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10815:"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.
9120:. 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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9284:, 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:
9972:, 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.
13807:...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.
9022:
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
2421:
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
9296:" (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.
9226:, 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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6862:(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
9633:. 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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9035:. 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
4338:
2360:, 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
1989:
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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
6833:
3773:, 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
9319:. 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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2413:. 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.
9519:
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.
9095:
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".
7509:
9845:. 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
2155:
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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9346:. 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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7991:. 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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12873:(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
9896:, 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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2217:
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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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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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2409:, 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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2352:. 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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2012:
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Schrödinger, Erwin (1980) . Trimmer, John (ed.). ""Die gegenwärtige Situation in der Quantenmechanik."" [The Present Situation in Quantum Mechanics].
11925:"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"
8009:. 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,
5273:
4224:
1811:
9968:
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
2296:
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
9801:
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.
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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.
7348:
9757:, 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
4492:{\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,}
3729:(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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13637:
D. Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories",
15737:
12394:
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:
408:
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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 "
9256:. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as
12792:
Ballentine, L. E. (1973). "Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation".
10620:(2015). "Quantum Information Science: Emerging No More". In Kelley, Paul; Agrawal, Govind; Bass, Mike; Hecht, Jeff; Stroud, Carlos (eds.).
9685:, 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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13909:
11195:
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8080:. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule
909:
13787:"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4"
5968:
2312:, ...). 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
15439:
9828:
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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14407:. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics.
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9744:. 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
15705:
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14693:
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389:
7281:, 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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14358:(Vol. I), English translation from French by G.M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.
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7643:{\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}
6287:
495:
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3987:: 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".
13790:
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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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9467:. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the
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2672:{\displaystyle \sigma _{X}={\textstyle {\sqrt {\left\langle X^{2}\right\rangle -\left\langle X\right\rangle ^{2}}}},}
155:
33:
17:
9645:'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure.
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1043:
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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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3737:. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.
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9475:. 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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2015:
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902:
351:
331:
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9358:, 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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13935:, 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.),
11377:
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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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9814:
9810:
9554:
9538:
9331:
state of the string. In string theory, one of the many vibrational states of the string corresponds to the
9223:
8989:
5776:
4915:
smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making
3027:
2293:
1892:
1776:
1132:
1027:
567:
562:
533:
384:
165:
14590:
13287:. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. 69–83.
10069:(1926). "Zur Quantenmechanik der Stoßvorgänge" [On the Quantum Mechanics of Collision Processes].
9042:
When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was
3818:
3026:
Another consequence of the canonical commutation relation is that the position and momentum operators are
2422:
momentum are observables, meaning that they are represented by Hermitian operators. The position operator
2163:
1568:. When an observable is measured, the result will be one of its eigenvalues with probability given by the
15790:
15198:
15105:
13660:
9877:
9047:
6228:
4609:
3695:{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}
3430:
3132:
2379:
1748:
is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the
1541:
1469:
1181:
954:
601:
346:
336:
12547:
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
11786:
9491:
published an argument that the principle of locality implies the incompleteness of quantum mechanics, a
3219:
3188:
3016:{\displaystyle \sigma _{A}\sigma _{B}\geq {\tfrac {1}{2}}\left|{\bigl \langle }{\bigr \rangle }\right|.}
2754:{\displaystyle \sigma _{P}={\sqrt {\left\langle P^{2}\right\rangle -\left\langle P\right\rangle ^{2}}}.}
15965:
15839:
15621:
15594:
15474:
15236:
15044:
15015:
12715:
11898:
9523:
8073:
3712:
If the state for a composite system is entangled, it is impossible to describe either component system
1896:
1782:
1653:
1424:
895:
547:
518:
11671:
11470:
Paris, M. G. A. (1999). "Entanglement and visibility at the output of a Mach–Zehnder interferometer".
11153:. This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1
10731:
6569:
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
15901:
15258:
15125:
15049:
15010:
14947:
14926:
14883:
14776:
14556:
13376:
13253:
9468:
9445:
9432:
9113:
7892:{\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}
7770:{\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}
3770:
3060:
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:
12567:
11005:
Drake, Gordon W. F. (2023). "High Precision Calculations for Helium". In Drake, Gordon W. F. (ed.).
7653:
and the probabilities that it will be detected at the right or at the top are given respectively by
7076:
1342:
15727:
15325:
15305:
15295:
15285:
15241:
14801:
13553:
9948:
9943:
9634:
9591:
9024:
8809:
8531:
8283:
7913:
7443:
A photon that enters the interferometer from the left will then be acted upon with a beam splitter
7316:
7284:
2074:
1459:
1260:
1248:
1030:
from theories to explain observations that could not be reconciled with classical physics, such as
636:
9459:, was troubled by its apparent failure to respect some cherished metaphysical principles, such as
7971:
7420:
1724:
15020:
14931:
12622:
12011:
Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory
11837:
11385:
10798:
9938:
9829:
9630:
9380:
9121:
8982:
8965:
8586:
8501:
4176:
3138:
2320:
modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp
1104:
805:
523:
431:
15434:
14982:
12166:
10778:
there's no way to understand the interpretation of quantum mechanics without also being able to
9842:
8279:
5881:
4653:
2454:
2425:
1137:
481:
379:
145:
15811:
15673:
15345:
15120:
15100:
15025:
14878:
14495:
13318:"Ăśber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"
12915:
12695:
12562:
12453:
11045:
9703:
9650:
9471:. Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids
9404:. The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with
9384:
9245:
9036:
8819:
8799:
8669:
8045:
6188:{\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}
4580:
3721:
3565:
3538:
3491:
3464:
3354:
3327:
2490:
2399:
2333:
1267:
1160:
1108:
988:
810:
528:
356:
326:
289:
14921:
13320:[On a heuristic point of view concerning the production and transformation of light].
11052:. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons.
10846:
10406:
9420:
once said, "I think I can safely say that nobody understands quantum mechanics." According to
3034:
2259:
2230:
1758:
1575:
436:
15816:
15785:
15648:
15569:
15370:
14868:
14848:
13877:
11809:
11446:
11438:
10867:
10460:
10433:
9637:
had been that they were indivisible units – the word "atom" deriving from the
9480:
9464:
9424:, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."
9405:
9140:
9080:
8464:
6199:
2873:
2070:
2049:
1565:
1545:
1291:
1279:
1144:
1083:
1020:
279:
264:
14626:
13601:
13591:
2348:
Some wave functions produce probability distributions that are independent of time, such as
2025:
15775:
15599:
15559:
15401:
15330:
15275:
15005:
14813:
14771:
14110:
13944:
13922:
13885:
13669:
13385:
13331:
13044:
12995:
12924:
12888:
12801:
12758:
12554:
12515:
12472:
12413:
12352:
12225:
12084:
11856:
11629:
11546:
11491:
11339:
11129:
10969:
10904:
10893:"Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems"
10639:
10319:
10258:
10205:
10170:
10078:
9969:
9918:
9766:
9754:
9669:
9599:
9472:
9416:. Perhaps the only consensus that exists about these issues is that there is no consensus.
9339:
9293:
9237:
9112:
involved the replacement of the Schrödinger equation with a covariant equation such as the
9043:
8596:
8474:
8459:
8449:
8201:{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }
8049:
5861:
5586:
5227:
4172:
3786:
3706:
3066:
2313:
1900:
1830:
1404:
1315:
1236:
1156:
1047:
1035:
992:
946:
596:
508:
234:
191:
13786:
13500:
11834:
8704:
8323:
5945:
5485:
3897:
3762:
3753:, 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:
15750:
15609:
15604:
15589:
15564:
15541:
15517:
15386:
15355:
15300:
15280:
15188:
15145:
15000:
14911:
14838:
14828:
14740:
13374:(1917). "Zur Quantentheorie der Strahlung" [On the Quantum Theory of Radiation].
13035:
13030:
10625:
9869:
9622:
9413:
9409:
9244:
have also been developed. The quantum field theory of the strong nuclear force is called
9072:
9055:
9028:
9019:
8719:
8546:
8536:
8484:
8479:
8454:
8353:
8077:
8022:
7417:, which means that if the photon is on the "upper" path it will gain a relative phase of
6757:{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}
6624:
6246:
6238:
5747:
5721:
5606:
5560:
5534:
3734:
2220:
1868:{\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}
1749:
1164:
695:
503:
421:
249:
229:
181:
14114:
13673:
13389:
13335:
13048:
12928:
12892:
12805:
12762:
12636:
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12558:
12519:
12476:
12417:
12356:
12330:
12229:
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11860:
11633:
11550:
11495:
11343:
11133:
10973:
10908:
10643:
10323:
10270:
10262:
10209:
10174:
10082:
9027:, a heuristic which states that the predictions of quantum mechanics reduce to those of
8804:
6281:
As in the classical case, the potential for the quantum harmonic oscillator is given by
5922:
5381:{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}
4322:{\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
15913:
15877:
15806:
15683:
15678:
15631:
15446:
15315:
15213:
14906:
14853:
14745:
14578:
14445:
14227:
Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
14180:
14165:
14142:
14126:
13928:
13464:
13405:
13322:
13221:
13128:
Evidence for the existence of "cathode-rays" was first found by PlĂĽcker and Hittorf ...
12967:
12948:
12878:
12817:
12774:
12748:
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10959:
10629:
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10102:
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9580:
9492:
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9257:
9241:
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9129:
9109:
8929:
8879:
8829:
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8371:
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4010:
3990:
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3946:
3926:
3877:
3794:
3705:
is a valid joint state that is not separable. States that are not separable are called
3592:
3518:
2846:
2826:
2593:
2575:
2555:
2341:
1997:
1884:
1549:
1204:
1096:
1051:
977:
950:
416:
341:
274:
186:
14359:
12770:
9439:
feature which will eventually be replaced by a deterministic theory, but is instead a
9069:
studies the relationship between classical and quantum descriptions in these systems.
8619:
7274:{\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
3059:
factor) to taking the derivative according to the position, since in Fourier analysis
15908:
15821:
15770:
15715:
15581:
15456:
15365:
15335:
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15226:
15221:
15203:
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12015:
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11373:
11359:
11294:
11284:
11241:
11210:
11200:
11162:
11149:
11120:
Ballentine, Leslie E. (1970), "The Statistical Interpretation of Quantum Mechanics",
11101:
11091:
11053:
11018:
10987:
10928:
10920:
10871:
10836:
10804:
10727:
10706:
10698:
10670:
10647:
10547:
10496:
10466:
10439:
10412:
10380:
10335:
10274:
10225:
10134:
10106:
10094:
9901:
9846:
9822:
9778:
9515:
9505:
9500:
9343:
9076:
8889:
8749:
8659:
8654:
8343:
8318:
8053:
8041:
8033:
8026:
8006:
6233:
6208:
4956:
4885:{\displaystyle {\hat {\psi }}(k,0)={\sqrt{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}
3758:
2387:
2383:
1463:
1271:
1256:
1252:
1244:
1213:
1208:
1063:
969:
965:
961:
942:
926:
850:
825:
765:
760:
660:
626:
606:
204:
14130:
14013:
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13195:
12952:
12492:
12433:
12380:
12315:
12198:
11884:
11511:
10286:
10195:
9646:
9587:
proposed a wave theory of light based on experimental observations. In 1803 English
9435:". According to these views, the probabilistic nature of quantum mechanics is not a
8814:
2405:
However, there are techniques for finding approximate solutions. One method, called
15865:
15853:
15760:
15700:
15626:
15183:
15178:
15035:
14916:
14440:
14330:
14270:
14118:
13984:
13755:
13677:
13651:
13401:
13339:
13288:
13052:
12932:
12906:
12809:
12778:
12766:
12691:
12631:
12572:
12523:
12480:
12421:
12360:
12295:
12233:
12178:
12151:
12131:
12092:
11868:
11864:
11641:
11637:
11554:
11499:
11347:
11306:
11276:
11137:
11081:
11010:
10977:
10912:
10688:
10679:
10617:
10537:
10529:
10331:
10327:
10266:
10213:
10178:
10086:
9857:
9838:
9673:
9626:
9324:
9316:
9308:
9227:
9125:
8944:
8919:
8894:
8884:
8839:
8709:
8516:
8496:
8491:
8348:
6266:
4183:
3774:
3754:
3725:
density matrices specify the state of a subsystem of a larger system, analogously,
2592:, and moreover for powers of them. Defining the uncertainty for an observable by a
2410:
2329:
1455:
1004:
855:
845:
835:
735:
715:
700:
670:
538:
426:
12598:
Einstein, Albert (1949). "Autobiographical Notes". In Schilpp, Paul Arthur (ed.).
11351:
10892:
10594:
10128:
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.
15780:
15690:
15636:
15536:
15413:
15340:
15320:
15290:
15253:
15248:
15153:
14977:
14665:
14646:
14276:
14247:
14237:
14169:
14138:
13988:
13950:
13904:
13782:
13597:
13504:
13371:
13141:
12671:
12448:
12364:
12035:
12009:
11980:
11231:
11186:
11077:
11014:
11009:. Springer Handbooks. Cham: Springer International Publishing. pp. 199–216.
10982:
10947:
10861:
10736:
10533:
10372:
10032:
9741:
9654:
9642:
9614:
9452:
9421:
9417:
9285:
9269:
9265:
9233:
9136:
9097:
8939:
8934:
8854:
8794:
8699:
8674:
8664:
8629:
8624:
8591:
8521:
8444:
8303:
8069:
3314:{\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}
2552:
Given a quantum state, the Born rule lets us compute expectation values for both
2356:
is pictured classically as a particle moving in a circular trajectory around the
2219:
is known as the time-evolution operator, and has the crucial property that it is
2043:
1333:
1255:. Contrary to popular misconception, entanglement does not allow sending signals
1039:
880:
750:
730:
476:
316:
14560:
14365:
13292:
13079:
12576:
12425:
10782: – to understand the theory, you need to be able to use it (and vice versa)
9793:. Never complete or self-consistent, the old quantum theory was rather a set of
9541:
appeared in the late 1990s as a modern derivative of Copenhagen-type ideas, and
7410:{\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}
5267:
The general solutions of the Schrödinger equation for the particle in a box are
1917:
The time evolution of a quantum state is described by the Schrödinger equation:
15937:
15925:
15710:
15665:
15643:
15531:
15391:
15360:
15350:
14972:
14952:
14781:
14402:
14396:
14351:
14255:
14222:
14057:
13960:
13859:
13587:
12096:
11952:
11920:
11558:
11087:
10161:
Jaeger, Gregg (September 2014). "What in the (quantum) world is macroscopic?".
