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10811:"For most physics students, (the "mathematical underpinning" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.
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1176:, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected
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superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule, with no consensus on whether they have been successful.
9122:. 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 quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory,
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9286:, their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. 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 particular applications. However, the lack of a correct theory of
6563:
9974:, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.
13803:...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation.
9024:
acting on vectors in that space – although they do not tell us which
Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices
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We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes
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is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave
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such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency. In his paper "On the
Quantum Theory of Radiation", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of
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One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and
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It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the
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that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as
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Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings
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shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The
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of each other, so that a description of an object according to its momentum is the
Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an
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occur in a multiverse composed of mostly independent parallel universes. This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum
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As we let the
Gaussian wave packet evolve in time, we see that its center moves 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 and more
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As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as
9298:" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.
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Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as
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nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr, Heisenberg, Schrödinger, Feynman, and
Zeilinger as well as 21st-century researchers in quantum foundations.
9228:, a charged particle is modeled as a quantum system, while the background magnetic field is described classically. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by
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6864:(MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the
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The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its
Hamiltonian consists only of its kinetic energy:
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put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de
Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and
9635:. The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics. While the early conception of atoms from
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in a hydrogen atom at different energy levels. Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations. The brighter areas represent a higher probability of finding the
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The published form of the EPR argument was due to
Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete
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says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its
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in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation. The word
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9037:. One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as
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1270:", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide. A collection of results, most significantly
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6558:{\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }
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renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the
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beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to
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The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere
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A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a
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2358:, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an
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It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also
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of the radiation. In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert
Einstein interpreted Planck's quantum hypothesis
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energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation, which became the basis of the laser.
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showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as
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can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling
6835:
3771:, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the
1274:, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of
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for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The
9321:. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its
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become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations. Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching
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2411:. Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.
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evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.
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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 the interaction of
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The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat
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Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian
Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (September 2019). "Quantum superposition of molecules beyond 25 kDa".
7511:
9847:. Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926. Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth
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Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by
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1298:, and other more advanced subjects. Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.
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will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.
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It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian
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9348:. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric "woven" of finite loops called
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7993:. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.
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12869:(2010). "One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation". In S. Saunders; J. Barrett; A. Kent; D. Wallace (eds.).
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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
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This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The
9898:, and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some
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The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.
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Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and
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can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if
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Tatsumi Aoyama; Masashi Hayakawa; Toichiro Kinoshita; Makiko Nio (2012). "Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant".
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1239:: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...
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have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.
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The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)
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Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators
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1007:, and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both
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1099:, and its application to the universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of
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1462:. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex
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2065:
1019:), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the
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2407:, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak
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One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between different measurable quantities. The most famous form of this
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2350:. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited
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2010:
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Schrödinger, Erwin (1980) . Trimmer, John (ed.). ""Die gegenwärtige Situation in der Quantenmechanik."" [The Present Situation in Quantum Mechanics].
11921:"Black Holes May Hide a Mind-Bending Secret About Our Universe – Take gravity, add quantum mechanics, stir. What do you get? Just maybe, a holographic cosmos"
8011:. Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons,
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4222:
1811:
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A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a
6203:
function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the
2294:
corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom:
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for "uncuttable" – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was
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corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a
9803:
to modern quantum mechanics. Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and
1140:
relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.
1131:
can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a
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in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the
980:) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic/microscopic) scale.
7350:
9759:, in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a
1243:
characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables
4490:{\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}
3727:(POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.
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72:
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1887:. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer
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D. Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories",
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Schlosshauer, Maximilian; Kofler, Johannes; Zeilinger, Anton (1 August 2013). "A snapshot of foundational attitudes toward quantum mechanics".
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In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include
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The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.
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Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are
1307:
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Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "
9258:. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as
12788:
Ballentine, L. E. (1973). "Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation".
10616:(2015). "Quantum Information Science: Emerging No More". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.).
9687:, meaning "how great" or "how much". According to Planck, quantities of energy could be thought of as divided into "elements" whose size (
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13905:
11191:
10824:
8082:. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule
909:
13783:"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4"
5970:
2310:, ...). Denser areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to
1184:
demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as
15414:
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In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist
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atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in
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14403:. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics.
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9746:. Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the
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larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.
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bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by
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13195:
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389:
7283:, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of
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1899:). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become
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572:
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Wallace, David (2003). "Everettian Rationality: defending Deutsch's approach to probability in the Everett interpretation".
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14354:(Vol. I), English translation from French by G.M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.
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7645:{\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}
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3985:: its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator
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Haack, G. R.; Förster, H.; Büttiker, M. (2010). "Parity detection and entanglement with a Mach-Zehnder interferometer".
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Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated
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There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "
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In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector
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9469:. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the
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2670:{\displaystyle \sigma _{X}={\textstyle {\sqrt {\left\langle X^{2}\right\rangle -\left\langle X\right\rangle ^{2}}}},}
155:
33:
9647:'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure.
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Fan, X.; Myers, T. G.; Sukra, B. A. D.; Gabrielse, G. (2023-02-13). "Measurement of the Electron Magnetic Moment".
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straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector
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244:
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3735:. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.
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9477:. He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how
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of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have
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2013:
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331:
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9360:, approximately 1.616Ă—10 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.
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passes through one slit (as would a classical particle), and not through both slits (as would a wave). However,
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Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong
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13931:, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using
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Martin, Andre (1986), "Cathode Ray Tubes for Industrial and Military Applications", in Hawkes, Peter (ed.),
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When two different quantum systems are considered together, the Hilbert space of the combined system is the
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is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by
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9816:
9812:
9556:
9540:
9333:
state of the string. In string theory, one of the many vibrational states of the string corresponds to the
9225:
8991:
5778:
4913:
smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making
3025:
2291:
1892:
1776:
1132:
1027:
567:
562:
533:
384:
165:
14582:
13283:. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83.
10065:(1926). "Zur Quantenmechanik der Stoßvorgänge" [On the Quantum Mechanics of Collision Processes].
9044:
When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was
3816:
3024:
Another consequence of the canonical commutation relation is that the position and momentum operators are
2420:
momentum are observables, meaning that they are represented by Hermitian operators. The position operator
2161:
1568:. When an observable is measured, the result will be one of its eigenvalues with probability given by the
15765:
15173:
15080:
13656:
9879:
9049:
6228:
4607:
3693:{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}
3428:
3130:
2377:
1748:
is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the
1541:
1469:
1181:
954:
601:
346:
336:
12543:
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
11782:
9493:
published an argument that the principle of locality implies the incompleteness of quantum mechanics, a
3217:
3186:
3014:{\displaystyle \sigma _{A}\sigma _{B}\geq {\tfrac {1}{2}}\left|{\bigl \langle }{\bigr \rangle }\right|.}
2752:{\displaystyle \sigma _{P}={\sqrt {\left\langle P^{2}\right\rangle -\left\langle P\right\rangle ^{2}}}.}
15940:
15814:
15596:
15569:
15449:
15211:
15019:
14990:
12711:
11894:
9525:
8075:
3710:
If the state for a composite system is entangled, it is impossible to describe either component system
1896:
1782:
1653:
1424:
895:
547:
518:
11667:
11466:
Paris, M. G. A. (1999). "Entanglement and visibility at the output of a Mach–Zehnder interferometer".
11149:. This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1
10727:
6571:
6249:(A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a
1511:
1363:. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys
15876:
15233:
15100:
15024:
14985:
14942:
14916:
14873:
14766:
14548:
13372:
13249:
9470:
9447:
9434:
9115:
7894:{\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}
7772:{\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}
3768:
3058:
1880:
1185:
1103:. For example, the refinement of quantum mechanics for the interaction of light and matter, known as
1086:, about what measurements of a particle's energy, momentum, and other physical properties may yield.
1016:
552:
513:
466:
441:
364:
224:
12563:
11001:
Drake, Gordon W. F. (2023). "High Precision Calculations for Helium". In Drake, Gordon W. F. (ed.).
7655:
and the probabilities that it will be detected at the right or at the top are given respectively by
7078:
1342:
15702:
15300:
15280:
15270:
15260:
15216:
14791:
13549:
9950:
9945:
9636:
9593:
9026:
8811:
8533:
8285:
7915:
7445:
A photon that enters the interferometer from the left will then be acted upon with a beam splitter
7318:
7286:
2072:
1459:
1260:
1248:
1030:
from theories to explain observations that could not be reconciled with classical physics, such as
636:
9461:, was troubled by its apparent failure to respect some cherished metaphysical principles, such as
7973:
7422:
1724:
14995:
14921:
12618:
12007:
Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory
11833:
11381:
10794:
9940:
9831:
9632:
9382:
9123:
8984:
8967:
8588:
8503:
4174:
3136:
2318:
modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp
1104:
805:
523:
431:
15409:
14957:
12162:
10774:
there's no way to understand the interpretation of quantum mechanics without also being able to
9844:
8281:
5883:
4651:
2452:
2423:
1137:
481:
379:
145:
15786:
15648:
15320:
15095:
15075:
15000:
14868:
14491:
13314:"Ăśber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"
12911:
12691:
12558:
12449:
11041:
9705:
9652:
9473:. Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids
9406:. The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with
9386:
9247:
9038:
8821:
8801:
8671:
8047:
6189:{\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}
4578:
3719:
3563:
3536:
3489:
3462:
3352:
3325:
2488:
2397:
2331:
1267:
1160:
1108:
988:
810:
528:
356:
326:
289:
14911:
13316:[On a heuristic point of view concerning the production and transformation of light].
11048:. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons.
10842:
10402:
9639:
had been that they were indivisible units – the word "atom" deriving from the
9422:
once said, "I think I can safely say that nobody understands quantum mechanics." According to
3032:
2257:
2228:
1758:
1575:
436:
15791:
15760:
15623:
15544:
15345:
14858:
14838:
13873:
11805:
11442:
11434:
10863:
10456:
10429:
9482:
9466:
9426:, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."
9407:
9142:
9082:
8466:
6199:
2871:
2068:
2047:
1565:
1545:
1291:
1279:
1144:
1083:
1020:
279:
264:
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13597:
13587:
2346:
Some wave functions produce probability distributions that are independent of time, such as
2023:
15750:
15574:
15534:
15376:
15305:
15250:
14980:
14803:
14761:
14106:
13940:
13918:
13881:
13665:
13381:
13327:
13040:
12991:
12920:
12884:
12797:
12754:
12550:
12511:
12468:
12409:
12348:
12221:
12080:
11852:
11625:
11542:
11487:
11335:
11125:
10965:
10900:
10889:"Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems"
10635:
10315:
10254:
10201:
10166:
10074:
9971:
9920:
9768:
9756:
9671:
9601:
9474:
9418:. Perhaps the only consensus that exists about these issues is that there is no consensus.
9341:
9295:
9239:
9114:
involved the replacement of the Schrödinger equation with a covariant equation such as the
9045:
8598:
8476:
8461:
8451:
8203:{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }
8051:
5863:
5588:
5229:
4170:
3784:
3704:
3064:
2311:
1900:
1830:
1404:
1315:
1236:
1156:
1047:
1035:
992:
946:
596:
508:
234:
191:
13782:
13496:
11830:
8706:
8325:
5947:
5487:
3895:
3760:
3751:, which unifies and generalizes the two earliest formulations of quantum mechanics –
1883:, in which the two scientists attempted to clarify these fundamental principles by way of
1454:
represent the same physical system. In other words, the possible states are points in the
1401:, and it is well-defined up to a complex number of modulus 1 (the global phase), that is,
1059:
840:
8:
15725:
15584:
15579:
15564:
15539:
15516:
15492:
15361:
15330:
15275:
15255:
15163:
15120:
14975:
14901:
14828:
14818:
14730:
13370:(1917). "Zur Quantentheorie der Strahlung" [On the Quantum Theory of Radiation].
13031:
13026:
10621:
9871:
9624:
9415:
9411:
9246:
have also been developed. The quantum field theory of the strong nuclear force is called
9074:
9057:
9030:
9021:
8721:
8548:
8538:
8486:
8481:
8456:
8355:
8079:
8024:
7419:, which means that if the photon is on the "upper" path it will gain a relative phase of
6759:{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}
6626:
6246:
6238:
5749:
5723:
5608:
5562:
5536:
3732:
2218:
1868:{\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}
1749:
1164:
695:
503:
421:
249:
229:
181:
14110:
13669:
13385:
13331:
13044:
12924:
12888:
12801:
12758:
12632:
12613:
12554:
12515:
12472:
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12326:
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12084:
11856:
11629:
11546:
11491:
11339:
11129:
10969:
10904:
10639:
10319:
10266:
10258:
10205:
10170:
10078:
9029:, a heuristic which states that the predictions of quantum mechanics reduce to those of
8806:
6281:
As in the classical case, the potential for the quantum harmonic oscillator is given by
5924:
5383:{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}
4320:{\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}
1155:
Another consequence of the mathematical rules of quantum mechanics is the phenomenon of
15888:
15852:
15781:
15658:
15653:
15606:
15421:
15290:
15188:
14896:
14843:
14735:
14570:
14441:
14223:
Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
14176:
14161:
14138:
14122:
13924:
13460:
13401:
13318:
13217:
13124:
Evidence for the existence of "cathode-rays" was first found by PlĂĽcker and Hittorf ...
12963:
12944:
12874:
12813:
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12744:
12484:
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10482:
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10217:
10098:
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9582:
9494:
9458:
9291:
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9259:
9243:
9221:
9131:
9111:
8931:
8881:
8831:
8573:
8568:
8508:
8373:
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7448:
6043:
5843:
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5394:
5249:
4971:
4916:
4896:
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4128:
4108:
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4048:
4028:
4008:
3988:
3964:
3944:
3924:
3875:
3792:
3703:
is a valid joint state that is not separable. States that are not separable are called
3590:
3516:
2844:
2824:
2591:
2573:
2553:
2339:
1995:
1884:
1549:
1204:
1096:
1051:
977:
950:
416:
341:
274:
186:
14355:
12766:
9441:
feature which will eventually be replaced by a deterministic theory, but is instead a
9071:
studies the relationship between classical and quantum descriptions in these systems.
8621:
7276:{\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}
6211:
effect that plays an important role in the performance of modern technologies such as
3057:
factor) to taking the derivative according to the position, since in Fourier analysis
15883:
15796:
15745:
15690:
15556:
15431:
15340:
15310:
15238:
15201:
15196:
15178:
15143:
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14813:
14796:
14651:
14556:
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14470:
14448:
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14016:
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13863:
13713:
13706:
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13225:
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11446:
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11369:
11355:
11290:
11280:
11237:
11206:
11196:
11158:
11145:
11116:
Ballentine, Leslie E. (1970), "The Statistical Interpretation of Quantum Mechanics",
11097:
11087:
11049:
11014:
10983:
10924:
10916:
10867:
10832:
10800:
10723:
10702:
10694:
10666:
10643:
10543:
10492:
10462:
10435:
10408:
10376:
10331:
10270:
10221:
10130:
10102:
10090:
9903:
9848:
9824:
9780:
9517:
9507:
9502:
9345:
9078:
8891:
8751:
8661:
8656:
8345:
8320:
8055:
8043:
8035:
8028:
8008:
6233:
6208:
4954:
4883:{\displaystyle {\hat {\psi }}(k,0)={\sqrt{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}
3756:
2385:
2381:
1463:
1271:
1256:
1252:
1244:
1213:
1208:
1063:
969:
965:
961:
942:
926:
850:
825:
765:
760:
660:
626:
606:
204:
14126:
14009:
13756:
13191:
12948:
12488:
12429:
12376:
12311:
12194:
11880:
11507:
10282:
10191:
9648:
9589:
proposed a wave theory of light based on experimental observations. In 1803 English
9437:". According to these views, the probabilistic nature of quantum mechanics is not a
8816:
2403:
However, there are techniques for finding approximate solutions. One method, called
15840:
15828:
15735:
15675:
15601:
15158:
15153:
15010:
14906:
14436:
14326:
14266:
14114:
13980:
13751:
13673:
13647:
13397:
13335:
13284:
13048:
12928:
12902:
12805:
12774:
12762:
12687:
12627:
12568:
12519:
12476:
12417:
12356:
12291:
12229:
12174:
12147:
12127:
12088:
11864:
11860:
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11633:
11550:
11495:
11343:
11302:
11272:
11133:
11077:
11006:
10973:
10908:
10684:
10675:
10613:
10533:
10525:
10327:
10323:
10262:
10209:
10174:
10082:
9859:
9840:
9675:
9628:
9326:
9318:
9310:
9229:
9127:
8946:
8921:
8896:
8886:
8841:
8711:
8518:
8498:
8493:
8350:
6266:
4181:
3772:
3752:
3723:
density matrices specify the state of a subsystem of a larger system, analogously,
2590:, and moreover for powers of them. Defining the uncertainty for an observable by a
2408:
2327:
1455:
1004:
855:
845:
835:
735:
715:
700:
670:
538:
426:
12594:
Einstein, Albert (1949). "Autobiographical Notes". In Schilpp, Paul Arthur (ed.).
11347:
10888:
10590:
10124:
1903:
so that the original quantum system ceases to exist as an independent entity (see
1200:
are all found to exhibit the same dual behavior when fired towards a double slit.
15755:
15665:
15611:
15511:
15388:
15315:
15295:
15265:
15228:
15223:
15128:
14952:
14655:
14636:
14272:
14243:
14233:
14165:
14134:
13984:
13946:
13900:
13778:
13593:
13500:
13367:
13137:
12667:
12444:
12360:
12031:
12005:
11976:
11227:
11182:
11073:
11010:
11005:. Springer Handbooks. Cham: Springer International Publishing. pp. 199–216.
10978:
10943:
10857:
10732:
10529:
10368:
10029:
9743:
9656:
9644:
9616:
9454:
9423:
9419:
9287:
9271:
9267:
9235:
9138:
9099:
8941:
8936:
8856:
8796:
8701:
8676:
8666:
8631:
8626:
8593:
8523:
8446:
8305:
8071:
3312:{\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}
2550:
Given a quantum state, the Born rule lets us compute expectation values for both
2354:
is pictured classically as a particle moving in a circular trajectory around the
2217:
is known as the time-evolution operator, and has the crucial property that it is
2041:
1333:
1255:. Contrary to popular misconception, entanglement does not allow sending signals
1039:
880:
750:
730:
476:
316:
14552:
14361:
13288:
13075:
12572:
12421:
10778: – to understand the theory, you need to be able to use it (and vice versa)
9795:. Never complete or self-consistent, the old quantum theory was rather a set of
9543:
appeared in the late 1990s as a modern derivative of Copenhagen-type ideas, and
7412:{\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}
5269:
The general solutions of the Schrödinger equation for the particle in a box are
1915:
The time evolution of a quantum state is described by the Schrödinger equation:
15912:
15900:
15685:
15640:
15618:
15506:
15366:
15335:
15325:
14947:
14937:
14771:
14398:
14392:
14347:
14251:
14218:
14053:
13956:
13855:
13583:
12092:
11948:
11916:
11554:
11083:
10157:
Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?".
9883:
9875:
9836:
9684:
9640:
9586:
9486:
9478:
9306:
9150:
9119:
9053:
9034:
8926:
8911:
8871:
8861:
8836:
8781:
8731:
8726:
8563:
8543:
8513:
8398:
8335:
8330:
8097:
8067:
8039:
6258:
6242:
3172:
2359:
2355:
2286:
1505:
1287:
1217:
1116:
1111:
to within 1 part in 10 when predicting the magnetic properties of an electron.
973:
815:
775:
755:
725:
705:
655:
621:
471:
461:
254:
14226:
14118:
13357:
See also "Einstein's early work on the quantum hypothesis", ibid. pp. 134–148.
13011:
12932:
12648:
12480:
12447:(2010). "Einstein, incompleteness, and the epistemic view of quantum states".
12178:
11276:
11137:
10829:
The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics
10213:
9987:
for some of the technological applications which use quantum mechanics, e.g.,
9659:
carried on and improved upon Faraday's work, leading to the identification of
6852:
1050:. These early attempts to understand microscopic phenomena, now known as the "
964:
cannot. Classical physics can describe many aspects of nature at an ordinary (
15934:
15864:
15695:
15285:
15138:
15029:
14863:
14833:
14786:
14307:
13968:
13687:
13569:
13545:
13434:
13340:
13313:
13254:
13163:
13060:
12940:
12580:
12368:
12268:
12186:
12139:
11598:
11576:
11499:
11294:
11210:
10987:
10920:
10698:
10094:
9895:
9891:
9855:
9808:
9764:
9752:
9664:
9620:
9560:
9357:
9344:(LQG), which describes quantum properties of gravity and is thus a theory of
9302:
9146:
9087:
9068:
9017:
8901:
8866:
8826:
8791:
8771:
8766:
8761:
8756:
8736:
8681:
8616:
8063:
7905:
7209:
6262:
5072:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}
4198:
2389:
1337:
1079:
875:
870:
800:
770:
740:
611:
557:
284:
259:
137:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
43:
15460:
14484:
14385:
14200:
14170:
Compendium of quantum physics, Concepts, experiments, history and philosophy
10758:
9433:, Werner Heisenberg and other physicists are often grouped together as the "
9016:
The rules of quantum mechanics assert that the state space of a system is a
8776:
4778:{\displaystyle \psi (x,0)={\frac {1}{\sqrt{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}
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2298:= 1, 2, 3, ...) and angular momenta (increasing across from left to right:
2017:
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865:
860:
795:
780:
745:
239:
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9081:, and thus become incapable of displaying many typically quantum effects:
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1508:
of a single proton is simply the space of two-dimensional complex vectors
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14632:
14574:
13425:
13281:
Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle
13245:
12866:
12838:. In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.).
12749:
12699:
11722:
11482:
10942:
Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01).
10586:
10431:
Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral
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8906:
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8651:
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and this provides the lower bound on the product of standard deviations:
2222:
1982:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t).}
1876:
1553:
984:
830:
785:
720:
675:
39:
32:
For a more accessible and less technical introduction to this topic, see
14172:, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics.
13822:. Physics Department, University of California, Berkeley. Archived from
12538:
10689:
10670:
10563:"Developing more energy-efficient transistors through quantum tunneling"
9870:. It has since permeated many disciplines, including quantum chemistry,
9827:'s extension of the Bohr model to include special-relativistic effects.
9667:
found to consist of subatomic particles that would be called electrons.
1054:", led to the full development of quantum mechanics in the mid-1920s by
15740:
15730:
14578:
14536:
14239:
14208:
14004:
13989:. Cambridge, Massachusetts: Belknap Press of Harvard University Press.
