by discrete "energy elements", such that each energy element [I]E[/I] is proportional to its [URL="http://en.wikipedia.org/wiki/Frequency"]frequency[/URL] [I]ν[/I]: [IMG]http://upload.wikimedia.org/math/7/2/a/72a186f3866ae93fc82a18d65b7b7644.png[/IMG] where [I]h[/I] is [URL="http://en.wikipedia.org/wiki/Planck_constant"]Planck's action constant[/URL]. Planck insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-7"][8][/URL] However, at that time, this appeared not to explain the [URL="http://en.wikipedia.org/wiki/Photoelectric_effect"]photoelectric effect[/URL] (1839), i.e. that shining light on certain materials can eject electrons from the material. In 1905, basing his work on Planck's quantum hypothesis, [URL="http://en.wikipedia.org/wiki/Albert_Einstein"]Albert Einstein[/URL] postulated that [URL="http://en.wikipedia.org/wiki/Light"]light[/URL] itself consists of individual quanta.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-8"][9][/URL] In the mid-1920s, developments in quantum mechanics quickly led to it becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the [URL="http://en.wikipedia.org/wiki/Old_quantum_theory"]"Old Quantum Theory"[/URL]. Light quanta came to be called [URL="http://en.wikipedia.org/wiki/Photons"]photons[/URL] (1926). From Einstein's simple postulation was born a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics, leading to its wider acceptance at the Fifth [URL="http://en.wikipedia.org/wiki/Solvay_Conference"]Solvay Conference[/URL] in 1927. [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=3"]edit[/URL]] Quantum mechanics and classical physics[/B] Predictions of quantum mechanics have been verified experimentally to a very high degree of accuracy. Thus, the current logic of [URL="http://en.wikipedia.org/wiki/Correspondence_principle"]correspondence principle[/URL] between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems (or a statistical quantum mechanics of a large collection of particles). Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large [URL="http://en.wikipedia.org/wiki/Quantum_number"]quantum numbers[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-9"][10][/URL] However, [URL="http://en.wikipedia.org/wiki/Chaos_theory"]chaotic systems[/URL] do not have good quantum numbers, and [URL="http://en.wikipedia.org/wiki/Quantum_chaos"]quantum chaos[/URL] studies the relationship between classical and quantum descriptions in these systems. The main differences between classical and quantum theories have already been mentioned above in the remarks on the [URL="http://en.wikipedia.org/wiki/EPR_paradox"]Einstein-Podolsky-Rosen paradox[/URL]. Essentially the difference boils down to the statement that quantum mechanics is [URL="http://en.wikipedia.org/wiki/Quantum_coherence"]coherent[/URL] (addition of [I][URL="http://en.wikipedia.org/wiki/Probability_amplitude"]amplitudes[/URL][/I]), whereas classical theories are [URL="http://en.wikipedia.org/wiki/Coherence_%28physics%29"]incoherent[/URL] (addition of [I]intensities[/I]). Thus, such quantities as [I]coherence lengths[/I] and [I]coherence times[/I] come into play. For microscopic bodies the extension of the system is certainly much smaller than the [URL="http://en.wikipedia.org/wiki/Coherence_length"]coherence length[/URL]; for macroscopic bodies one expects that it should be the other way round.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-10"][11][/URL] An exception to this rule can occur at extremely low temperatures, when quantum behavior can manifest itself on more macroscopic scales (see [URL="http://en.wikipedia.org/wiki/Bose-Einstein_condensate"]Bose-Einstein condensate[/URL]). This is in accordance with the following observations: Many macroscopic properties of classical systems are direct consequences of quantum behavior of its parts. For example, the stability of bulk matter (which consists of atoms and [URL="http://en.wikipedia.org/wiki/Molecule"]molecules[/URL] which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of interaction of [URL="http://en.wikipedia.org/wiki/Electric_charge"]electric charges[/URL] under the rules of quantum mechanics.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-11"][12][/URL] While the seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with extremely fast-moving or extremely tiny particles, the laws of classical Newtonian physics remain accurate in predicting the behavior of large objects—of the order of the size of large molecules and bigger—at velocities much smaller than the [URL="http://en.wikipedia.org/wiki/Speed_of_light"]velocity of light[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-12"][13][/URL] [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=4"]edit[/URL]] Theory[/B] There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the [URL="http://en.wikipedia.org/wiki/Transformation_theory_%28quantum_mechanics%29"]transformation theory[/URL] proposed by Cambridge [URL="http://en.wikipedia.org/wiki/Theoretical_physics"]theoretical physicist[/URL] [URL="http://en.wikipedia.org/wiki/Paul_Dirac"]Paul Dirac[/URL], which unifies and generalizes the two earliest formulations of quantum mechanics, [URL="http://en.wikipedia.org/wiki/Matrix_mechanics"]matrix mechanics[/URL] (invented by [URL="http://en.wikipedia.org/wiki/Werner_Heisenberg"]Werner Heisenberg[/URL])[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-13"][14][/URL][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-14"][15][/URL] and [URL="http://en.wikipedia.org/wiki/Wave_mechanics"]wave mechanics[/URL] (invented by [URL="http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger"]Erwin Schrödinger[/URL]).[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-15"][16][/URL] In this formulation, the [URL="http://en.wikipedia.org/wiki/Quantum_state"]instantaneous state of a quantum system[/URL] encodes the probabilities of its measurable properties, or "[URL="http://en.wikipedia.org/wiki/Observable"]observables[/URL]". Examples of observables include [URL="http://en.wikipedia.org/wiki/Energy"]energy[/URL], [URL="http://en.wikipedia.org/wiki/Position_operator"]position[/URL], [URL="http://en.wikipedia.org/wiki/Momentum_operator"]momentum[/URL], and [URL="http://en.wikipedia.org/wiki/Angular_momentum"]angular momentum[/URL]. Observables can be either [URL="http://en.wikipedia.org/wiki/Continuous_function"]continuous[/URL] (e.g., the position of a particle) or [URL="http://en.wikipedia.org/wiki/Discrete_mathematics"]discrete[/URL] (e.g., the energy of an electron bound to a hydrogen atom).