9881:
9873:
9834:
9682:
9638:
9584:
9484:
9476:
9304:
9148:
9117:
9051:
9032:
8924:
8909:
8869:
8859:
8834:
8779:
8729:
8724:
8561:
8541:
8511:
8396:
8333:
8328:
8095:
8065:
8037:
6258:
6242:
3174:
2361:
2357:
2288:
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:
14230:
14122:
13361:
See also "Einstein's early work on the quantum hypothesis", ibid. pp. 134–148.
13015:
12936:
12652:
12484:
12451:(2010). "Einstein, incompleteness, and the epistemic view of quantum states".
12182:
11280:
11141:
10833:
The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics
10217:
9985:
for some of the technological applications which use quantum mechanics, e.g.,
9657:
carried on and improved upon Faraday's work, leading to the identification of
6850:
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 (
15959:
15889:
15720:
15310:
15163:
15054:
14957:
14873:
14843:
14796:
14311:
13972:
13691:
13573:
13549:
13438:
13344:
13317:
13258:
13167:
13064:
12944:
12584:
12372:
12272:
12190:
12143:
11602:
11580:
11503:
11298:
11214:
10991:
10924:
10702:
10098:
9893:
9889:
9853:
9806:
9762:
9750:
9662:
9618:
9558:
9355:
9342:(LQG), which describes quantum properties of gravity and is thus a theory of
9300:
9144:
9085:
9066:
9015:
8899:
8864:
8824:
8789:
8769:
8764:
8759:
8754:
8734:
8679:
8614:
8061:
7903:
7207:
6262:
5074:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}
4200:
2391:
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:
15485:
14488:
14389:
14204:
14174:
Compendium of quantum physics, Concepts, experiments, history and philosophy
10762:
9431:, Werner Heisenberg and other physicists are often grouped together as the "
9014:
The rules of quantum mechanics assert that the state space of a system is a
8774:
4780:{\displaystyle \psi (x,0)={\frac {1}{\sqrt{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}
15949:
15745:
15193:
14791:
14786:
14098:
14080:
14034:
13968:
13940:
13881:
13705:
13352:
13280:
12836:
12707:
12699:
11876:
11756:
11649:
11190:
11105:
10916:
10758:
10710:
10551:
10339:
10278:
9818:
9658:
9576:
9488:
9347:
9062:
8914:
8739:
8714:
8689:
8644:
8639:
8576:
8556:
6270:
6212:
4168:
2300:= 1, 2, 3, ...) and angular momenta (increasing across from left to right:
2019:
1295:
1225:
865:
860:
795:
780:
745:
239:
10932:
9079:, and thus become incapable of displaying many typically quantum effects:
4947:
1508:
of a single proton is simply the space of two-dimensional complex vectors
15231:
14642:
14582:
13429:
13285:
Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle
13249:
12870:
12842:. In Weinert, F.; Hentschel, K.; Greenberger, D.; Falkenburg, B. (eds.).
12753:
12703:
11726:
11486:
10946:
Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01).
10590:
10435:
Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral
9897:
9802:
9773:
9610:
9563:
9460:
9261:
8904:
8874:
8844:
8649:
8406:
6839:
4689:
4210:
2926:
and this provides the lower bound on the product of standard deviations:
2224:
1984:{\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
14176:, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics.
13826:. Physics Department, University of California, Berkeley. Archived from
12542:
10693:
10674:
10567:"Developing more energy-efficient transistors through quantum tunneling"
9868:. It has since permeated many disciplines, including quantum chemistry,
9825:'s extension of the Bohr model to include special-relativistic effects.
9665:
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
15765:
15755:
14586:
14542:
14243:
14212:
14008:
13993:. Cambridge, Massachusetts: Belknap Press of Harvard University Press.
13359:(in German). Vol. 2. Princeton University Press. pp. 149–166.
12813:
12299:
11412:
Feynman, Richard P.; Hibbs, Albert R. (2005). Steyer, Daniel F. (ed.).
11073:
10948:"Comparing many-body approaches against the helium atom exact solution"
10863:
Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature
10794:
10090:
9986:
9905:
9758:
9598:. This experiment played a major role in the general acceptance of the
9567:
9428:
9397:
8784:
8694:
8634:
8551:
8401:
8338:
8308:
8298:
8271:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
8057:
7345:. The phase shifter on the upper arm is modelled as the unitary matrix
6356:
4503:
4007:
can generate a family of unitary operators parameterized by a variable
3750:
2864:
2349:
1561:
1557:
1071:
1055:
1031:
820:
790:
710:
685:
680:
665:
15932:
13682:
13655:
13056:
12307:
10182:
9880:. It also provides a useful framework for many features of the modern
9797:
corrections to classical mechanics. The theory is now understood as a
9139:
particles as quantum mechanical objects being acted on by a classical
7902:
One can therefore use the Mach–Zehnder interferometer to estimate the
6838:
This is another example illustrating the discretization of energy for
4205:
3815:
of time evolution, since it defines a unitary time-evolution operator
15616:
15085:
14766:
14571:
12991:
12238:
12213:
9852:
By 1930 quantum mechanics had been further unified and formalized by
9794:
9690:
9606:
9509:
9351:
9328:
8849:
8581:
6254:
4966:
a certain region, and therefore infinite potential energy everywhere
2325:
2317:
1569:
1197:
1189:
1120:
311:
14678:
14603:
13624:
David Edwards, "The Mathematical Foundations of Quantum Mechanics",
12120:"Who Invented the 'Copenhagen Interpretation'? A Study in Mythology"
7313:, 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
4990:
direction, the time-independent Schrödinger equation may be written
27:
Description of physical properties at the atomic and subatomic scale
15872:
13953:, 1990, "Spooky actions at a distance: mysteries of the QT" in his
12987:
12214:"The Quantum Postulate and the Recent Development of Atomic Theory"
12135:
11334:
10964:
10314:
10253:
10066:
9782:
9588:
9371:
9332:
9092:
8684:
8421:
8313:
3063:. 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:
14063:
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
12883:
12467:
12408:
12347:
12079:
11851:
11541:
11271:
10634:
9765:
of hydrogen. Einstein further developed this idea to show that an
9335:, a quantum mechanical particle that carries gravitational force.
6028:{\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}
3247:, respectively. The Hilbert space of the composite system is then
15884:
13936:
13712:
A Tale of Two Continents: A Physicist's Life in a Turbulent World
10377:
Mastering Quantum Mechanics: Essentials, Theory, and Applications
9248:, and describes the interactions of subnuclear particles such as
8411:
8014:
4939:
uncertain. The uncertainty in momentum, however, stays constant.
4790:
which has Fourier transform, and therefore momentum distribution
3766:
1235:
When quantum systems interact, the result can be the creation of
972:) scale, but is not sufficient for describing them at very small
13821:"Berkeley Experiments on Superfluid Macroscopic Quantum Effects"
13485:
12543:"An Einstein manuscript on the EPR paradox for spin observables"
12329:
Ma, Xiao-song; Kofler, Johannes; Zeilinger, Anton (2016-03-03).
11320:
Schlosshauer, Maximilian (October 2019). "Quantum decoherence".
5482:
The infinite potential walls of the box determine the values of
1038:
problem, and the correspondence between energy and frequency in
14514:
13031:"Bringing one of the great moments of science to the classroom"
9046:
classical mechanics. For instance, the well-known model of the
8391:
8386:
8018:
8010:
6269:"). Each standing-wave frequency is proportional to a possible
4951:
1-dimensional potential energy box (or infinite potential well)
4577:, which are eigenstates of the momentum operator with momentum
3061:
differentiation corresponds to multiplication in the dual space
2813:{\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}
2395:
2321:
1177:
996:
930:
14055:
13965:
Timeless Reality: Symmetry, Simplicity, and Multiple Universes
13143:
Flash of the Cathode Rays: A History of J J Thomson's Electron
12257:. World perspectives (3 ed.). London: Allen & Unwin.
6828:{\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}
5858:
cannot be zero as this would conflict with the postulate that
4650:, which is the Fourier transform of the initial quantum state
3177:
of the Hilbert spaces of the two components. For example, let
10002:
9997:, which are follow-on technology in solid-state physics (vol
9885:
9542:
9253:
9249:
8416:
8376:
4332:
The general solution of the Schrödinger equation is given by
3135:
the momentum-squared term is replaced with a Laplacian times
2374:
Analytic solutions of the Schrödinger equation are known for
1643:{\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}
1193:
1168:
14538:
14465:
13887:
The Quantum Universe: Everything That Can Happen Does Happen
12506:
Howard, D. (1985). "Einstein on locality and separability".
12331:"Delayed-choice gedanken experiments and their realizations"
11700:(5th ed.). W.H. Freeman and Company. pp. 160–161.
7014:{\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}
6958:{\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}}
14101:(1957). "Relative State Formulation of Quantum Mechanics".
14087:, Princeton Series in Physics, Princeton University Press.
14033:
13396:
Einstein, A. (1967). "On the Quantum Theory of Radiation".
13078:
Feynman, Richard; Leighton, Robert; Sands, Matthew (1964).
12672:"Everett's Relative-State Formulation of Quantum Mechanics"
12393:
11007:
Springer Handbook of Atomic, Molecular, and Optical Physics
10127:
Feynman, Richard; Leighton, Robert; Sands, Matthew (1964).
9320:
9132:, one of the most accurate physical theories ever devised.
8426:
8381:
6250:
3726:
2353:
2256:– it makes a definite prediction of what the quantum state
1266:
Another possibility opened by entanglement is testing for "
1012:
934:
15944:
14242:
13921:. Four elementary lectures on quantum electrodynamics and
12690:
9864:, the statistical nature of our knowledge of reality, and
9508:, that can be violated by entangled particles. Since then
9212:{\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)}
13226:
Black-body theory and the quantum discontinuity 1894–1912
12968:"Quantum-Bayesian and Pragmatist Views of Quantum Theory"
12065:
Weinberg, Steven (2012). "Collapse of the state vector".
9103:
8081:
7440:, and it will stay unchanged if it is in the lower path.
4167:. This implies a quantum version of the result proven by
1119:, known as a probability amplitude. This is known as the
13434:"A century ago Einstein sparked the notion of the laser"
12255:
Physics and philosophy: the revolution in modern science
12167:"Constructing the Myth of the Copenhagen Interpretation"
10800:
The Demon-Haunted World: Science as a Candle in the Dark
10518:"Quantum tunnelling to the origin and evolution of life"
10239:
Bojowald, Martin (2015). "Quantum cosmology: a review".
9354:. The characteristic length scale of a spin foam is the
9350:. The evolution of a spin network over time is called a
9054:
of the oscillator, and is thus a quantum version of the
8087:
2227:
in the sense that – given an initial quantum state
2069:
is introduced so that the Hamiltonian is reduced to the
1247:
and is part of quantum communication protocols, such as
13114:
Advances in Electronics and Electron Physics, Volume 67
11199:(2nd ed.). Cambridge: Cambridge University Press.
11044:
10945:
9050:
uses an explicitly non-relativistic expression for the
3133:
non-relativistic Schrödinger equation in position space
3124:{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}
2080:
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
14572:
Modern Physics: With waves, thermodynamics, and optics
13967:. Buffalo, NY: Prometheus Books. Chpts. 5–8. Includes
13071:
11378:"Erwin Schrödinger and the creation of wave mechanics"
10299:
9841:
and the Austrian physicist Erwin Schrödinger invented
9158:
7567:
7363:
7240:
7066:{\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}}
6990:
6934:
5143:{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}
3765:). An alternative formulation of quantum mechanics is
3620:
2960:
2620:
2340:
discrete values like those shown. (As is the case for
1814:
15837:
13716:. Princeton, New Jersey: Princeton University Press.
13628:, Volume 42, Number 1/September, 1979, pp. 1–70.
9706:
9157:
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:
6368:
6345:{\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}
6290:
6067:
6044:
5971:
5948:
5925:
5884:
5864:
5844:
5779:
5750:
5724:
5638:
5609:
5589:
5563:
5537:
5517:
5488:
5404:
5276:
5250:
5230:
5162:
5093:
4999:
4976:
4921:
4901:
4799:
4701:
4656:
4612:
4583:
4511:
4341:
4227:
4153:
4133:
4113:
4093:
4073:
4053:
4033:
4013:
3993:
3969:
3949:
3929:
3900:
3880:
3821:
3797:
3618:
3595:
3568:
3541:
3521:
3494:
3467:
3433:
3387:
3357:
3330:
3256:
3222:
3191:
3168:
3141:
3096:
3069:
3037:
2935:
2876:
2849:
2829:
2773:
2691:
2605:
2578:
2558:
2493:
2457:
2428:
2262:
2233:
2166:
2089:
2052:
2028:
2000:
1926:
1785:
1761:
1727:
1685:
1656:
1598:
1578:
1514:
1472:
1427:
1407:
1369:
1345:
1318:
75:
14137:
13077:
11524:
10126:
9929:
List of textbooks on classical and quantum mechanics
6058:
implies a constraint on the energy levels, yielding
5264:
coincident with the kinetic energy of the particle.
5214:{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}
4570:{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}
1912:
1089:
14179:
14085:
The Many-Worlds Interpretation of Quantum Mechanics
13084:. Vol. 1. California Institute of Technology.
12712:
The Many-Worlds Interpretation of Quantum Mechanics
12508:
Studies in History and Philosophy of Science Part A
12396:
Studies in History and Philosophy of Science Part B
11950:
11761:"The Math That Takes Newton Into the Quantum World"
10133:. Vol. 3. California Institute of Technology.
9143:. For example, the elementary quantum model of the
3780:
3740:
1078:. In one of them, a mathematical entity called the
14500:Facts and Mysteries in Elementary Particle Physics
14444:
14275:
14012:
13739:"Von Neumann's contributions to quantum mechanics"
13709:
13463:
11437:Mathews, Piravonu Mathews; Venkatesan, K. (1976).
11040:
11038:
11036:
11034:
9724:
9479:is valid, but the fundamental theory behind it is
9211:
9075:is a mechanism through which quantum systems lose
8270:
8200:
7983:
7960:
7891:
7769:
7642:
7495:
7475:
7455:
7432:
7409:
7337:
7305:
7273:
7195:
7129:
7091:
7065:
7013:
6957:
6901:
6827:
6756:
6605:
6557:
6344:
6187:
6050:
6027:
5954:
5934:
5911:
5870:
5850:
5827:
5762:
5736:
5707:
5621:
5595:
5575:
5549:
5523:
5503:
5471:
5380:
5256:
5236:
5213:
5142:
5073:
4982:
4927:
4907:
4884:
4779:
4677:
4642:
4598:
4569:
4491:
4321:
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:
2443:
2277:
2248:
2211:
2149:
2061:
2034:
2006:
1983:
1867:
1800:
1767:
1740:
1713:
1671:
1642:
1592:is non-degenerate and the probability is given by
1584:
1529:
1496:
1446:
1413:
1393:
1355:
1324:
136:
14439:
14329:
14316:The mathematical foundations of quantum mechanics
13983:
13548:
12328:
12322:
11957:String theory and M-theory: A modern introduction
11436:
11416:(Emended ed.). McGraw-Hill. pp. v–vii.