13355:(in German). Vol. 2. Princeton University Press. pp. 149–166.
12809:
12295:
11408:
Feynman, Richard P.; Hibbs, Albert R. (2005). Steyer, Daniel F. (ed.).
11069:
10944:"Comparing many-body approaches against the helium atom exact solution"
10859:
Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature
10790:
10086:
9988:
9907:
9760:
9600:. This experiment played a major role in the general acceptance of the
9569:
9430:
9399:
8786:
8696:
8636:
8553:
8403:
8340:
8310:
8300:
8273:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
8059:
7347:. The phase shifter on the upper arm is modelled as the unitary matrix
6356:
4501:
4005:
can generate a family of unitary operators parameterized by a variable
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2347:
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685:
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10178:
9882:. It also provides a useful framework for many features of the modern
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corrections to classical mechanics. The theory is now understood as a
9141:
particles as quantum mechanical objects being acted on by a classical
7904:
One can therefore use the Mach–Zehnder interferometer to estimate the
6840:
This is another example illustrating the discretization of energy for
4203:
3813:
of time evolution, since it defines a unitary time-evolution operator
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15060:
14756:
14563:
12987:
12234:
12209:
9854:
By 1930 quantum mechanics had been further unified and formalized by
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a certain region, and therefore infinite potential energy everywhere
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1197:
1189:
1120:
311:
14668:
14595:
13620:
David Edwards, "The Mathematical Foundations of Quantum Mechanics",
12116:"Who Invented the 'Copenhagen Interpretation'? A Study in Mythology"
7315:, or be reflected to the other path with a probability amplitude of
6261:
shown in red. Some of the trajectories (such as C, D, E, and F) are
4988:
direction, the time-independent Schrödinger equation may be written
27:
Description of physical properties at the atomic and subatomic scale
15847:
13949:, 1990, "Spooky actions at a distance: mysteries of the QT" in his
12983:
12210:"The Quantum Postulate and the Recent Development of Atomic Theory"
12131:
11330:
10960:
10310:
10249:
10062:
9784:
9590:
9373:
9334:
9094:
8686:
8423:
8315:
3061:. This is why in quantum equations in position space, the momentum
1203:
Another non-classical phenomenon predicted by quantum mechanics is
1149:
1128:
1124:
1100:
1067:
1008:
1000:
690:
47:
14059:
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
12879:
12463:
12404:
12343:
12075:
11847:
11537:
11267:
10630:
9767:
of hydrogen. Einstein further developed this idea to show that an
9337:, a quantum mechanical particle that carries gravitational force.
6030:{\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}
3245:, respectively. The Hilbert space of the composite system is then
15859:
13932:
13708:
A Tale of Two Continents: A Physicist's Life in a Turbulent World
10373:
Mastering Quantum Mechanics: Essentials, Theory, and Applications
9250:, and describes the interactions of subnuclear particles such as
8413:
8016:
4937:
uncertain. The uncertainty in momentum, however, stays constant.
4788:
which has Fourier transform, and therefore momentum distribution
3764:
1235:
When quantum systems interact, the result can be the creation of
972:) scale, but is not sufficient for describing them at very small
13817:"Berkeley Experiments on Superfluid Macroscopic Quantum Effects"
13481:
12539:"An Einstein manuscript on the EPR paradox for spin observables"
12325:
Ma, Xiao-song; Kofler, Johannes; Zeilinger, Anton (2016-03-03).
11316:
Schlosshauer, Maximilian (October 2019). "Quantum decoherence".
5484:
The infinite potential walls of the box determine the values of
1038:
problem, and the correspondence between energy and frequency in
14508:
13027:"Bringing one of the great moments of science to the classroom"
9048:
classical mechanics. For instance, the well-known model of the
8393:
8388:
8020:
8012:
6269:"). Each standing-wave frequency is proportional to a possible
4949:
1-dimensional potential energy box (or infinite potential well)
4575:, which are eigenstates of the momentum operator with momentum
3059:
differentiation corresponds to multiplication in the dual space
2811:{\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}
2393:
2319:
1177:
996:
930:
14051:
13961:
Timeless Reality: Symmetry, Simplicity, and Multiple Universes
13139:
Flash of the Cathode Rays: A History of J J Thomson's Electron
12253:. World perspectives (3 ed.). London: Allen & Unwin.
6830:{\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}
5860:
cannot be zero as this would conflict with the postulate that
4648:, which is the Fourier transform of the initial quantum state
3175:
of the Hilbert spaces of the two components. For example, let
10004:
9999:, which are follow-on technology in solid-state physics (vol
9887:
9544:
9255:
9251:
8418:
8378:
4330:
The general solution of the Schrödinger equation is given by
3133:
the momentum-squared term is replaced with a Laplacian times
2372:
Analytic solutions of the Schrödinger equation are known for
1643:{\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}
1193:
1168:
14532:
14461:
13883:
The Quantum Universe: Everything That Can Happen Does Happen
12502:
Howard, D. (1985). "Einstein on locality and separability".
12327:"Delayed-choice gedanken experiments and their realizations"
11696:(5th ed.). W.H. Freeman and Company. pp. 160–161.
7016:{\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}
6960:{\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}}
14097:(1957). "Relative State Formulation of Quantum Mechanics".
14083:, Princeton Series in Physics, Princeton University Press.
14029:
13392:
Einstein, A. (1967). "On the Quantum Theory of Radiation".
13074:
Feynman, Richard; Leighton, Robert; Sands, Matthew (1964).
12668:"Everett's Relative-State Formulation of Quantum Mechanics"
12389:
11003:
Springer Handbook of Atomic, Molecular, and Optical Physics
10123:
Feynman, Richard; Leighton, Robert; Sands, Matthew (1964).
9322:
9134:, one of the most accurate physical theories ever devised.
8428:
8383:
6250:
3724:
2351:
2254:– it makes a definite prediction of what the quantum state
1266:
Another possibility opened by entanglement is testing for "
1012:
934:
15919:
14238:
13917:. Four elementary lectures on quantum electrodynamics and
12686:
9866:, the statistical nature of our knowledge of reality, and
9510:, that can be violated by entangled particles. Since then
9214:{\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)}
13222:
Black-body theory and the quantum discontinuity 1894–1912
12964:"Quantum-Bayesian and Pragmatist Views of Quantum Theory"
12061:
Weinberg, Steven (2012). "Collapse of the state vector".
9105:
8083:
7442:, and it will stay unchanged if it is in the lower path.
4165:. This implies a quantum version of the result proven by
1119:, known as a probability amplitude. This is known as the
13430:"A century ago Einstein sparked the notion of the laser"
12251:
Physics and philosophy: the revolution in modern science
12163:"Constructing the Myth of the Copenhagen Interpretation"
10796:
The Demon-Haunted World: Science as a Candle in the Dark
10514:"Quantum tunnelling to the origin and evolution of life"
10235:
Bojowald, Martin (2015). "Quantum cosmology: a review".
9356:. The characteristic length scale of a spin foam is the
9352:. The evolution of a spin network over time is called a
9056:
of the oscillator, and is thus a quantum version of the
8089:
2225:
in the sense that – given an initial quantum state
2067:
is introduced so that the Hamiltonian is reduced to the
1247:
and is part of quantum communication protocols, such as
13110:
Advances in Electronics and Electron Physics, Volume 67
11195:(2nd ed.). Cambridge: Cambridge University Press.
11040:
10941:
9052:
uses an explicitly non-relativistic expression for the
3131:
non-relativistic Schrödinger equation in position space
3122:{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}
2078:
The solution of this differential equation is given by
1891:
have been formulated that do away with the concept of "
1714:{\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }
1552:
acting on the Hilbert space. A quantum state can be an
1074:
and others. The modern theory is formulated in various
14564:
Modern Physics: With waves, thermodynamics, and optics
13963:. Buffalo, NY: Prometheus Books. Chpts. 5–8. Includes
13067:
11374:"Erwin Schrödinger and the creation of wave mechanics"
10295:
9843:
and the Austrian physicist Erwin Schrödinger invented
9160:
7569:
7365:
7242:
7068:{\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}}
6992:
6936:
5141:{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}
3763:). An alternative formulation of quantum mechanics is
3618:
2958:
2618:
2338:
discrete values like those shown. (As is the case for
1814:
15812:
13712:. Princeton, New Jersey: Princeton University Press.
13624:, Volume 42, Number 1/September, 1979, pp. 1–70.
9708:
9159:
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8109:
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7425:
7353:
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7218:
7145:
7107:
7081:
7029:
6973:
6917:
6882:
6778:
6637:
6574:
6368:
6345:{\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}
6290:
6068:
6046:
5973:
5950:
5927:
5886:
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3139:
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2164:
2087:
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1998:
1924:
1785:
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1472:
1427:
1407:
1369:
1345:
1318:
75:
14133:
13073:
11520:
10122:
9931:
List of textbooks on classical and quantum mechanics
6060:
implies a constraint on the energy levels, yielding
5266:
coincident with the kinetic energy of the particle.
5216:{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}
4568:{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}
1910:
1089:
14175:
14081:
The Many-Worlds Interpretation of Quantum Mechanics
13080:. Vol. 1. California Institute of Technology.
12708:
The Many-Worlds Interpretation of Quantum Mechanics
12504:
Studies in History and Philosophy of Science Part A
12392:
Studies in History and Philosophy of Science Part B
11946:
11757:"The Math That Takes Newton Into the Quantum World"
10129:. Vol. 3. California Institute of Technology.
9145:. For example, the elementary quantum model of the
3778:
3738:
1078:. In one of them, a mathematical entity called the
14496:Facts and Mysteries in Elementary Particle Physics
14440:
14271:
14008:
13735:"Von Neumann's contributions to quantum mechanics"
13705:
13459:
11433:Mathews, Piravonu Mathews; Venkatesan, K. (1976).
11036:
11034:
11032:
11030:
9726:
9481:is valid, but the fundamental theory behind it is
9213:
9077:is a mechanism through which quantum systems lose
8272:
8202:
7985:
7962:
7893:
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7644:
7497:
7477:
7457:
7434:
7411:
7339:
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6758:
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5872:
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2147:
2059:
2032:
2004:
1981:
1867:
1800:
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1740:
1713:
1671:
1642:
1592:is non-degenerate and the probability is given by
1584:
1529:
1496:
1446:
1413:
1393:
1355:
1324:
136:
14435:
14325:
14312:The mathematical foundations of quantum mechanics
13979:
13544:
12324:
12318:
11953:String theory and M-theory: A modern introduction
11432:
11412:(Emended ed.). McGraw-Hill. pp. v–vii.
10831:(2nd ed.). Jones and Bartlett. p. 215.
9277:
5470:
4968:that region. For the one-dimensional case in the
4207:Position space probability density of a Gaussian
2148:{\displaystyle \psi (t)=e^{-iHt/\hbar }\psi (0).}
1835:
1775:was obtained, the quantum state is postulated to
960:Quantum mechanics can describe many systems that
15932:
13494:
12972:. Metaphysics Research Lab, Stanford University.
12871:Many Worlds? Everett, Quantum Theory and Reality
12676:. Metaphysics Research Lab, Stanford University.
12655:. Metaphysics Research Lab, Stanford University.
12442:
11614:"Essay: Fifty Years of Condensed Matter Physics"
11435:"The Schrödinger Equation and Stationary States"
10822:
10599:. Metaphysics Research Lab, Stanford University.
10401:Lederman, Leon M.; Hill, Christopher T. (2011).
3322:If the state for the first system is the vector
1556:of an observable, in which case it is called an
14213:The Conceptual Development of Quantum Mechanics
13495:Bokulich, Alisa; Bokulich, Peter (2020-08-13).
13162:
12976:
11691:
11605:
11068:
11027:
10458:Niels Bohr and Complementarity: An Introduction
9572:is considered the father of the quantum theory.
9110:Early attempts to merge quantum mechanics with
9090:quantum behavior may manifest macroscopically.
5474:{\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}
4180:of a Hamiltonian, there exists a corresponding
14467:What is Quantum Mechanics? A Physics Adventure
13921:, yet containing many insights for the expert.
12781:
12710:. Princeton Series in Physics. Princeton, NJ:
12003:
11803:
11108:
10718:
10716:
10427:
10363:
10361:
10359:
10357:
10355:
10353:
10351:
10349:
10347:
10345:
9868:philosophical speculation about the 'observer'
9532:the possibilities described by quantum theory
9126:, provides a fully quantum description of the
9067:, which do not have good quantum numbers, and
6847:
5710:{\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}
4145:is conserved under the evolution generated by
15490:
15476:
14684:
14360:
13743:Bulletin of the American Mathematical Society
13548:; Napolitano, J. (2014). "Quantum Dynamics".
11181:
10421:
9363:
9294:and the search by physicists for an elegant "
8992:
7132:{\displaystyle \langle \psi ,\psi \rangle =1}
2998:
2976:
2374:very few relatively simple model Hamiltonians
1394:{\displaystyle \langle \psi ,\psi \rangle =1}
1308:Mathematical formulation of quantum mechanics
1163:. In the basic version of this experiment, a
903:
13771:
13589:Introduction to the Theory of Ferromagnetism
12901:
11909:
11717:
11715:
11713:
11407:
11315:
11261:(2nd ed.). Cambridge University Press.
11226:
10661:
10659:
10400:
8197:
8146:
7842:
7807:
7720:
7685:
7198:{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}
7120:
7108:
6911:that is a superposition of the "lower" path
4604:. The coefficients of the superposition are
3376:, then the state of the composite system is
3183:be two quantum systems, with Hilbert spaces
1860:
1838:
1708:
1686:
1625:
1604:
1382:
1370:
131:
105:
14528:Introduction to Quantum Theory at Quantiki.
14413:
14003:
13906:QED: The Strange Theory of Light and Matter
12728:
12281:
12004:Rovelli, Carlo; Vidotto, Francesca (2014).
11581:Introduction to Quantum Information Science
11192:Quantum Computation and Quantum Information
10887:Zhang, Ruiqin; Deng, Conghao (1993-01-01).
10823:Greenstein, George; Zajonc, Arthur (2006).
10713:
10505:
10342:
9375:
3415:{\displaystyle \psi _{A}\otimes \psi _{B}.}
1301:
1076:specially developed mathematical formalisms
15483:
15469:
14691:
14677:
14619:Imperial College Quantum Mechanics Course.
14248:Quantum Mechanics: Non-Relativistic Theory
13872:
13453:
13451:
12787:
12248:
11368:
11115:
11079:Quantum Mechanics: Non-Relativistic Theory
10454:
10448:
10396:
10394:
10392:
9977:
9819:that classical physics cannot account for
9485:. In 1935, Einstein and his collaborators
8999:
8985:
6872:, and in studies of quantum entanglement.
6856:Schematic of a Mach–Zehnder interferometer
5151:the previous equation is evocative of the
5082:With the differential operator defined by
3718:by a state vector. One can instead define
3425:Not all states in the joint Hilbert space
1127:. For example, a quantum particle like an
910:
896:
13755:
13677:
13646:
13627:
13614:
13339:
13250:"Max Planck: the reluctant revolutionary"
12878:
12748:
12646:
12631:
12562:
12462:
12403:
12342:
12233:
12160:
12074:
11846:
11804:Hawking, Stephen; Penrose, Roger (2010).
11710:
11536:
11514:
11481:
11459:
11329:
11266:
11157:, Cambridge: Cambridge University Press,
11150:
10977:
10959:
10886:
10816:
10722:
10688:
10656:
10629:
10537:
10481:
10309:
10248:
10118:
10116:
10114:
10112:
9329:, and other properties determined by the
9130:. Quantum electrodynamics is, along with
7101:. In order to respect the postulate that
6904:{\displaystyle \psi \in \mathbb {C} ^{2}}
6891:
6408:
4500:which is a superposition of all possible
1517:
1487:
14469:. Language Research Foundation, Boston.
13732:
13694:
13457:
13391:
13366:
13311:
13024:
12982:
12859:
12830:
12680:
12614:"On the Einstein Podolsky Rosen paradox"
12593:
12436:
12201:
12060:
11975:
11233:Quantum Computing: A Gentle Introduction
10855:
10728:"Experiment Reaffirms Quantum Weirdness"
10608:
10606:
10367:
10234:
9774:
9564:
9389:", give rise to the reality we perceive?
6851:
6769:and the corresponding energy levels are
6232:
4944:
4202:
2414:
2285:
1278:hidden variables, then the results of a
38:
14225:, 3rd ed. Singapore: World Scientific.
14093:
13814:
13582:
13576:
13448:
13353:The Collected Papers of Albert Einstein
13347:
13275:
12734:
12665:
12030:
11915:
11692:Tipler, Paul; Llewellyn, Ralph (2008).
11062:
10671:"Death by experiment for local realism"
10665:
10511:
10389:
10055:
9787:was the fifth world physics conference.
9631:and others built upon to establish the
9153:of the hydrogen atom using a classical
9020:and that observables of the system are
3349:and the state for the second system is
2478:do not commute, but rather satisfy the
1458:of a Hilbert space, usually called the
1095:raises philosophical problems, such as
14:
15933:
15681:Atomic, molecular, and optical physics
14533:Quantum Physics Made Relatively Simple
13943:can be passed over on a first reading.
13107:
12961:
12833:"The Born rule and its interpretation"
12702:; Van Vechten, D.; Graham, N. (1973).
12596:Albert Einstein: Philosopher-Scientist
12501:
12495:
12113:
11685:
11575:
11569:
11044:; Diu, Bernard; Laloë, Franck (2005).
10560:
10156:
10109:
9894:and the flow of electrons in computer
9106:Special relativity and electrodynamics
9102:under the rules of quantum mechanics.
9011:
8031:are dependent upon quantum mechanics.
6222:
2762:The uncertainty principle states that
2016:, the observable corresponding to the
1572:: in the simplest case the eigenvalue
1159:, which is often illustrated with the
15464:
14698:
14672:
13953:. Cambridge University Press: 110–76.
13244:
13238:
12536:
11721:
11668:"What Is Quantum Mechanics Good for?"
11611:
11465:
11256:
11250:
11222:
11220:
11177:
11175:
11173:
11000:
10789:
10612:
10603:
8090:Relation to other scientific theories
6253:(called the wave function), with the
5830:{\displaystyle \psi (L)=0=C\sin(kL),}
4211:moving in one dimension in free space
1082:provides information, in the form of
14553:PHYS 201: Fundamentals of Physics II
13700:
13424:
13216:
13135:
12865:
12608:
12207:
11751:
11728:Quantum Theory: Concepts and Methods
11410:Quantum Mechanics and Path Integrals
11154:Measuring the Quantum State of Light
10753:
10434:. US: World Scientific. p. 14.
10061:
9957:
9370:Interpretations of quantum mechanics
4940:
3865:{\displaystyle U(t)=e^{-iHt/\hbar }}
3587:are both possible states for system
3513:are both possible states for system
2210:{\displaystyle U(t)=e^{-iHt/\hbar }}
1889:interpretations of quantum mechanics
1808:, in the non-degenerate case, or to
1207:: a particle that goes up against a
14661:Stanford Encyclopedia of Philosophy
14642:Stanford Encyclopedia of Philosophy
13777:
13506:Stanford Encyclopedia of Philosophy
12969:Stanford Encyclopedia of Philosophy
12955:
12673:Stanford Encyclopedia of Philosophy
12653:Stanford Encyclopedia of Philosophy
12633:10.1103/PhysicsPhysiqueFizika.1.195
12383:
10763:University of California, Riverside
10596:Stanford Encyclopedia of Philosophy
10585:
9393:(more unsolved problems in physics)
7908:by estimating these probabilities.
7212:are modelled as the unitary matrix
4641:{\displaystyle {\hat {\psi }}(k,0)}
4025:. Under the evolution generated by
3452:{\displaystyle {\mathcal {H}}_{AB}}
1497:{\displaystyle L^{2}(\mathbb {C} )}
1220:in stars, and applications such as
24:
14630:
13845:
13787:California Institute of Technology
13767:from the original on Jan 20, 2024.
13402:10.1016/b978-0-08-012102-4.50018-8
11665:
11217:
11170:
8233:
7980:
7977:
7879:
7876:
7757:
7754:
7617:
7614:
7587:
7584:
7551:
7548:
7429:
7426:
7396:
7393:
6207:, which furnishes a model for the
4477:
4389:
4384:
3759:) and wave mechanics (invented by
3435:
3295:
3278:
3258:
3238:{\displaystyle {\mathcal {H}}_{B}}
3224:
3207:{\displaystyle {\mathcal {H}}_{A}}
3193:
3167:Composite systems and entanglement
3110:
3106:
1937:
1933:
1504:, while the Hilbert space for the
1348:
442:Sum-over-histories (path integral)
128:
102:
58:Part of a series of articles about
25:
15952:
14649:
14521:J. O'Connor and E. F. Robertson:
14515:
14182:Introduction to Quantum Mechanics
14147:. Vol. 1–3. Addison-Wesley.
13652:"Max Born and the quantum theory"
13497:"Bohr's Correspondence Principle"
13466:. Pergamon Press. pp. 3–75.
13198:from the original on Oct 26, 2012
12537:Sauer, Tilman (1 December 2007).
11895:"The Nobel Prize in Physics 1979"
10799:. Ballantine Books. p. 249.
10488:Introduction to Quantum Mechanics
10428:MĂĽller-Kirsten, H. J. W. (2006).
9886:, and describes the behaviors of
9691:) would be proportional to their
9528:, formulated in 1956, holds that
9457:, himself one of the founders of
8155:
8003:Applications of quantum mechanics
6792:
6537:
6500:
6441:
5120:
4588:
3941:, it follows that any observable
3857:
3725:positive operator-valued measures
3101:
3044:
2531:
2202:
2125:
2054:
1928:
1911:Time evolution of a quantum state
1801:{\displaystyle {\vec {\lambda }}}
1755:After the measurement, if result
1672:{\displaystyle {\vec {\lambda }}}
1447:{\displaystyle e^{i\alpha }\psi }
1090:Overview and fundamental concepts
79:
34:Introduction to quantum mechanics
15918:
15906:
15894:
15882:
15870:
15858:
15846:
15834:
15822:
15445:
15444:
14401:: Its Story and Its Significance
14035:The Physics of Quantum Mechanics
13808:
13793:from the original on 15 Dec 2016
13726:
13635:International J. of Theor. Phys.
12598:. Open Court Publishing Company.
12161:Camilleri, Kristian (May 2009).
10776:solve quantum mechanics problems
10461:. US: Springer. pp. 75–76.
9763:that successfully predicted the
9547:was developed some years later.