[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-16"][17][/URL] Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions using [URL="http://en.wikipedia.org/wiki/Probability_distribution"]probability distributions[/URL]; that is, the probability of obtaining possible outcomes from measuring an observable. Often these results are skewed by many causes, such as dense [URL="http://en.wikipedia.org/w/index.php?title=Probability_clouds&action=edit&redlink=1"]probability clouds[/URL][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-17"][18][/URL] or quantum state nuclear attraction.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-18"][19][/URL][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-19"][20][/URL] Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. There are, however, certain states that are associated with a definite value of a particular observable. These are known as [URL="http://en.wikipedia.org/wiki/Eigenstate"]eigenstates[/URL] of the observable ("eigen" can be translated from [URL="http://en.wikipedia.org/wiki/German_language"]German[/URL] as inherent or as a characteristic).[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-20"][21][/URL] In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle for its position and momentum (since they are [URL="http://en.wikipedia.org/wiki/Conjugate_variables"]conjugate pairs[/URL]) or its energy and time (since they too are conjugate pairs); rather, it only provides a range of probabilities of where that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having [I][URL="http://en.wikipedia.org/wiki/Uncertainty_principle"]uncertain[/URL][/I] values and states having [I]definite[/I] values (eigenstate). [URL="http://en.wikipedia.org/wiki/File:QuantumDot_wf.gif"][IMG]http://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/QuantumDot_wf.gif/600px-QuantumDot_wf.gif[/IMG][/URL] [URL="http://en.wikipedia.org/wiki/File:QuantumDot_wf.gif"][IMG]http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png[/IMG][/URL] 3D confined electron wave functions for each eigenstate in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ‘s-type’ and ‘p-type’. However, in a triangular dot the wave functions are mixed due to confinement symmetry. For example, consider a [URL="http://en.wikipedia.org/wiki/Free_particle"]free particle[/URL]. In quantum mechanics, there is [URL="http://en.wikipedia.org/wiki/Wave-particle_duality"]wave-particle duality[/URL] so the properties of the particle can be described as the properties of a wave. Therefore, its [URL="http://en.wikipedia.org/wiki/Quantum_state"]quantum state[/URL] can be represented as a [URL="http://en.wikipedia.org/wiki/Wave"]wave[/URL] of arbitrary shape and extending over space as a [URL="http://en.wikipedia.org/wiki/Wave_function"]wave function[/URL]. The position and momentum of the particle are [URL="http://en.wikipedia.org/wiki/Observables"]observables[/URL]. The [URL="http://en.wikipedia.org/wiki/Uncertainty_Principle"]Uncertainty Principle[/URL] states that both the position and the momentum cannot simultaneously be measured with full precision at the same time. However, one can measure the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large (a [URL="http://en.wikipedia.org/wiki/Dirac_delta"]Dirac delta[/URL]) at a particular position [I]x[/I] and zero everywhere else. If one performs a position measurement on such a wavefunction, the result [I]x[/I] will be obtained with 100% probability (full certainty). This is called an eigenstate of position (mathematically more precise: a [I]generalized position eigenstate ([URL="http://en.wikipedia.org/wiki/Distribution_%28mathematics%29"]eigendistribution[/URL])[/I]). If the particle is in an eigenstate of position then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum then its position is completely unknown.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-21"][22][/URL] In an eigenstate of momentum having a [URL="http://en.wikipedia.org/wiki/Plane_wave"]plane wave[/URL] form, it can be shown that the [URL="http://en.wikipedia.org/wiki/Wavelength"]wavelength[/URL] is equal to [I]h/p[/I], where [I]h[/I] is [URL="http://en.wikipedia.org/wiki/Planck%27s_constant"]Planck's constant[/URL] and [I]p[/I] is the momentum of the [URL="http://en.wikipedia.org/wiki/Eigenstate"]eigenstate[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-22"][23][/URL] Usually, a system will not be in an [URL="http://en.wikipedia.org/wiki/Eigenstate"]eigenstate[/URL] of the observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as [URL="http://en.wikipedia.org/wiki/Wavefunction_collapse"]wavefunction collapse[/URL], a debatable process.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-23"][24][/URL] It involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a [URL="http://en.wikipedia.org/wiki/Wave_packet"]wave packet[/URL] centered around some mean position [I]x[/I]0, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-24"][25][/URL] It is probable, but not certain, that it will be near [I]x[/I]0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result [I]x[/I], the wave function collapses into a position eigenstate centered at [I]x[/I].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-25"][26][/URL] Wave functions can change as time progresses. An equation known as the [URL="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation"]Schrödinger equation[/URL] describes how wave functions change in time, a role similar to [URL="http://en.wikipedia.org/wiki/Newton%27s_second_law"]Newton's second law[/URL] in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-26"][27][/URL] Some wave functions produce probability distributions that are constant or independent of time, such as when in a [URL="http://en.wikipedia.org/wiki/Eigenstate#Schr.C3.B6dinger_equation"]stationary state[/URL] of constant energy, time drops out of the absolute square of the wave function. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single [URL="http://en.wikipedia.org/wiki/Electron"]electron[/URL] in an unexcited [URL="http://en.wikipedia.org/wiki/Atom"]atom[/URL] is pictured classically as a particle moving in a circular trajectory around the [URL="http://en.wikipedia.org/wiki/Atomic_nucleus"]atomic nucleus[/URL], whereas in quantum mechanics it is described by a static, [URL="http://en.wikipedia.org/wiki/Spherical_coordinate_system"]spherically symmetric[/URL] wavefunction surrounding the nucleus ([URL="http://en.wikipedia.org/wiki/File:HAtomOrbitals.png"]Fig. 1[/URL]). (Note that only the lowest angular momentum states, labeled [I]s[/I], are spherically symmetric).[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-27"][28][/URL] The [URL="http://en.wikipedia.org/wiki/Time_evolution"]time evolution[/URL] of wave functions is [URL="http://en.wikipedia.org/wiki/Determinism"]deterministic[/URL] in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-28"][29][/URL] During a [URL="http://en.wikipedia.org/wiki/Quantum_measurement"]measurement[/URL], the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., [URL="http://en.wikipedia.org/wiki/Random"]random[/URL]. A time-evolution simulation can be seen here.[URL="http://demonstrations.wolfram.com/TimeEvolutionOfAWavepacketInASquareWell/"][1][/URL] The [URL="http://en.wikipedia.org/wiki/Probability"]probabilistic[/URL] nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous [URL="http://en.wikipedia.org/wiki/Bohr-Einstein_debates"]Bohr-Einstein debates[/URL], in which the two scientists attempted to clarify these fundamental principles by way of [URL="http://en.wikipedia.org/wiki/Thought_experiment"]thought experiments[/URL]. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. [URL="http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics"]Interpretations of quantum mechanics[/URL] have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the [URL="http://en.wikipedia.org/wiki/Relative_state_interpretation"]relative state interpretation[/URL]. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become [URL="http://en.wikipedia.org/wiki/Quantum_Entanglement"]entangled[/URL], so that the original quantum system ceases to exist as an independent entity. For details, see the article on [URL="http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics"]measurement in quantum mechanics[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-29"][30][/URL] [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=5"]edit[/URL]] Mathematical formulation[/B] Main article: [URL="http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics"]Mathematical formulation of quantum mechanics[/URL] See also: [URL="http://en.wikipedia.org/wiki/Quantum_logic"]Quantum logic[/URL] In the mathematically rigorous formulation of quantum mechanics, developed by [URL="http://en.wikipedia.org/wiki/Paul_Dirac"]Paul Dirac[/URL][URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-30"][31][/URL] and [URL="http://en.wikipedia.org/wiki/John_von_Neumann"]John von Neumann[/URL],[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-31"][32][/URL] the possible states of a quantum mechanical system are represented by [URL="http://en.wikipedia.org/wiki/Unit_vector"]unit vectors[/URL] (called "state vectors") residing in a [URL="http://en.wikipedia.org/wiki/Complex_number"]complex[/URL] [URL="http://en.wikipedia.org/wiki/Separable_space"]separable[/URL] [URL="http://en.wikipedia.org/wiki/Hilbert_space"]Hilbert space[/URL] (variously called the "[URL="http://en.wikipedia.org/wiki/State_space_%28physics%29"]state space[/URL]" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the [URL="http://en.wikipedia.org/wiki/Projective_space"]projectivization[/URL] of a Hilbert space, usually called the [URL="http://en.wikipedia.org/wiki/Complex_projective_space"]complex projective space[/URL]. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of [URL="http://en.wikipedia.org/wiki/Square-integrable"]square-integrable[/URL] functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally [URL="http://en.wikipedia.org/wiki/Hermitian_adjoint"]Hermitian[/URL] (precisely: by a [URL="http://en.wikipedia.org/wiki/Self-adjoint_operator"]self-adjoint[/URL]) linear [URL="http://en.wikipedia.org/wiki/Operator"]operator[/URL] acting on the state space. Each eigenstate of an observable corresponds to an [URL="http://en.wikipedia.org/wiki/Eigenvector"]eigenvector[/URL] of the operator, and the associated [URL="http://en.wikipedia.org/wiki/Eigenvalue"]eigenvalue[/URL] corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. The time evolution of a quantum state is described by the [URL="http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation"]Schrödinger equation[/URL], in which the [URL="http://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29"]Hamiltonian[/URL], the [URL="http://en.wikipedia.org/wiki/Operator_%28physics%29"]operator[/URL] corresponding to the [URL="http://en.wikipedia.org/wiki/Total_energy"]total energy[/URL] of the system, generates time evolution. The [URL="http://en.wikipedia.org/wiki/Inner_product"]inner product[/URL] between two state vectors is a complex number known as a [URL="http://en.wikipedia.org/wiki/Probability_amplitude"]probability amplitude[/URL]. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the [URL="http://en.wikipedia.org/wiki/Absolute_value"]absolute value[/URL] of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator — which explains the choice of [I]Hermitian[/I] operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the [URL="http://en.wikipedia.org/wiki/Spectral_theorem"]spectral decomposition[/URL] of the corresponding operator. Heisenberg's [URL="http://en.wikipedia.org/wiki/Uncertainty_principle"]uncertainty principle[/URL] is represented by the statement that the operators corresponding to certain observables do not [URL="http://en.wikipedia.org/wiki/Commutator"]commute[/URL]. The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its [URL="http://en.wikipedia.org/wiki/Phase_%28waves%29"]phase[/URL] encodes information about the [URL="http://en.wikipedia.org/wiki/Interference_%28wave_propagation%29"]interference[/URL] between quantum states. This gives rise to the wave-like behavior of quantum states. It turns out that analytic solutions of Schrödinger's equation are only available for [URL="http://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions"]a small number of model Hamiltonians[/URL], of which the [URL="http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator"]quantum harmonic oscillator[/URL], the [URL="http://en.wikipedia.org/wiki/Particle_in_a_box"]particle in a box[/URL], the [URL="http://en.wikipedia.org/wiki/Hydrogen_molecular_ion"]hydrogen molecular ion[/URL] and the [URL="http://en.wikipedia.org/wiki/Hydrogen_atom"]hydrogen atom[/URL] are the most important representatives. Even the [URL="http://en.wikipedia.org/wiki/Helium"]helium[/URL] atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as [URL="http://en.wikipedia.org/wiki/Perturbation_theory_%28quantum_mechanics%29"]perturbation theory[/URL] one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak [URL="http://en.wikipedia.org/wiki/Potential_energy"]potential energy[/URL]. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of [URL="http://en.wikipedia.org/wiki/Quantum_chaos"]quantum chaos[/URL]. An alternative formulation of quantum mechanics is [URL="http://en.wikipedia.org/wiki/Feynman"]Feynman[/URL]'s [URL="http://en.wikipedia.org/wiki/Path_integral_formulation"]path integral formulation[/URL], in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of [URL="http://en.wikipedia.org/wiki/Action_principle"]action principles[/URL] in classical mechanics. [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=6"]edit[/URL]] Interactions with other scientific theories[/B] The fundamental rules of quantum mechanics are very deep. They assert that the state space of a system is a [URL="http://en.wikipedia.org/wiki/Hilbert_space"]Hilbert space[/URL] and the observables are [URL="http://en.wikipedia.org/wiki/Hermitian_operators"]Hermitian operators[/URL] acting on that space, but do not tell us which Hilbert space or which operators, or if it even exists. These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the [URL="http://en.wikipedia.org/wiki/Correspondence_principle"]correspondence principle[/URL], which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or equivalently, larger quantum numbers. In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the [I]classical[/I] or [I]correspondence limit[/I]. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit. [URL="http://en.wikipedia.org/wiki/Unsolved_problems_in_physics"]Unsolved problems in physics[/URL] [I]In the [URL="http://en.wikipedia.org/wiki/Correspondence_limit"]correspondence limit[/URL] of [B]quantum mechanics[/B]: Is there a preferred interpretation of quantum mechanics? How does the quantum description of [URL="http://en.wikipedia.org/wiki/Reality"]reality[/URL], which includes elements such as the "[URL="http://en.wikipedia.org/wiki/Superposition_principle"]superposition[/URL] of states" and "[URL="http://en.wikipedia.org/wiki/Wavefunction_collapse"]wavefunction collapse[/URL]", give rise to the reality we [URL="http://en.wikipedia.org/wiki/Perception"]perceive[/URL]?[/I] [URL="http://en.wikipedia.org/wiki/File:Question_mark2.svg"][IMG]http://upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Question_mark2.svg/40px-Question_mark2.svg.png[/IMG][/URL] When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was [URL="http://en.wikipedia.org/wiki/Theory_of_relativity"]non-relativistic[/URL] [URL="http://en.wikipedia.org/wiki/Classical_mechanics"]classical mechanics[/URL]. For instance, the well-known model of the [URL="http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator"]quantum harmonic oscillator[/URL] uses an explicitly non-relativistic expression for the [URL="http://en.wikipedia.org/wiki/Kinetic_energy"]kinetic energy[/URL] of the oscillator, and is thus a quantum version of the [URL="http://en.wikipedia.org/wiki/Harmonic_oscillator"]classical harmonic oscillator[/URL]. Early attempts to merge quantum mechanics with [URL="http://en.wikipedia.org/wiki/Special_relativity"]special relativity[/URL] involved the replacement of the Schrödinger equation with a covariant equation such as the [URL="http://en.wikipedia.org/wiki/Klein-Gordon_equation"]Klein-Gordon equation[/URL] or the [URL="http://en.wikipedia.org/wiki/Dirac_equation"]Dirac equation[/URL]. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of [URL="http://en.wikipedia.org/wiki/Quantum_field_theory"]quantum field theory[/URL], which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, [URL="http://en.wikipedia.org/wiki/Quantum_electrodynamics"]quantum electrodynamics[/URL], provides a fully quantum description of the [URL="http://en.wikipedia.org/wiki/Electromagnetism"]electromagnetic interaction[/URL]. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat [URL="http://en.wikipedia.org/wiki/Electric_charge"]charged[/URL] particles as quantum mechanical objects being acted on by a classical [URL="http://en.wikipedia.org/wiki/Electromagnetic_field"]electromagnetic field[/URL]. For example, the elementary quantum model of the [URL="http://en.wikipedia.org/wiki/Hydrogen_atom"]hydrogen atom[/URL] describes the [URL="http://en.wikipedia.org/wiki/Electric_field"]electric field[/URL] of the hydrogen atom using a classical [IMG]http://upload.wikimedia.org/math/f/8/2/f828cf54da1642bc4affbb6a8778724a.png[/IMG] [URL="http://en.wikipedia.org/wiki/Electric_potential"]Coulomb potential[/URL]. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of [URL="http://en.wikipedia.org/wiki/Photon"]photons[/URL] by [URL="http://en.wikipedia.org/wiki/Charged_particle"]charged particles[/URL]. [URL="http://en.wikipedia.org/wiki/Field_%28physics%29"]Quantum field[/URL] theories for the [URL="http://en.wikipedia.org/wiki/Strong_nuclear_force"]strong nuclear force[/URL] and the [URL="http://en.wikipedia.org/wiki/Weak_nuclear_force"]weak nuclear force[/URL] have been developed. The quantum field theory of the strong nuclear force is called [URL="http://en.wikipedia.org/wiki/Quantum_chromodynamics"]quantum chromodynamics[/URL], and describes the interactions of the subnuclear particles: [URL="http://en.wikipedia.org/wiki/Quark"]quarks[/URL] and [URL="http://en.wikipedia.org/wiki/Gluon"]gluons[/URL]. The [URL="http://en.wikipedia.org/wiki/Weak_nuclear_force"]weak nuclear force[/URL] and the [URL="http://en.wikipedia.org/wiki/Electromagnetic_force"]electromagnetic force[/URL] were unified, in their quantized forms, into a single quantum field theory known as [URL="http://en.wikipedia.org/wiki/Electroweak_theory"]electroweak theory[/URL], by the physicists [URL="http://en.wikipedia.org/wiki/Abdus_Salam"]Abdus Salam[/URL], [URL="http://en.wikipedia.org/wiki/Sheldon_Glashow"]Sheldon Glashow[/URL] and [URL="http://en.wikipedia.org/wiki/Steven_Weinberg"]Steven Weinberg[/URL]. These three men shared the Nobel Prize in Physics in 1979 for this work.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-32"][33][/URL] It has proven difficult to construct quantum models of [URL="http://en.wikipedia.org/wiki/Gravity"]gravity[/URL], the remaining [URL="http://en.wikipedia.org/wiki/Fundamental_force"]fundamental force[/URL]. Semi-classical approximations are workable, and have led to predictions such as [URL="http://en.wikipedia.org/wiki/Hawking_radiation"]Hawking radiation[/URL]. However, the formulation of a complete theory of [URL="http://en.wikipedia.org/wiki/Quantum_gravity"]quantum gravity[/URL] is hindered by apparent incompatibilities between [URL="http://en.wikipedia.org/wiki/General_relativity"]general relativity[/URL], the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as [URL="http://en.wikipedia.org/wiki/String_theory"]string theory[/URL] are among the possible candidates for a future theory of quantum gravity. In the 21st century classical mechanics has been extended into the [URL="http://en.wikipedia.org/wiki/Complex_domain"]complex domain[/URL] and [URL="http://en.wikipedia.org/w/index.php?title=Complex_classical_mechanics&action=edit&redlink=1"]complex classical mechanics[/URL] exhibits behaviours very similar to quantum mechanics.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-33"][34][/URL] [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=7"]edit[/URL]] Example[/B] [URL="http://en.wikipedia.org/wiki/File:Infinite_potential_well.svg"][IMG]http://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Infinite_potential_well.svg/220px-Infinite_potential_well.svg.png[/IMG][/URL] [URL="http://en.wikipedia.org/wiki/File:Infinite_potential_well.svg"][IMG]http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png[/IMG][/URL] 1-dimensional potential energy box (or infinite potential well) Main article: [URL="http://en.wikipedia.org/wiki/Particle_in_a_box"]Particle in a box[/URL] The particle in a 1-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as zero potential energy inside a certain interval and infinite everywhere outside that interval. For the 1-dimensional case in the [I]x[/I] direction, the time-independent Schrödinger equation can be written as:[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-34"][35][/URL] [IMG]http://upload.wikimedia.org/math/3/9/0/39053c61c19246d2197dc26df468cf4c.png[/IMG] Writing the differential operator [IMG]http://upload.wikimedia.org/math/1/3/8/1388c8db0749e17c4d673b86ae72bd05.png[/IMG] the previous equation can be seen to be evocative of the [URL="http://en.wikipedia.org/wiki/Kinetic_energy#Kinetic_energy_of_rigid_bodies"]classic analogue[/URL] [IMG]http://upload.wikimedia.org/math/b/8/3/b83283f4b14a213280136cfe25f3bf13.png[/IMG] with [I]E[/I] as the energy for the state ψ, in this case coinciding with the kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are: [IMG]http://upload.wikimedia.org/math/b/3/d/b3db706e83d4fc31bd3fac4cef6f26ec.png[/IMG] or, from [URL="http://en.wikipedia.org/wiki/Euler%27s_formula"]Euler's formula[/URL], [IMG]http://upload.wikimedia.org/math/a/6/e/a6e6b07e7ce12709ab98c2ba47cca5af.png[/IMG] The presence of the walls of the box determines the values of [I]C[/I], [I]D[/I], and [I]k[/I]. At each wall ([I]x[/I] = 0 and [I]x[/I] = [I]L[/I]), [I]ψ[/I] = 0. Thus when [I]x[/I] = 0, [IMG]http://upload.wikimedia.org/math/3/5/0/3501417b5587efe9e1355368fc5e640b.png[/IMG] and so [I]D[/I] = 0. When [I]x[/I] = [I]L[/I], [IMG]http://upload.wikimedia.org/math/2/c/5/2c5e1eb6d34b55cfc8d117d347ec2bc8.png[/IMG] [I]C[/I] cannot be zero, since this would conflict with the Born interpretation. Therefore sin [I]kL[/I] = 0, and so it must be that [I]kL[/I] is an integer multiple of π. Therefore, [IMG]http://upload.wikimedia.org/math/3/d/2/3d23973128a8a07d48f0ba1f8585f724.png[/IMG] The quantization of energy levels follows from this constraint on [I]k[/I], since [IMG]http://upload.wikimedia.org/math/2/f/d/2fd009416ed37cc1420e759f5fb8296a.png[/IMG] [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=8"]edit[/URL]] Attempts at a unified field theory[/B] Main article: [URL="http://en.wikipedia.org/wiki/Grand_unified_theory"]Grand unified theory[/URL] As of 2010 the quest for unifying the [URL="http://en.wikipedia.org/wiki/Fundamental_force"]fundamental forces[/URL] through quantum mechanics is still ongoing. [URL="http://en.wikipedia.org/wiki/Quantum_electrodynamics"]Quantum electrodynamics[/URL] (or "quantum electromagnetism"), which is currently (in the perturbative regime at least) the most accurately tested physical theory,[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-35"][36][/URL] has been successfully merged with the weak nuclear force into the [URL="http://en.wikipedia.org/wiki/Electroweak_force"]electroweak force[/URL] and work is currently being done to merge the electroweak and strong force into the [URL="http://en.wikipedia.org/wiki/Electrostrong_force"]electrostrong force[/URL]. Current predictions state that at around 1014 GeV the three aforementioned forces are fused into a single unified