10835:(2nd ed.). Jones and Bartlett. p. 215.
9275:
5468:
4970:that region. For the one-dimensional case in the
4209:Position space probability density of a Gaussian
2150:{\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
15957:
13498:
12976:. Metaphysics Research Lab, Stanford University.
12875:Many Worlds? Everett, Quantum Theory and Reality
12680:. Metaphysics Research Lab, Stanford University.
12659:. Metaphysics Research Lab, Stanford University.
12446:
11618:"Essay: Fifty Years of Condensed Matter Physics"
11439:"The Schrödinger Equation and Stationary States"
10826:
10603:. Metaphysics Research Lab, Stanford University.
10405:Lederman, Leon M.; Hill, Christopher T. (2011).
3324:If the state for the first system is the vector
1556:of an observable, in which case it is called an
14217:The Conceptual Development of Quantum Mechanics
13499:Bokulich, Alisa; Bokulich, Peter (2020-08-13).
13166:
12980:
11695:
11609:
11072:
11031:
10462:Niels Bohr and Complementarity: An Introduction
9570:is considered the father of the quantum theory.
9108:Early attempts to merge quantum mechanics with
9088:quantum behavior may manifest macroscopically.
5472:{\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}
4182:of a Hamiltonian, there exists a corresponding
14471:What is Quantum Mechanics? A Physics Adventure
13925:, yet containing many insights for the expert.
12785:
12714:. Princeton Series in Physics. Princeton, NJ:
12007:
11807:
11112:
10722:
10720:
10431:
10367:
10365:
10363:
10361:
10359:
10357:
10355:
10353:
10351:
10349:
9866:philosophical speculation about the 'observer'
9530:the possibilities described by quantum theory
9124:, provides a fully quantum description of the
9065:, which do not have good quantum numbers, and
6845:
5708:{\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}
4147:is conserved under the evolution generated by
15515:
15501:
14694:
14364:
13747:Bulletin of the American Mathematical Society
13552:; Napolitano, J. (2014). "Quantum Dynamics".
11185:
10425:
9361:
9292:and the search by physicists for an elegant "
8990:
7130:{\displaystyle \langle \psi ,\psi \rangle =1}
3000:
2978:
2376: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:
13775:
13593:Introduction to the Theory of Ferromagnetism
12905:
11913:
11721:
11719:
11717:
11411:
11319:
11265:(2nd ed.). Cambridge University Press.
11230:
10665:
10663:
10404:
8195:
8144:
7840:
7805:
7718:
7683:
7196:{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}
7118:
7106:
6909:that is a superposition of the "lower" path
4606:. The coefficients of the superposition are
3378:, then the state of the composite system is
3185:be two quantum systems, with Hilbert spaces
1860:
1838:
1708:
1686:
1625:
1604:
1382:
1370:
131:
105:
14534:Introduction to Quantum Theory at Quantiki.
14417:
14007:
13910:QED: The Strange Theory of Light and Matter
12732:
12285:
12008:Rovelli, Carlo; Vidotto, Francesca (2014).
11585:Introduction to Quantum Information Science
11196:Quantum Computation and Quantum Information
10891:Zhang, Ruiqin; Deng, Conghao (1993-01-01).
10827:Greenstein, George; Zajonc, Arthur (2006).
10717:
10509:
10346:
9373:
3417:{\displaystyle \psi _{A}\otimes \psi _{B}.}
1301:
1076:specially developed mathematical formalisms
15508:
15494:
14701:
14687:
14627:Imperial College Quantum Mechanics Course.
14252:Quantum Mechanics: Non-Relativistic Theory
13876:
13457:
13455:
12791:
12252:
11372:
11119:
11083:Quantum Mechanics: Non-Relativistic Theory
10458:
10452:
10400:
10398:
10396:
9975:
9817:that classical physics cannot account for
9483:. In 1935, Einstein and his collaborators
8997:
8983:
6870:, and in studies of quantum entanglement.
6854:Schematic of a Mach–Zehnder interferometer
5150:the previous equation is evocative of the
5084:With the differential operator defined by
3720:by a state vector. One can instead define
3427:Not all states in the joint Hilbert space
1127:. For example, a quantum particle like an
910:
896:
13759:
13681:
13650:
13631:
13618:
13343:
13254:"Max Planck: the reluctant revolutionary"
12882:
12752:
12650:
12635:
12566:
12466:
12407:
12346:
12237:
12164:
12078:
11850:
11808:Hawking, Stephen; Penrose, Roger (2010).
11714:
11540:
11518:
11485:
11463:
11333:
11270:
11161:, Cambridge: Cambridge University Press,
11154:
10981:
10963:
10890:
10820:
10726:
10692:
10660:
10633:
10541:
10485:
10313:
10252:
10122:
10120:
10118:
10116:
9327:, and other properties determined by the
9128:. Quantum electrodynamics is, along with
7099:. In order to respect the postulate that
6902:{\displaystyle \psi \in \mathbb {C} ^{2}}
6889:
6408:
4502:which is a superposition of all possible
1517:
1487:
14473:. Language Research Foundation, Boston.
13736:
13698:
13461:
13395:
13370:
13315:
13028:
12986:
12863:
12834:
12684:
12618:"On the Einstein Podolsky Rosen paradox"
12597:
12440:
12205:
12064:
11979:
11237:Quantum Computing: A Gentle Introduction
10859:
10732:"Experiment Reaffirms Quantum Weirdness"
10612:
10610:
10371:
10238:
9772:
9562:
9387:", give rise to the reality we perceive?
6849:
6767:and the corresponding energy levels are
6232:
4946:
4204:
2416:
2287:
1278:hidden variables, then the results of a
38:
14229:, 3rd ed. Singapore: World Scientific.
14097:
13818:
13586:
13580:
13452:
13357:The Collected Papers of Albert Einstein
13351:
13279:
12738:
12669:
12034:
11919:
11696:Tipler, Paul; Llewellyn, Ralph (2008).
11066:
10675:"Death by experiment for local realism"
10669:
10515:
10393:
10059:
9785:was the fifth world physics conference.
9629:and others built upon to establish the
9151:of the hydrogen atom using a classical
9018:and that observables of the system are
3351:and the state for the second system is
2480:do not commute, but rather satisfy the
1458:of a Hilbert space, usually called the
1095:raises philosophical problems, such as
14:
15958:
15706:Atomic, molecular, and optical physics
14539:Quantum Physics Made Relatively Simple
13947:can be passed over on a first reading.
13111:
12965:
12837:"The Born rule and its interpretation"
12706:; Van Vechten, D.; Graham, N. (1973).
12600:Albert Einstein: Philosopher-Scientist
12505:
12499:
12117:
11689:
11579:
11573:
11048:; Diu, Bernard; Laloë, Franck (2005).
10564:
10160:
10113:
9892:and the flow of electrons in computer
9104:Special relativity and electrodynamics
9100:under the rules of quantum mechanics.
9009:
8029:are dependent upon quantum mechanics.
6222:
2764:The uncertainty principle states that
2018:, the observable corresponding to the
1572:: in the simplest case the eigenvalue
1159:, which is often illustrated with the
15489:
14708:
14682:
13957:. Cambridge University Press: 110–76.
13248:
13242:
12540:
11725:
11672:"What Is Quantum Mechanics Good for?"
11615:
11469:
11260:
11254:
11226:
11224:
11181:
11179:
11177:
11004:
10793:
10616:
10607:
8088:Relation to other scientific theories
6253:(called the wave function), with the
5828:{\displaystyle \psi (L)=0=C\sin(kL),}
4213:moving in one dimension in free space
1082:provides information, in the form of
14561:PHYS 201: Fundamentals of Physics II
13704:
13428:
13220:
13139:
12869:
12612:
12211:
11755:
11732:Quantum Theory: Concepts and Methods
11414:Quantum Mechanics and Path Integrals
11158:Measuring the Quantum State of Light
10757:
10438:. US: World Scientific. p. 14.
10065:
9955:
9368:Interpretations of quantum mechanics
4942:
3867:{\displaystyle U(t)=e^{-iHt/\hbar }}
3589:are both possible states for system
3515:are both possible states for system
2212:{\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
14671:Stanford Encyclopedia of Philosophy
14652:Stanford Encyclopedia of Philosophy
13781:
13510:Stanford Encyclopedia of Philosophy
12973:Stanford Encyclopedia of Philosophy
12959:
12677:Stanford Encyclopedia of Philosophy
12657:Stanford Encyclopedia of Philosophy
12637:10.1103/PhysicsPhysiqueFizika.1.195
12387:
10767:University of California, Riverside
10600:Stanford Encyclopedia of Philosophy
10589:
9391:(more unsolved problems in physics)
7906:by estimating these probabilities.
7210:are modelled as the unitary matrix
4643:{\displaystyle {\hat {\psi }}(k,0)}
4027:. Under the evolution generated by
3454:{\displaystyle {\mathcal {H}}_{AB}}
1497:{\displaystyle L^{2}(\mathbb {C} )}
1220:in stars, and applications such as
24:
14640:
13849:
13791:California Institute of Technology
13771:from the original on Jan 20, 2024.
13406:10.1016/b978-0-08-012102-4.50018-8
11669:
11221:
11174:
8231:
7978:
7975:
7877:
7874:
7755:
7752:
7615:
7612:
7585:
7582:
7549:
7546:
7427:
7424:
7394:
7391:
6207:, which furnishes a model for the
4479:
4391:
4386:
3761:) and wave mechanics (invented by
3437:
3297:
3280:
3260:
3240:{\displaystyle {\mathcal {H}}_{B}}
3226:
3209:{\displaystyle {\mathcal {H}}_{A}}
3195:
3169:Composite systems and entanglement
3112:
3108:
1939:
1935:
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:
15977:
14659:
14527:J. O'Connor and E. F. Robertson:
14521:
14186:Introduction to Quantum Mechanics
14151:. Vol. 1–3. Addison-Wesley.
13656:"Max Born and the quantum theory"
13501:"Bohr's Correspondence Principle"
13470:. Pergamon Press. pp. 3–75.
13202:from the original on Oct 26, 2012
12541:Sauer, Tilman (1 December 2007).
11899:"The Nobel Prize in Physics 1979"
10803:. Ballantine Books. p. 249.
10492:Introduction to Quantum Mechanics
10432:MĂĽller-Kirsten, H. J. W. (2006).
9884:, and describes the behaviors of
9689:) would be proportional to their
9526:, formulated in 1956, holds that
9455:, himself one of the founders of
8153:
8001:Applications of quantum mechanics
6790:
6537:
6500:
6441:
5122:
4590:
3943:, it follows that any observable
3859:
3727:positive operator-valued measures
3103:
3046:
2533:
2204:
2127:
2056:
1930:
1913: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
15943:
15931:
15919:
15907:
15895:
15883:
15871:
15859:
15847:
15470:
15469:
14405:: Its Story and Its Significance
14039:The Physics of Quantum Mechanics
13812:
13797:from the original on 15 Dec 2016
13730:
13639:International J. of Theor. Phys.
12602:. Open Court Publishing Company.
12165:Camilleri, Kristian (May 2009).
10780:solve quantum mechanics problems
10465:. US: Springer. pp. 75–76.
9761:that successfully predicted the
9545:was developed some years later.
6606:{\displaystyle n=0,1,2,\ldots .}
4194:
4107:is conserved by evolution under
4087:will be conserved. Moreover, if
3781:Symmetries and conservation laws
3741:Equivalence between formulations
1906:Measurement in quantum mechanics
1530:{\displaystyle \mathbb {C} ^{2}}
15827:Timeline of physics discoveries
14662:"Measurement in Quantum Theory"
14529:A history of quantum mechanics.
14423:Principles of Quantum Mechanics
14371:Understanding Quantum Mechanics
14337:. Wiley, John & Sons, Inc.
14189:(2nd ed.). Prentice Hall.
14148:The Feynman Lectures on Physics
14083:, R. Neill Graham, eds., 1973.
13761:10.1090/s0002-9904-1958-10206-2
13644:
13542:
13517:
13492:
13422:
13364:
13309:
13283:(2009). "Bohr and the Photon".
13273:
13214:
13196:"Quantum – Definition and More"
13188:
13160:
13133:
13116:, Academic Press, p. 183,
13105:
13081:The Feynman Lectures on Physics
13029:Scheider, Walter (April 1986).
13022:
12899:
12828:
12663:
12644:
12606:
12591:
12534:
12279:
12246:
12158:
12111:
12058:
12028:
12001:
11985:A First Course in String Theory
11973:
11944:
11891:
11828:
11801:
11779:
11749:
11663:
11443:A Textbook of Quantum Mechanics
11430:
11405:
11366:
11313:
10998:
10939:
10884:
10853:
10787:
10763:"How to Learn Math and Physics"
10751:
10583:
10558:
10479:
10130:The Feynman Lectures on Physics
10038:
10012:
9983:the Feynman Lectures on Physics
9962:
9605:During the early 19th century,
9497:Einstein–Podolsky–Rosen paradox
7994:
6554:
5994:
5993:
5942:must be an integer multiple of
5337:
5336:
5152:classic kinetic energy analogue
2682:and likewise for the momentum:
929:that describes the behavior of
15419:Relativistic quantum mechanics
14451:. Princeton University Press.
14374:. Princeton University Press.
14281:Introductory Quantum Mechanics
14254:. Vol. 3 (3rd ed.).
14209:A standard undergraduate text.
13933:Sneaking a Look at God's Cards
13913:, Princeton University Press.
13525:"Semi-classical approximation"
13400:. Elsevier. pp. 167–183.
13198:. Merriam-Webster Dictionary.
13000:. Cambridge University Press.
12014:. Cambridge University Press.
11987:. Cambridge University Press.
11959:. Cambridge University Press.
11869:10.1103/PhysRevLett.109.111807
11814:. Princeton University Press.
11642:10.1103/PhysRevLett.101.250001
11086:. Vol. 3 (3rd ed.).