6608:{\displaystyle n=0,1,2,\ldots .}
4192:
4105:is conserved by evolution under
4085:will be conserved. Moreover, if
3779:Symmetries and conservation laws
3739:Equivalence between formulations
1905:measurement in quantum mechanics
1530:{\displaystyle \mathbb {C} ^{2}}
15802:Timeline of physics discoveries
14652:"Measurement in Quantum Theory"
14523:A history of quantum mechanics.
14419:Principles of Quantum Mechanics
14367:Understanding Quantum Mechanics
14333:. Wiley, John & Sons, Inc.
14185:(2nd ed.). Prentice Hall.
14144:The Feynman Lectures on Physics
14079:, R. Neill Graham, eds., 1973.
13757:10.1090/s0002-9904-1958-10206-2
13640:
13538:
13513:
13488:
13418:
13360:
13305:
13279:(2009). "Bohr and the Photon".
13269:
13210:
13192:"Quantum – Definition and More"
13184:
13156:
13129:
13112:, Academic Press, p. 183,
13101:
13077:The Feynman Lectures on Physics
13025:Scheider, Walter (April 1986).
13018:
12895:
12824:
12659:
12640:
12602:
12587:
12530:
12275:
12242:
12154:
12107:
12054:
12024:
11997:
11981:A First Course in String Theory
11969:
11940:
11887:
11824:
11797:
11775:
11745:
11659:
11439:A Textbook of Quantum Mechanics
11426:
11401:
11362:
11309:
10994:
10935:
10880:
10849:
10783:
10759:"How to Learn Math and Physics"
10747:
10579:
10554:
10475:
10126:The Feynman Lectures on Physics
10034:
10014:
9985:the Feynman Lectures on Physics
9964:
9607:During the early 19th century,
9499:Einstein–Podolsky–Rosen paradox
7996:
6554:
5996:
5995:
5944:must be an integer multiple of
5339:
5338:
5153:classic kinetic energy analogue
2680:and likewise for the momentum:
929:that describes the behavior of
15394:Relativistic quantum mechanics
14447:. Princeton University Press.
14370:. Princeton University Press.
14277:Introductory Quantum Mechanics
14250:. Vol. 3 (3rd ed.).
14205:A standard undergraduate text.
13929:Sneaking a Look at God's Cards
13909:, Princeton University Press.
13521:"Semi-classical approximation"
13396:. Elsevier. pp. 167–183.
13194:. Merriam-Webster Dictionary.
12996:. Cambridge University Press.
12010:. Cambridge University Press.
11983:. Cambridge University Press.
11955:. Cambridge University Press.
11865:10.1103/PhysRevLett.109.111807
11810:. Princeton University Press.
11638:10.1103/PhysRevLett.101.250001
11082:. Vol. 3 (3rd ed.).
10328:10.1103/PhysRevLett.130.071801
10289:
10237:Reports on Progress in Physics
10228:
10185:
10150:
9926:Einstein's thought experiments
9278:Relation to general relativity
9207:
9179:
8529:Atomic, molecular, and optical
8194:
8188:
8174:
8143:
8137:
8123:
8116:
7943:
7937:
7928:
7922:
7847:
7803:
7796:
7790:
7725:
7681:
7674:
7668:
7628:
7611:
7598:
7581:
7179:
7170:
7156:
7147:
7094:{\displaystyle \alpha ,\beta }
6670:
6660:
6654:
6648:
6385:
6379:
6300:
6294:
5902:
5893:
5821:
5812:
5791:
5785:
5698:
5692:
5677:
5671:
5650:
5644:
5464:
5455:
5440:
5431:
5416:
5410:
5288:
5282:
5187:
5099:
4822:
4810:
4804:
4715:
4703:
4670:
4658:
4635:
4623:
4617:
4560:
4518:
4471:
4429:
4418:
4406:
4400:
4355:
4343:
3908:
3902:
3829:
3823:
2993:
2981:
2887:
2875:
2522:
2516:
2501:
2492:
2480:canonical commutation relation
2462:
2433:
2348:eigenstates of the Hamiltonian
2270:
2264:
2241:
2235:
2174:
2168:
2139:
2133:
2097:
2091:
1973:
1967:
1955:
1949:
1792:
1663:
1630:
1613:
1600:
1537:with the usual inner product.
1491:
1483:
1356:{\displaystyle {\mathcal {H}}}
1230:tunnel field-effect transistor
1222:scanning tunnelling microscopy
1109:shown to agree with experiment
937:. It is the foundation of all
592:Relativistic quantum mechanics
124:
117:
98:
13:
1:
15372:Quantum statistical mechanics
15149:Quantum differential calculus
15071:Delayed-choice quantum eraser
14854:Symmetry in quantum mechanics
14614:Examples in Quantum Mechanics
13170:. New York: Springer-Verlag.
13142:. CRC Press. pp. 47–57.
12966:. In Zalta, Edward N. (ed.).
12840:Compendium of Quantum Physics
12767:10.1016/S1355-2198(03)00036-4
12670:. In Zalta, Edward N. (ed.).
12114:Howard, Don (December 2004).
12037:The Character of Physical Law
11348:10.1016/j.physrep.2019.10.001
11230:; Polak, Wolfgang H. (2011).
10593:. In Zalta, Edward N. (ed.).
10561:Phifer, Arnold (2012-03-27).
10267:10.1088/0034-4885/78/2/023901
10048:
10022:macroscopic quantum phenomena
9936:Macroscopic quantum phenomena
9301:One proposal for doing so is
9058:classical harmonic oscillator
7970:, independently of the phase
7963:{\displaystyle p(u)=p(l)=1/2}
7505:, and so end up in the state
7340:{\displaystyle i/{\sqrt {2}}}
7308:{\displaystyle 1/{\sqrt {2}}}
6866:delayed choice quantum eraser
6217:scanning tunneling microscopy
6205:rectangular potential barrier
5880:has norm 1. Therefore, since
3892:. From this relation between
632:Quantum statistical mechanics
14463:Transnational College of Lex
12524:10.1016/0039-3681(85)90001-9
12361:10.1103/RevModPhys.88.015005
11807:The Nature of Space and Time
11785:. Academic.brooklyn.cuny.edu
11441:. Tata McGraw-Hill. p.
11011:10.1007/978-3-030-73893-8_12
10979:10.21468/SciPostPhys.6.4.040
10530:10.2174/13852728113179990083
9813:Hendrika Johanna van Leeuwen
9801:semi-classical approximation
9557:History of quantum mechanics
9541:Relational quantum mechanics
8072:medical and research imaging
7986:{\displaystyle \Delta \Phi }
7485:, and another beam splitter
7435:{\displaystyle \Delta \Phi }
1741:{\displaystyle P_{\lambda }}
1133:probability density function
7:
15766:Quantum information science
15174:Quantum stochastic calculus
15164:Quantum measurement problem
15086:Mach–Zehnder interferometer
14037:. Oxford University Press.
13951:Boojums All the Way Through
13860:Primer of Quantum Mechanics
13657:American Journal of Physics
13525:Encyclopedia of Mathematics
13289:10.1007/978-1-4020-9107-0_5
13224:. Oxford: Clarendon Press.
12873:. Oxford University Press.
12737:Stud. Hist. Phil. Mod. Phys
12706:; Graham, R. Neill (eds.).
12647:Goldstein, Sheldon (2017).
12573:10.1016/j.shpsb.2007.03.002
12422:10.1016/j.shpsb.2013.04.004
12249:Heisenberg, Werner (1971).
11583:. Oxford University Press.
10455:Plotnitsky, Arkady (2012).
10159:American Journal of Physics
9913:
9880:quantum information science
9791:This phase is known as the
9755:and used it to explain the
9376:Unsolved problem in physics
9128:electromagnetic interaction
9050:quantum harmonic oscillator
6870:Elitzur–Vaidman bomb tester
6862:Mach–Zehnder interferometer
6848:Mach–Zehnder interferometer
6241:(i.e. a ball attached to a
6229:Quantum harmonic oscillator
5246:in this case having energy
4187:
3156:{\displaystyle -\hbar ^{2}}
3129:, and in particular in the
2378:quantum harmonic oscillator
2283:will be at any later time.
1875:, in the general case. The
1034:'s solution in 1900 to the
955:quantum information science
602:Quantum information science
10:
15957:
15597:Classical electromagnetism
14535:: three video lectures by
14297:. London: Pergamon Press.
14168:, F. Weinert, eds., 2009.
12712:Princeton University Press
12093:10.1103/PhysRevA.85.062116
12040:. MIT Press. p. 129.
11555:10.1103/PhysRevB.82.155303
11259:Quantum Information Theory
9884:periodic table of elements
9761:model of the hydrogen atom
9674:problem was discovered by
9554:
9550:
9526:many-worlds interpretation
9367:
9364:Philosophical implications
9340:Another popular theory is
8076:magnetic resonance imaging
8000:
6226:
5914:{\displaystyle \sin(kL)=0}
4952:
4676:{\displaystyle \psi (x,0)}
4196:
3782:
2865:of these two operators is
2471:{\displaystyle {\hat {P}}}
2442:{\displaystyle {\hat {X}}}
1897:many-worlds interpretation
1305:
933:at and below the scale of
31:
15774:
15711:
15639:
15555:
15527:
15499:
15440:
15402:
15354:
15234:Quantum complexity theory
15212:Quantum cellular automata
15187:
15119:
15053:
14966:
14930:
14917:Path integral formulation
14884:
14749:
14706:
14141:; Sands, Matthew (1965).
14119:10.1103/RevModPhys.29.454
14099:Reviews of Modern Physics
14033:; Skinner, David (2008).
13815:Packard, Richard (2006).
13373:Physikalische Zeitschrift
13166:; Rechenberg, H. (1982).
12933:10.1007/s10701-009-9326-5
12666:Barrett, Jeffrey (2018).
12481:10.1007/s10701-009-9347-0
12331:Reviews of Modern Physics
12179:10.1162/posc.2009.17.1.26
11612:Cohen, Marvin L. (2008).
11277:10.1017/9781316809976.001
11138:10.1103/RevModPhys.42.358
11118:Reviews of Modern Physics
10856:Weinberg, Steven (2010).
10518:Current Organic Chemistry
10404:Quantum Physics for Poets
10214:10.1038/s41567-019-0663-9
9862:with greater emphasis on
9727:{\displaystyle E=h\nu \ }
9435:Copenhagen interpretation
9290:is an important issue in
9063:Complications arise with
4597:{\displaystyle p=\hbar k}
3769:path integral formulation
3580:{\displaystyle \phi _{B}}
3553:{\displaystyle \psi _{B}}
3506:{\displaystyle \phi _{A}}
3479:{\displaystyle \psi _{A}}
3369:{\displaystyle \psi _{B}}
3342:{\displaystyle \psi _{A}}
2540:{\displaystyle =i\hbar .}
2221:. This time evolution is
1895:" (see, for example, the
1259:, as demonstrated by the
15703:Condensed matter physics
15301:Quantum machine learning
15281:Quantum key distribution
15271:Quantum image processing
15261:Quantum error correction
15111:Wheeler's delayed choice
14443:Einstein and the Quantum
13637:, Vol. 20, No. 7 (1981).
13551:Modern Quantum Mechanics
13341:10.1002/andp.19053220607
12962:Healey, Richard (2016).
12831:Landsman, N. P. (2008).
11951:; Schwarz, John (2007).
11500:10.1103/PhysRevA.59.1615
10862:. Random House. p.
10407:. US: Prometheus Books.
10026:Bose–Einstein condensate
9972:Dirac delta distribution
9951:Two-state quantum system
9946:Regularization (physics)
9623:of matter, an idea that
9305:, which posits that the
9226:Stern–Gerlach experiment
9027:correspondence principle
8286:Einstein field equations
3775:in classical mechanics.
3720:reduced density matrices
3050:{\displaystyle i/\hbar }
2276:{\displaystyle \psi (t)}
2247:{\displaystyle \psi (0)}
2073:correspondence principle
1768:{\displaystyle \lambda }
1585:{\displaystyle \lambda }
1460:complex projective space
1302:Mathematical formulation
1261:no-communication theorem
1249:quantum key distribution
1188:. In addition to light,
1123:, named after physicist
637:Quantum machine learning
390:Wheeler's delayed-choice
15217:Quantum finite automata
14062:(2nd ed.). Wiley.
13733:Van Hove, Leon (1958).
12619:Physics Physique Fizika
12167:Perspectives on Science
11834:Physical Review Letters
11618:Physical Review Letters
11382:Acta Physica Polonica B
11257:Wilde, Mark M. (2017).
11151:Leonhardt, Ulf (1997),
11042:Cohen-Tannoudji, Claude
10298:Physical Review Letters
9941:Phase-space formulation
9633:kinetic theory of gases
9124:quantum electrodynamics
9033:in the regime of large
8589:Theories of gravitation
8504:Electroweak interaction
8048:superconducting magnets
6237:Some trajectories of a
5605:must be zero. Thus, at
4893:We see that as we make
4173:) mechanics: for every
2914:{\displaystyle =AB-BA,}
2060:{\displaystyle i\hbar }
1105:quantum electrodynamics
347:Leggett–Garg inequality
15787:Nobel Prize in Physics
15649:Relativistic mechanics
15321:Quantum neural network
14492:Veltman, Martinus J.G.
14314:. Dover Publications.
14293:Gunther Ludwig, 1968.
14015:. Dover Publications.
13462:The Old Quantum Theory
13394:The Old Quantum Theory
12912:Foundations of Physics
12790:Foundations of Physics
12450:Foundations of Physics
10913:10.1103/PhysRevA.47.71
10591:"Quantum entanglement"
10067:Zeitschrift fĂĽr Physik
9788:
9728:
9653:Johann Wilhelm Hittorf
9598:double-slit experiment
9573:
9387:wave function collapse
9248:quantum chromodynamics
9215:
9083:quantum superpositions
8274:
8204:
7987:
7964:
7895:
7773:
7646:
7499:
7479:
7459:
7436:
7413:
7341:
7309:
7277:
7199:
7133:
7095:
7069:
7017:
6961:
6905:
6857:
6831:
6760:
6609:
6559:
6346:
6278:
6257:shown in blue and the
6190:
6054:
6031:
5958:
5938:
5915:
5874:
5854:
5831:
5766:
5740:
5711:
5625:
5599:
5579:
5553:
5527:
5507:
5475:
5384:
5260:
5240:
5217:
5142:
5073:
4982:
4950:
4927:
4907:
4884:
4779:
4677:
4642:
4598:
4569:
4491:
4321:
4212:
4159:
4139:
4119:
4099:
4079:
4059:
4039:
4019:
3999:
3975:
3955:
3935:
3915:
3886:
3866:
3803:
3694:
3601:
3581:
3554:
3527:
3507:
3480:
3453:
3416:
3370:
3343:
3313:
3239:
3208:
3157:
3123:
3082:
3051:
3015:
2915:
2855:
2835:
2812:
2753:
2671:
2584:
2564:
2541:
2472:
2449:and momentum operator
2443:
2343:
2277:
2248:
2211:
2149:
2061:
2034:
2033:{\displaystyle \hbar }
2006:
1983:
1893:wave function collapse
1869:
1802:
1769:
1742:
1715:
1673:
1644:
1586:
1531:
1498:
1448:
1415:
1395:
1357:
1326:
1292:differential equations
1161:double-slit experiment
1084:probability amplitudes
1046:, which explained the
138:
52:
15792:Philosophy of physics
15346:Quantum teleportation
14874:Wave–particle duality
14567:– an online textbook.
14559:, Yale OpenCourseware
14227:Draft of 4th edition.
13458:ter Haar, D. (1967).
13312:Einstein, A. (1905).
13136:Dahl, Per F. (1997).
12120:Philosophy of Science
10618:OSA Century of Optics
9778:
9729:
9596:described the famous
9568:
9483:statistical mechanics
9471:Bohr–Einstein debates
9408:wavefunction collapse
9262:), by the physicists
9216:
9143:electromagnetic field
9116:Klein–Gordon equation
8467:Philosophy of physics
8275:
8205:
8052:light-emitting diodes
7988:
7965:
7896:
7774:
7647:
7500:
7480:
7460:
7437:
7414:
7342:
7310:
7278:
7200:
7134:
7096:
7070:
7018:
6967:and the "upper" path
6962:
6906:
6855:
6832:
6761:
6610:
6560:
6347:
6236:
6200:finite potential well
6191:
6055:
6032:
5959:
5939:
5916:
5875:
5873:{\displaystyle \psi }
5855:
5832:
5767:
5741:
5712:
5626:
5600:
5598:{\displaystyle \psi }
5580:
5554:
5528:
5508:
5476:
5385:
5261:
5241:
5239:{\displaystyle \psi }
5218:
5143:
5074:
4983:
4948:
4928:
4908:
4885:
4780:
4678:
4643:
4599:
4570:
4492:
4322:
4206:
4160:
4140:
4120:
4100:
4080:
4060:
4040:
4020:
4000:
3976:
3956:
3936:
3916:
3887:
3867:
3804:
3745:transformation theory
3695:
3602:
3582:
3555:
3528:
3508:
3481:
3454:
3417:
3371:
3344:
3314:
3240:
3209:
3158:
3124:
3083:
3081:{\displaystyle p_{i}}
3052:
3016:
2916:
2856:
2836:
2813:
2754:
2672:
2585:
2565:
2542:
2473:
2444:
2415:Uncertainty principle
2322:and thus, a definite
2292:Probability densities
2289:
2278:
2249:
2212:
2150:
2069:classical Hamiltonian
2062:
2035:
2007:
1984:
1881:Bohr–Einstein debates
1870:
1803:
1770:
1743:
1716:
1674:
1645:
1587:
1566:quantum superposition
1560:, and the associated
1532:
1499:
1449:
1416:
1414:{\displaystyle \psi }
1396:
1358:
1327:
1325:{\displaystyle \psi }
1186:wave–particle duality
1165:coherent light source
1145:uncertainty principle
1021:uncertainty principle
1017:wave–particle duality
983:Quantum systems have
970:(optical) microscopic
332:Elitzur–Vaidman
322:Davisson–Germer
139:
42:
15751:Mathematical physics
15377:Quantum field theory
15306:Quantum metamaterial
15251:Quantum cryptography
14981:Consistent histories
13919:quantum field theory
12993:Principles of Optics
12903:Van Fraassen, Bas C.
12443:Harrigan, Nicholas;
10624:. pp. 320–323.
10512:Trixler, F. (2013).
9811:of solids, Bohr and
9769:electromagnetic wave
9757:photoelectric effect
9706:
9672:black-body radiation
9602:wave theory of light
9475:action at a distance
9342:loop quantum gravity
9307:point-like particles
9296:Theory of Everything
9240:strong nuclear force
9157:
8599:Theory of everything
8477:Quantum field theory
8214:
8107:
7974:
7916:
7784:
7662:
7512:
7489:
7469:
7449:
7423:
7351:
7319:
7287:
7216:
7143:
7105:
7079:
7027:
6971:
6915:
6880:
6776:
6635:
6572:
6366:
6288:
6066:
6044:
5971:
5957:{\displaystyle \pi }
5948:
5925:
5884:
5864:
5844:
5779:
5750:
5724:
5638:
5609:
5589:
5563:
5537:
5517:
5506:{\displaystyle C,D,}
5488:
5404:
5276:
5250:
5230:
5162:
5089:
4995:
4972:
4917:
4897:
4795:
4697:
4652:
4608:
4579:
4507:
4337:
4223:
4149:
4129:
4109:
4089:
4069:
4049:
4029:
4009:
3989:
3965:
3945:
3925:
3914:{\displaystyle U(t)}
3896:
3876:
3817:
3793:
3614:
3591:
3564:
3537:
3517:
3490:
3463:
3429:
3383:
3353:
3326:
3252:
3218:
3187:
3137:
3092:
3065:
3033:
2931:
2872:
2845:
2825:
2769:
2687:
2601:
2574:
2554:
2489:
2453:
2424:
2340:resonant frequencies
2258:
2229:
2162:
2085:
2048:
2024:
1996:
1922:
1812:
1783:
1759:
1725:
1683:
1654:
1596:
1576:
1512:
1470:
1425:
1405:
1367:
1343:
1316:
1237:quantum entanglement
1157:quantum interference
1138:Schrödinger equation
1048:photoelectric effect
1036:black-body radiation
947:quantum field theory
597:Quantum field theory
509:Consistent histories
146:Schrödinger equation
73:
15726:Atmospheric physics
15565:Classical mechanics
15493:branches of physics
15362:Quantum fluctuation
15331:Quantum programming
15291:Quantum logic gates
15276:Quantum information
15256:Quantum electronics
14731:Classical mechanics
14633:"Quantum Mechanics"
14177:Griffiths, David J.
14139:Leighton, Robert B.
14135:Feynman, Richard P.
14111:1957RvMP...29..454E
13925:Ghirardi, GianCarlo
13829:on 25 November 2015
13750:(3): Part 2:95–99.
13670:2005AmJPh..73..999B
13386:1917PhyZ...18..121E
13332:1905AnP...322..132E
13248:(1 December 2000).
13045:1986PhTea..24..217S
13032:The Physics Teacher
12925:2010FoPh...40..390V
12889:2009arXiv0905.0624K
12802:1973FoPh....3..229B
12759:2003SHPMP..34..415W
12649:"Bohmian Mechanics"
12612:(1 November 1964).
12555:2007SHPMP..38..879S
12516:1985SHPSA..16..171H
12473:2010FoPh...40..125H
12445:Spekkens, Robert W.
12414:2013SHPMP..44..222S
12353:2016RvMP...88a5005M
12284:Naturwissenschaften
12226:1928Natur.121..580B
12085:2012PhRvA..85f2116W
11919:(10 October 2022).
11857:2012PhRvL.109k1807A
11783:"Atomic Properties"
11672:Scientific American
11630:2008PhRvL.101y0001C
11547:2010PhRvB..82o5303H
11492:1999PhRvA..59.1615P
11340:2019PhR...831....1S
11228:Rieffel, Eleanor G.
11183:Nielsen, Michael A.
11130:1970RvMP...42..358B
10970:2019ScPP....6...40L
10905:1993PhRvA..47...71Z
10726:(7 February 2017).
10690:10.1038/nature15631
10640:2013arXiv1302.1864C
10622:The Optical Society
10483:Griffiths, David J.