field,[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-36"][37][/URL] Beyond this "grand unification", it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 1019 GeV. However — and while special relativity is parsimoniously incorporated into quantum electrodynamics — the expanded [URL="http://en.wikipedia.org/wiki/General_relativity"]general relativity[/URL], currently the best theory describing the gravitation force, has not been fully incorporated into quantum theory. [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=9"]edit[/URL]] Relativity and quantum mechanics[/B] [I]Main articles: [URL="http://en.wikipedia.org/wiki/Quantum_gravity"]Quantum gravity[/URL] and [URL="http://en.wikipedia.org/wiki/Theory_of_everything"]Theory of everything[/URL][/I] Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated [URL="http://en.wikipedia.org/wiki/Empirical_research"]empirical evidence[/URL] and while they do not directly contradict each other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one cohesive model.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-37"][38][/URL] Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to the field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as the lack of deterministic [URL="http://en.wikipedia.org/wiki/Causality"]causality[/URL] and the assertion that a single subatomic particle can occupy numerous areas of space at one time. He also was the first to notice some of the apparently exotic consequences of [URL="http://en.wikipedia.org/wiki/Quantum_entanglement"]entanglement[/URL] and used them to formulate the [URL="http://en.wikipedia.org/wiki/Einstein-Podolsky-Rosen_paradox"]Einstein-Podolsky-Rosen paradox[/URL], in the hope of showing that quantum mechanics had unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see [URL="http://en.wikipedia.org/wiki/Bell_inequality"]Bell inequality[/URL]) that Einstein's assumption was correct, but had to be completed by [I]hidden variables[/I] and thus based on wrong philosophical assumptions. According to the paper of J. Bell and the [URL="http://en.wikipedia.org/wiki/Copenhagen_interpretation"]Copenhagen interpretation[/URL] (the common interpretation of quantum mechanics by physicists since 1927), and contrary to Einstein's ideas, quantum mechanics was not at the same time [LIST] [*]a "realistic" theory [/LIST] [LIST] [*]and a [I][URL="http://en.wikipedia.org/wiki/Principle_of_locality"]local[/URL][/I] theory [/LIST] The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate [URL="http://en.wikipedia.org/wiki/Causality"]causality[/URL], since no transfer of information happens. These experiments are the basis of some of the most topical applications of the theory, [URL="http://en.wikipedia.org/wiki/Quantum_cryptography"]quantum cryptography[/URL], which has been on the market since 2004 and works well, although at small distances of typically [IMG]http://upload.wikimedia.org/math/5/d/4/5d487c224cbe0c2f3324982c01dddb27.png[/IMG] 1000 km.[[I][URL="http://en.wikipedia.org/wiki/Wikipedia:Citation_needed"]citation needed[/URL][/I]] Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those applications. However, the lack of a correct theory of [URL="http://en.wikipedia.org/wiki/Quantum_gravity"]quantum gravity[/URL] is an important issue in [URL="http://en.wikipedia.org/wiki/Cosmology"]cosmology[/URL] and physicists' search for an elegant "[URL="http://en.wikipedia.org/wiki/Theory_of_everything"]theory of everything[/URL]". Thus, resolving the inconsistencies between both theories has been a major goal of twentieth- and twenty-first-century physics. Many prominent physicists, including [URL="http://en.wikipedia.org/wiki/Stephen_Hawking"]Stephen Hawking[/URL], have labored in the attempt to discover a theory underlying [I]everything[/I], combining not only different models of subatomic physics, but also deriving the universe's four forces —the [URL="http://en.wikipedia.org/wiki/Strong_interaction"]strong force[/URL], [URL="http://en.wikipedia.org/wiki/Electromagnetism"]electromagnetism[/URL], [URL="http://en.wikipedia.org/wiki/Weak_interaction"]weak force[/URL], and [URL="http://en.wikipedia.org/wiki/Gravity"]gravity[/URL]— from a single force or phenomenon. One of the leaders in this field is [URL="http://en.wikipedia.org/wiki/Edward_Witten"]Edward Witten[/URL], a theoretical physicist who formulated the groundbreaking [URL="http://en.wikipedia.org/wiki/M-theory"]M-theory[/URL], which is an attempt at describing the supersymmetrical based [URL="http://en.wikipedia.org/wiki/String_theory"]string theory[/URL]. [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=10"]edit[/URL]] Applications[/B] Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behaviour of the subatomic particles that make up all forms of [URL="http://en.wikipedia.org/wiki/Matter"]matter[/URL]—[URL="http://en.wikipedia.org/wiki/Electron"]electrons[/URL], [URL="http://en.wikipedia.org/wiki/Proton"]protons[/URL], [URL="http://en.wikipedia.org/wiki/Neutron"]neutrons[/URL], [URL="http://en.wikipedia.org/wiki/Photon"]photons[/URL] and others—can often only be satisfactorily described using quantum mechanics. Quantum mechanics has strongly influenced [URL="http://en.wikipedia.org/wiki/String_theory"]string theory[/URL], a candidate for a [URL="http://en.wikipedia.org/wiki/Theory_of_everything"]theory of everything[/URL] (see [URL="http://en.wikipedia.org/wiki/Reductionism"]reductionism[/URL]) and the [URL="http://en.wikipedia.org/wiki/Multiverse"]multiverse[/URL] hypothesis. It is also related to [URL="http://en.wikipedia.org/wiki/Statistical_mechanics"]statistical mechanics[/URL]. Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or molecules. The application of quantum mechanics to [URL="http://en.wikipedia.org/wiki/Chemistry"]chemistry[/URL] is known as [URL="http://en.wikipedia.org/wiki/Quantum_chemistry"]quantum chemistry[/URL]. (Relativistic) quantum mechanics can in principle mathematically describe most of chemistry. Quantum mechanics can provide quantitative insight into [URL="http://en.wikipedia.org/wiki/Ionic_bond"]ionic[/URL] and [URL="http://en.wikipedia.org/wiki/Covalent_bonding"]covalent bonding[/URL] processes by explicitly showing which molecules are energetically favorable to which others, and by approximately how much.