10332:10.1103/PhysRevLett.130.071801
10293:
10241:Reports on Progress in Physics
10232:
10189:
10154:
9924:Einstein's thought experiments
9276:Relation to general relativity
9205:
9177:
8527:Atomic, molecular, and optical
8192:
8186:
8172:
8141:
8135:
8121:
8114:
7941:
7935:
7926:
7920:
7845:
7801:
7794:
7788:
7723:
7679:
7672:
7666:
7626:
7609:
7596:
7579:
7177:
7168:
7154:
7145:
7092:{\displaystyle \alpha ,\beta }
6668:
6658:
6652:
6646:
6385:
6379:
6300:
6294:
5900:
5891:
5819:
5810:
5789:
5783:
5696:
5690:
5675:
5669:
5648:
5642:
5462:
5453:
5438:
5429:
5414:
5408:
5286:
5280:
5185:
5101:
4824:
4812:
4806:
4717:
4705:
4672:
4660:
4637:
4625:
4619:
4562:
4520:
4473:
4431:
4420:
4408:
4402:
4357:
4345:
3910:
3904:
3831:
3825:
2995:
2983:
2889:
2877:
2524:
2518:
2503:
2494:
2482:canonical commutation relation
2464:
2435:
2350:eigenstates of the Hamiltonian
2272:
2266:
2243:
2237:
2176:
2170:
2141:
2135:
2099:
2093:
1975:
1969:
1957:
1951:
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:
15397:Quantum statistical mechanics
15174:Quantum differential calculus
15096:Delayed-choice quantum eraser
14864:Symmetry in quantum mechanics
14622:Examples in Quantum Mechanics
13174:. New York: Springer-Verlag.
13146:. CRC Press. pp. 47–57.
12970:. In Zalta, Edward N. (ed.).
12844:Compendium of Quantum Physics
12771:10.1016/S1355-2198(03)00036-4
12674:. In Zalta, Edward N. (ed.).
12118:Howard, Don (December 2004).
12041:The Character of Physical Law
11352:10.1016/j.physrep.2019.10.001
11234:; Polak, Wolfgang H. (2011).
10597:. In Zalta, Edward N. (ed.).
10565:Phifer, Arnold (2012-03-27).
10271:10.1088/0034-4885/78/2/023901
10052:
10021:Macroscopic quantum phenomena
9934:Macroscopic quantum phenomena
9299:One proposal for doing so is
9056:classical harmonic oscillator
7968:, independently of the phase
7961:{\displaystyle p(u)=p(l)=1/2}
7503:, and so end up in the state
7338:{\displaystyle i/{\sqrt {2}}}
7306:{\displaystyle 1/{\sqrt {2}}}
6864:delayed choice quantum eraser
6217:scanning tunneling microscopy
6205:rectangular potential barrier
5878:has norm 1. Therefore, since
3894:. From this relation between
632:Quantum statistical mechanics
14467:Transnational College of Lex
12528:10.1016/0039-3681(85)90001-9
12365:10.1103/RevModPhys.88.015005
11811:The Nature of Space and Time
11789:. Academic.brooklyn.cuny.edu
11445:. Tata McGraw-Hill. p.
11015:10.1007/978-3-030-73893-8_12
10983:10.21468/SciPostPhys.6.4.040
10534:10.2174/13852728113179990083
9811:Hendrika Johanna van Leeuwen
9799:semi-classical approximation
9555:History of quantum mechanics
9539:Relational quantum mechanics
8070:medical and research imaging
7984:{\displaystyle \Delta \Phi }
7483:, and another beam splitter
7433:{\displaystyle \Delta \Phi }
1741:{\displaystyle P_{\lambda }}
1133:probability density function
7:
15791:Quantum information science
15199:Quantum stochastic calculus
15189:Quantum measurement problem
15111:Mach–Zehnder interferometer
14041:. Oxford University Press.
13955:Boojums All the Way Through
13864:Primer of Quantum Mechanics
13661:American Journal of Physics
13529:Encyclopedia of Mathematics
13293:10.1007/978-1-4020-9107-0_5
13228:. Oxford: Clarendon Press.
12877:. Oxford University Press.
12741:Stud. Hist. Phil. Mod. Phys
12710:; Graham, R. Neill (eds.).
12651:Goldstein, Sheldon (2017).
12577:10.1016/j.shpsb.2007.03.002
12426:10.1016/j.shpsb.2013.04.004
12253:Heisenberg, Werner (1971).
11587:. Oxford University Press.
10459:Plotnitsky, Arkady (2012).
10163:American Journal of Physics
9911:
9878:quantum information science
9789:This phase is known as the
9753:and used it to explain the
9374:Unsolved problem in physics
9126:electromagnetic interaction
9048:quantum harmonic oscillator
6868:Elitzur–Vaidman bomb tester
6860:Mach–Zehnder interferometer
6846:Mach–Zehnder interferometer
6241:(i.e. a ball attached to a
6229:Quantum harmonic oscillator
5244:in this case having energy
4189:
3158:{\displaystyle -\hbar ^{2}}
3131:, and in particular in the
2380:quantum harmonic oscillator
2285: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:
15982:
15622:Classical electromagnetism
14541:: three video lectures by
14301:. London: Pergamon Press.
14172:, F. Weinert, eds., 2009.
12716:Princeton University Press
12097:10.1103/PhysRevA.85.062116
12044:. MIT Press. p. 129.
11559:10.1103/PhysRevB.82.155303
11263:Quantum Information Theory
9882:periodic table of elements
9759:model of the hydrogen atom
9672:problem was discovered by
9552:
9548:
9524:many-worlds interpretation
9365:
9362:Philosophical implications
9338:Another popular theory is
8074:magnetic resonance imaging
7998:
6226:
5912:{\displaystyle \sin(kL)=0}
4954:
4678:{\displaystyle \psi (x,0)}
4198:
3784:
2867:of these two operators is
2473:{\displaystyle {\hat {P}}}
2444:{\displaystyle {\hat {X}}}
1897:many-worlds interpretation
1305:
933:at and below the scale of
31:
15799:
15736:
15664:
15580:
15552:
15524:
15465:
15427:
15379:
15259:Quantum complexity theory
15237:Quantum cellular automata
15212:
15144:
15078:
14991:
14940:
14927:Path integral formulation
14894:
14759:
14716:
14145:; Sands, Matthew (1965).
14123:10.1103/RevModPhys.29.454
14103:Reviews of Modern Physics
14037:; Skinner, David (2008).
13819:Packard, Richard (2006).
13377:Physikalische Zeitschrift
13170:; Rechenberg, H. (1982).
12937:10.1007/s10701-009-9326-5
12670:Barrett, Jeffrey (2018).
12485:10.1007/s10701-009-9347-0
12335:Reviews of Modern Physics
12183:10.1162/posc.2009.17.1.26
11616:Cohen, Marvin L. (2008).
11281:10.1017/9781316809976.001
11142:10.1103/RevModPhys.42.358
11122:Reviews of Modern Physics
10860:Weinberg, Steven (2010).
10522:Current Organic Chemistry
10408:Quantum Physics for Poets
10218:10.1038/s41567-019-0663-9
9860:with greater emphasis on
9725:{\displaystyle E=h\nu \ }
9433:Copenhagen interpretation
9288:is an important issue in
9061:Complications arise with
4599:{\displaystyle p=\hbar k}
3771:path integral formulation
3582:{\displaystyle \phi _{B}}
3555:{\displaystyle \psi _{B}}
3508:{\displaystyle \phi _{A}}
3481:{\displaystyle \psi _{A}}
3371:{\displaystyle \psi _{B}}
3344:{\displaystyle \psi _{A}}
2542:{\displaystyle =i\hbar .}
2223:. This time evolution is
1895:" (see, for example, the
1259:, as demonstrated by the
15728:Condensed matter physics
15326:Quantum machine learning
15306:Quantum key distribution
15296:Quantum image processing
15286:Quantum error correction
15136:Wheeler's delayed choice
14447:Einstein and the Quantum
13641:, Vol. 20, No. 7 (1981).
13555:Modern Quantum Mechanics
13345:10.1002/andp.19053220607
12966:Healey, Richard (2016).
12835:Landsman, N. P. (2008).
11955:; Schwarz, John (2007).
11504:10.1103/PhysRevA.59.1615
10866:. Random House. p.
10411:. US: Prometheus Books.
10027:Bose–Einstein condensate
9970:Dirac delta distribution
9949:Two-state quantum system
9944:Regularization (physics)
9621:of matter, an idea that
9303:, which posits that the
9224:Stern–Gerlach experiment
9025:correspondence principle
8284:Einstein field equations
3777:in classical mechanics.
3722:reduced density matrices
3052:{\displaystyle i/\hbar }
2278:{\displaystyle \psi (t)}
2249:{\displaystyle \psi (0)}
2075: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
15242:Quantum finite automata
14066:(2nd ed.). Wiley.
13737:Van Hove, Leon (1958).
12623:Physics Physique Fizika
12171:Perspectives on Science
11838:Physical Review Letters
11622:Physical Review Letters
11386:Acta Physica Polonica B
11261:Wilde, Mark M. (2017).
11155:Leonhardt, Ulf (1997),
11046:Cohen-Tannoudji, Claude
10302:Physical Review Letters
9939:Phase-space formulation
9631:kinetic theory of gases
9122:quantum electrodynamics
9031:in the regime of large
8587:Theories of gravitation
8502:Electroweak interaction
8046:superconducting magnets
6237:Some trajectories of a
5603:must be zero. Thus, at
4895:We see that as we make
4175:) mechanics: for every
2916:{\displaystyle =AB-BA,}
2062:{\displaystyle i\hbar }
1105:quantum electrodynamics
347:Leggett–Garg inequality
15812:Nobel Prize in Physics
15674:Relativistic mechanics
15346:Quantum neural network
14496:Veltman, Martinus J.G.
14318:. Dover Publications.
14297:Gunther Ludwig, 1968.
14019:. Dover Publications.
13466:The Old Quantum Theory
13398:The Old Quantum Theory
12916:Foundations of Physics
12794:Foundations of Physics
12454:Foundations of Physics
10917:10.1103/PhysRevA.47.71
10595:"Quantum entanglement"
10071:Zeitschrift fĂĽr Physik
9786:
9726:
9651:Johann Wilhelm Hittorf
9596:double-slit experiment
9571:
9385:wave function collapse
9246:quantum chromodynamics
9213:
9081:quantum superpositions
8272:
8202:
7985:
7962:
7893:
7771:
7644:
7497:
7477:
7457:
7434:
7411:
7339:
7307:
7275:
7197:
7131:
7093:
7067:
7015:
6959:
6903:
6855:
6829:
6758:
6607:
6559:
6346:
6278:
6257:shown in blue and the
6189:
6052:
6029:
5956:
5936:
5913:
5872:
5852:
5829:
5764:
5738:
5709:
5623:
5597:
5577:
5551:
5525:
5505:
5473:
5382:
5258:
5238:
5215:
5144:
5075:
4984:
4952:
4929:
4909:
4886:
4781:
4679:
4644:
4600:
4571:
4493:
4323:
4214:
4161:
4141:
4121:
4101:
4081:
4061:
4041:
4021:
4001:
3977:
3957:
3937:
3917:
3888:
3868:
3805:
3696:
3603:
3583:
3556:
3529:
3509:
3482:
3455:
3418:
3372:
3345:
3315:
3241:
3210:
3159:
3125:
3084:
3053:
3017:
2917:
2857:
2837:
2814:
2755:
2673:
2586:
2566:
2543:
2474:
2451:and momentum operator
2445:
2345:
2279:
2250:
2213:
2151:
2063:
2036:
2035:{\displaystyle \hbar }
2008:
1985:
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:
15817:Philosophy of physics
15371:Quantum teleportation
14884:Wave–particle duality
14575:– an online textbook.
14567:, Yale OpenCourseware
14231:Draft of 4th edition.
13462:ter Haar, D. (1967).
13316:Einstein, A. (1905).
13140:Dahl, Per F. (1997).
12124:Philosophy of Science
10622:OSA Century of Optics
9776:
9727:
9594:described the famous
9566:
9481:statistical mechanics
9469:Bohr–Einstein debates
9406:wavefunction collapse
9260:), by the physicists
9214:
9141:electromagnetic field
9114:Klein–Gordon equation
8465:Philosophy of physics
8273:
8203:
8050:light-emitting diodes
7986:
7963:
7894:
7772:
7645:
7498:
7478:
7458:
7435:
7412:
7340:
7308:
7276:
7198:
7132:
7094:
7068:
7016:
6965:and the "upper" path
6960:
6904:
6853:
6830:
6759:
6608:
6560:
6347:
6236:
6200:finite potential well
6190:
6053:
6030:
5957:
5937:
5914:
5873:
5871:{\displaystyle \psi }
5853:
5830:
5765:
5739:
5710:
5624:
5598:
5596:{\displaystyle \psi }
5578:
5552:
5526:
5506:
5474:
5383:
5259:
5239:
5237:{\displaystyle \psi }
5216:
5145:
5076:
4985:
4950:
4930:
4910:
4887:
4782:
4680:
4645:
4601:
4572:
4494:
4324:
4208:
4162:
4142:
4122:
4102:
4082:
4062:
4042:
4022:
4002:
3978:
3958:
3938:
3918:
3889:
3869:
3806:
3747:transformation theory
3697:
3604:
3584:
3557:
3530:
3510:
3483:
3456:
3419:
3373:
3346:
3316:
3242:
3211:
3160:
3126:
3085:
3083:{\displaystyle p_{i}}
3054:
3018:
2918:
2858:
2838:
2815:
2756:
2674:
2587:
2567:
2544:
2475:
2446:
2417:Uncertainty principle
2324:and thus, a definite
2294:Probability densities
2291:
2280:
2251:
2214:
2152:
2071:classical Hamiltonian
2064:
2037:
2009:
1986:
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:
15776:Mathematical physics
15402:Quantum field theory
15331:Quantum metamaterial
15276:Quantum cryptography
15006:Consistent histories
13923:quantum field theory
12997:Principles of Optics
12907:Van Fraassen, Bas C.
12447:Harrigan, Nicholas;
10628:. pp. 320–323.
10516:Trixler, F. (2013).
9809:of solids, Bohr and
9767:electromagnetic wave
9755:photoelectric effect
9704:
9670:black-body radiation
9600:wave theory of light
9473:action at a distance
9340:loop quantum gravity
9305:point-like particles
9294:Theory of Everything
9238:strong nuclear force
9155:
8597:Theory of everything
8475:Quantum field theory
8212:
8105:
7972:
7914:
7782:
7660:
7510:
7487:
7467:
7447:
7421:
7349:
7317:
7285:
7214:
7141:
7103:
7077:
7025:
6969:
6913:
6878:
6774:
6633:
6570:
6366:
6288:
6065:
6042:
5969:
5955:{\displaystyle \pi }
5946:
5923:
5882:
5862:
5842:
5777:
5748:
5722:
5636:
5607:
5587:
5561:
5535:
5515:
5504:{\displaystyle C,D,}
5486:
5402:
5274:
5248:
5228:
5160:
5091:
4997:
4974:
4919:
4899:
4797:
4699:
4654:
4610:
4581:
4509:
4339:
4225:
4151:
4131:
4111:
4091:
4071:
4051:
4031:
4011:
3991:
3967:
3947:
3927:
3916:{\displaystyle U(t)}
3898:
3878:
3819:
3795:
3616:
3593:
3566:
3539:
3519:
3492:
3465:
3431:
3385:
3355:
3328:
3254:
3220:
3189:
3139:
3094:
3067:
3035:
2933:
2874:
2847:
2827:
2771:
2689:
2603:
2576:
2556:
2491:
2455:
2426:
2342:resonant frequencies
2260:
2231:
2164:
2087:
2050:
2026:
1998:
1924:
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:
15751:Atmospheric physics
15590:Classical mechanics
15518:branches of physics
15387:Quantum fluctuation
15356:Quantum programming
15316:Quantum logic gates
15301:Quantum information
15281:Quantum electronics
14741:Classical mechanics
14643:"Quantum Mechanics"
14181:Griffiths, David J.