10320:2023PhRvL.130g1801F
10259:2015RPPh...78b3901B
10206:2019NatPh..15.1242F
10171:2014AmJPh..82..896J
10079:1926ZPhy...37..863B
9997:integrated circuits
9872:quantum electronics
9625:James Clerk Maxwell
9619:lent weight to the
9512:several experiments
9416:quantum nonlocality
9412:measurement problem
9075:Quantum decoherence
9031:classical mechanics
9022:Hermitian operators
9012:Classical mechanics
8549:Non-linear dynamics
8487:Quantum computation
8482:Quantum information
8356:Satyendra Nath Bose
8080:electron microscopy
8058:and the laser, the
8025:Solid-state physics
6627:Hermite polynomials
6247:classical mechanics
6239:harmonic oscillator
6223:Harmonic oscillator
6040:This constraint on
5765:{\displaystyle x=L}
5739:{\displaystyle D=0}
5624:{\displaystyle x=0}
5578:{\displaystyle x=L}
5552:{\displaystyle x=0}
5203:
4393:
4065:that commutes with
3961:that commutes with
3733:quantum decoherence
2405:perturbation theory
2020:of the system, and
1885:thought experiments
1750:probability density
385:Stern–Gerlach
182:Classical mechanics
15877:History of science
15782:History of physics
15415:in popular culture
15197:Quantum algorithms
15045:Von Neumann–Wigner
15025:Objective collapse
14736:Old quantum theory
14571:MIT OpenCourseWare
14509:This Quantum World
14279:. Addison-Wesley.
14273:Liboff, Richard L.
14232:2008-06-15 at the
13319:Annalen der Physik
12810:10.1007/BF00708440
12296:10.1007/BF01491987
11926:The New York Times
11897:. Nobel Foundation
11762:Nautilus Quarterly
11370:Rechenberg, Helmut
10724:Wolchover, Natalie
10087:10.1007/BF01397477
9983:See, for example,
9902:phenomena such as
9793:old quantum theory
9789:
9724:
9583:Christiaan Huygens
9574:
9495:thought experiment
9292:physical cosmology
9284:empirical evidence
9260:electroweak theory
9244:weak nuclear force
9211:
9210:
9132:general relativity
9112:special relativity
8574:General relativity
8569:Special relativity
8509:Strong interaction
8270:
8200:
7983:
7960:
7891:
7769:
7642:
7633:
7495:
7475:
7465:, a phase shifter
7455:
7432:
7409:
7403:
7337:
7305:
7273:
7267:
7195:
7129:
7091:
7065:
7013:
7007:
6957:
6951:
6901:
6858:
6827:
6756:
6605:
6555:
6342:
6279:
6186:
6050:
6027:
5954:
5937:{\displaystyle kL}
5934:
5911:
5870:
5850:
5827:
5762:
5736:
5707:
5621:
5595:
5575:
5549:
5523:
5503:
5471:
5380:
5256:
5236:
5213:
5180:
5138:
5069:
4978:
4951:
4923:
4903:
4880:
4775:
4673:
4638:
4594:
4565:
4487:
4376:
4317:
4213:
4155:
4135:
4115:
4095:
4075:
4055:
4035:
4015:
3995:
3971:
3951:
3931:
3911:
3882:
3872:for each value of
3862:
3799:
3690:
3629:
3597:
3577:
3550:
3523:
3503:
3476:
3449:
3412:
3366:
3339:
3309:
3235:
3204:
3153:
3119:
3078:
3047:
3026:Fourier transforms
3011:
2967:
2911:
2851:
2831:
2808:
2749:
2667:
2662:
2592:standard deviation
2580:
2560:
2537:
2468:
2439:
2344:
2273:
2244:
2207:
2145:
2057:
2030:
2002:
1979:
1865:
1798:
1765:
1738:
1711:
1669:
1640:
1582:
1527:
1494:
1444:
1411:
1391:
1353:
1322:
1205:quantum tunnelling
1052:old quantum theory
1026:Quantum mechanics
951:quantum technology
573:Von Neumann–Wigner
553:Objective-collapse
352:Mach–Zehnder
342:Leggett inequality
337:Franck–Hertz
187:Old quantum theory
134:
53:
15941:Quantum mechanics
15810:
15809:
15797:Physics education
15746:Materials science
15713:Interdisciplinary
15671:Quantum mechanics
15458:
15457:
15432:Quantum mysticism
15410:Schrödinger's cat
15341:Quantum simulator
15311:Quantum metrology
15239:Quantum computing
15202:Quantum amplifier
15179:Quantum spacetime
15144:Quantum cosmology
15134:Quantum chemistry
14849:Scattering theory
14797:Zero-point energy
14792:Degenerate levels
14700:Quantum mechanics
14557:Ramamurti Shankar
14549:Quantum Cook Book
14476:978-0-9643504-1-0
14454:978-0-691-13968-5
14437:Stone, A. Douglas
14428:978-0-306-44790-7
14377:978-0-691-00435-8
14352:Quantum Mechanics
14340:978-0-471-88702-7
14331:Quantum Mechanics
14327:Merzbacher, Eugen
14286:978-0-8053-8714-8
14261:978-0-08-020940-1
14192:978-0-13-111892-8
14154:978-0-7382-0008-8
14069:978-0-471-87373-0
14052:Eisberg, Robert;
14044:978-0-19-968857-9
14022:978-0-486-65969-5
13996:978-0-674-03541-6
13981:Bernstein, Jeremy
13893:978-1-84614-432-5
13679:10.1119/1.2060717
13650:(November 2005).
13648:Bernstein, Jeremy
13561:978-1-292-02410-3
13473:978-0-08-012101-7
13411:978-0-08-012102-4
13298:978-1-4020-9106-3
13231:978-0-19-502383-1
13177:978-0-387-90642-3
13149:978-0-7503-0453-5
13119:978-0-08-057733-3
13087:978-0-201-50064-6
13053:10.1119/1.2341987
12907:"Rovelli's World"
12849:978-3-540-70622-9
12260:978-0-04-530016-7
12220:(3050): 580–590.
12208:Bohr, N. (1928).
12063:Physical Review A
12017:978-1-316-14811-2
11990:978-0-521-88032-9
11962:978-0-521-86069-7
11817:978-1-4008-3474-7
11703:978-0-7167-7550-8
11590:978-0-19-921570-6
11524:Physical Review B
11469:Physical Review A
11452:978-0-07-096510-2
11419:978-0-486-47722-0
11286:978-1-107-17616-4
11243:978-0-262-01506-6
11202:978-1-107-00217-3
11093:978-0-08-020940-1
11046:Quantum Mechanics
11020:978-3-030-73892-1
10893:Physical Review A
10873:978-1-4070-6396-6
10838:978-0-7637-2470-2
10757:(20 March 2020).
10683:(7575): 649–650.
10649:978-1-943580-04-0
10614:Caves, Carlton M.
10524:(16): 1758–1770.
10491:. Prentice Hall.
10468:978-1-4614-4517-3
10441:978-981-256-691-1
10414:978-1-61614-281-0
10382:978-0-262-04613-8
10200:(12): 1242–1245.
10179:10.1119/1.4878358
10136:978-0-201-50064-6
9995:, pp. 14–11 ff),
9958:Explanatory notes
9858:, Paul Dirac and
9849:Solvay Conference
9825:Arnold Sommerfeld
9781:Solvay Conference
9744:Planck's constant
9723:
9683:derives from the
9518:Bohmian mechanics
9508:Bell inequalities
9497:later termed the
9402:debates and many
9346:quantum spacetime
9238:theories for the
9230:charged particles
9224:. Likewise, in a
9222:Coulomb potential
9009:
9008:
8472:Quantum mechanics
8346:Ernest Rutherford
8326:Erwin Schrödinger
8321:Werner Heisenberg
8171:
8119:
8056:optical amplifier
8044:quantum computing
8036:quantum chemistry
8029:materials science
8009:classical methods
7886:
7764:
7498:{\displaystyle B}
7478:{\displaystyle P}
7458:{\displaystyle B}
7335:
7303:
7235:
7234:
6817:
6723:
6541:
6540:
6504:
6445:
6417:
6416:
6314:
6267:stationary states
6209:quantum tunneling
6181:
6134:
6053:{\displaystyle k}
5993:
5853:{\displaystyle C}
5526:{\displaystyle k}
5378:
5259:{\displaystyle E}
5190:
5178:
5136:
5102:
5055:
5021:
4981:{\displaystyle x}
4955:Particle in a box
4941:Particle in a box
4926:{\displaystyle a}
4906:{\displaystyle a}
4873:
4843:
4837:
4807:
4771:
4741:
4740:
4620:
4555:
4466:
4403:
4374:
4373:
4312:
4283:
4245:
4158:{\displaystyle B}
4138:{\displaystyle A}
4118:{\displaystyle A}
4098:{\displaystyle B}
4078:{\displaystyle A}
4058:{\displaystyle B}
4045:, any observable
4038:{\displaystyle A}
4018:{\displaystyle t}
3998:{\displaystyle A}
3974:{\displaystyle H}
3954:{\displaystyle A}
3934:{\displaystyle H}
3885:{\displaystyle t}
3802:{\displaystyle H}
3785:Noether's theorem
3761:Erwin Schrödinger
3757:Werner Heisenberg
3628:
3627:
3600:{\displaystyle B}
3526:{\displaystyle A}
3117:
2966:
2854:{\displaystyle B}
2834:{\displaystyle A}
2803:
2744:
2660:
2583:{\displaystyle P}
2563:{\displaystyle X}
2519:
2504:
2465:
2436:
2386:dihydrogen cation
2382:particle in a box
2312:Chladni's figures
2005:{\displaystyle H}
1944:
1863:
1795:
1666:
1616:
1544:(more precisely,
1464:square-integrable
1257:faster than light
1253:superdense coding
1245:quantum computing
1214:radioactive decay
1209:potential barrier
1064:Werner Heisenberg
1060:Erwin Schrödinger
962:classical physics
943:quantum chemistry
941:, which includes
925:is a fundamental
923:Quantum mechanics
920:
919:
627:Scattering theory
607:Quantum computing
380:Schrödinger's cat
312:Bell's inequality
120:
95:
64:Quantum mechanics
16:(Redirected from
15948:
15923:
15922:
15911:
15910:
15899:
15898:
15887:
15886:
15875:
15874:
15873:
15863:
15862:
15851:
15850:
15839:
15838:
15837:
15827:
15826:
15825:
15818:
15736:Chemical physics
15676:Particle physics
15602:Classical optics
15485:
15478:
15471:
15462:
15461:
15448:
15447:
15159:Quantum geometry
15154:Quantum dynamics
15011:Superdeterminism
14907:Matrix mechanics
14762:Bra–ket notation
14693:
14686:
14679:
14670:
14669:
14665:
14656:Zalta, Edward N.
14646:
14637:Zalta, Edward N.
14631:Ismael, Jenann.
14613:
14611:
14610:
14607:
14604:
14600:
14488:
14458:
14446:
14432:
14389:
14344:
14290:
14265:
14204:
14158:
14130:
14073:
14048:
14026:
14014:
14000:
13975:More technical:
13941:bra–ket notation
13897:
13839:
13838:
13836:
13834:
13828:
13821:
13812:
13806:
13805:
13800:
13798:
13779:Feynman, Richard
13775:
13769:
13768:
13766:
13759:
13739:
13730:
13724:
13723:
13711:
13698:
13692:
13691:
13681:
13664:(11): 999–1008.
13644:
13638:
13631:
13625:
13618:
13612:
13611:
13580:
13574:
13573:
13542:
13536:
13535:
13533:
13531:
13517:
13511:
13510:
13501:Zalta, Edward N.
13492:
13486:
13485:
13465:
13455:
13446:
13445:
13443:
13442:
13422:
13416:
13415:
13389:
13368:Einstein, Albert
13364:
13358:
13356:
13345:
13343:
13309:
13303:
13302:
13273:
13267:
13266:
13264:
13262:
13242:
13236:
13235:
13214:
13208:
13207:
13205:
13203:
13188:
13182:
13181:
13160:
13154:
13153:
13133:
13127:
13126:
13105:
13099:
13098:
13096:
13094:
13071:
13065:
13064:
13022:
13016:
13015:
12980:
12974:
12973:
12959:
12953:
12952:
12899:
12893:
12892:
12882:
12863:
12857:
12856:
12837:
12828:
12822:
12821:
12785:
12779:
12778:
12752:
12750:quant-ph/0303050
12732:
12726:
12725:
12684:
12678:
12677:
12663:
12657:
12656:
12644:
12638:
12637:
12635:
12606:
12600:
12599:
12591:
12585:
12584:
12566:
12534:
12528:
12527:
12499:
12493:
12492:
12466:
12440:
12434:
12433:
12407:
12387:
12381:
12380:
12346:
12322:
12316:
12315:
12279:
12273:
12272:
12246:
12240:
12239:
12237:
12235:10.1038/121580a0
12205:
12199:
12198:
12158:
12152:
12151:
12111:
12105:
12104:
12078:
12058:
12052:
12051:
12032:Feynman, Richard
12028:
12022:
12021:
12001:
11995:
11994:
11977:Zwiebach, Barton
11973:
11967:
11966:
11947:Becker, Katrin;
11944:
11938:
11937:
11935:
11933:
11913:
11907:
11906:
11904:
11902:
11891:
11885:
11884:
11850:
11828:
11822:
11821:
11801:
11795:
11794:
11792:
11790:
11779:
11773:
11772:
11770:
11769:
11749:
11743:
11742:
11719:
11708:
11707:
11689:
11683:
11682:
11680:
11678:
11663:
11657:
11656:
11654:
11652:
11609:
11603:
11602:
11573:
11567:
11566:
11540:
11518:
11512:
11511:
11485:
11483:quant-ph/9811078
11476:(2): 1615–1621.
11463:
11457:
11456:
11430:
11424:
11423:
11405:
11399:
11398:
11396:
11394:
11378:
11366:
11360:
11359:
11333:
11313:
11307:
11306:
11270:
11254:
11248:
11247:
11224:
11215:
11214:
11187:Chuang, Isaac L.
11179:
11168:
11167:
11148:
11112:
11106:
11105:
11066:
11060:
11059:
11038:
11025:
11024:
10998:
10992:
10991:
10981:
10963:
10939:
10933:
10932:
10884:
10878:
10877:
10853:
10847:
10846:
10841:. Archived from
10820:
10814:
10813:
10787:
10781:
10780:
10771:
10769:
10751:
10745:
10744:
10742:
10740:
10720:
10711:
10710:
10692:
10669:(October 2015).
10663:
10654:
10653:
10633:
10610:
10601:
10600:
10583:
10577:
10576:
10574:
10573:
10558:
10552:
10551:
10541:
10509:
10503:
10502:
10479:
10473:
10472:
10452:
10446:
10445:
10425:
10419:
10418:
10398:
10387:
10386:
10369:Zwiebach, Barton
10365:
10340:
10339:
10313:
10293:
10287:
10286:
10252:
10232:
10226:
10225:
10189:
10183:
10182:
10154:
10148:
10147:
10145:
10143:
10120:
10107:
10106:
10059:
10042:
10038:
10032:
10018:
10012:
10003:, pp. 8–6), and
9981:
9975:
9968:
9921:Bra–ket notation
9892:chemical bonding
9860:John von Neumann
9841:matrix mechanics
9832:Louis de Broglie
9748:physical reality
9733:
9731:
9730:
9725:
9721:
9676:Gustav Kirchhoff
9637:Greek philosophy
9629:Ludwig Boltzmann
9410:and the related
9385:of states" and "
9377:
9313:are replaced by
9311:particle physics
9220:
9218:
9217:
9212:
9203:
9202:
9201:
9200:
9178:
9173:
9172:
9100:electric charges
9046:non-relativistic
9001:
8994:
8987:
8652:Skłodowska-Curie
8534:Condensed matter
8499:Weak interaction
8494:Electromagnetism
8351:Louis de Broglie
8279:
8277:
8276:
8271:
8269:
8268:
8256:
8248:
8247:
8229:
8228:
8209:
8207:
8206:
8201:
8187:
8186:
8177:
8172:
8170:
8159:
8136:
8135:
8126:
8121:
8120:
8112:
8094:
8093:
7992:
7990:
7989:
7984:
7969:
7967:
7966:
7961:
7956:
7900:
7898:
7897:
7892:
7887:
7882:
7874:
7869:
7868:
7856:
7855:
7850:
7841:
7840:
7819:
7818:
7806:
7778:
7776:
7775:
7770:
7765:
7760:
7752:
7747:
7746:
7734:
7733:
7728:
7719:
7718:
7697:
7696:
7684:
7651:
7649:
7648:
7643:
7638:
7637:
7624:
7594:
7563:
7562:
7558:
7533:
7532:
7504:
7502:
7501:
7496:
7484:
7482:
7481:
7476:
7464:
7462:
7461:
7456:
7441:
7439:
7438:
7433:
7418:
7416:
7415:
7410:
7408:
7407:
7400:
7399:
7346:
7344:
7343:
7338:
7336:
7331:
7329:
7314:
7312:
7311:
7306:
7304:
7299:
7297:
7282:
7280:
7279:
7274:
7272:
7271:
7236:
7230:
7226:
7204:
7202:
7201:
7196:
7188:
7187:
7182:
7173:
7165:
7164:
7159:
7150:
7139:we require that
7138:
7136:
7135:
7130:
7100:
7098:
7097:
7092:
7074:
7072:
7071:
7066:
7064:
7063:
7048:
7047:
7022:
7020:
7019:
7014:
7012:
7011:
6983:
6982:
6966:
6964:
6963:
6958:
6956:
6955:
6927:
6926:
6910:
6908:
6907:
6902:
6900:
6899:
6894:
6836:
6834:
6833:
6828:
6823:
6819:
6818:
6810:
6788:
6787:
6765:
6763:
6762:
6757:
6752:
6748:
6747:
6746:
6745:
6724:
6722:
6721:
6720:
6707:
6706:
6697:
6695:
6694:
6693:
6692:
6678:
6677:
6647:
6646:
6614:
6612:
6611:
6606:
6564:
6562:
6561:
6556:
6550:
6546:
6542:
6536:
6528:
6527:
6520:
6519:
6507:
6506:
6505:
6503:
6495:
6494:
6493:
6477:
6464:
6463:
6459:
6450:
6446:
6444:
6436:
6428:
6418:
6415:
6407:
6406:
6393:
6392:
6378:
6377:
6351:
6349:
6348:
6343:
6338:
6337:
6328:
6327:
6315:
6307:
6195:
6193:
6192:
6187:
6182:
6180:
6179:
6178:
6162:
6161:
6160:
6151:
6150:
6140:
6135:
6133:
6132:
6131:
6115:
6114:
6113:
6104:
6103:
6094:
6093:
6083:
6078:
6077:
6059:
6057:
6056:
6051:
6036:
6034:
6033:
6028:
5994:
5989:
5981:
5963:
5961:
5960:
5955:
5943:
5941:
5940:
5935:
5920:
5918:
5917:
5912:
5879:
5877:
5876:
5871:
5859:
5857:
5856:
5851:
5836:
5834:
5833:
5828:
5771:
5769:
5768:
5763:
5745:
5743:
5742:
5737:
5716:
5714:
5713:
5708:
5630:
5628:
5627:
5622:
5604:
5602:
5601:
5596:
5584:
5582:
5581:
5576:
5558:
5556:
5555:
5550:
5532:
5530:
5529:
5524:
5512:
5510:
5509:
5504:
5480:
5478:
5477:
5472:
5389:
5387:
5386:
5381:
5379:
5377:
5369:
5368:
5367:
5358:
5357:
5347:
5337:
5336:
5312:
5311:
5265:
5263:
5262:
5257:
5245:
5243:
5242:
5237:
5222:
5220:
5219:
5214:
5202:
5197:
5192:
5191:
5183:
5179:
5177:
5166:
5147:
5145:
5144:
5139:
5137:
5135:
5124:
5110:
5109:
5104:
5103:
5095:
5078:
5076:
5075:
5070:
5056:
5054:
5053:
5052:
5039:
5035:
5034:
5024:
5022:
5020:
5012:
5011:
5002:
4987:
4985:
4984:
4979:
4932:
4930:
4929:
4924:
4912:
4910:
4909:
4904:
4889:
4887:
4886:
4881:
4876:
4875:
4874:
4869:
4868:
4867:
4854:
4844:
4842:
4830:
4829:
4809:
4808:
4800:
4784:
4782:
4781:
4776:
4774:
4773:
4772:
4770:
4762:
4761:
4752:
4742:
4739:
4734:
4726:
4722:
4682:
4680:
4679:
4674:
4647:
4645:
4644:
4639:
4622:
4621:
4613:
4603:
4601:
4600:
4595:
4574:
4572:
4571:
4566:
4564:
4563:
4556:
4554:
4546:
4545:
4544:
4531:
4496:
4494:
4493:
4488:
4480:
4475:
4474:
4467:
4465:
4457:
4456:
4455:
4442:
4405:
4404:
4396:
4392:
4387:
4375:
4366:
4362:
4326:
4324:
4323:
4318:
4313:
4311:
4310:
4309:
4296:
4295:
4286:
4284:
4282:
4274:
4273:
4264:
4256:
4255:
4246:
4244:
4233:
4182:conservation law
4164:
4162:
4161:
4156:
4144:
4142:
4141:
4136:
4124:
4122:
4121:
4116:
4104:
4102:
4101:
4096:
4084:
4082:
4081:
4076:
4064:
4062:
4061:
4056:
4044:
4042:
4041:
4036:
4024:
4022:
4021:
4016:
4004:
4002:
4001:
3996:
3980:
3978:
3977:
3972:
3960:
3958:
3957:
3952:
3940:
3938:
3937:
3932:
3920:
3918:
3917:
3912:
3891:
3889:
3888:
3883:
3871:
3869:
3868:
3863:
3861:
3860:
3856:
3809:is known as the
3808:
3806:
3805:
3800:
3789:The Hamiltonian
3773:action principle
3753:matrix mechanics
3717:
3713:
3699:
3697:
3696:
3691:
3689:
3685:
3684:
3683:
3671:
3670:
3658:
3657:
3645:
3644:
3630:
3623:
3619:
3606:
3604:
3603:
3598:
3586:
3584:
3583:
3578:
3576:
3575:
3559:
3557:
3556:
3551:
3549:
3548:
3532:
3530:
3529:
3524:
3512:
3510:
3509:
3504:
3502:
3501:
3485:
3483:
3482:
3477:
3475:
3474:
3458:
3456:
3455:
3450:
3448:
3447:
3439:
3438:
3421:
3419:
3418:
3413:
3408:
3407:
3395:
3394:
3375:
3373:
3372:
3367:
3365:
3364:
3348:
3346:
3345:
3340:
3338:
3337:
3318:
3316:
3315:
3310:
3305:
3304:
3299:
3298:
3288:
3287:
3282:
3281:
3271:
3270:
3262:
3261:
3244:
3242:
3241:
3236:
3234:
3233:
3228:
3227:
3213:
3211:
3210:
3205:
3203:
3202:
3197:
3196:
3182:
3178:
3162:
3160:
3159:
3154:
3152:
3151:
3128:
3126:
3125:
3120:
3118:
3116:
3105:
3087:
3085:
3084:
3079:
3077:
3076:
3056:
3054:
3053:
3048:
3043:
3020:
3018:
3017:
3012:
3007:
3003:
3002:
3001:
2980:
2979:
2968:
2959:
2953:
2952:
2943:
2942:
2920:
2918:
2917:
2912:
2860:
2858:
2857:
2852:
2840:
2838:
2837:
2832:
2817:
2815:
2814:
2809:
2804:
2796:
2791:
2790:
2781:
2780:
2758:
2756:
2755:
2750:
2745:
2743:
2742:
2737:
2722:
2718:
2717:
2704:
2699:
2698:
2676:
2674:
2673:
2668:
2663:
2661:
2659:
2658:
2653:
2638:
2634:
2633:
2620:
2613:
2612:
2589:
2587:
2586:
2581:
2569:
2567:
2566:
2561:
2546:
2544:
2543:
2538:
2521:
2520:
2512:
2506:
2505:
2497:
2477:
2475:
2474:
2469:
2467:
2466:
2458:
2448:
2446:
2445:
2440:
2438:
2437:
2429:
2409:potential energy
2328:angular momentum
2282:
2280:
2279:
2274:
2253:
2251:
2250:
2245:
2216:
2214:
2213:
2208:
2206:
2205:
2201:
2154:
2152:
2151:
2146:
2129:
2128:
2124:
2066:
2064:
2063:
2058:
2039:
2037:
2036:
2031:
2011:
2009:
2008:
2003:
1988:
1986:
1985:
1980:
1945:
1943:
1932:
1874:
1872:
1871:
1866:
1864:
1856:
1855:
1837:
1834:
1833:
1824:
1823:
1807:
1805:
1804:
1799:
1797:
1796:
1788:
1774:
1772:
1771:
1766:
1747:
1745:
1744:
1739:
1737:
1736:
1720:
1718:
1717:
1712:
1704:
1703:
1678:
1676:
1675:
1670:
1668:
1667:
1659:
1649:
1647:
1646:
1641:
1639:
1638:
1633:
1618:
1617:
1609:
1603:
1591:
1589:
1588:
1583:
1536:
1534:
1533:
1528:
1526:
1525:
1520:
1503:
1501:
1500:
1495:
1490:
1482:
1481:
1456:projective space
1453:
1451:
1450:
1445:
1440:
1439:
1420:
1418:
1417:
1412:
1400:
1398:
1397:
1392:
1362:
1360:
1359:
1354:
1352:
1351:
1332:belonging to a (
1331:
1329:
1328:
1323:
1268:hidden variables
1182:such experiments
1107:(QED), has been
1005:angular momentum
987:states that are
912:
905:
898:
539:Superdeterminism
192:Bra–ket notation
143:
141:
140:
135:
127:
122:
121:
113:
101:
96:
94:
83:
55:
54:
21:
15956:
15955:
15951:
15950:
15949:
15947:
15946:
15945:
15931:
15930:
15929:
15917:
15905:
15893:
15881:
15871:
15869:
15857:
15845:
15835:
15833:
15823:
15821:
15813:
15811:
15806:
15770:
15756:Medical physics
15707:
15666:Nuclear physics
15635:
15629:Non-equilibrium
15551:
15523:
15495:
15489:
15459:
15454:
15436:
15422:Wigner's friend
15398:
15389:Quantum gravity
15350:
15336:Quantum sensing
15316:Quantum network
15296:Quantum machine
15266:Quantum imaging
15229:Quantum circuit
15224:Quantum channel
15183:
15129:Quantum biology
15115:
15091:Elitzur–Vaidman
15066:Davisson–Germer
15049:
15001:Hidden-variable
14991:de Broglie–Bohm
14968:Interpretations
14962:
14926:
14880:
14767:Complementarity
14745:
14702:
14697:
14608:
14605:
14602:
14601:
14598:
14596:
14543:Course material
14518:
14513:
14477:
14455:
14429:
14393:Scerri, Eric R.