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-38"][39][/URL] Most of the calculations performed in [URL="http://en.wikipedia.org/wiki/Computational_chemistry"]computational chemistry[/URL] rely on quantum mechanics.[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-39"][40][/URL] [URL="http://en.wikipedia.org/wiki/File:Rtd_seq_v3.gif"][IMG]http://upload.wikimedia.org/wikipedia/commons/thumb/5/51/Rtd_seq_v3.gif/350px-Rtd_seq_v3.gif[/IMG][/URL] [URL="http://en.wikipedia.org/wiki/File:Rtd_seq_v3.gif"][IMG]http://bits.wikimedia.org/skins-1.5/common/images/magnify-clip.png[/IMG][/URL] A working mechanism of a [URL="http://en.wikipedia.org/wiki/Resonant_tunneling_diode"]resonant tunneling diode[/URL] device, based on the phenomenon of quantum tunneling through the potential barriers. Much of modern [URL="http://en.wikipedia.org/wiki/Technology"]technology[/URL] operates at a scale where quantum effects are significant. Examples include the [URL="http://en.wikipedia.org/wiki/Laser"]laser[/URL], the [URL="http://en.wikipedia.org/wiki/Transistor"]transistor[/URL] (and thus the [URL="http://en.wikipedia.org/wiki/Integrated_circuit"]microchip[/URL]), the [URL="http://en.wikipedia.org/wiki/Electron_microscope"]electron microscope[/URL], and [URL="http://en.wikipedia.org/wiki/Magnetic_Resonance_Imaging"]magnetic resonance imaging[/URL]. The study of semiconductors led to the invention of the [URL="http://en.wikipedia.org/wiki/Diode"]diode[/URL] and the [URL="http://en.wikipedia.org/wiki/Transistor"]transistor[/URL], which are indispensable for modern [URL="http://en.wikipedia.org/wiki/Electronics"]electronics[/URL]. Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop [URL="http://en.wikipedia.org/wiki/Quantum_cryptography"]quantum cryptography[/URL], which will allow guaranteed secure transmission of [URL="http://en.wikipedia.org/wiki/Information"]information[/URL]. A more distant goal is the development of [URL="http://en.wikipedia.org/wiki/Quantum_computer"]quantum computers[/URL], which are expected to perform certain computational tasks exponentially faster than classical [URL="http://en.wikipedia.org/wiki/Computer"]computers[/URL]. Another active research topic is [URL="http://en.wikipedia.org/wiki/Quantum_teleportation"]quantum teleportation[/URL], which deals with techniques to transmit quantum states over arbitrary distances. [URL="http://en.wikipedia.org/wiki/Quantum_tunneling"]Quantum tunneling[/URL] is vital in many devices, even in the simple [URL="http://en.wikipedia.org/wiki/Light_switch"]light switch[/URL], as otherwise the electrons in the [URL="http://en.wikipedia.org/wiki/Electric_current"]electric current[/URL] could not penetrate the potential barrier made up of a layer of oxide. [URL="http://en.wikipedia.org/wiki/Flash_memory"]Flash memory[/URL] chips found in [URL="http://en.wikipedia.org/wiki/USB_drive"]USB drives[/URL] use quantum tunneling to erase their memory cells. QM primarily applies to the atomic regimes of matter and energy, but some systems exhibit [URL="http://en.wikipedia.org/wiki/Mechanics#Classical_versus_quantum"]quantum mechanical effects[/URL] on a large scale; [URL="http://en.wikipedia.org/wiki/Superfluidity"]superfluidity[/URL] (the frictionless flow of a liquid at temperatures near absolute zero) is one well-known example. Quantum theory also provides accurate descriptions for many previously unexplained phenomena such as [URL="http://en.wikipedia.org/wiki/Black_body_radiation"]black body radiation[/URL] and the stability of [URL="http://en.wikipedia.org/wiki/Atomic_orbital"]electron orbitals[/URL]. It has also given insight into the workings of many different [URL="http://en.wikipedia.org/wiki/Biological_systems"]biological systems[/URL], including [URL="http://en.wikipedia.org/wiki/Smell_receptors"]smell receptors[/URL] and [URL="http://en.wikipedia.org/wiki/Protein_structure"]protein structures[/URL].[URL="http://en.wikipedia.org/wiki/Quantum_mechanics#cite_note-40"][41][/URL] Even so, [URL="http://en.wikipedia.org/wiki/Classical_physics"]classical physics[/URL] often can be a good approximation to results otherwise obtained by [B]quantum physics[/B], typically in circumstances with large numbers of particles or large quantum numbers. (However, some open questions remain in the field of [URL="http://en.wikipedia.org/wiki/Quantum_chaos"]quantum chaos[/URL].) [B][[URL="http://en.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit&section=11"]edit[/URL]] Philosophical consequences[/B] Main article: [URL="http://en.wikipedia.org/wiki/Interpretation_of_quantum_mechanics"]Interpretation of quantum mechanics[/URL] Since its inception, the many [URL="http://en.wikipedia.org/wiki/Counter-intuitive"]counter-intuitive[/URL] results of quantum mechanics have provoked strong [URL="http://en.wikipedia.org/wiki/Philosophy"]philosophical[/URL] debate and many [URL="http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics"]interpretations[/URL]. Even fundamental issues such as [URL="http://en.wikipedia.org/wiki/Max_Born"]Max Born[/URL]'s basic [URL="http://en.wikipedia.org/wiki/Born_rule"]rules[/URL] concerning [URL="http://en.wikipedia.org/wiki/Probability_amplitude"]probability amplitudes[/URL] and [URL="http://en.wikipedia.org/wiki/Probability_distribution"]probability distributions[/URL] took decades to be appreciated. The [URL="http://en.wikipedia.org/wiki/Copenhagen_interpretation"]Copenhagen interpretation[/URL], due largely to the Danish theoretical physicist [URL="http://en.wikipedia.org/wiki/Niels_Bohr"]Niels Bohr[/URL], is the interpretation of quantum mechanical formalism most widely accepted amongst physicists. According to it, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but instead must be considered to be a final renunciation of the classical ideal of causality. In this interpretation, it is believed that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the [URL="http://en.wikipedia.org/wiki/Complementary"]complementarity[/URL]