14143:Leighton, Robert B.
14139:Feynman, Richard P.
14115:1957RvMP...29..454E
13929:Ghirardi, GianCarlo
13833:on 25 November 2015
13754:(3): Part 2:95–99.
13674:2005AmJPh..73..999B
13390:1917PhyZ...18..121E
13336:1905AnP...322..132E
13252:(1 December 2000).
13049:1986PhTea..24..217S
13036:The Physics Teacher
12929:2010FoPh...40..390V
12893:2009arXiv0905.0624K
12806:1973FoPh....3..229B
12763:2003SHPMP..34..415W
12653:"Bohmian Mechanics"
12616:(1 November 1964).
12559:2007SHPMP..38..879S
12520:1985SHPSA..16..171H
12477:2010FoPh...40..125H
12449:Spekkens, Robert W.
12418:2013SHPMP..44..222S
12357:2016RvMP...88a5005M
12288:Naturwissenschaften
12230:1928Natur.121..580B
12089:2012PhRvA..85f2116W
11923:(10 October 2022).
11861:2012PhRvL.109k1807A
11787:"Atomic Properties"
11676:Scientific American
11634:2008PhRvL.101y0001C
11551:2010PhRvB..82o5303H
11496:1999PhRvA..59.1615P
11344:2019PhR...831....1S
11232:Rieffel, Eleanor G.
11187:Nielsen, Michael A.
11134:1970RvMP...42..358B
10974:2019ScPP....6...40L
10909:1993PhRvA..47...71Z
10730:(7 February 2017).
10694:10.1038/nature15631
10644:2013arXiv1302.1864C
10626:The Optical Society
10487:Griffiths, David J.
10324:2023PhRvL.130g1801F
10263:2015RPPh...78b3901B
10210:2019NatPh..15.1242F
10175:2014AmJPh..82..896J
10083:1926ZPhy...37..863B
9995:integrated circuits
9870:quantum electronics
9623:James Clerk Maxwell
9617:lent weight to the
9510:several experiments
9414:quantum nonlocality
9410:measurement problem
9073:Quantum decoherence
9029:classical mechanics
9020:Hermitian operators
9010:Classical mechanics
8547:Non-linear dynamics
8485:Quantum computation
8480:Quantum information
8354:Satyendra Nath Bose
8078:electron microscopy
8056:and the laser, the
8023:Solid-state physics
6625:Hermite polynomials
6247:classical mechanics
6239:harmonic oscillator
6223:Harmonic oscillator
6038:This constraint on
5763:{\displaystyle x=L}
5737:{\displaystyle D=0}
5622:{\displaystyle x=0}
5576:{\displaystyle x=L}
5550:{\displaystyle x=0}
5201:
4395:
4067:that commutes with
3963:that commutes with
3735:quantum decoherence
2407:perturbation theory
2022:of the system, and
1885:thought experiments
1750:probability density
385:Stern–Gerlach
182:Classical mechanics
15902:History of science
15807:History of physics
15440:in popular culture
15222:Quantum algorithms
15070:Von Neumann–Wigner
15050:Objective collapse
14746:Old quantum theory
14579:MIT OpenCourseWare
14515:This Quantum World
14283:. Addison-Wesley.
14277:Liboff, Richard L.
14236:2008-06-15 at the
13323:Annalen der Physik
12814:10.1007/BF00708440
12300:10.1007/BF01491987
11930:The New York Times
11901:. Nobel Foundation
11766:Nautilus Quarterly
11374:Rechenberg, Helmut
10728:Wolchover, Natalie
10091:10.1007/BF01397477
9981:See, for example,
9900:phenomena such as
9791:old quantum theory
9787:
9722:
9581:Christiaan Huygens
9572:
9493:thought experiment
9290:physical cosmology
9282:empirical evidence
9258:electroweak theory
9242:weak nuclear force
9209:
9208:
9130:general relativity
9110:special relativity
8572:General relativity
8567:Special relativity
8507:Strong interaction
8268:
8198:
7981:
7958:
7889:
7767:
7640:
7631:
7493:
7473:
7463:, a phase shifter
7453:
7430:
7407:
7401:
7335:
7303:
7271:
7265:
7193:
7127:
7089:
7063:
7011:
7005:
6955:
6949:
6899:
6856:
6825:
6754:
6603:
6555:
6342:
6279:
6185:
6048:
6025:
5952:
5935:{\displaystyle kL}
5932:
5909:
5868:
5848:
5825:
5760:
5734:
5705:
5619:
5593:
5573:
5547:
5521:
5501:
5469:
5378:
5254:
5234:
5211:
5178:
5140:
5071:
4980:
4953:
4925:
4905:
4882:
4777:
4675:
4640:
4596:
4567:
4489:
4378:
4319:
4215:
4157:
4137:
4117:
4097:
4077:
4057:
4037:
4017:
3997:
3973:
3953:
3933:
3913:
3884:
3874:for each value of
3864:
3801:
3692:
3631:
3599:
3579:
3552:
3525:
3505:
3478:
3451:
3414:
3368:
3341:
3311:
3237:
3206:
3155:
3121:
3080:
3049:
3028:Fourier transforms
3013:
2969:
2913:
2853:
2833:
2810:
2751:
2669:
2664:
2594:standard deviation
2582:
2562:
2539:
2470:
2441:
2346:
2275:
2246:
2209:
2147:
2059:
2032:
2004:
1981:
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:
15966:Quantum mechanics
15835:
15834:
15822:Physics education
15771:Materials science
15738:Interdisciplinary
15696:Quantum mechanics
15483:
15482:
15457:Quantum mysticism
15435:Schrödinger's cat
15366:Quantum simulator
15336:Quantum metrology
15264:Quantum computing
15227:Quantum amplifier
15204:Quantum spacetime
15169:Quantum cosmology
15159:Quantum chemistry
14859:Scattering theory
14807:Zero-point energy
14802:Degenerate levels
14710:Quantum mechanics
14565:Ramamurti Shankar
14557:Quantum Cook Book
14480:978-0-9643504-1-0
14458:978-0-691-13968-5
14441:Stone, A. Douglas
14432:978-0-306-44790-7
14381:978-0-691-00435-8
14356:Quantum Mechanics
14344:978-0-471-88702-7
14335:Quantum Mechanics
14331:Merzbacher, Eugen
14290:978-0-8053-8714-8
14265:978-0-08-020940-1
14196:978-0-13-111892-8
14158:978-0-7382-0008-8
14073:978-0-471-87373-0
14056:Eisberg, Robert;
14048:978-0-19-968857-9
14026:978-0-486-65969-5
14000:978-0-674-03541-6
13985:Bernstein, Jeremy
13897:978-1-84614-432-5
13683:10.1119/1.2060717
13654:(November 2005).
13652:Bernstein, Jeremy
13565:978-1-292-02410-3
13477:978-0-08-012101-7
13415:978-0-08-012102-4
13302:978-1-4020-9106-3
13235:978-0-19-502383-1
13181:978-0-387-90642-3
13153:978-0-7503-0453-5
13123:978-0-08-057733-3
13091:978-0-201-50064-6
13057:10.1119/1.2341987
12911:"Rovelli's World"
12853:978-3-540-70622-9
12264:978-0-04-530016-7
12224:(3050): 580–590.
12212:Bohr, N. (1928).
12067:Physical Review A
12021:978-1-316-14811-2
11994:978-0-521-88032-9
11966:978-0-521-86069-7
11821:978-1-4008-3474-7
11707:978-0-7167-7550-8
11594:978-0-19-921570-6
11528:Physical Review B
11473:Physical Review A
11456:978-0-07-096510-2
11423:978-0-486-47722-0
11290:978-1-107-17616-4
11247:978-0-262-01506-6
11206:978-1-107-00217-3
11097:978-0-08-020940-1
11050:Quantum Mechanics
11024:978-3-030-73892-1
10897:Physical Review A
10877:978-1-4070-6396-6
10842:978-0-7637-2470-2
10761:(20 March 2020).
10687:(7575): 649–650.
10653:978-1-943580-04-0
10618:Caves, Carlton M.
10528:(16): 1758–1770.
10495:. Prentice Hall.
10472:978-1-4614-4517-3
10445:978-981-256-691-1
10418:978-1-61614-281-0
10386:978-0-262-04613-8
10204:(12): 1242–1245.
10183:10.1119/1.4878358
10140:978-0-201-50064-6
9993:, pp. 14–11 ff),
9956:Explanatory notes
9856:, Paul Dirac and
9847:Solvay Conference
9823:Arnold Sommerfeld
9779:Solvay Conference
9721:
9681:derives from the
9516:Bohmian mechanics
9506:Bell inequalities
9495:later termed the
9400:debates and many
9344:quantum spacetime
9236:theories for the
9228:charged particles
9222:. Likewise, in a
9220:Coulomb potential
9007:
9006:
8470:Quantum mechanics
8344:Ernest Rutherford
8324:Erwin Schrödinger
8319:Werner Heisenberg
8169:
8117:
8054:optical amplifier
8042:quantum computing
8034:quantum chemistry
8027:materials science
8007:classical methods
7884:
7762:
7496:{\displaystyle B}
7476:{\displaystyle P}
7456:{\displaystyle B}
7333:
7301:
7233:
7232:
6815:
6721:
6541:
6540:
6504:
6445:
6417:
6416:
6314:
6267:stationary states
6209:quantum tunneling
6180:
6133:
6051:{\displaystyle k}
5991:
5851:{\displaystyle C}
5524:{\displaystyle k}
5376:
5257:{\displaystyle E}
5188:
5176:
5138:
5104:
5057:
5023:
4983:{\displaystyle x}
4957:Particle in a box
4943:Particle in a box
4928:{\displaystyle a}
4908:{\displaystyle a}
4875:
4845:
4839:
4809:
4773:
4743:
4742:
4622:
4557:
4468:
4405:
4376:
4375:
4314:
4285:
4247:
4160:{\displaystyle B}
4140:{\displaystyle A}
4120:{\displaystyle A}
4100:{\displaystyle B}
4080:{\displaystyle A}
4060:{\displaystyle B}
4047:, any observable
4040:{\displaystyle A}
4020:{\displaystyle t}
4000:{\displaystyle A}
3976:{\displaystyle H}
3956:{\displaystyle A}
3936:{\displaystyle H}
3887:{\displaystyle t}
3804:{\displaystyle H}
3787:Noether's theorem
3763:Erwin Schrödinger
3759:Werner Heisenberg
3630:
3629:
3602:{\displaystyle B}
3528:{\displaystyle A}
3119:
2968:
2856:{\displaystyle B}
2836:{\displaystyle A}
2805:
2746:
2662:
2585:{\displaystyle P}
2565:{\displaystyle X}
2521:
2506:
2467:
2438:
2388:dihydrogen cation
2384:particle in a box
2314:Chladni's figures
2007:{\displaystyle H}
1946:
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
18:Quantum physicist
16:(Redirected from
15973:
15948:
15947:
15936:
15935:
15924:
15923:
15912:
15911:
15900:
15899:
15898:
15888:
15887:
15876:
15875:
15864:
15863:
15862:
15852:
15851:
15850:
15843:
15761:Chemical physics
15701:Particle physics
15627:Classical optics
15510:
15503:
15496:
15487:
15486:
15473:
15472:
15184:Quantum geometry
15179:Quantum dynamics
15036:Superdeterminism
14968:Rarita–Schwinger
14917:Matrix mechanics
14772:Bra–ket notation
14703:
14696:
14689:
14680:
14679:
14675:
14666:Zalta, Edward N.
14656:
14647:Zalta, Edward N.
14641:Ismael, Jenann.
14621:
14619:
14618:
14615:
14612:
14608:
14492:
14462:
14450:
14436:
14393:
14348:
14294:
14269:
14208:
14162:
14134:
14077:
14052:
14030:
14018:
14004:
13979:More technical:
13945:bra–ket notation
13901:
13843:
13842:
13840:
13838:
13832:
13825:
13816:
13810:
13809:
13804:
13802:
13783:Feynman, Richard
13779:
13773:
13772:
13770:
13763:
13743:
13734:
13728:
13727:
13715:
13702:
13696:
13695:
13685:
13668:(11): 999–1008.
13648:
13642:
13635:
13629:
13622:
13616:
13615:
13584:
13578:
13577:
13546:
13540:
13539:
13537:
13535:
13521:
13515:
13514:
13505:Zalta, Edward N.
13496:
13490:
13489:
13469:
13459:
13450:
13449:
13447:
13446:
13426:
13420:
13419:
13393:
13372:Einstein, Albert
13368:
13362:
13360:
13349:
13347:
13313:
13307:
13306:
13277:
13271:
13270:
13268:
13266:
13246:
13240:
13239:
13218:
13212:
13211:
13209:
13207:
13192:
13186:
13185:
13164:
13158:
13157:
13137:
13131:
13130:
13109:
13103:
13102:
13100:
13098:
13075:
13069:
13068:
13026:
13020:
13019:
12984:
12978:
12977:
12963:
12957:
12956:
12903:
12897:
12896:
12886:
12867:
12861:
12860:
12841:
12832:
12826:
12825:
12789:
12783:
12782:
12756:
12754:quant-ph/0303050
12736:
12730:
12729:
12688:
12682:
12681:
12667:
12661:
12660:
12648:
12642:
12641:
12639:
12610:
12604:
12603:
12595:
12589:
12588:
12570:
12538:
12532:
12531:
12503:
12497:
12496:
12470:
12444:
12438:
12437:
12411:
12391:
12385:
12384:
12350:
12326:
12320:
12319:
12283:
12277:
12276:
12250:
12244:
12243:
12241:
12239:10.1038/121580a0
12209:
12203:
12202:
12162:
12156:
12155:
12115:
12109:
12108:
12082:
12062:
12056:
12055:
12036:Feynman, Richard
12032:
12026:
12025:
12005:
11999:
11998:
11981:Zwiebach, Barton
11977:
11971:
11970:
11951:Becker, Katrin;
11948:
11942:
11941:
11939:
11937:
11917:
11911:
11910:
11908:
11906:
11895:
11889:
11888:
11854:
11832:
11826:
11825:
11805:
11799:
11798:
11796:
11794:
11783:
11777:
11776:
11774:
11773:
11753:
11747:
11746:
11723:
11712:
11711:
11693:
11687:
11686:
11684:
11682:
11667:
11661:
11660:
11658:
11656:
11613:
11607:
11606:
11577:
11571:
11570:
11544:
11522:
11516:
11515:
11489:
11487:quant-ph/9811078
11480:(2): 1615–1621.