14378:
14341:
14287:
14262:
14234:Wayback Machine
14193:
14155:
14070:
14054:Resnick, Robert
14045:
14023:
13997:
13971:considerations.
13947:N. David Mermin
13901:Richard Feynman
13894:
13856:Chester, Marvin
13848:
13846:Further reading
13843:
13842:
13832:
13830:
13826:
13819:
13813:
13809:
13796:
13794:
13776:
13772:
13764:
13737:
13731:
13727:
13720:
13699:
13695:
13645:
13641:
13632:
13628:
13619:
13615:
13608:
13594:Clarendon Press
13584:Aharoni, Amikam
13581:
13577:
13562:
13543:
13539:
13529:
13527:
13519:
13518:
13514:
13493:
13489:
13474:
13456:
13449:
13440:
13438:
13423:
13419:
13412:
13365:
13361:
13310:
13306:
13299:
13274:
13270:
13260:
13258:
13243:
13239:
13232:
13215:
13211:
13201:
13199:
13190:
13189:
13185:
13178:
13161:
13157:
13150:
13134:
13130:
13120:
13106:
13102:
13092:
13090:
13088:
13072:
13068:
13023:
13019:
13004:
12981:
12977:
12960:
12956:
12900:
12896:
12864:
12860:
12850:
12835:
12829:
12825:
12786:
12782:
12733:
12729:
12722:
12685:
12681:
12664:
12660:
12645:
12641:
12607:
12603:
12592:
12588:
12564:10.1.1.571.6089
12535:
12531:
12500:
12496:
12441:
12437:
12388:
12384:
12323:
12319:
12290:(50): 844–849.
12280:
12276:
12261:
12247:
12243:
12206:
12202:
12159:
12155:
12112:
12108:
12059:
12055:
12048:
12029:
12025:
12018:
12002:
11998:
11991:
11974:
11970:
11963:
11949:Becker, Melanie
11945:
11941:
11931:
11929:
11917:Overbye, Dennis
11914:
11910:
11900:
11898:
11893:
11892:
11888:
11829:
11825:
11818:
11802:
11798:
11788:
11786:
11781:
11780:
11776:
11767:
11765:
11750:
11746:
11739:
11720:
11711:
11704:
11690:
11686:
11676:
11674:
11664:
11660:
11650:
11648:
11610:
11606:
11591:
11574:
11570:
11519:
11515:
11464:
11460:
11453:
11431:
11427:
11420:
11406:
11402:
11392:
11390:
11376:
11367:
11363:
11318:Physics Reports
11314:
11310:
11287:
11255:
11251:
11244:
11225:
11218:
11203:
11180:
11171:
11165:
11114:Section 3.2 of
11113:
11109:
11094:
11074:Lifschitz, E.M.
11067:
11063:
11056:
11039:
11028:
11021:
10999:
10995:
10948:SciPost Physics
10940:
10936:
10885:
10881:
10874:
10854:
10850:
10839:
10825:"8 Measurement"
10821:
10817:
10807:
10788:
10784:
10767:
10765:
10752:
10748:
10738:
10736:
10733:Quanta Magazine
10721:
10714:
10667:Wiseman, Howard
10664:
10657:
10650:
10611:
10604:
10584:
10580:
10571:
10569:
10567:Notre Dame News
10559:
10555:
10510:
10506:
10499:
10480:
10476:
10469:
10453:
10449:
10442:
10426:
10422:
10415:
10399:
10390:
10383:
10366:
10343:
10294:
10290:
10233:
10229:
10190:
10186:
10155:
10151:
10141:
10139:
10137:
10121:
10110:
10073:(12): 863–867.
10060:
10056:
10051:
10046:
10045:
10039:
10035:
10030:Quantum machine
10019:
10015:
9982:
9978:
9969:
9965:
9960:
9955:
9916:
9904:superconductors
9807:'s work on the
9707:
9704:
9703:
9657:Eugen Goldstein
9645:Michael Faraday
9617:Amedeo Avogadro
9563:
9555:Main articles:
9553:
9455:Albert Einstein
9424:Steven Weinberg
9420:Richard Feynman
9404:interpretations
9396:
9395:
9390:
9379:
9372:
9366:
9317:objects called
9315:one-dimensional
9288:quantum gravity
9280:
9272:Steven Weinberg
9268:Sheldon Glashow
9196:
9193:
9192:
9188:
9174:
9168:
9164:
9158:
9155:
9154:
9108:
9065:chaotic systems
9035:quantum numbers
9014:
9005:
8976:
8975:
8972:
8971:
8970:
8960:
8952:
8951:
8612:
8604:
8603:
8594:Quantum gravity
8544:Complex systems
8442:
8434:
8433:
8369:
8361:
8360:
8306:Albert Einstein
8296:
8288:
8261:
8257:
8252:
8240:
8236:
8221:
8217:
8215:
8212:
8211:
8210:
8182:
8178:
8173:
8163:
8158:
8131:
8127:
8122:
8111:
8110:
8108:
8105:
8104:
8092:
8023:, and others).
8005:
7999:
7975:
7972:
7971:
7952:
7917:
7914:
7913:
7875:
7873:
7864:
7860:
7851:
7846:
7845:
7836:
7832:
7814:
7810:
7802:
7785:
7782:
7781:
7753:
7751:
7742:
7738:
7729:
7724:
7723:
7714:
7710:
7692:
7688:
7680:
7663:
7660:
7659:
7632:
7631:
7620:
7602:
7601:
7590:
7565:
7564:
7554:
7544:
7540:
7528:
7524:
7513:
7510:
7509:
7490:
7487:
7486:
7470:
7467:
7466:
7450:
7447:
7446:
7424:
7421:
7420:
7402:
7401:
7389:
7385:
7383:
7377:
7376:
7371:
7361:
7360:
7352:
7349:
7348:
7330:
7325:
7320:
7317:
7316:
7298:
7293:
7288:
7285:
7284:
7266:
7265:
7260:
7254:
7253:
7248:
7238:
7237:
7225:
7217:
7214:
7213:
7183:
7178:
7177:
7169:
7160:
7155:
7154:
7146:
7144:
7141:
7140:
7106:
7103:
7102:
7080:
7077:
7076:
7059:
7055:
7043:
7039:
7028:
7025:
7024:
7006:
7005:
6999:
6998:
6988:
6987:
6978:
6974:
6972:
6969:
6968:
6950:
6949:
6943:
6942:
6932:
6931:
6922:
6918:
6916:
6913:
6912:
6895:
6890:
6889:
6881:
6878:
6877:
6850:
6809:
6802:
6798:
6783:
6779:
6777:
6774:
6773:
6741:
6737:
6733:
6729:
6725:
6716:
6712:
6708:
6702:
6698:
6696:
6688:
6684:
6683:
6679:
6673:
6669:
6642:
6638:
6636:
6633:
6632:
6623:
6573:
6570:
6569:
6529:
6526:
6525:
6521:
6515:
6511:
6496:
6489:
6485:
6478:
6476:
6472:
6468:
6455:
6451:
6437:
6429:
6427:
6423:
6422:
6402:
6398:
6397:
6391:
6373:
6369:
6367:
6364:
6363:
6333:
6329:
6323:
6319:
6306:
6289:
6286:
6285:
6231:
6225:
6174:
6170:
6163:
6156:
6152:
6146:
6142:
6141:
6139:
6127:
6123:
6116:
6109:
6105:
6099:
6095:
6089:
6085:
6084:
6082:
6073:
6069:
6067:
6064:
6063:
6045:
6042:
6041:
5982:
5980:
5972:
5969:
5968:
5949:
5946:
5945:
5926:
5923:
5922:
5885:
5882:
5881:
5865:
5862:
5861:
5845:
5842:
5841:
5780:
5777:
5776:
5751:
5748:
5747:
5725:
5722:
5721:
5639:
5636:
5635:
5610:
5607:
5606:
5590:
5587:
5586:
5564:
5561:
5560:
5538:
5535:
5534:
5518:
5515:
5514:
5489:
5486:
5485:
5405:
5402:
5401:
5395:Euler's formula
5370:
5363:
5359:
5353:
5349:
5348:
5346:
5323:
5319:
5301:
5297:
5277:
5274:
5273:
5251:
5248:
5247:
5231:
5228:
5227:
5198:
5193:
5182:
5181:
5170:
5165:
5163:
5160:
5159:
5128:
5123:
5105:
5094:
5093:
5092:
5090:
5087:
5086:
5048:
5044:
5040:
5030:
5026:
5025:
5023:
5013:
5007:
5003:
5001:
4996:
4993:
4992:
4973:
4970:
4969:
4957:
4943:
4918:
4915:
4914:
4898:
4895:
4894:
4863:
4859:
4855:
4853:
4849:
4845:
4838:
4828:
4799:
4798:
4796:
4793:
4792:
4763:
4757:
4753:
4751:
4747:
4743:
4735:
4727:
4721:
4698:
4695:
4694:
4653:
4650:
4649:
4612:
4611:
4609:
4606:
4605:
4580:
4577:
4576:
4547:
4540:
4536:
4532:
4530:
4514:
4510:
4508:
4505:
4504:
4476:
4458:
4451:
4447:
4443:
4441:
4425:
4421:
4395:
4394:
4388:
4380:
4361:
4338:
4335:
4334:
4305:
4301:
4297:
4291:
4287:
4285:
4275:
4269:
4265:
4263:
4251:
4247:
4237:
4232:
4224:
4221:
4220:
4201:
4195:
4190:
4150:
4147:
4146:
4130:
4127:
4126:
4110:
4107:
4106:
4090:
4087:
4086:
4070:
4067:
4066:
4050:
4047:
4046:
4030:
4027:
4026:
4010:
4007:
4006:
3990:
3987:
3986:
3966:
3963:
3962:
3946:
3943:
3942:
3926:
3923:
3922:
3897:
3894:
3893:
3877:
3874:
3873:
3852:
3839:
3835:
3818:
3815:
3814:
3794:
3791:
3790:
3787:
3781:
3741:
3715:
3711:
3679:
3675:
3666:
3662:
3653:
3649:
3640:
3636:
3635:
3631:
3617:
3615:
3612:
3611:
3592:
3589:
3588:
3571:
3567:
3565:
3562:
3561:
3544:
3540:
3538:
3535:
3534:
3533:, and likewise
3518:
3515:
3514:
3497:
3493:
3491:
3488:
3487:
3470:
3466:
3464:
3461:
3460:
3440:
3434:
3433:
3432:
3430:
3427:
3426:
3403:
3399:
3390:
3386:
3384:
3381:
3380:
3360:
3356:
3354:
3351:
3350:
3333:
3329:
3327:
3324:
3323:
3300:
3294:
3293:
3292:
3283:
3277:
3276:
3275:
3263:
3257:
3256:
3255:
3253:
3250:
3249:
3229:
3223:
3222:
3221:
3219:
3216:
3215:
3198:
3192:
3191:
3190:
3188:
3185:
3184:
3180:
3176:
3169:
3147:
3143:
3138:
3135:
3134:
3109:
3104:
3093:
3090:
3089:
3088:is replaced by
3072:
3068:
3066:
3063:
3062:
3039:
3034:
3031:
3030:
2997:
2996:
2975:
2974:
2973:
2969:
2957:
2948:
2944:
2938:
2934:
2932:
2929:
2928:
2873:
2870:
2869:
2846:
2843:
2842:
2826:
2823:
2822:
2795:
2786:
2782:
2776:
2772:
2770:
2767:
2766:
2738:
2727:
2726:
2713:
2709:
2705:
2703:
2694:
2690:
2688:
2685:
2684:
2654:
2643:
2642:
2629:
2625:
2621:
2619:
2617:
2608:
2604:
2602:
2599:
2598:
2575:
2572:
2571:
2555:
2552:
2551:
2511:
2510:
2496:
2495:
2490:
2487:
2486:
2457:
2456:
2454:
2451:
2450:
2428:
2427:
2425:
2422:
2421:
2417:
2330:and energy are
2259:
2256:
2255:
2230:
2227:
2226:
2197:
2184:
2180:
2163:
2160:
2159:
2120:
2107:
2103:
2086:
2083:
2082:
2049:
2046:
2045:
2044:. The constant
2042:Planck constant
2040:is the reduced
2025:
2022:
2021:
1997:
1994:
1993:
1936:
1931:
1923:
1920:
1919:
1913:
1851:
1847:
1836:
1829:
1828:
1819:
1815:
1813:
1810:
1809:
1787:
1786:
1784:
1781:
1780:
1760:
1757:
1756:
1732:
1728:
1726:
1723:
1722:
1699:
1695:
1684:
1681:
1680:
1658:
1657:
1655:
1652:
1651:
1634:
1629:
1628:
1608:
1607:
1599:
1597:
1594:
1593:
1577:
1574:
1573:
1521:
1516:
1515:
1513:
1510:
1509:
1486:
1477:
1473:
1471:
1468:
1467:
1432:
1428:
1426:
1423:
1422:
1406:
1403:
1402:
1368:
1365:
1364:
1347:
1346:
1344:
1341:
1340:
1317:
1314:
1313:
1310:
1304:
1097:Wigner's friend
1092:
1040:Albert Einstein
1028:arose gradually
993:discrete values
939:quantum physics
916:
887:
886:
885:
650:
642:
641:
587:
586:Advanced topics
579:
578:
577:
529:Hidden-variable
519:de Broglie–Bohm
498:
496:Interpretations
488:
487:
486:
456:
448:
447:
446:
404:
396:
395:
394:
361:
317:CHSH inequality
306:
298:
297:
296:
225:Complementarity
219:
211:
210:
209:
177:
148:
123:
112:
111:
97:
87:
82:
74:
71:
70:
37:
28:
23:
22:
18:Quantum Physics
15:
12:
11:
5:
15954:
15944:
15943:
15928:
15927:
15915:
15903:
15891:
15879:
15867:
15855:
15843:
15831:
15808:
15807:
15805:
15804:
15799:
15794:
15789:
15784:
15778:
15776:
15772:
15771:
15769:
15768:
15763:
15758:
15753:
15748:
15743:
15738:
15733:
15728:
15723:
15717:
15715:
15709:
15708:
15706:
15705:
15700:
15699:
15698:
15693:
15688:
15678:
15673:
15668:
15663:
15662:
15661:
15656:
15645:
15643:
15637:
15636:
15634:
15633:
15632:
15631:
15626:
15619:Thermodynamics
15616:
15615:
15614:
15609:
15599:
15594:
15589:
15588:
15587:
15582:
15577:
15572:
15561:
15559:
15553:
15552:
15550:
15549:
15548:
15547:
15537:
15531:
15529:
15525:
15524:
15522:
15521:
15520:
15519:
15509:
15503:
15501:
15497:
15496:
15488:
15487:
15480:
15473:
15465:
15456:
15455:
15453:
15452:
15441:
15438:
15437:
15435:
15434:
15429:
15424:
15419:
15418:
15417:
15406:
15404:
15400:
15399:
15397:
15396:
15391:
15386:
15385:
15384:
15374:
15369:
15367:Casimir effect
15364:
15358:
15356:
15352:
15351:
15349:
15348:
15343:
15338:
15333:
15328:
15326:Quantum optics
15323:
15318:
15313:
15308:
15303:
15298:
15293:
15288:
15283:
15278:
15273:
15268:
15263:
15258:
15253:
15248:
15247:
15246:
15236:
15231:
15226:
15221:
15220:
15219:
15209:
15204:
15199:
15193:
15191:
15185:
15184:
15182:
15181:
15176:
15171:
15166:
15161:
15156:
15151:
15146:
15141:
15136:
15131:
15125:
15123:
15117:
15116:
15114:
15113:
15108:
15103:
15101:Quantum eraser
15098:
15093:
15088:
15083:
15078:
15073:
15068:
15063:
15057:
15055:
15051:
15050:
15048:
15047:
15042:
15037:
15032:
15027:
15022:
15017:
15016:
15015:
15014:
15013:
14998:
14993:
14988:
14983:
14978:
14972:
14970:
14964:
14963:
14961:
14960:
14955:
14950:
14945:
14940:
14934:
14932:
14928:
14927:
14925:
14924:
14919:
14914:
14909:
14904:
14899:
14894:
14888:
14886:
14882:
14881:
14879:
14878:
14877:
14876:
14871:
14861:
14856:
14851:
14846:
14841:
14836:
14831:
14826:
14821:
14816:
14811:
14806:
14801:
14800:
14799:
14794:
14789:
14784:
14774:
14772:Density matrix
14769:
14764:
14759:
14753:
14751:
14747:
14746:
14744:
14743:
14738:
14733:
14728:
14727:
14726:
14716:
14710:
14708:
14704:
14703:
14696:
14695:
14688:
14681:
14673:
14667:
14666:
14650:Krips, Henry.
14647:
14627:
14626:
14622:
14621:
14616:
14593:
14568:
14560:
14545:
14544:
14540:
14539:
14530:
14525:
14517:
14516:External links
14514:
14512:
14511:
14500:
14499:
14489:
14475:
14459:
14453:
14433:
14427:
14411:
14399:Periodic Table
14390:
14376:
14358:
14348:Albert Messiah
14345:
14339:
14323:
14305:
14295:Wave Mechanics
14291:
14285:
14269:
14260:
14252:Pergamon Press
14236:
14219:Hagen Kleinert
14216:
14215:. McGraw Hill.
14206:
14191:
14173:
14162:D. Greenberger
14159:
14153:
14131:
14105:(3): 454–462.
14091:
14074:
14068:
14049:
14043:
14027:
14021:
14011:Quantum Theory
14001:
13995:
13973:
13972:
13957:Victor Stenger
13954:
13944:
13922:
13898:
13892:
13886:. Allen Lane.
13870:
13862:. John Wiley.
13849:
13847:
13844:
13841:
13840:
13807:
13770:
13725:
13718:
13693:
13639:
13626:
13613:
13606:
13575:
13560:
13546:Sakurai, J. J.
13537:
13512:
13487:
13472:
13447:
13428:(2017-08-31).
13417:
13410:
13390:Translated in
13359:
13351:, ed. (1989).
13326:(6): 132–148.
13304:
13297:
13268:
13237:
13230:
13209:
13183:
13176:
13155:
13148:
13128:
13118:
13100:
13086:
13066:
13039:(4): 217–219.
13017:
13002:
12975:
12954:
12919:(4): 390–417.
12905:(April 2010).