Why the fuck did you copy and paste from wikipedia? [editline]02:49PM[/editline] This doesn't make you smart.

Massive useless wall of text.

oh god, even all the images are hotlinked

[QUOTE=Wakka;22449305]Why the fuck did you copy and paste from wikipedia? [editline]02:49PM[/editline] This doesn't make you smart.[/QUOTE] read "Tracer Ammo" and come back

Can't see any use for this thread

If we would know that we'll just search it ourself. And why did you copy this?

[QUOTE=Armyis1337;22449343]read "Tracer Ammo" and come back[/QUOTE] How about you leave and never come back.

I have a strong feeling that OP doesn't even understand this article

Oh god, even the edit links to the wikipedia edit page.

[QUOTE=Wakka;22449390]How about you leave and never come back.[/QUOTE] Good one, I haven't thought of that before. Also, I don't need to understand it for it to be a SHITPOST PARODY.

[QUOTE=Wakka;22449415]Oh god, even the edit links to the wikipedia edit page.[/QUOTE] Ctrl+A, Ctrl+C, Ctrl+V

It'd be funnier if something happened recently that you're parodying. Also, calling everyone who criticizes you a 12 year old doesn't make it so.

[QUOTE=Bassplaya7;22449486]It'd be funnier if something happened recently that you're parodying. Also, calling everyone who criticizes you a 12 year old doesn't make it so.[/QUOTE] It happened in another thread like 5 hours ago, don't you people live in GD.

heh this is what i do at uni.

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