11467:
11461:
11460:
11434:
11428:
11427:
11409:
11403:
11402:
11400:
11398:
11382:
11370:
11364:
11363:
11337:
11317:
11311:
11310:
11274:
11258:
11252:
11251:
11228:
11219:
11218:
11191:Chuang, Isaac L.
11183:
11172:
11171:
11152:
11116:
11110:
11109:
11070:
11064:
11063:
11042:
11029:
11028:
11002:
10996:
10995:
10985:
10967:
10943:
10937:
10936:
10888:
10882:
10881:
10857:
10851:
10850:
10845:. Archived from
10824:
10818:
10817:
10791:
10785:
10784:
10775:
10773:
10755:
10749:
10748:
10746:
10744:
10724:
10715:
10714:
10696:
10673:(October 2015).
10667:
10658:
10657:
10637:
10614:
10605:
10604:
10587:
10581:
10580:
10578:
10577:
10562:
10556:
10555:
10545:
10513:
10507:
10506:
10483:
10477:
10476:
10456:
10450:
10449:
10429:
10423:
10422:
10402:
10391:
10390:
10373:Zwiebach, Barton
10369:
10344:
10343:
10317:
10297:
10291:
10290:
10256:
10236:
10230:
10229:
10193:
10187:
10186:
10158:
10152:
10151:
10149:
10147:
10124:
10111:
10110:
10063:
10046:
10042:
10036:
10016:
10010:
10001:, pp. 8–6), and
9979:
9973:
9966:
9919:Bra–ket notation
9890:chemical bonding
9858:John von Neumann
9839:matrix mechanics
9830:Louis de Broglie
9746:physical reality
9731:
9729:
9728:
9723:
9719:
9674:Gustav Kirchhoff
9635:Greek philosophy
9627:Ludwig Boltzmann
9408:and the related
9383:of states" and "
9375:
9311:are replaced by
9309:particle physics
9218:
9216:
9215:
9210:
9201:
9200:
9199:
9198:
9176:
9171:
9170:
9098:electric charges
9044:non-relativistic
8999:
8992:
8985:
8650:Skłodowska-Curie
8532:Condensed matter
8497:Weak interaction
8492:Electromagnetism
8349:Louis de Broglie
8277:
8275:
8274:
8269:
8267:
8266:
8254:
8246:
8245:
8227:
8226:
8207:
8205:
8204:
8199:
8185:
8184:
8175:
8170:
8168:
8157:
8134:
8133:
8124:
8119:
8118:
8110:
8092:
8091:
7990:
7988:
7987:
7982:
7967:
7965:
7964:
7959:
7954:
7898:
7896:
7895:
7890:
7885:
7880:
7872:
7867:
7866:
7854:
7853:
7848:
7839:
7838:
7817:
7816:
7804:
7776:
7774:
7773:
7768:
7763:
7758:
7750:
7745:
7744:
7732:
7731:
7726:
7717:
7716:
7695:
7694:
7682:
7649:
7647:
7646:
7641:
7636:
7635:
7622:
7592:
7561:
7560:
7556:
7531:
7530:
7502:
7500:
7499:
7494:
7482:
7480:
7479:
7474:
7462:
7460:
7459:
7454:
7439:
7437:
7436:
7431:
7416:
7414:
7413:
7408:
7406:
7405:
7398:
7397:
7344:
7342:
7341:
7336:
7334:
7329:
7327:
7312:
7310:
7309:
7304:
7302:
7297:
7295:
7280:
7278:
7277:
7272:
7270:
7269:
7234:
7228:
7224:
7202:
7200:
7199:
7194:
7186:
7185:
7180:
7171:
7163:
7162:
7157:
7148:
7137:we require that
7136:
7134:
7133:
7128:
7098:
7096:
7095:
7090:
7072:
7070:
7069:
7064:
7062:
7061:
7046:
7045:
7020:
7018:
7017:
7012:
7010:
7009:
6981:
6980:
6964:
6962:
6961:
6956:
6954:
6953:
6925:
6924:
6908:
6906:
6905:
6900:
6898:
6897:
6892:
6834:
6832:
6831:
6826:
6821:
6817:
6816:
6808:
6786:
6785:
6763:
6761:
6760:
6755:
6750:
6746:
6745:
6744:
6743:
6722:
6720:
6719:
6718:
6705:
6704:
6695:
6693:
6692:
6691:
6690:
6676:
6675:
6645:
6644:
6612:
6610:
6609:
6604:
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:
6194:
6192:
6191:
6186:
6181:
6179:
6178:
6177:
6161:
6160:
6159:
6150:
6149:
6139:
6134:
6132:
6131:
6130:
6114:
6113:
6112:
6103:
6102:
6093:
6092:
6082:
6077:
6076:
6057:
6055:
6054:
6049:
6034:
6032:
6031:
6026:
5992:
5987:
5979:
5961:
5959:
5958:
5953:
5941:
5939:
5938:
5933:
5918:
5916:
5915:
5910:
5877:
5875:
5874:
5869:
5857:
5855:
5854:
5849:
5834:
5832:
5831:
5826:
5769:
5767:
5766:
5761:
5743:
5741:
5740:
5735:
5714:
5712:
5711:
5706:
5628:
5626:
5625:
5620:
5602:
5600:
5599:
5594:
5582:
5580:
5579:
5574:
5556:
5554:
5553:
5548:
5530:
5528:
5527:
5522:
5510:
5508:
5507:
5502:
5478:
5476:
5475:
5470:
5387:
5385:
5384:
5379:
5377:
5375:
5367:
5366:
5365:
5356:
5355:
5345:
5335:
5334:
5310:
5309:
5263:
5261:
5260:
5255:
5243:
5241:
5240:
5235:
5220:
5218:
5217:
5212:
5200:
5195:
5190:
5189:
5181:
5177:
5175:
5164:
5149:
5147:
5146:
5141:
5139:
5137:
5126:
5112:
5111:
5106:
5105:
5097:
5080:
5078:
5077:
5072:
5058:
5056:
5055:
5054:
5041:
5037:
5036:
5026:
5024:
5022:
5014:
5013:
5004:
4989:
4987:
4986:
4981:
4934:
4932:
4931:
4926:
4914:
4912:
4911:
4906:
4891:
4889:
4888:
4883:
4878:
4877:
4876:
4871:
4870:
4869:
4856:
4846:
4844:
4832:
4831:
4811:
4810:
4802:
4786:
4784:
4783:
4778:
4776:
4775:
4774:
4772:
4764:
4763:
4754:
4744:
4741:
4736:
4728:
4724:
4684:
4682:
4681:
4676:
4649:
4647:
4646:
4641:
4624:
4623:
4615:
4605:
4603:
4602:
4597:
4576:
4574:
4573:
4568:
4566:
4565:
4558:
4556:
4548:
4547:
4546:
4533:
4498:
4496:
4495:
4490:
4482:
4477:
4476:
4469:
4467:
4459:
4458:
4457:
4444:
4407:
4406:
4398:
4394:
4389:
4377:
4368:
4364:
4328:
4326:
4325:
4320:
4315:
4313:
4312:
4311:
4298:
4297:
4288:
4286:
4284:
4276:
4275:
4266:
4258:
4257:
4248:
4246:
4235:
4184:conservation law
4166:
4164:
4163:
4158:
4146:
4144:
4143:
4138:
4126:
4124:
4123:
4118:
4106:
4104:
4103:
4098:
4086:
4084:
4083:
4078:
4066:
4064:
4063:
4058:
4046:
4044:
4043:
4038:
4026:
4024:
4023:
4018:
4006:
4004:
4003:
3998:
3982:
3980:
3979:
3974:
3962:
3960:
3959:
3954:
3942:
3940:
3939:
3934:
3922:
3920:
3919:
3914:
3893:
3891:
3890:
3885:
3873:
3871:
3870:
3865:
3863:
3862:
3858:
3811:is known as the
3810:
3808:
3807:
3802:
3791:The Hamiltonian
3775:action principle
3755:matrix mechanics
3719:
3715:
3701:
3699:
3698:
3693:
3691:
3687:
3686:
3685:
3673:
3672:
3660:
3659:
3647:
3646:
3632:
3625:
3621:
3608:
3606:
3605:
3600:
3588:
3586:
3585:
3580:
3578:
3577:
3561:
3559:
3558:
3553:
3551:
3550:
3534:
3532:
3531:
3526:
3514:
3512:
3511:
3506:
3504:
3503:
3487:
3485:
3484:
3479:
3477:
3476:
3460:
3458:
3457:
3452:
3450:
3449:
3441:
3440:
3423:
3421:
3420:
3415:
3410:
3409:
3397:
3396:
3377:
3375:
3374:
3369:
3367:
3366:
3350:
3348:
3347:
3342:
3340:
3339:
3320:
3318:
3317:
3312:
3307:
3306:
3301:
3300:
3290:
3289:
3284:
3283:
3273:
3272:
3264:
3263:
3246:
3244:
3243:
3238:
3236:
3235:
3230:
3229:
3215:
3213:
3212:
3207:
3205:
3204:
3199:
3198:
3184:
3180:
3164:
3162:
3161:
3156:
3154:
3153:
3130:
3128:
3127:
3122:
3120:
3118:
3107:
3089:
3087:
3086:
3081:
3079:
3078:
3058:
3056:
3055:
3050:
3045:
3022:
3020:
3019:
3014:
3009:
3005:
3004:
3003:
2982:
2981:
2970:
2961:
2955:
2954:
2945:
2944:
2922:
2920:
2919:
2914:
2862:
2860:
2859:
2854:
2842:
2840:
2839:
2834:
2819:
2817:
2816:
2811:
2806:
2798:
2793:
2792:
2783:
2782:
2760:
2758:
2757:
2752:
2747:
2745:
2744:
2739:
2724:
2720:
2719:
2706:
2701:
2700:
2678:
2676:
2675:
2670:
2665:
2663:
2661:
2660:
2655:
2640:
2636:
2635:
2622:
2615:
2614:
2591:
2589:
2588:
2583:
2571:
2569:
2568:
2563:
2548:
2546:
2545:
2540:
2523:
2522:
2514:
2508:
2507:
2499:
2479:
2477:
2476:
2471:
2469:
2468:
2460:
2450:
2448:
2447:
2442:
2440:
2439:
2431:
2411:potential energy
2330:angular momentum
2284:
2282:
2281:
2276:
2255:
2253:
2252:
2247:
2218:
2216:
2215:
2210:
2208:
2207:
2203:
2156:
2154:
2153:
2148:
2131:
2130:
2126:
2068:
2066:
2065:
2060:
2041:
2039:
2038:
2033:
2013:
2011:
2010:
2005:
1990:
1988:
1987:
1982:
1947:
1945:
1934:
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:
15981:
15980:
15976:
15975:
15974:
15972:
15971:
15970:
15956:
15955:
15954:
15942:
15930:
15918:
15906:
15896:
15894:
15882:
15870:
15860:
15858:
15848:
15846:
15838:
15836:
15831:
15795:
15781:Medical physics
15732:
15691:Nuclear physics
15660:
15654:Non-equilibrium
15576:
15548:
15520:
15514:
15484:
15479:
15461:
15447:Wigner's friend
15423:
15414:Quantum gravity
15375:
15361:Quantum sensing
15341:Quantum network
15321:Quantum machine
15291:Quantum imaging
15254:Quantum circuit
15249:Quantum channel
15208:
15154:Quantum biology
15140:
15116:Elitzur–Vaidman
15091:Davisson–Germer
15074:
15026:Hidden-variable
15016:de Broglie–Bohm
14993:Interpretations
14987:
14936:
14890:
14777:Complementarity
14755:
14712:
14707:
14616:
14613:
14610:
14609:
14606:
14604:
14549:Course material
14524:
14519:
14481:
14459:
14433:
14397:Scerri, Eric R.
14382:
14345:
14291:
14266:
14238:Wayback Machine
14197:
14159:
14074:
14058:Resnick, Robert
14049:
14027:
14001:
13975:considerations.
13951:N. David Mermin
13905:Richard Feynman
13898:
13860:Chester, Marvin
13852:
13850:Further reading
13847:
13846:
13836:
13834:
13830:
13823:
13817:
13813:
13800:
13798:
13780:
13776:
13768:
13741:
13735:
13731:
13724:
13703:
13699:
13649:
13645:
13636:
13632:
13623:
13619:
13612:
13598:Clarendon Press
13588:Aharoni, Amikam
13585:
13581:
13566:
13547:
13543:
13533:
13531:
13523:
13522:
13518:
13497:
13493:
13478:
13460:
13453:
13444:
13442:
13427:
13423:
13416:
13369:
13365:
13314:
13310:
13303:
13278:
13274:
13264:
13262:
13247:
13243:
13236:
13219:
13215:
13205:
13203:
13194:
13193:
13189:
13182:
13165:
13161:
13154:
13138:
13134:
13124:
13110:
13106:
13096:
13094:
13092:
13076:
13072:
13027:
13023:
13008:
12985:
12981:
12964:
12960:
12904:
12900:
12868:
12864:
12854:
12839:
12833:
12829:
12790:
12786:
12737:
12733:
12726:
12689:
12685:
12668:
12664:
12649:
12645:
12611:
12607:
12596:
12592:
12568:10.1.1.571.6089
12539:
12535:
12504:
12500:
12445:
12441:
12392:
12388:
12327:
12323:
12294:(50): 844–849.
12284:
12280:
12265:
12251:
12247:
12210:
12206:
12163:
12159:
12116:
12112:
12063:
12059:
12052:
12033:
12029:
12022:
12006:
12002:
11995:
11978:
11974:
11967:
11953:Becker, Melanie
11949:
11945:
11935:
11933:
11921:Overbye, Dennis
11918:
11914:
11904:
11902:
11897:
11896:
11892:
11833:
11829:
11822:
11806:
11802:
11792:
11790:
11785:
11784:
11780:
11771:
11769:
11754:
11750:
11743:
11724:
11715:
11708:
11694:
11690:
11680:
11678:
11668:
11664:
11654:
11652:
11614:
11610:
11595:
11578:
11574:
11523:
11519:
11468:
11464:
11457:
11435:
11431:
11424:
11410:
11406:
11396:
11394:
11380:
11371:
11367:
11322:Physics Reports
11318:
11314:
11291:
11259:
11255:
11248:
11229:
11222:
11207:
11184:
11175:
11169:
11118:Section 3.2 of
11117:
11113:
11098:
11078:Lifschitz, E.M.