12894:
12858:
12848:
12823:
12796:(2): 229–240.
12780:
12743:(3): 415–438.
12727:
12720:
12692:Wheeler, J. A.
12679:
12658:
12639:
12626:(3): 195–200.
12601:
12586:
12549:(4): 879–887.
12529:
12510:(3): 171–201.
12494:
12435:
12398:(3): 222–230.
12382:
12317:
12274:
12259:
12241:
12200:
12153:
12132:10.1086/425941
12126:(5): 669–682.
12106:
12053:
12046:
12023:
12016:
11996:
11989:
11968:
11961:
11939:
11908:
11886:
11841:(11): 111807.
11823:
11816:
11796:
11774:
11755:(2019-02-26).
11744:
11737:
11709:
11702:
11694:Modern Physics
11684:
11666:Matson, John.
11658:
11624:(25): 250001.
11604:
11589:
11577:Vedral, Vlatko
11568:
11531:(15): 155303.
11513:
11458:
11451:
11425:
11418:
11400:
11361:
11308:
11285:
11249:
11242:
11216:
11201:
11169:
11163:
11124:(4): 358–381,
11107:
11092:
11084:Pergamon Press
11061:
11054:
11026:
11019:
10993:
10934:
10879:
10872:
10848:
10845:on 2023-01-02.
10837:
10815:
10805:
10782:
10746:
10712:
10655:
10648:
10602:
10578:
10553:
10504:
10497:
10474:
10467:
10447:
10440:
10420:
10413:
10388:
10381:
10341:
10288:
10227:
10194:Nature Physics
10184:
10165:(9): 896–905.
10149:
10135:
10108:
10053:
10052:
10050:
10047:
10044:
10043:
10033:
10013:
9976:
9962:
9961:
9959:
9956:
9954:
9953:
9948:
9943:
9938:
9933:
9928:
9923:
9917:
9915:
9912:
9896:semiconductors
9876:quantum optics
9845:wave mechanics
9837:Pascual Jordan
9765:spectral lines
9736:
9735:
9720:
9717:
9714:
9711:
9649:Julius PlĂĽcker
9587:Leonhard Euler
9552:
9549:
9534:simultaneously
9487:Boris Podolsky
9479:thermodynamics
9459:quantum theory
9391:
9380:
9374:
9368:Main article:
9365:
9362:
9279:
9276:
9209:
9206:
9199:
9195:
9191:
9187:
9184:
9181:
9177:
9171:
9167:
9163:
9151:electric field
9149:describes the
9120:Dirac equation
9107:
9104:
9054:kinetic energy
9013:
9010:
9007:
9006:
9004:
9003:
8996:
8989:
8981:
8978:
8977:
8974:
8973:
8968:Modern physics
8966:
8965:
8961:
8958:
8957:
8954:
8953:
8950:
8949:
8944:
8939:
8934:
8929:
8924:
8919:
8914:
8909:
8904:
8899:
8894:
8889:
8884:
8879:
8874:
8869:
8864:
8859:
8854:
8849:
8844:
8839:
8834:
8829:
8824:
8819:
8814:
8809:
8804:
8802:P. W. Anderson
8799:
8794:
8789:
8784:
8779:
8774:
8769:
8764:
8759:
8754:
8749:
8744:
8739:
8734:
8729:
8724:
8719:
8714:
8709:
8704:
8699:
8694:
8689:
8684:
8679:
8674:
8669:
8664:
8659:
8654:
8649:
8644:
8639:
8634:
8629:
8624:
8619:
8613:
8610:
8609:
8606:
8605:
8602:
8601:
8596:
8591:
8586:
8581:
8576:
8571:
8566:
8564:Plasma physics
8561:
8556:
8551:
8546:
8541:
8536:
8531:
8526:
8521:
8516:
8511:
8506:
8501:
8496:
8491:
8490:
8489:
8484:
8479:
8469:
8464:
8459:
8454:
8449:
8443:
8440:
8439:
8436:
8435:
8432:
8431:
8426:
8421:
8416:
8411:
8406:
8401:
8396:
8391:
8386:
8381:
8376:
8370:
8367:
8366:
8363:
8362:
8359:
8358:
8353:
8348:
8343:
8338:
8336:Wolfgang Pauli
8333:
8331:Pascual Jordan
8328:
8323:
8318:
8313:
8308:
8303:
8297:
8294:
8293:
8290:
8289:
8280:
8267:
8264:
8260:
8255:
8251:
8246:
8243:
8239:
8235:
8232:
8227:
8224:
8220:
8199:
8196:
8193:
8190:
8185:
8181:
8176:
8169:
8166:
8162:
8157:
8154:
8151:
8148:
8145:
8142:
8139:
8134:
8130:
8125:
8118:
8115:
8101:
8100:
8098:Modern physics
8091:
8088:
8068:microprocessor
8064:semiconductors
8040:quantum optics
8001:Main article:
7998:
7995:
7982:
7979:
7959:
7955:
7951:
7948:
7945:
7942:
7939:
7936:
7933:
7930:
7927:
7924:
7921:
7902:
7901:
7890:
7885:
7881:
7878:
7872:
7867:
7863:
7859:
7854:
7849:
7844:
7839:
7835:
7831:
7828:
7825:
7822:
7817:
7813:
7809:
7805:
7801:
7798:
7795:
7792:
7789:
7779:
7768:
7763:
7759:
7756:
7750:
7745:
7741:
7737:
7732:
7727:
7722:
7717:
7713:
7709:
7706:
7703:
7700:
7695:
7691:
7687:
7683:
7679:
7676:
7673:
7670:
7667:
7653:
7652:
7641:
7636:
7630:
7627:
7623:
7619:
7616:
7613:
7610:
7607:
7604:
7603:
7600:
7597:
7593:
7589:
7586:
7583:
7580:
7577:
7574:
7571:
7570:
7568:
7561:
7557:
7553:
7550:
7547:
7543:
7539:
7536:
7531:
7527:
7523:
7520:
7517:
7494:
7474:
7454:
7431:
7428:
7406:
7398:
7395:
7392:
7388:
7384:
7382:
7379:
7378:
7375:
7372:
7370:
7367:
7366:
7364:
7359:
7356:
7334:
7328:
7324:
7302:
7296:
7292:
7270:
7264:
7261:
7259:
7256:
7255:
7252:
7249:
7247:
7244:
7243:
7241:
7233:
7229:
7224:
7221:
7210:beam splitters
7194:
7191:
7186:
7181:
7176:
7172:
7168:
7163:
7158:
7153:
7149:
7128:
7125:
7122:
7119:
7116:
7113:
7110:
7090:
7087:
7084:
7062:
7058:
7054:
7051:
7046:
7042:
7038:
7035:
7032:
7010:
7004:
7001:
7000:
6997:
6994:
6993:
6991:
6986:
6981:
6977:
6954:
6948:
6945:
6944:
6941:
6938:
6937:
6935:
6930:
6925:
6921:
6898:
6893:
6888:
6885:
6849:
6846:
6838:
6837:
6826:
6822:
6816:
6813:
6808:
6805:
6801:
6797:
6794:
6791:
6786:
6782:
6767:
6766:
6755:
6751:
6744:
6740:
6736:
6732:
6728:
6719:
6715:
6711:
6705:
6701:
6691:
6687:
6682:
6676:
6672:
6668:
6665:
6662:
6659:
6656:
6653:
6650:
6645:
6641:
6621:
6616:
6615:
6604:
6601:
6598:
6595:
6592:
6589:
6586:
6583:
6580:
6577:
6566:
6565:
6553:
6549:
6545:
6539:
6535:
6532:
6524:
6518:
6514:
6510:
6502:
6499:
6492:
6488:
6484:
6481:
6475:
6471:
6467:
6462:
6458:
6454:
6449:
6443:
6440:
6435:
6432:
6426:
6421:
6414:
6411:
6405:
6401:
6396:
6390:
6387:
6384:
6381:
6376:
6372:
6353:
6352:
6341:
6336:
6332:
6326:
6322:
6318:
6313:
6310:
6305:
6302:
6299:
6296:
6293:
6263:standing waves
6259:imaginary part
6227:Main article:
6224:
6221:
6185:
6177:
6173:
6169:
6166:
6159:
6155:
6149:
6145:
6138:
6130:
6126:
6122:
6119:
6112:
6108:
6102:
6098:
6092:
6088:
6081:
6076:
6072:
6049:
6038:
6037:
6026:
6023:
6020:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5992:
5988:
5985:
5979:
5976:
5953:
5933:
5930:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5869:
5849:
5838:
5837:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5805:
5802:
5799:
5796:
5793:
5790:
5787:
5784:
5761:
5758:
5755:
5735:
5732:
5729:
5718:
5717:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5646:
5643:
5620:
5617:
5614:
5594:
5574:
5571:
5568:
5548:
5545:
5542:
5522:
5502:
5499:
5496:
5493:
5482:
5481:
5469:
5466:
5463:
5460:
5457:
5454:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5391:
5390:
5376:
5373:
5366:
5362:
5356:
5352:
5345:
5342:
5335:
5332:
5329:
5326:
5322:
5318:
5315:
5310:
5307:
5304:
5300:
5296:
5293:
5290:
5287:
5284:
5281:
5255:
5235:
5224:
5223:
5212:
5209:
5206:
5201:
5196:
5189:
5186:
5176:
5173:
5169:
5149:
5148:
5134:
5131:
5127:
5122:
5119:
5116:
5113:
5108:
5101:
5098:
5080:
5079:
5068:
5065:
5062:
5059:
5051:
5047:
5043:
5038:
5033:
5029:
5019:
5016:
5010:
5006:
5000:
4977:
4953:Main article:
4942:
4939:
4922:
4902:
4891:
4890:
4879:
4872:
4866:
4862:
4858:
4852:
4848:
4841:
4836:
4833:
4827:
4824:
4821:
4818:
4815:
4812:
4806:
4803:
4786:
4785:
4769:
4766:
4760:
4756:
4750:
4746:
4738:
4733:
4730:
4725:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4672:
4669:
4666:
4663:
4660:
4657:
4637:
4634:
4631:
4628:
4625:
4619:
4616:
4593:
4590:
4587:
4584:
4562:
4559:
4553:
4550:
4543:
4539:
4535:
4529:
4526:
4523:
4520:
4517:
4513:
4498:
4497:
4486:
4483:
4479:
4473:
4470:
4464:
4461:
4454:
4450:
4446:
4440:
4437:
4434:
4431:
4428:
4424:
4420:
4417:
4414:
4411:
4408:
4402:
4399:
4391:
4386:
4383:
4379:
4372:
4369:
4365:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4328:
4327:
4316:
4308:
4304:
4300:
4294:
4290:
4281:
4278:
4272:
4268:
4262:
4259:
4254:
4250:
4243:
4240:
4236:
4231:
4228:
4197:Main article:
4194:
4191:
4189:
4186:
4175:differentiable
4169:in classical (
4154:
4134:
4114:
4094:
4074:
4054:
4034:
4014:
3994:
3970:
3950:
3930:
3910:
3907:
3904:
3901:
3881:
3859:
3855:
3851:
3848:
3845:
3842:
3838:
3834:
3831:
3828:
3825:
3822:
3798:
3783:Main article:
3780:
3777:
3747:" proposed by
3740:
3737:
3701:
3700:
3688:
3682:
3678:
3674:
3669:
3665:
3661:
3656:
3652:
3648:
3643:
3639:
3634:
3626:
3622:
3596:
3574:
3570:
3547:
3543:
3522:
3500:
3496:
3473:
3469:
3446:
3443:
3437:
3423:
3422:
3411:
3406:
3402:
3398:
3393:
3389:
3363:
3359:
3336:
3332:
3320:
3319:
3308:
3303:
3297:
3291:
3286:
3280:
3274:
3269:
3266:
3260:
3232:
3226:
3201:
3195:
3173:tensor product
3168:
3165:
3150:
3146:
3142:
3115:
3112:
3108:
3103:
3100:
3097:
3075:
3071:
3046:
3042:
3038:
3022:
3021:
3010:
3006:
3000:
2995:
2992:
2989:
2986:
2983:
2978:
2972:
2965:
2962:
2956:
2951:
2947:
2941:
2937:
2922:
2921:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2850:
2830:
2819:
2818:
2807:
2802:
2799:
2794:
2789:
2785:
2779:
2775:
2760:
2759:
2748:
2741:
2736:
2733:
2730:
2725:
2721:
2716:
2712:
2708:
2702:
2697:
2693:
2678:
2677:
2666:
2657:
2652:
2649:
2646:
2641:
2637:
2632:
2628:
2624:
2616:
2611:
2607:
2579:
2559:
2548:
2547:
2536:
2533:
2530:
2527:
2524:
2518:
2515:
2509:
2503:
2500:
2494:
2464:
2461:
2435:
2432:
2416:
2413:
2376:including the
2356:atomic nucleus
2342:in acoustics.)
2272:
2269:
2266:
2263:
2243:
2240:
2237:
2234:
2204:
2200:
2196:
2193:
2190:
2187:
2183:
2179:
2176:
2173:
2170:
2167:
2156:
2155:
2144:
2141:
2138:
2135:
2132:
2127:
2123:
2119:
2116:
2113:
2110:
2106:
2102:
2099:
2096:
2093:
2090:
2056:
2053:
2029:
2001:
1990:
1989:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1942:
1939:
1935:
1930:
1927:
1912:
1909:
1862:
1859:
1854:
1850:
1846:
1843:
1840:
1832:
1827:
1822:
1818:
1794:
1791:
1764:
1735:
1731:
1710:
1707:
1702:
1698:
1694:
1691:
1688:
1665:
1662:
1637:
1632:
1627:
1624:
1621:
1615:
1612:
1606:
1602:
1581:
1524:
1519:
1493:
1489:
1485:
1480:
1476:
1443:
1438:
1435:
1431:
1410:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1350:
1321:
1306:Main article:
1303:
1300:
1288:linear algebra
1272:Bell's theorem
1218:nuclear fusion
1117:complex number
1091:
1088:
974:submicroscopic
918:
917:
915:
914:
907:
900:
892:
889:
888:
884:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
652:
651:
648:
647:
644:
643:
640:
639:
634:
629:
624:
622:Density matrix
619:
614:
609:
604:
599:
594:
588:
585:
584:
581:
580:
576:
575:
570:
565:
560:
555:
550:
545:
544:
543:
542:
541:
526:
521:
516:
511:
506:
500:
499:
494:
493:
490:
489:
485:
484:
479:
474:
469:
464:
458:
457:
454:
453:
450:
449:
445:
444:
439:
434:
429:
424:
419:
413:
412:
411:
405:
402:
401:
398:
397:
393:
392:
387:
382:
376:
375:
374:
373:
372:
370:Delayed-choice
365:Quantum eraser
360:
359:
354:
349:
344:
339:
334:
329:
324:
319:
314:
308:
307:
304:
303:
300:
299:
295:
294:
293:
292:
282:
277:
272:
267:
262:
257:
255:Quantum number
252:
247:
242:
237:
232:
227:
221:
220:
217:
216:
213:
212:
208:
207:
202:
196:
195:
194:
189:
184:
178:
175:
174:
171:
170:
169:
168:
163:
158:
150:
149:
144:
133:
130:
126:
119:
116:
110:
107:
104:
100:
93:
90:
86:
81:
78:
67:
66:
60:
59:
44:Wave functions
26:
9:
6:
4:
3:
2:
15953:
15942:
15939:
15938:
15936:
15926:
15921:
15916:
15914:
15909:
15904:
15902:
15897:
15892:
15890:
15885:
15880:
15878:
15868:
15866:
15861:
15856:
15854:
15849:
15844:
15842:
15832:
15830:
15820:
15819:
15816:
15803:
15800:
15798:
15795:
15793:
15790:
15788:
15785:
15783:
15780:
15779:
15777:
15773:
15767:
15764:
15762:
15761:Ocean physics
15759:
15757:
15754:
15752:
15749:
15747:
15744:
15742:
15739:
15737:
15734:
15732:
15729:
15727:
15724:
15722:
15719:
15718:
15716:
15714:
15710:
15704:
15701:
15697:
15696:Modern optics
15694:
15692:
15689:
15687:
15684:
15683:
15682:
15679:
15677:
15674:
15672:
15669:
15667:
15664:
15660:
15657:
15655:
15652:
15651:
15650:
15647:
15646:
15644:
15642:
15638:
15630:
15627:
15625:
15622:
15621:
15620:
15617:
15613:
15610:
15608:
15605:
15604:
15603:
15600:
15598:
15595:
15593:
15590:
15586:
15583:
15581:
15578:
15576:
15573:
15571:
15568:
15567:
15566:
15563:
15562:
15560:
15558:
15554:
15546:
15545:Computational
15543:
15542:
15541:
15538:
15536:
15533:
15532:
15530:
15526:
15518:
15515:
15514:
15513:
15510:
15508:
15505:
15504:
15502:
15498:
15494:
15486:
15481:
15479:
15474:
15472:
15467:
15466:
15463:
15451:
15443:
15442:
15439:
15433:
15430:
15428:
15425:
15423:
15420:
15416:
15413:
15412:
15411:
15408:
15407:
15405:
15401:
15395:
15392:
15390:
15387:
15383:
15380:
15379:
15378:
15375:
15373:
15370:
15368:
15365:
15363:
15360:
15359:
15357:
15353:
15347:
15344:
15342:
15339:
15337:
15334:
15332:
15329:
15327:
15324:
15322:
15319:
15317:
15314:
15312:
15309:
15307:
15304:
15302:
15299:
15297:
15294:
15292:
15289:
15287:
15286:Quantum logic
15284:
15282:
15279:
15277:
15274:
15272:
15269:
15267:
15264:
15262:
15259:
15257:
15254:
15252:
15249:
15245:
15242:
15241:
15240:
15237:
15235:
15232:
15230:
15227:
15225:
15222:
15218:
15215:
15214:
15213:
15210:
15208:
15205:
15203:
15200:
15198:
15195:
15194:
15192:
15190:
15186:
15180:
15177:
15175:
15172:
15170:
15167:
15165:
15162:
15160:
15157:
15155:
15152:
15150:
15147:
15145:
15142:
15140:
15139:Quantum chaos
15137:
15135:
15132:
15130:
15127:
15126:
15124:
15122:
15118:
15112:
15109:
15107:
15106:Stern–Gerlach
15104:
15102:
15099:
15097:
15094:
15092:
15089:
15087:
15084:
15082:
15079:
15077:
15074:
15072:
15069:
15067:
15064:
15062:
15059:
15058:
15056:
15052:
15046:
15043:
15041:
15040:Transactional
15038:
15036:
15033:
15031:
15030:Quantum logic
15028:
15026:
15023:
15021:
15018:
15012:
15009:
15008:
15007:
15004:
15003:
15002:
14999:
14997:
14994:
14992:
14989:
14987:
14984:
14982:
14979:
14977:
14974:
14973:
14971:
14969:
14965:
14959:
14956:
14954:
14951:
14949:
14946:
14944:
14941:
14939:
14936:
14935:
14933:
14929:
14923:
14920:
14918:
14915:
14913:
14910:
14908:
14905:
14903:
14900:
14898:
14895:
14893:
14890:
14889:
14887:
14883:
14875:
14872:
14870:
14867:
14866:
14865:
14864:Wave function
14862:
14860:
14857:
14855:
14852:
14850:
14847:
14845:
14842:
14840:
14839:Superposition
14837:
14835:
14834:Quantum state
14832:
14830:
14827:
14825:
14822:
14820:
14817:
14815:
14812:
14810:
14807:
14805:
14802:
14798:
14795:
14793:
14790:
14788:
14787:Excited state
14785:
14783:
14780:
14779:
14778:
14775:
14773:
14770:
14768:
14765:
14763:
14760:
14758:
14755:
14754:
14752:
14748:
14742:
14739:
14737:
14734:
14732:
14729:
14725:
14722:
14721:
14720:
14717:
14715:
14712:
14711:
14709:
14705:
14701:
14694:
14689:
14687:
14682:
14680:
14675:
14674:
14671:
14663:
14662:
14657:
14653:
14648:
14644:
14643:
14638:
14634:
14629:
14628:
14624:
14623:
14620:
14617:
14615:
14594:
14592:
14588:
14584:
14580:
14576:
14572:
14569:
14566:
14565:
14561:
14558:
14554:
14550:
14547:
14546:
14542:
14541:
14538:
14534:
14531:
14529:
14526:
14524:
14520:
14519:
14510:
14507:
14506:
14505:
14504:
14497:
14493:
14490:
14486:
14482:
14478:
14472:
14468:
14464:
14460:
14456:
14450:
14445:
14444:
14438:
14434:
14430:
14424:
14420:
14416:
14412:
14410:
14409:0-19-530573-6
14406:
14402:
14400:
14394:
14391:
14387:
14383:
14379:
14373:
14369:
14368:
14363:
14362:Omnès, Roland
14359:
14357:
14353:
14349:
14346:
14342:
14336:
14332:
14328:
14324:
14321:
14320:0-486-43517-2
14317:
14313:
14309:
14308:George Mackey
14306:
14304:
14303:0-08-203204-1
14300:
14296:
14292:
14288:
14282:
14278:
14274:
14270:
14268:
14263:
14257:
14253:
14249:
14245:
14244:E.M. Lifshitz
14241:
14237:
14235:
14231:
14228:
14224:
14220:
14217:
14214:
14210:
14207:
14202:
14198:
14194:
14188:
14184:
14183:
14178:
14174:
14171:
14167:
14163:
14160:
14156:
14150:
14146:
14145:
14140:
14136:
14132:
14128:
14124:
14120:
14116:
14112:
14108:
14104:
14100:
14096:
14095:Everett, Hugh
14092:
14090:
14089:0-691-08131-X
14086:
14082:
14078:
14075:
14071:
14065:
14061:
14060:
14055:
14050:
14046:
14040:
14036:
14032:
14031:Binney, James
14028:
14024:
14018:
14013:
14012:
14006:
14002:
13998:
13992:
13988:
13987:
13986:Quantum Leaps
13982:
13978:
13977:
13976:
13970:
13969:philosophical
13966:
13962:
13958:
13955:
13952:
13948:
13945:
13942:
13938:
13934:
13930:
13926:
13923:
13920:
13916:
13915:0-691-08388-6
13912:
13908:
13907:
13902:
13899:
13895:
13889:
13885:
13884:
13879:
13878:Forshaw, Jeff
13875:
13871:
13869:
13868:0-486-42878-8
13865:
13861:
13857:
13854:
13853:
13852:
13825:
13818:
13811:
13804:
13792:
13788:
13784:
13780:
13774:
13763:
13758:
13753:
13749:
13745:
13744:
13736:
13729:
13721:
13719:0-691-01243-1
13715:
13710:
13709:
13703:
13702:Pais, Abraham
13697:
13689:
13685:
13680:
13675:
13671:
13667:
13663:
13659:
13658:
13653:
13649:
13643:
13636:
13630:
13623:
13617:
13609:
13607:0-19-851791-2
13603:
13599:
13595:
13591:
13590:
13585:
13579:
13571:
13567:
13563:
13557:
13553:
13552:
13547:
13541:
13526:
13522:
13516:
13508:
13507:
13502:
13498:
13491:
13483:
13479:
13475:
13469:
13464:
13463:
13454:
13452:
13437:
13436:
13435:Physics World
13431:
13427:
13421:
13413:
13407:
13403:
13399:
13395:
13387:
13383:
13379:
13376:(in German).
13375:
13374:
13369:
13363:
13354:
13350:
13349:Stachel, John
13346:Reprinted in
13342:
13337:
13333:
13329:
13325:
13321:
13320:
13315:
13308:
13300:
13294:
13290:
13286:
13282:
13278:
13277:Stachel, John
13272:
13257:
13256:
13255:Physics World
13251:
13247:
13241:
13233:
13227:
13223:
13219:
13213:
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13050:
13046:
13042:
13038:
13034:
13033:
13028:
13021:
13013:
13009:
13005:
13003:0-521-64222-1
12999:
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12979:
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12965:
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12950:
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12868:
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12845:
12841:
12834:
12827:
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12815:
12811:
12807:
12803:
12799:
12795:
12791:
12784:
12776:
12772:
12768:
12764:
12760:
12756:
12751:
12746:
12742:
12738:
12731:
12723:
12721:0-691-08131-X
12717:
12714:. p. v.
12713:
12709:
12705:
12704:DeWitt, Bryce
12701:
12700:Cooper, L. N.
12697:
12696:DeWitt, B. S.
12693:
12689:
12688:Everett, Hugh
12683:
12675:
12674:
12669:
12662:
12654:
12650:
12643:
12634:
12629:
12625:
12621:
12620:
12615:
12611:
12605:
12597:
12590:
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12578:
12574:
12570:
12565:
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12556:
12552:
12548:
12544:
12540:
12533:
12525:
12521:
12517:
12513:
12509:
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12490:
12486:
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12474:
12470:
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12456:
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12439:
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12419:
12415:
12411:
12406:
12401:
12397:
12393:
12386:
12378:
12374:
12370:
12366:
12362:
12358:
12354:
12350:
12345:
12340:
12337:(1): 015005.
12336:
12332:
12328:
12321:
12313:
12309:
12305:
12301:
12297:
12293:
12289:
12285:
12278:
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12125:
12121:
12117:
12110:
12102:
12098:
12094:
12090:
12086:
12082:
12077:
12072:
12069:(6): 062116.
12068:
12064:
12057:
12049:
12047:0-262-56003-8
12043:
12039:
12038:
12033:
12027:
12019:
12013:
12009:
12008:
12000:
11992:
11986:
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11849:
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11836:
11835:
11827:
11819:
11813:
11809:
11808:
11800:
11784:
11778:
11764:
11763:
11758:
11754:
11753:Baez, John C.
11748:
11740:
11738:0-7923-2549-4
11734:
11730:
11729:
11724:
11718:
11716:
11714:
11705:
11699:
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11526:
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11475:
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11470:
11462:
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11415:
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11404:
11388:
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11349:
11345:
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11332:
11327:
11323:
11319:
11312:
11304:
11300:
11296:
11292:
11288:
11282:
11278:
11274:
11269:
11264:
11260:
11253:
11245:
11239:
11236:. MIT Press.
11235:
11234:
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11212:
11208:
11204:
11198:
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11188:
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11166:
11164:0-521-49730-2
11160:
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11131:
11127:
11123:
11119:
11111:
11103:
11099:
11095:
11089:
11085:
11081:
11080:
11075:
11071:
11065:
11057:
11055:0-471-16433-X
11051:
11047:
11043:
11037:
11035:
11033:
11031:
11022:
11016:
11012:
11008:
11004:
10997:
10989:
10985:
10980:
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10971:
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10962:
10957:
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10949:
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10938:
10930:
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10922:
10918:
10914:
10910:
10906:
10902:
10898:
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10890:
10883:
10875:
10869:
10865:
10861:
10860:
10852:
10844:
10840:
10834:
10830:
10826:
10819:
10812:
10808:
10806:0-345-40946-9
10802:
10798:
10797:
10792:
10786:
10779:
10777:
10764:
10760:
10756:
10755:Baez, John C.
10750:
10735:
10734:
10729:
10725:
10719:
10717:
10708:
10704:
10700:
10696:
10691:
10686:
10682:
10678:
10677:
10672:
10668:
10662:
10660:
10651:
10645:
10641:
10637:
10632:
10627:
10623:
10619:
10615:
10609:
10607:
10598:
10597:
10592:
10588:
10582:
10568:
10564:
10557:
10549:
10545:
10540:
10535:
10531:
10527:
10523:
10519:
10515:
10508:
10500:
10498:0-13-124405-1
10494:
10490:
10489:
10484:
10478:
10470:
10464:
10460:
10459:
10451:
10443:
10437:
10433:
10432:
10424:
10416:
10410:
10406:
10405:
10397:
10395:
10393:
10384:
10378:
10375:. MIT Press.
10374:
10370:
10364:
10362:
10360:
10358:
10356:
10354:
10352:
10350:
10348:
10346:
10337:
10333:
10329:
10325:
10321:
10317:
10312:
10307:
10304:(7): 071801.
10303:
10299:
10292:
10284:
10280:
10276:
10272:
10268:
10264:
10260:
10256:
10251:
10246:
10243:(2): 023901.
10242:
10238:
10231:
10223:
10219:
10215:
10211:
10207:
10203:
10199:
10195:
10188:
10180:
10176:
10172:
10168:
10164:
10160:
10153:
10138:
10132:
10128:
10127:
10119:
10117:
10115:
10113:
10104:
10100:
10096:
10092:
10088:
10084:
10080:
10076:
10072:
10068:
10064:
10058:
10054:
10037:
10031:
10027:
10023:
10017:
10010:
10006:
10002:
9998:
9994:
9990:
9986:
9980:
9973:
9967:
9963:
9952:
9949:
9947:
9944:
9942:
9939:
9937:
9934:
9932:
9929:
9927:
9924:
9922:
9919:
9918:
9911:
9909:
9905:
9901:
9897:
9893:
9889:
9885:
9881:
9877:
9873:
9869:
9865:
9861:
9857:
9856:David Hilbert
9852:
9850:
9846:
9842:
9838:
9833:
9828:
9826:
9822:
9818:
9814:
9810:
9809:specific heat
9806:
9802:
9798:
9794:
9786:
9782:
9777:
9773:
9770:
9766:
9762:
9758:
9754:
9753:realistically
9749:
9745:
9741:
9718:
9715:
9712:
9709:
9702:
9701:
9700:
9698:
9694:
9690:
9686:
9682:
9677:
9673:
9668:
9666:
9665:J. J. Thomson
9662:
9658:
9654:
9650:
9646:
9642:
9638:
9634:
9630:
9626:
9622:
9621:atomic theory
9618:
9614:
9610:
9605:
9603:
9599:
9595:
9592:
9588:
9584:
9580:
9571:
9567:
9562:
9561:Atomic theory
9558:
9548:
9546:
9542:
9538:
9535:
9531:
9527:
9522:
9519:
9515:
9513:
9509:
9504:
9500:
9496:
9492:
9488:
9484:
9480:
9476:
9472:
9468:
9464:
9460:
9456:
9452:
9449:
9448:complementary
9444:
9440:
9436:
9432:
9429:The views of
9427:
9425:
9421:
9417:
9413:
9409:
9405:
9401:
9400:philosophical
9394:
9388:
9384:
9383:superposition
9371:
9361:
9359:
9358:Planck length
9355:
9351:
9350:spin networks
9347:
9343:
9338:
9336:
9332:
9328:
9324:
9320:
9316:
9312:
9308:
9304:
9303:string theory
9299:
9297:
9293:
9289:
9285:
9275:
9273:
9269:
9265:
9261:
9257:
9253:
9249:
9245:
9241:
9237:
9236:Quantum field
9233:
9231:
9227:
9223:
9204:
9197:
9194:
9189:
9185:
9182:
9175:
9169:
9165:
9161:
9152:
9148:
9147:hydrogen atom
9144:
9140:
9135:
9133:
9129:
9125:
9121:
9117:
9113:
9103:
9101:
9096:
9091:
9089:
9088:absolute zero
9084:
9080:
9076:
9072:
9070:
9069:quantum chaos
9066:
9061:
9059:
9055:
9051:
9047:
9042:
9040:
9036:
9032:
9028:
9023:
9019:
9018:Hilbert space
9002:
8997:
8995:
8990:
8988:
8983:
8982:
8980:
8979:
8969:
8963:
8962:
8956:
8955:
8948:
8945:
8943:
8940:
8938:
8935:
8933:
8930:
8928:
8925:
8923:
8920:
8918:
8915:
8913:
8910:
8908:
8905:
8903:
8902:J. J. Thomson
8900:
8898:
8895:
8893:
8890:
8888:
8885:
8883:
8880:
8878:
8875:
8873:
8870:
8868:
8865:
8863:
8860:
8858:
8855:
8853:
8850:
8848:
8845:
8843:
8840:
8838:
8835:
8833:
8830:
8828:
8825:
8823:
8820:
8818:
8815:
8813:
8810:
8808:
8805:
8803:
8800:
8798:
8795:
8793:
8790:
8788:
8785:
8783:
8780:
8778:
8775:
8773:
8770:
8768:
8765:
8763:
8760:
8758:
8755:
8753:
8750:
8748:
8747:van der Waals
8745:
8743:
8740:
8738:
8735:
8733:
8730:
8728:
8725:
8723:
8720:
8718:
8715:
8713:
8710:
8708:
8705:
8703:
8700:
8698:
8695:
8693:
8690:
8688:
8685:
8683:
8680:
8678:
8675:
8673:
8670:
8668:
8665:
8663:
8660:
8658:
8655:
8653:
8650:
8648:
8645:
8643:
8640:
8638:
8635:
8633:
8630:
8628:
8625:
8623:
8620:
8618:
8615:
8614:
8608:
8607:
8600:
8597:
8595:
8592:
8590:
8587:
8585:
8582:
8580:
8577:
8575:
8572:
8570:
8567:
8565:
8562:
8560:
8557:
8555:
8552:
8550:
8547:
8545:
8542:
8540:
8537:
8535:
8532:
8530:
8527:
8525:
8522:
8520:
8517:
8515:
8512:
8510:
8507:
8505:
8502:
8500:
8497:
8495:
8492:
8488:
8485:
8483:
8480:
8478:
8475:
8474:
8473:
8470:
8468:
8465:
8463:
8460:
8458:
8455:
8453:
8450:
8448:
8445:
8444:
8438:
8437:
8430:
8427:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8395:
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8371:
8365:
8364:
8357:
8354:
8352:
8349:
8347:
8344:
8342:
8339:
8337:
8334:
8332:
8329:
8327:
8324:
8322:
8319:
8317:
8314:
8312:
8309:
8307:
8304:
8302:
8299:
8298:
8292:
8291:
8287:
8283:
8265:
8262:
8258:
8253:
8249:
8244:
8241:
8237:
8230:
8225:
8222:
8218:
8191:
8183:
8179:
8167:
8164:
8160:
8152:
8149:
8140:
8132:
8128:
8113:
8103:
8102:
8099:
8096:
8095:
8087:
8085:
8081:
8077:
8073:
8069:
8065:
8061:
8057:
8053:
8049:
8045:
8041:
8037:
8032:
8030:
8026:
8022:
8018:
8014:
8010:
8004:
7994:
7957:
7953:
7949:
7946:
7940:
7934:
7931:
7925:
7919:
7909:
7907:
7888:
7883:
7870:
7865:
7861:
7857:
7852:
7837:
7833:
7829:
7826:
7823:
7820:
7815:
7811:
7799:
7793:
7787:
7780:
7766:
7761:
7748:
7743:
7739:
7735:
7730:
7715:
7711:
7707:
7704:
7701:
7698:
7693:
7689:
7677:
7671:
7665:
7658:
7657:
7656:
7639:
7634:
7625:
7621:
7608:
7605:
7595:
7591:
7578:
7575:
7572:
7566:
7559:
7555:
7545:
7541:
7537:
7534:
7529:
7525:
7521:
7518:
7515:
7508:
7507:
7506:
7492:
7472:
7452:
7443:
7404:
7390:
7386:
7380:
7373:
7368:
7362:
7357:
7354:
7332:
7326:
7322:
7300:
7294:
7290:
7268:
7262:
7257:
7250:
7245:
7239:
7231:
7227:
7222:
7219:
7211:
7206:
7192:
7189:
7184:
7174:
7166:
7161:
7151:
7126:
7123:
7117:
7114:
7111:
7088:
7085:
7082:
7060:
7056:
7052:
7049:
7044:
7040:
7036:
7033:
7030:
7008:
7002:
6995:
6989:
6984:
6979:
6975:
6952:
6946:
6939:
6933:
6928:
6923:
6919:
6896:
6886:
6883:
6873:
6871:
6867:
6863:
6854:
6845:
6843:
6824:
6820:
6814:
6811:
6806:
6803:
6799:
6795:
6789:
6784:
6780:
6772:
6771:
6770:
6753:
6749:
6742:
6738:
6734:
6730:
6726:
6717:
6713:
6709:
6703:
6699:
6689:
6685:
6680:
6674:
6666:
6663:
6657:
6651:
6643:
6639:
6631:
6630:
6629:
6628:
6624:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6568:
6567:
6551:
6547:
6543:
6533:
6530:
6522:
6516:
6512:
6508:
6497:
6490:
6486:
6482:
6479:
6473:
6469:
6465:
6460:
6456:
6452:
6447:
6438:
6433:
6430:
6424:
6419:
6412:
6409:
6403:
6399:
6394:
6388:
6382:
6374:
6370:
6362:
6361:
6360:
6359:are given by
6358:
6339:
6334:
6330:
6324:
6320:
6316:
6311:
6308:
6303:
6297:
6291:
6284:
6283:
6282:
6276:
6272:
6268:
6264:
6260:
6256:
6252:
6248:
6244:
6240:
6235:
6230:
6220:
6218:
6214:
6210:
6206:
6201:
6196:
6183:
6175:
6171:
6167:
6164:
6157:
6153:
6147:
6143:
6136:
6128:
6124:
6120:
6117:
6110:
6106:
6100:
6096:
6090:
6086:
6079:
6074:
6070:
6061:
6047:
6024:
6021:
6018:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5990:
5986:
5983:
5977:
5974:
5967:
5966:
5965:
5951:
5931:
5928:
5908:
5905:
5899:
5896:
5890:
5887:
5867:
5847:
5824:
5818:
5815:
5809:
5806:
5803:
5800:
5797:
5794:
5788:
5782:
5775:
5774:
5773:
5759:
5756:
5753:
5733:
5730:
5727:
5704:
5701:
5695:
5689:
5686:
5683:
5680:
5674:
5668:
5665:
5662:
5659:
5656:
5653:
5647:
5641:
5634:
5633:
5632:
5618:
5615:
5612:
5592:
5572:
5569:
5566:
5546:
5543:
5540:
5520:
5500:
5497:
5494:
5491:
5467:
5461:
5458:
5452:
5449:
5446:
5443:
5437:
5434:
5428:
5425:
5422:
5419:
5413:
5407:
5400:
5399:
5398:
5396:
5374:
5371:
5364:
5360:
5354:
5350:
5343:
5340:
5333:
5330:
5327:
5324:
5320:
5316:
5313:
5308:
5305:
5302:
5298:
5294:
5291:
5285:
5279:
5272:
5271:
5270:
5267:
5253:
5233:
5210:
5207:
5204:
5199:
5194:
5184:
5174:
5171:
5167:
5158:
5157:
5156:
5154:
5132:
5129:
5125:
5117:
5114:
5111:
5106:
5096:
5085:
5084:
5083:
5066:
5063:
5060:
5057:
5049:
5045:
5041:
5036:
5031:
5027:
5017:
5014:
5008:
5004:
4998:
4991:
4990:
4989:
4975:
4967:
4963:
4956:
4947:
4938:
4934:
4920:
4900:
4877:
4870:
4864:
4860:
4856:
4850:
4846:
4839:
4834:
4831:
4825:
4819:
4816:
4813:
4801:
4791:
4790:
4789:
4767:
4764:
4758:
4754:
4748:
4744:
4736:
4731:
4728:
4723:
4718:
4712:
4709:
4706:
4700:
4693:
4692:
4691:
4689:
4684:
4667:
4664:
4661:
4655:
4632:
4629:
4626:
4614:
4591:
4585:
4582:
4557:
4551:
4548:
4541:
4537:
4533:
4527:
4524:
4521:
4515:
4511:
4503:
4484:
4481:
4468:
4462:
4459:
4452:
4448:
4444:
4438:
4435:
4432:
4426:
4422:
4415:
4412:
4409:
4397:
4381:
4377:
4370:
4367:
4363:
4358:
4352:
4349:
4346:
4340:
4333:
4332:
4331:
4314:
4306:
4302:
4298:
4292:
4288:
4279:
4276:
4270:
4266:
4260:
4257:
4252:
4248:
4241:
4238:
4234:
4229:
4226:
4219:
4218:
4217:
4210:
4205:
4200:
4199:Free particle
4193:Free particle
4185:
4183:
4179:
4176:
4172:
4168:
4152:
4132:
4112:
4092:
4072:
4052:
4032:
4012:
3992:
3984:
3968:
3948:
3928:
3905:
3899:
3879:
3853:
3849:
3846:
3843:
3840:
3836:
3832:
3826:
3820:
3812:
3796:
3786:
3776:
3774:
3770:
3766:
3762:
3758:
3755:(invented by
3754:
3750:
3746:
3736:
3734:
3728:
3726:
3721:
3708:
3706:
3686:
3680:
3676:
3672:
3667:
3663:
3659:
3654:
3650:
3646:
3641:
3637:
3632:
3624:
3620:
3610:
3609:
3608:
3594:
3572:
3568:
3545:
3541:
3520:
3498:
3494:
3471:
3467:
3444:
3441:
3409:
3404:
3400:
3396:
3391:
3387:
3379:
3378:
3377:
3361:
3357:
3334:
3330:
3306:
3301:
3289:
3284:
3272:
3267:
3264:
3248:
3247:
3246:
3230:
3199:
3174:
3164:
3148:
3144:
3140:
3132:
3113:
3098:
3095:
3073:
3069:
3060:
3040:
3036:
3027:
3008:
3004:
2990:
2987:
2984:
2970:
2963:
2960:
2954:
2949:
2945:
2939:
2935:
2927:
2926:
2925:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2884:
2881:
2878:
2868:
2867:
2866:
2864:
2848:
2828:
2805:
2800:
2797:
2792:
2787:
2783:
2777:
2773:
2765:
2764:
2763:
2746:
2739:
2734:
2731:
2728:
2723:
2719:
2714:
2710:
2706:
2700:
2695:
2691:
2683:
2682:
2681:
2664:
2655:
2650:
2647:
2644:
2639:
2635:
2630:
2626:
2622:
2614:
2609:
2605:
2597:
2596:
2595:
2593:
2577:
2557:
2534:
2528:
2525:
2513:
2507:
2498:
2485:
2484:
2483:
2481:
2459:
2430:
2412:
2410:
2406:
2401:
2399:
2395:
2391:
2390:hydrogen atom
2387:
2383:
2379:
2375:
2370:
2368:
2364:
2362:
2357:
2353:
2349:
2341:
2337:
2333:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2288:
2284:
2267:
2261:
2238:
2232:
2224:
2223:deterministic
2220:
2198:
2194:
2191:
2188:
2185:
2181:
2177:
2171:
2165:
2158:The operator
2142:
2136:
2130:
2121:
2117:
2114:
2111:
2108:
2104:
2100:
2094:
2088:
2081:
2080:
2079:
2076:
2074:
2070:
2051:
2043:
2027:
2019:
2015:
1999:
1976:
1970:
1964:
1961:
1958:
1952:
1946:
1940:
1925:
1918:
1917:
1916:
1908:
1906:
1902:
1898:
1894:
1890:
1886:
1882:
1878:
1877:probabilistic
1857:
1852:
1848:
1844:
1841:
1825:
1820:
1816:
1789:
1778:
1762:
1753:
1751:
1733:
1729:
1705:
1700:
1696:
1692:
1689:
1660:
1635:
1622:
1619:
1610:
1579:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1538:
1522:
1507:
1478:
1474:
1465:
1461:
1457:
1441:
1436:
1433:
1429:
1408:
1388:
1385:
1379:
1376:
1373:
1339:
1338:Hilbert space
1335:
1319:
1309:
1299:
1297:
1293:
1289:
1283:
1281:
1277:
1273:
1269:
1264:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1233:
1231:
1227:
1223:
1219:
1215:
1210:
1206:
1201:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1170:
1166:
1162:
1158:
1153:
1151:
1146:
1141:
1139:
1134:
1130:
1126:
1122:
1118:
1112:
1110:
1106:
1102:
1098:
1087:
1085:
1081:
1080:wave function
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1024:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
981:
979:
975:
971:
967:
963:
958:
956:
952:
948:
944:
940:
936:
932:
928:
924:
913:
908:
906:
901:
899:
894:
893:
891:
890:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
653:
646:
645:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
612:Quantum chaos
610:
608:
605:
603:
600:
598:
595:
593:
590:
589:
583:
582:
574:
571:
569:
568:Transactional
566:
564:
561:
559:
558:Quantum logic
556:
554:
551:
549:
546:
540:
537:
536:
535:
532:
531:
530:
527:
525:
522:
520:
517:
515:
512:
510:
507:
505:
502:
501:
497:
492:
491:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
459:
452:
451:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
414:
410:
407:
406:
400:
399:
391:
388:
386:
383:
381:
378:
377:
371:
368:
367:
366:
363:
362:
358:
355:
353:
350:
348:
345:
343:
340:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
309:
302:
301:
291:
288:
287:
286:
285:Wave function
283:
281:
278:
276:
273:
271:
268:
266:
265:Superposition
263:
261:
258:
256:
253:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
222:
215:
214:
206:
203:
201:
198:
197:
193:
190:
188:
185:
183:
180:
179:
173:
172:
167:
164:
162:
159:
157:
154:
153:
152:
151:
147:
114:
108:
91:
88:
84:
76:
69:
68:
65:
62:
61:
57:
56:
49:
45:
41:
35:
30:
19:
15721:Astrophysics
15670:
15535:Experimental
15169:Quantum mind
15081:Franck–Hertz
14943:Klein–Gordon
14892:Formulations
14885:Formulations
14814:Interference
14804:Entanglement
14782:Ground state
14777:Energy level
14750:Fundamentals
14714:Introduction
14699:
14659:
14640:
14562:
14503:On Wikibooks
14502:
14501:
14495:
14466:
14442:
14421:. Springer.