11071:
11067:
11060:
11043:
11032:
11025:
11003:
10999:
10952:SciPost Physics
10944:
10940:
10889:
10885:
10878:
10858:
10854:
10843:
10829:"8 Measurement"
10825:
10821:
10811:
10792:
10788:
10771:
10769:
10756:
10752:
10742:
10740:
10737:Quanta Magazine
10725:
10718:
10671:Wiseman, Howard
10668:
10661:
10654:
10615:
10608:
10588:
10584:
10575:
10573:
10571:Notre Dame News
10563:
10559:
10514:
10510:
10503:
10484:
10480:
10473:
10457:
10453:
10446:
10430:
10426:
10419:
10403:
10394:
10387:
10370:
10347:
10298:
10294:
10237:
10233:
10194:
10190:
10159:
10155:
10145:
10143:
10141:
10125:
10114:
10077:(12): 863–867.
10064:
10060:
10055:
10050:
10049:
10043:
10039:
10033:Quantum machine
10017:
10013:
9980:
9976:
9967:
9963:
9958:
9953:
9914:
9902:superconductors
9805:'s work on the
9742:Planck constant
9705:
9702:
9701:
9655:Eugen Goldstein
9643:Michael Faraday
9615:Amedeo Avogadro
9561:
9553:Main articles:
9551:
9453:Albert Einstein
9422:Steven Weinberg
9418:Richard Feynman
9402:interpretations
9394:
9393:
9388:
9377:
9370:
9364:
9315:objects called
9313:one-dimensional
9286:quantum gravity
9278:
9270:Steven Weinberg
9266:Sheldon Glashow
9194:
9191:
9190:
9186:
9172:
9166:
9162:
9156:
9153:
9152:
9106:
9063:chaotic systems
9033:quantum numbers
9012:
9003:
8974:
8973:
8970:
8969:
8968:
8958:
8950:
8949:
8610:
8602:
8601:
8592:Quantum gravity
8542:Complex systems
8440:
8432:
8431:
8367:
8359:
8358:
8304:Albert Einstein
8294:
8286:
8259:
8255:
8250:
8238:
8234:
8219:
8215:
8213:
8210:
8209:
8208:
8180:
8176:
8171:
8161:
8156:
8129:
8125:
8120:
8109:
8108:
8106:
8103:
8102:
8090:
8021:, and others).
8003:
7997:
7973:
7970:
7969:
7950:
7915:
7912:
7911:
7873:
7871:
7862:
7858:
7849:
7844:
7843:
7834:
7830:
7812:
7808:
7800:
7783:
7780:
7779:
7751:
7749:
7740:
7736:
7727:
7722:
7721:
7712:
7708:
7690:
7686:
7678:
7661:
7658:
7657:
7630:
7629:
7618:
7600:
7599:
7588:
7563:
7562:
7552:
7542:
7538:
7526:
7522:
7511:
7508:
7507:
7488:
7485:
7484:
7468:
7465:
7464:
7448:
7445:
7444:
7422:
7419:
7418:
7400:
7399:
7387:
7383:
7381:
7375:
7374:
7369:
7359:
7358:
7350:
7347:
7346:
7328:
7323:
7318:
7315:
7314:
7296:
7291:
7286:
7283:
7282:
7264:
7263:
7258:
7252:
7251:
7246:
7236:
7235:
7223:
7215:
7212:
7211:
7181:
7176:
7175:
7167:
7158:
7153:
7152:
7144:
7142:
7139:
7138:
7104:
7101:
7100:
7078:
7075:
7074:
7057:
7053:
7041:
7037:
7026:
7023:
7022:
7004:
7003:
6997:
6996:
6986:
6985:
6976:
6972:
6970:
6967:
6966:
6948:
6947:
6941:
6940:
6930:
6929:
6920:
6916:
6914:
6911:
6910:
6893:
6888:
6887:
6879:
6876:
6875:
6848:
6807:
6800:
6796:
6781:
6777:
6775:
6772:
6771:
6739:
6735:
6731:
6727:
6723:
6714:
6710:
6706:
6700:
6696:
6694:
6686:
6682:
6681:
6677:
6671:
6667:
6640:
6636:
6634:
6631:
6630:
6621:
6571:
6568:
6567:
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:
6173:
6169:
6162:
6155:
6151:
6145:
6141:
6140:
6138:
6126:
6122:
6115:
6108:
6104:
6098:
6094:
6088:
6084:
6083:
6081:
6072:
6068:
6066:
6063:
6062:
6043:
6040:
6039:
5980:
5978:
5970:
5967:
5966:
5947:
5944:
5943:
5924:
5921:
5920:
5883:
5880:
5879:
5863:
5860:
5859:
5843:
5840:
5839:
5778:
5775:
5774:
5749:
5746:
5745:
5723:
5720:
5719:
5637:
5634:
5633:
5608:
5605:
5604:
5588:
5585:
5584:
5562:
5559:
5558:
5536:
5533:
5532:
5516:
5513:
5512:
5487:
5484:
5483:
5403:
5400:
5399:
5393:Euler's formula
5368:
5361:
5357:
5351:
5347:
5346:
5344:
5321:
5317:
5299:
5295:
5275:
5272:
5271:
5249:
5246:
5245:
5229:
5226:
5225:
5196:
5191:
5180:
5179:
5168:
5163:
5161:
5158:
5157:
5130:
5125:
5107:
5096:
5095:
5094:
5092:
5089:
5088:
5050:
5046:
5042:
5032:
5028:
5027:
5025:
5015:
5009:
5005:
5003:
4998:
4995:
4994:
4975:
4972:
4971:
4959:
4945:
4920:
4917:
4916:
4900:
4897:
4896:
4865:
4861:
4857:
4855:
4851:
4847:
4840:
4830:
4801:
4800:
4798:
4795:
4794:
4765:
4759:
4755:
4753:
4749:
4745:
4737:
4729:
4723:
4700:
4697:
4696:
4655:
4652:
4651:
4614:
4613:
4611:
4608:
4607:
4582:
4579:
4578:
4549:
4542:
4538:
4534:
4532:
4516:
4512:
4510:
4507:
4506:
4478:
4460:
4453:
4449:
4445:
4443:
4427:
4423:
4397:
4396:
4390:
4382:
4363:
4340:
4337:
4336:
4307:
4303:
4299:
4293:
4289:
4287:
4277:
4271:
4267:
4265:
4253:
4249:
4239:
4234:
4226:
4223:
4222:
4203:
4197:
4192:
4152:
4149:
4148:
4132:
4129:
4128:
4112:
4109:
4108:
4092:
4089:
4088:
4072:
4069:
4068:
4052:
4049:
4048:
4032:
4029:
4028:
4012:
4009:
4008:
3992:
3989:
3988:
3968:
3965:
3964:
3948:
3945:
3944:
3928:
3925:
3924:
3899:
3896:
3895:
3879:
3876:
3875:
3854:
3841:
3837:
3820:
3817:
3816:
3796:
3793:
3792:
3789:
3783:
3743:
3717:
3713:
3681:
3677:
3668:
3664:
3655:
3651:
3642:
3638:
3637:
3633:
3619:
3617:
3614:
3613:
3594:
3591:
3590:
3573:
3569:
3567:
3564:
3563:
3546:
3542:
3540:
3537:
3536:
3535:, and likewise
3520:
3517:
3516:
3499:
3495:
3493:
3490:
3489:
3472:
3468:
3466:
3463:
3462:
3442:
3436:
3435:
3434:
3432:
3429:
3428:
3405:
3401:
3392:
3388:
3386:
3383:
3382:
3362:
3358:
3356:
3353:
3352:
3335:
3331:
3329:
3326:
3325:
3302:
3296:
3295:
3294:
3285:
3279:
3278:
3277:
3265:
3259:
3258:
3257:
3255:
3252:
3251:
3231:
3225:
3224:
3223:
3221:
3218:
3217:
3200:
3194:
3193:
3192:
3190:
3187:
3186:
3182:
3178:
3171:
3149:
3145:
3140:
3137:
3136:
3111:
3106:
3095:
3092:
3091:
3090:is replaced by
3074:
3070:
3068:
3065:
3064:
3041:
3036:
3033:
3032:
2999:
2998:
2977:
2976:
2975:
2971:
2959:
2950:
2946:
2940:
2936:
2934:
2931:
2930:
2875:
2872:
2871:
2848:
2845:
2844:
2828:
2825:
2824:
2797:
2788:
2784:
2778:
2774:
2772:
2769:
2768:
2740:
2729:
2728:
2715:
2711:
2707:
2705:
2696:
2692:
2690:
2687:
2686:
2656:
2645:
2644:
2631:
2627:
2623:
2621:
2619:
2610:
2606:
2604:
2601:
2600:
2577:
2574:
2573:
2557:
2554:
2553:
2513:
2512:
2498:
2497:
2492:
2489:
2488:
2459:
2458:
2456:
2453:
2452:
2430:
2429:
2427:
2424:
2423:
2419:
2332:and energy are
2261:
2258:
2257:
2232:
2229:
2228:
2199:
2186:
2182:
2165:
2162:
2161:
2122:
2109:
2105:
2088:
2085:
2084:
2051:
2048:
2047:
2046:. The constant
2044:Planck constant
2042:is the reduced
2027:
2024:
2023:
1999:
1996:
1995:
1938:
1933:
1925:
1922:
1921:
1915:
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:
15:
12:
11:
5:
15979:
15969:
15968:
15953:
15952:
15940:
15928:
15916:
15904:
15892:
15880:
15868:
15856:
15833:
15832:
15830:
15829:
15824:
15819:
15814:
15809:
15803:
15801:
15797:
15796:
15794:
15793:
15788:
15783:
15778:
15773:
15768:
15763:
15758:
15753:
15748:
15742:
15740:
15734:
15733:
15731:
15730:
15725:
15724:
15723:
15718:
15713:
15703:
15698:
15693:
15688:
15687:
15686:
15681:
15670:
15668:
15662:
15661:
15659:
15658:
15657:
15656:
15651:
15644:Thermodynamics
15641:
15640:
15639:
15634:
15624:
15619:
15614:
15613:
15612:
15607:
15602:
15597:
15586:
15584:
15578:
15577:
15575:
15574:
15573:
15572:
15562:
15556:
15554:
15550:
15549:
15547:
15546:
15545:
15544:
15534:
15528:
15526:
15522:
15521:
15513:
15512:
15505:
15498:
15490:
15481:
15480:
15478:
15477:
15466:
15463:
15462:
15460:
15459:
15454:
15449:
15444:
15443:
15442:
15431:
15429:
15425:
15424:
15422:
15421:
15416:
15411:
15410:
15409:
15399:
15394:
15392:Casimir effect
15389:
15383:
15381:
15377:
15376:
15374:
15373:
15368:
15363:
15358:
15353:
15351:Quantum optics
15348:
15343:
15338:
15333:
15328:
15323:
15318:
15313:
15308:
15303:
15298:
15293:
15288:
15283:
15278:
15273:
15272:
15271:
15261:
15256:
15251:
15246:
15245:
15244:
15234:
15229:
15224:
15218:
15216:
15210:
15209:
15207:
15206:
15201:
15196:
15191:
15186:
15181:
15176:
15171:
15166:
15161:
15156:
15150:
15148:
15142:
15141:
15139:
15138:
15133:
15128:
15126:Quantum eraser
15123:
15118:
15113:
15108:
15103:
15098:
15093:
15088:
15082:
15080:
15076:
15075:
15073:
15072:
15067:
15062:
15057:
15052:
15047:
15042:
15041:
15040:
15039:
15038:
15023:
15018:
15013:
15008:
15003:
14997:
14995:
14989:
14988:
14986:
14985:
14980:
14975:
14970:
14965:
14960:
14955:
14950:
14944:
14942:
14938:
14937:
14935:
14934:
14929:
14924:
14919:
14914:
14909:
14904:
14898:
14896:
14892:
14891:
14889:
14888:
14887:
14886:
14881:
14871:
14866:
14861:
14856:
14851:
14846:
14841:
14836:
14831:
14826:
14821:
14816:
14811:
14810:
14809:
14804:
14799:
14794:
14784:
14782:Density matrix
14779:
14774:
14769:
14763:
14761:
14757:
14756:
14754:
14753:
14748:
14743:
14738:
14737:
14736:
14726:
14720:
14718:
14714:
14713:
14706:
14705:
14698:
14691:
14683:
14677:
14676:
14660:Krips, Henry.
14657:
14637:
14636:
14634:
14630:
14629:
14624:
14601:
14576:
14568:
14553:
14552:
14550:
14546:
14545:
14536:
14531:
14523:
14522:External links
14520:
14518:
14517:
14511:
14510:
14508:
14504:
14503:
14493:
14479:
14463:
14457:
14437:
14431:
14415:
14403:Periodic Table
14394:
14380:
14362:
14352:Albert Messiah
14349:
14343:
14327:
14309:
14299:Wave Mechanics
14295:
14289:
14273:
14264:
14256:Pergamon Press
14240:
14223:Hagen Kleinert
14220:
14219:. McGraw Hill.
14210:
14195:
14177:
14166:D. Greenberger
14163:
14157:
14135:
14109:(3): 454–462.
14095:
14078:
14072:
14053:
14047:
14031:
14025:
14015:Quantum Theory
14005:
13999:
13977:
13976:
13961:Victor Stenger
13958:
13948:
13926:
13902:
13896:
13890:. Allen Lane.
13874:
13866:. John Wiley.
13853:
13851:
13848:
13845:
13844:
13811:
13774:
13729:
13722:
13697:
13643:
13630:
13617:
13610:
13579:
13564:
13550:Sakurai, J. J.
13541:
13516:
13491:
13476:
13451:
13432:(2017-08-31).
13421:
13414:
13394:Translated in
13363:
13355:, ed. (1989).
13330:(6): 132–148.
13308:
13301:
13272:
13241:
13234:
13213:
13187:
13180:
13159:
13152:
13132:
13122:
13104:
13090:
13070:
13043:(4): 217–219.
13021:
13006:
12979:
12958:
12923:(4): 390–417.
12909:(April 2010).
12898:
12862:
12852:
12827:
12800:(2): 229–240.
12784:
12747:(3): 415–438.
12731:
12724:
12696:Wheeler, J. A.
12683:
12662:
12643:
12630:(3): 195–200.
12605:
12590:
12553:(4): 879–887.
12533:
12514:(3): 171–201.