14418:
14396:
14366:
14351:
14330:
14311:
14294:
14276:
14247:
14222:
14212:
14180:
14169:
14166:K. Hentschel
14143:
14102:
14098:
14080:
14077:Bryce DeWitt
14058:
14034:
14010:
13985:
13974:
13965:cosmological
13960:
13950:
13937:trigonometry
13928:
13904:
13882:
13859:
13850:
13831:. Retrieved
13824:the original
13810:
13802:
13795:. Retrieved
13773:
13747:
13741:
13728:
13707:
13696:
13661:
13655:
13642:
13634:
13629:
13621:
13616:
13588:
13578:
13550:
13540:
13528:. Retrieved
13524:
13515:
13504:
13490:
13461:
13439:. Retrieved
13433:
13426:Ball, Philip
13420:
13393:
13377:
13371:
13362:
13352:
13323:
13317:
13307:
13280:
13271:
13259:. Retrieved
13253:
13246:Kragh, Helge
13240:
13221:
13212:
13200:. Retrieved
13186:
13167:
13158:
13138:
13131:
13123:
13109:
13103:
13093:30 September
13091:. Retrieved
13076:
13069:
13036:
13030:
13020:
12992:
12978:
12967:
12957:
12916:
12910:
12897:
12870:
12867:Kent, Adrian
12861:
12853:
12842:. Springer.
12839:
12826:
12793:
12789:
12783:
12740:
12736:
12730:
12707:
12682:
12671:
12661:
12652:
12642:
12623:
12617:
12604:
12595:
12589:
12546:
12542:
12532:
12507:
12503:
12497:
12454:
12448:
12438:
12395:
12391:
12385:
12334:
12330:
12320:
12287:
12283:
12277:
12250:
12244:
12217:
12213:
12203:
12173:(1): 26–57.
12170:
12166:
12156:
12123:
12119:
12109:
12066:
12062:
12056:
12036:
12026:
12006:
11999:
11980:
11971:
11952:
11942:
11930:. Retrieved
11924:
11911:
11899:. Retrieved
11889:
11838:
11832:
11826:
11806:
11799:
11787:. Retrieved
11777:
11766:. Retrieved
11760:
11747:
11727:
11723:Peres, Asher
11693:
11687:
11675:. Retrieved
11671:
11661:
11649:. Retrieved
11621:
11617:
11607:
11580:
11571:
11528:
11522:
11516:
11473:
11467:
11461:
11438:
11428:
11409:
11403:
11391:. Retrieved
11389:(8): 683–695
11386:
11380:
11364:
11321:
11317:
11311:
11258:
11252:
11232:
11190:
11153:
11121:
11117:
11110:
11078:
11070:Landau, L.D.
11064:
11045:
11002:
10996:
10951:
10947:
10937:
10899:(1): 71–77.
10896:
10892:
10882:
10858:
10851:
10843:the original
10828:
10818:
10810:
10795:
10785:
10775:
10773:
10766:. Retrieved
10762:
10749:
10737:. Retrieved
10731:
10680:
10674:
10617:
10595:
10587:Bub, Jeffrey
10581:
10570:. Retrieved
10566:
10556:
10521:
10517:
10507:
10487:
10477:
10457:
10450:
10430:
10423:
10403:
10372:
10301:
10297:
10291:
10240:
10236:
10230:
10197:
10193:
10187:
10162:
10158:
10152:
10140:. Retrieved
10125:
10070:
10066:
10057:
10036:
10016:
10011:, pp. 9–13).
10008:
10000:
9992:
9979:
9966:
9853:
9829:
9821:diamagnetism
9790:
9747:
9739:
9737:
9696:
9688:
9680:
9669:
9661:cathode rays
9611:research by
9606:
9594:Thomas Young
9579:Robert Hooke
9575:
9539:
9533:
9529:
9523:
9516:
9491:Nathan Rosen
9453:
9442:
9438:
9428:
9397:
9339:
9300:
9281:
9234:
9136:
9109:
9092:
9073:
9062:
9043:
9039:quantization
9015:
8579:Astrophysics
8559:Neurophysics
8471:
8462:Mathematical
8452:Experimental
8066:such as the
8033:
8006:
7997:Applications
7910:
7903:
7654:
7444:
7207:
7075:for complex
6874:
6859:
6842:bound states
6839:
6768:
6619:
6617:
6354:
6280:
6274:
6271:energy level
6213:flash memory
6197:
6062:
6039:
5839:
5719:
5483:
5392:
5268:
5225:
5150:
5081:
4965:
4961:
4958:
4935:
4892:
4787:
4685:
4499:
4329:
4214:
4167:Emmy Noether
3982:
3810:
3788:
3742:
3729:
3709:
3702:
3424:
3321:
3170:
3023:
2923:
2820:
2761:
2679:
2549:
2418:
2402:
2371:
2360:
2345:
2335:
2307:
2303:
2299:
2295:
2157:
2077:
2018:total energy
2012:denotes the
1991:
1914:
1754:
1546:self-adjoint
1539:
1311:
1296:group theory
1284:
1275:
1265:
1240:
1234:
1226:tunnel diode
1202:
1167:, such as a
1154:
1142:
1113:
1093:
1025:
982:
976:(atomic and
959:
938:
922:
921:
467:Klein–Gordon
403:Formulations
240:Energy level
235:Entanglement
218:Fundamentals
205:Interference
156:Introduction
63:
29:
15889:Mathematics
15853:Electronics
15624:Statistical
15540:Theoretical
15517:Engineering
15427:EPR paradox
15207:Quantum bus
15076:Double-slit
15054:Experiments
15020:Many-worlds
14958:Schrödinger
14922:Phase space
14912:Schrödinger
14902:Interaction
14859:Uncertainty
14829:Nonlocality
14824:Measurement
14819:Decoherence
14809:Hamiltonian
14415:Shankar, R.
14267:Online copy
14240:L.D. Landau
14005:Bohm, David
13833:24 November
13797:24 November
13596:. pp.
13554:. Pearson.
13380:: 121–128.
13261:12 December
13218:Kuhn, T. S.
12610:Bell, J. S.
11901:16 December
10791:Sagan, Carl
10768:19 December
10142:19 December
9989:transistors
9908:superfluids
9900:macroscopic
9864:measurement
9805:Peter Debye
9613:John Dalton
9501:. In 1964,
9463:determinism
9331:vibrational
9264:Abdus Salam
8842:von Neumann
8707:Schrödinger
8539:Statistical
8457:Theoretical
8409:Information
8282:Schrödinger
7906:phase shift
7023:, that is,
6357:eigenstates
5226:with state
4688:wave packet
4502:plane waves
4209:wave packet
2398:closed form
2392:. Even the
2014:Hamiltonian
1554:eigenvector
966:macroscopic
856:von Neumann
841:Schrödinger
617:EPR paradox
548:Many-worlds
482:Schrödinger
437:Schrödinger
432:Phase-space
422:Interaction
327:Double-slit
305:Experiments
280:Uncertainty
250:Nonlocality
245:Measurement
230:Decoherence
200:Hamiltonian
15741:Geophysics
15731:Biophysics
15575:Analytical
15528:Approaches
15355:Extensions
15189:Technology
15035:Relational
14986:Copenhagen
14897:Heisenberg
14844:Tunnelling
14707:Background
14625:Philosophy
14537:Hans Bethe
14209:Max Jammer
13874:Cox, Brian
13530:1 February
13441:2024-03-23
13012:1151058062
12988:Wolf, Emil
12457:(2): 125.
11932:10 October
11768:2024-03-23
11731:. Kluwer.
11331:1911.06282
10961:1801.09977
10954:(4): 040.
10739:8 February
10572:2024-06-07
10311:2209.13084
10250:1501.04899
10049:References
9839:developed
9570:Max Planck
9524:Everett's
9431:Niels Bohr
8959:Categories
8752:Heisenberg
8712:de Broglie
8662:Rutherford
8657:Sommerfeld
8611:Scientists
8554:Biophysics
8404:Randomness
8341:Paul Dirac
8311:Niels Bohr
8301:Max Planck
8060:transistor
4171:Lagrangian
3749:Paul Dirac
3714:or system
2863:commutator
2594:, we have
2388:, and the
1562:eigenvalue
1558:eigenstate
1466:functions
1336:) complex
1072:Paul Dirac
1056:Niels Bohr
1044:1905 paper
1032:Max Planck
851:Sommerfeld
766:Heisenberg
761:Gutzwiller
701:de Broglie
649:Scientists
563:Relational
514:Copenhagen
417:Heisenberg
275:Tunnelling
176:Background
15841:Chemistry
15829:Astronomy
15691:Molecular
15592:Acoustics
15585:Continuum
15580:Celestial
15570:Newtonian
15557:Classical
15500:Divisions
15061:Bell test
14931:Equations
14757:Born rule
14575:Chemistry
13688:0002-9505
13570:929609283
13202:18 August
13164:Mehra, J.
13061:0031-921X
12984:Born, Max
12941:0015-9018
12880:0905.0624
12818:121747282
12581:1355-2198
12559:CiteSeerX
12464:0706.2661
12405:1301.1069
12369:0034-6861
12344:1407.2930
12269:743037461
12187:1063-6145
12140:0031-8248
12101:119273840
12076:1109.6462
11848:1205.5368
11789:18 August
11599:442351498
11563:119261326
11538:1005.3976
11356:208006050
11295:973404322
11268:1106.1445
11211:844974180
11146:120024263
10988:2542-4653
10921:1050-2947
10699:0028-0836
10631:1302.1864
10222:203638258
10103:119896026
10095:1434-6001
9851:in 1927.
9797:heuristic
9779:The 1927
9719:ν
9693:frequency
9503:John Bell
9439:temporary
9354:spin foam
9190:ϵ
9186:π
9162:−
9095:molecules
9079:coherence
8947:Uhlenbeck
8937:Zeilinger
8897:Gell-Mann
8627:Becquerel
8584:Cosmology
8266:ν
8263:μ
8254:κ
8245:ν
8242:μ
8234:Λ
8226:ν
8223:μ
8198:⟩
8180:ψ
8156:ℏ
8147:⟩
8129:ψ
8117:^
7981:Φ
7978:Δ
7880:Φ
7877:Δ
7871:
7843:⟩
7834:ψ
7812:ψ
7808:⟨
7758:Φ
7755:Δ
7749:
7721:⟩
7712:ψ
7690:ψ
7686:⟨
7618:Φ
7615:Δ
7609:
7588:Φ
7585:Δ
7579:
7573:−
7552:Φ
7549:Δ
7526:ψ
7430:Φ
7427:Δ
7397:Φ
7394:Δ
7175:β
7152:α
7121:⟩
7118:ψ
7112:ψ
7109:⟨
7089:β
7083:α
7057:ψ
7053:β
7041:ψ
7037:α
7031:ψ
6976:ψ
6920:ψ
6887:∈
6884:ψ
6796:ω
6793:ℏ
6735:−
6664:−
6600:…
6538:ℏ
6534:ω
6509:⋅
6501:ℏ
6483:ω
6474:−
6466:⋅
6442:ℏ
6439:π
6434:ω
6420:⋅
6371:ψ
6321:ω
6255:real part
6097:π
6087:ℏ
6022:…
5987:π
5952:π
5891:
5868:ψ
5840:in which
5810:
5783:ψ
5690:
5669:
5642:ψ
5593:ψ
5453:
5429:
5408:ψ
5393:or, from
5351:ℏ
5325:−
5280:ψ
5234:ψ
5188:^
5121:ℏ
5115:−
5100:^
5064:ψ
5037:ψ
5005:ℏ
4999:−
4851:−
4835:π
4805:^
4802:ψ
4749:−
4729:π
4701:ψ
4656:ψ
4618:^
4615:ψ
4589:ℏ
4534:ℏ
4528:−
4445:ℏ
4439:−
4401:^
4398:ψ
4390:∞
4385:∞
4382:−
4378:∫
4371:π
4341:ψ
4267:ℏ
4261:−
3983:conserved
3858:ℏ
3841:−
3811:generator
3705:entangled
3677:ϕ
3673:⊗
3664:ϕ
3651:ψ
3647:⊗
3638:ψ
3569:ϕ
3542:ψ
3495:ϕ
3468:ψ
3401:ψ
3397:⊗
3388:ψ
3358:ψ
3331:ψ
3290:⊗
3145:ℏ
3141:−
3111:∂
3107:∂
3102:ℏ
3096:−
3045:ℏ
2955:≥
2946:σ
2936:σ
2900:−
2798:ℏ
2793:≥
2784:σ
2774:σ
2724:−
2692:σ
2640:−
2606:σ
2532:ℏ
2517:^
2502:^
2463:^
2434:^
2334:and take
2332:quantized
2324:frequency
2262:ψ
2233:ψ
2203:ℏ
2186:−
2131:ψ
2126:ℏ
2109:−
2089:ψ
2055:ℏ
2028:ℏ
1965:ψ
1947:ψ
1938:∂
1934:∂
1929:ℏ
1901:entangled
1861:⟩
1858:ψ
1853:λ
1842:ψ
1839:⟨
1826:ψ
1821:λ
1793:→
1790:λ
1763:λ
1734:λ
1709:⟩
1706:ψ
1701:λ
1690:ψ
1687:⟨
1664:→
1661:λ
1626:⟩
1623:ψ
1614:→
1611:λ
1605:⟨
1580:λ
1570:Born rule
1550:operators
1548:) linear
1542:Hermitian
1442:ψ
1437:α
1409:ψ
1383:⟩
1380:ψ
1374:ψ
1371:⟨
1334:separable
1320:ψ
1280:Bell test
1198:molecules
1190:electrons
1174:interfere
1121:Born rule
1009:particles
989:quantized
978:subatomic
881:Zeilinger
726:Ehrenfest
455:Equations
132:⟩
129:Ψ
118:^
106:⟩
103:Ψ
80:ℏ
51:electron.
15935:Category
15450:Category
15244:Timeline
14996:Ensemble
14976:Bayesian
14869:Collapse
14741:Glossary
14724:Timeline
14597:5
14494:(2003),
14485:34661512
14465:(1996).
14439:(2013).
14417:(1994).
14395:, 2006.
14386:39849482
14364:(1999).
14350:, 1966.
14329:(1998).
14310:(2004).
14275:(2002).
14246:(1977).
14230:Archived
14221:, 2004.
14211:, 1966.
14201:40251748
14179:(2004).
14127:17178479
14056:(1985).
14007:(1989).
13983:(2009).
13959:, 2000.
13927:, 2004.
13903:, 1985.
13880:(2011).
13858:(1987).
13791:Archived
13762:Archived
13704:(1997).
13622:Synthese
13586:(1996).
13482:66-29628
13220:(1978).
13196:Archived
12990:(1999).
12949:17217776
12489:32755624
12430:55537196
12377:34901303
12312:22433857
12195:57559199
12034:(1967).
11979:(2009).
11881:14712017
11873:23005618
11725:(1993).
11651:31 March
11646:19113681
11579:(2006).
11508:13963928
11372:(1987).
11324:: 1–57.
11189:(2010).
11076:(1977).
10793:(1996).
10707:26503054
10589:(2019).
10548:24039543
10485:(1995).
10371:(2022).
10336:36867820
10283:18463042
10275:25582917
10063:Born, M.
9914:See also
9785:Brussels
9663:, which
9609:chemical
9591:polymath
9467:locality
9335:graviton
9242:and the
8942:Goudsmit
8932:Lawrence
8927:Chadwick
8922:Shockley
8882:'t Hooft
8832:Millikan
8817:Poincaré
8807:Lemaître
8677:Einstein
8519:Particle
8441:Branches
8424:Particle
8374:Topology
8368:Concepts
8316:Max Born
8295:Founders
8074:such as
8017:neutrons
6625:are the
4188:Examples
4178:symmetry
3981:will be
2999:⟩
2977:⟨
2735:⟩
2729:⟨
2720:⟩
2707:⟨
2651:⟩
2645:⟨
2636:⟩
2623:⟨
2316:acoustic
2290:Fig. 1:
1777:collapse
1721:, where
1650:, where
1150:momentum
1129:electron
1125:Max Born
1101:accuracy
1068:Max Born
1001:momentum
806:Millikan
731:Einstein
716:Davisson
671:Blackett
656:Aharonov
524:Ensemble
504:Bayesian
409:Overview
290:Collapse
270:Symmetry
161:Glossary
48:electron
15913:Science
15901:Physics
15815:Portals
15775:Related
15659:General
15654:Special
15512:Applied
15403:Related
15382:History
15121:Science
14953:Rydberg
14719:History
14658:(ed.).
14639:(ed.).
14612:
14579:Physics
14107:Bibcode
13933:algebra
13666:Bibcode
13503:(ed.).
13382:Bibcode
13328:Bibcode
13041:Bibcode
12921:Bibcode
12885:Bibcode
12798:Bibcode
12775:1921913
12755:Bibcode
12551:Bibcode
12512:Bibcode
12469:Bibcode
12410:Bibcode
12349:Bibcode
12222:Bibcode
12148:9454552
12081:Bibcode
11853:Bibcode
11626:Bibcode
11543:Bibcode
11488:Bibcode
11393:13 June
11336:Bibcode
11303:2515538
11126:Bibcode
11102:2284121
10966:Bibcode
10929:9908895
10901:Bibcode
10636:Bibcode
10539:3768233
10316:Bibcode
10255:Bibcode
10202:Bibcode
10167:Bibcode
10075:Bibcode
10041:theory.
9890:during
9681:quantum
9551:History
9319:strings
9139:charged
9118:or the
9025:is the
8917:Bardeen
8887:Veltman
8857:Feynman
8827:Penrose
8822:Wheeler
8812:Thomson
8797:Hawking
8772:Hilbert
8767:Moseley
8727:Compton
8702:Kramers
8682:Wilczek
8632:Lorentz
8622:Röntgen
8524:Nuclear
8447:Applied
8414:Entropy
8021:photons
8013:protons
6277:energy.
4966:outside
4125:, then
3765:Feynman
3607:, then
2363:orbital
2219:unitary
846:Simmons
836:Rydberg
801:Moseley
781:Kramers
771:Hilbert
756:Glauber
751:Feynman
736:Everett
706:Compton
477:Rydberg
166:History
46:of the
15865:Energy
15686:Atomic
15641:Modern
15491:Major
15096:Popper
14581:. See
14483:
14473:
14451:
14425:
14407:
14384:
14374:
14356:online
14337:
14318:
14301:
14283:
14258:
14199:
14189:
14151:
14125:
14087:
14066:
14041:
14019:
13993:
13939:, and
13913:
13890:
13866:
13716:
13686:
13604:
13568:
13558:
13480:
13470:
13408:
13295:
13228:
13174:
13146:
13116:
13084:
13059:
13010:
13000:
12947:
12939:
12846:
12816:
12773:
12718:
12579:
12561:
12487:
12428:
12375:
12367:
12310:
12304:986572
12302:
12267:
12257:
12214:Nature
12193:
12185:
12146:
12138:
12099:
12044:
12014:
11987:
11959:
11879:
11871:
11814:
11735:
11700:
11677:18 May
11644:
11597:
11587:
11561:
11506:
11449:
11416:
11354:
11301:
11293:
11283:
11240:
11209:
11199:
11161:
11144:
11100:
11090:
11052:
11017:
10986:
10927:
10919:
10870:
10835:
10803:
10705:
10697:
10676:Nature
10646:
10546:
10536:
10495:
10465:
10438:
10411:
10379:
10334:
10281:
10273:
10220:
10133:
10101:
10093:
10028:, and
10005:lasers
9878:, and
9823:, and
9738:where
9722:
9414:, and
9327:charge
9256:gluons
9252:quarks
8872:Lenard
8792:Wigner
8782:Jordan
8762:Zeeman
8737:Walton
8637:Planck
8617:Witten
8514:Atomic
8394:Matter
8389:Energy
8054:, the
6868:, the
6618:where
6243:spring
5585:where
4962:inside
2861:. The
2394:helium
2384:, the
2380:, the
2367:Fig. 1
2326:. The
2320:energy
1196:, and
1178:photon
997:energy
953:, and
931:nature
927:theory
876:Zeeman
871:Wigner
821:Planck
791:Landau
776:Jordan
427:Matrix
357:Popper
15925:Stars
15006:Local
14948:Pauli
14938:Dirac
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2991:B
2988:,
2985:A
2982:[
2971:|
2964:2
2961:1
2950:B
2940:A
2909:,
2906:A
2903:B
2897:B
2894:A
2891:=
2888:]
2885:B
2882:,
2879:A
2876:[
2849:B
2829:A
2806:.
2801:2
2788:P
2778:X
2747:.
2740:2
2732:P
2715:2
2711:P
2701:=
2696:P
2665:,
2656:2
2648:X
2631:2
2627:X
2615:=
2610:X
2578:P
2558:X
2535:.
2529:i
2526:=
2523:]
2514:P
2508:,
2499:X
2493:[
2460:P
2431:X
2365:(
2361:s
2308:d
2304:p
2300:s
2296:n
2271:)
2268:t
2265:(
2242:)
2239:0
2236:(
2199:/
2195:t
2192:H
2189:i
2182:e
2178:=
2175:)
2172:t
2169:(
2166:U
2143:.
2140:)
2137:0
2134:(
2122:/
2118:t
2115:H
2112:i
2105:e
2101:=
2098:)
2095:t
2092:(
2052:i
2000:H
1977:.
1974:)
1971:t
1968:(
1962:H
1959:=
1956:)
1953:t
1950:(
1941:t
1926:i
1849:P
1845:,
1831:/
1817:P
1730:P
1697:P
1693:,
1636:2
1631:|
1620:,
1601:|
1523:2
1518:C
1492:)
1488:C
1484:(
1479:2
1475:L
1434:i
1430:e
1389:1
1386:=
1377:,
1349:H
1015:(
911:e
904:t
897:v
125:|
115:H
109:=
99:|
92:t
89:d
85:d
77:i
36:.
20:)
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