12498:
12439:
12402:(3): 222–230.
12386:
12321:
12278:
12263:
12245:
12204:
12157:
12136:10.1086/425941
12130:(5): 669–682.
12110:
12057:
12050:
12027:
12020:
12000:
11993:
11972:
11965:
11943:
11912:
11890:
11845:(11): 111807.
11827:
11820:
11800:
11778:
11759:(2019-02-26).
11748:
11741:
11713:
11706:
11698:Modern Physics
11688:
11670:Matson, John.
11662:
11628:(25): 250001.
11608:
11593:
11581:Vedral, Vlatko
11572:
11535:(15): 155303.
11517:
11462:
11455:
11429:
11422:
11404:
11365:
11312:
11289:
11253:
11246:
11220:
11205:
11173:
11167:
11128:(4): 358–381,
11111:
11096:
11088:Pergamon Press
11065:
11058:
11030:
11023:
10997:
10938:
10883:
10876:
10852:
10849:on 2023-01-02.
10841:
10819:
10809:
10786:
10750:
10716:
10659:
10652:
10606:
10582:
10557:
10508:
10501:
10478:
10471:
10451:
10444:
10424:
10417:
10392:
10385:
10345:
10292:
10231:
10198:Nature Physics
10188:
10169:(9): 896–905.
10153:
10139:
10112:
10057:
10056:
10054:
10051:
10048:
10047:
10037:
10011:
9974:
9960:
9959:
9957:
9954:
9952:
9951:
9946:
9941:
9936:
9931:
9926:
9921:
9915:
9913:
9910:
9894:semiconductors
9874:quantum optics
9843:wave mechanics
9835:Pascual Jordan
9763:spectral lines
9734:
9733:
9718:
9715:
9712:
9709:
9647:Julius PlĂĽcker
9585:Leonhard Euler
9550:
9547:
9532:simultaneously
9485:Boris Podolsky
9477:thermodynamics
9457:quantum theory
9389:
9378:
9372:
9366:Main article:
9363:
9360:
9277:
9274:
9207:
9204:
9197:
9193:
9189:
9185:
9182:
9179:
9175:
9169:
9165:
9161:
9149:electric field
9147:describes the
9118:Dirac equation
9105:
9102:
9052:kinetic energy
9011:
9008:
9005:
9004:
9002:
9001:
8994:
8987:
8979:
8976:
8975:
8972:
8971:
8966:Modern physics
8964:
8963:
8959:
8956:
8955:
8952:
8951:
8948:
8947:
8942:
8937:
8932:
8927:
8922:
8917:
8912:
8907:
8902:
8897:
8892:
8887:
8882:
8877:
8872:
8867:
8862:
8857:
8852:
8847:
8842:
8837:
8832:
8827:
8822:
8817:
8812:
8807:
8802:
8800:P. W. Anderson
8797:
8792:
8787:
8782:
8777:
8772:
8767:
8762:
8757:
8752:
8747:
8742:
8737:
8732:
8727:
8722:
8717:
8712:
8707:
8702:
8697:
8692:
8687:
8682:
8677:
8672:
8667:
8662:
8657:
8652:
8647:
8642:
8637:
8632:
8627:
8622:
8617:
8611:
8608:
8607:
8604:
8603:
8600:
8599:
8594:
8589:
8584:
8579:
8574:
8569:
8564:
8562:Plasma physics
8559:
8554:
8549:
8544:
8539:
8534:
8529:
8524:
8519:
8514:
8509:
8504:
8499:
8494:
8489:
8488:
8487:
8482:
8477:
8467:
8462:
8457:
8452:
8447:
8441:
8438:
8437:
8434:
8433:
8430:
8429:
8424:
8419:
8414:
8409:
8404:
8399:
8394:
8389:
8384:
8379:
8374:
8368:
8365:
8364:
8361:
8360:
8357:
8356:
8351:
8346:
8341:
8336:
8334:Wolfgang Pauli
8331:
8329:Pascual Jordan
8326:
8321:
8316:
8311:
8306:
8301:
8295:
8292:
8291:
8288:
8287:
8278:
8265:
8262:
8258:
8253:
8249:
8244:
8241:
8237:
8233:
8230:
8225:
8222:
8218:
8197:
8194:
8191:
8188:
8183:
8179:
8174:
8167:
8164:
8160:
8155:
8152:
8149:
8146:
8143:
8140:
8137:
8132:
8128:
8123:
8116:
8113:
8099:
8098:
8096:Modern physics
8089:
8086:
8066:microprocessor
8062:semiconductors
8038:quantum optics
7999:Main article:
7996:
7993:
7980:
7977:
7957:
7953:
7949:
7946:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7919:
7900:
7899:
7888:
7883:
7879:
7876:
7870:
7865:
7861:
7857:
7852:
7847:
7842:
7837:
7833:
7829:
7826:
7823:
7820:
7815:
7811:
7807:
7803:
7799:
7796:
7793:
7790:
7787:
7777:
7766:
7761:
7757:
7754:
7748:
7743:
7739:
7735:
7730:
7725:
7720:
7715:
7711:
7707:
7704:
7701:
7698:
7693:
7689:
7685:
7681:
7677:
7674:
7671:
7668:
7665:
7651:
7650:
7639:
7634:
7628:
7625:
7621:
7617:
7614:
7611:
7608:
7605:
7602:
7601:
7598:
7595:
7591:
7587:
7584:
7581:
7578:
7575:
7572:
7569:
7568:
7566:
7559:
7555:
7551:
7548:
7545:
7541:
7537:
7534:
7529:
7525:
7521:
7518:
7515:
7492:
7472:
7452:
7429:
7426:
7404:
7396:
7393:
7390:
7386:
7382:
7380:
7377:
7376:
7373:
7370:
7368:
7365:
7364:
7362:
7357:
7354:
7332:
7326:
7322:
7300:
7294:
7290:
7268:
7262:
7259:
7257:
7254:
7253:
7250:
7247:
7245:
7242:
7241:
7239:
7231:
7227:
7222:
7219:
7208:beam splitters
7192:
7189:
7184:
7179:
7174:
7170:
7166:
7161:
7156:
7151:
7147:
7126:
7123:
7120:
7117:
7114:
7111:
7108:
7088:
7085:
7082:
7060:
7056:
7052:
7049:
7044:
7040:
7036:
7033:
7030:
7008:
7002:
6999:
6998:
6995:
6992:
6991:
6989:
6984:
6979:
6975:
6952:
6946:
6943:
6942:
6939:
6936:
6935:
6933:
6928:
6923:
6919:
6896:
6891:
6886:
6883:
6847:
6844:
6836:
6835:
6824:
6820:
6814:
6811:
6806:
6803:
6799:
6795:
6792:
6789:
6784:
6780:
6765:
6764:
6753:
6749:
6742:
6738:
6734:
6730:
6726:
6717:
6713:
6709:
6703:
6699:
6689:
6685:
6680:
6674:
6670:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6643:
6639:
6619:
6614:
6613:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
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:
6196:
6195:
6184:
6176:
6172:
6168:
6165:
6158:
6154:
6148:
6144:
6137:
6129:
6125:
6121:
6118:
6111:
6107:
6101:
6097:
6091:
6087:
6080:
6075:
6071:
6047:
6036:
6035:
6024:
6021:
6018:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5990:
5986:
5983:
5977:
5974:
5951:
5931:
5928:
5908:
5905:
5902:
5899:
5896:
5893:
5890:
5887:
5867:
5847:
5836:
5835:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5803:
5800:
5797:
5794:
5791:
5788:
5785:
5782:
5759:
5756:
5753:
5733:
5730:
5727:
5716:
5715:
5704:
5701:
5698:
5695:
5692:
5689:
5686:
5683:
5680:
5677:
5674:
5671:
5668:
5665:
5662:
5659:
5656:
5653:
5650:
5647:
5644:
5641:
5618:
5615:
5612:
5592:
5572:
5569:
5566:
5546:
5543:
5540:
5520:
5500:
5497:
5494:
5491:
5480:
5479:
5467:
5464:
5461:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5410:
5407:
5389:
5388:
5374:
5371:
5364:
5360:
5354:
5350:
5343:
5340:
5333:
5330:
5327:
5324:
5320:
5316:
5313:
5308:
5305:
5302:
5298:
5294:
5291:
5288:
5285:
5282:
5279:
5253:
5233:
5222:
5221:
5210:
5207:
5204:
5199:
5194:
5187:
5184:
5174:
5171:
5167:
5155:
5136:
5133:
5129:
5124:
5121:
5118:
5115:
5110:
5103:
5100:
5082:
5081:
5070:
5067:
5064:
5061:
5053:
5049:
5045:
5040:
5035:
5031:
5021:
5018:
5012:
5008:
5002:
4979:
4955:Main article:
4944:
4941:
4924:
4904:
4893:
4892:
4881:
4874:
4868:
4864:
4860:
4854:
4850:
4843:
4838:
4835:
4829:
4826:
4823:
4820:
4817:
4814:
4808:
4805:
4788:
4787:
4771:
4768:
4762:
4758:
4752:
4748:
4740:
4735:
4732:
4727:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4674:
4671:
4668:
4665:
4662:
4659:
4639:
4636:
4633:
4630:
4627:
4621:
4618:
4595:
4592:
4589:
4586:
4564:
4561:
4555:
4552:
4545:
4541:
4537:
4531:
4528:
4525:
4522:
4519:
4515:
4500:
4499:
4488:
4485:
4481:
4475:
4472:
4466:
4463:
4456:
4452:
4448:
4442:
4439:
4436:
4433:
4430:
4426:
4422:
4419:
4416:
4413:
4410:
4404:
4401:
4393:
4388:
4385:
4381:
4374:
4371:
4367:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4330:
4329:
4318:
4310:
4306:
4302:
4296:
4292:
4283:
4280:
4274:
4270:
4264:
4261:
4256:
4252:
4245:
4242:
4238:
4233:
4230:
4199:Main article:
4196:
4193:
4191:
4188:
4177:differentiable
4171:in classical (
4156:
4136:
4116:
4096:
4076:
4056:
4036:
4016:
3996:
3972:
3952:
3932:
3912:
3909:
3906:
3903:
3883:
3861:
3857:
3853:
3850:
3847:
3844:
3840:
3836:
3833:
3830:
3827:
3824:
3800:
3785:Main article:
3782:
3779:
3749:" proposed by
3742:
3739:
3703:
3702:
3690:
3684:
3680:
3676:
3671:
3667:
3663:
3658:
3654:
3650:
3645:
3641:
3636:
3628:
3624:
3598:
3576:
3572:
3549:
3545:
3524:
3502:
3498:
3475:
3471:
3448:
3445:
3439:
3425:
3424:
3413:
3408:
3404:
3400:
3395:
3391:
3365:
3361:
3338:
3334:
3322:
3321:
3310:
3305:
3299:
3293:
3288:
3282:
3276:
3271:
3268:
3262:
3234:
3228:
3203:
3197:
3175:tensor product
3170:
3167:
3152:
3148:
3144:
3117:
3114:
3110:
3105:
3102:
3099:
3077:
3073:
3048:
3044:
3040:
3024:
3023:
3012:
3008:
3002:
2997:
2994:
2991:
2988:
2985:
2980:
2974:
2967:
2964:
2958:
2953:
2949:
2943:
2939:
2924:
2923:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2852:
2832:
2821:
2820:
2809:
2804:
2801:
2796:
2791:
2787:
2781:
2777:
2762:
2761:
2750:
2743:
2738:
2735:
2732:
2727:
2723:
2718:
2714:
2710:
2704:
2699:
2695:
2680:
2679:
2668:
2659:
2654:
2651:
2648:
2643:
2639:
2634:
2630:
2626:
2618:
2613:
2609:
2581:
2561:
2550:
2549:
2538:
2535:
2532:
2529:
2526:
2520:
2517:
2511:
2505:
2502:
2496:
2466:
2463:
2437:
2434:
2418:
2415:
2378:including the
2358:atomic nucleus
2344:in acoustics.)
2274:
2271:
2268:
2265:
2245:
2242:
2239:
2236:
2206:
2202:
2198:
2195:
2192:
2189:
2185:
2181:
2178:
2175:
2172:
2169:
2158:
2157:
2146:
2143:
2140:
2137:
2134:
2129:
2125:
2121:
2118:
2115:
2112:
2108:
2104:
2101:
2098:
2095:
2092:
2058:
2055:
2031:
2003:
1992:
1991:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1944:
1941:
1937:
1932:
1929:
1914:
1911:
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:
15978:
15967:
15964:
15963:
15961:
15951:
15946:
15941:
15939:
15934:
15929:
15927:
15922:
15917:
15915:
15910:
15905:
15903:
15893:
15891:
15886:
15881:
15879:
15874:
15869:
15867:
15857:
15855:
15845:
15844:
15841:
15828:
15825:
15823:
15820:
15818:
15815:
15813:
15810:
15808:
15805:
15804:
15802:
15798:
15792:
15789:
15787:
15786:Ocean physics
15784:
15782:
15779:
15777:
15774:
15772:
15769:
15767:
15764:
15762:
15759:
15757:
15754:
15752:
15749:
15747:
15744:
15743:
15741:
15739:
15735:
15729:
15726:
15722:
15721:Modern optics
15719:
15717:
15714:
15712:
15709:
15708:
15707:
15704:
15702:
15699:
15697:
15694:
15692:
15689:
15685:
15682:
15680:
15677:
15676:
15675:
15672:
15671:
15669:
15667:
15663:
15655:
15652:
15650:
15647:
15646:
15645:
15642:
15638:
15635:
15633:
15630:
15629:
15628:
15625:
15623:
15620:
15618:
15615:
15611:
15608:
15606:
15603:
15601:
15598:
15596:
15593:
15592:
15591:
15588:
15587:
15585:
15583:
15579:
15571:
15570:Computational
15568:
15567:
15566:
15563:
15561:
15558:
15557:
15555:
15551:
15543:
15540:
15539:
15538:
15535:
15533:
15530:
15529:
15527:
15523:
15519:
15511:
15506:
15504:
15499:
15497:
15492:
15491:
15488:
15476:
15468:
15467:
15464:
15458:
15455:
15453:
15450:
15448:
15445:
15441:
15438:
15437:
15436:
15433:
15432:
15430:
15426:
15420:
15417:
15415:
15412:
15408:
15405:
15404:
15403:
15400:
15398:
15395:
15393:
15390:
15388:
15385:
15384:
15382:
15378:
15372:
15369:
15367:
15364:
15362:
15359:
15357:
15354:
15352:
15349:
15347:
15344:
15342:
15339:
15337:
15334:
15332:
15329:
15327:
15324:
15322:
15319:
15317:
15314:
15312:
15311:Quantum logic
15309:
15307:
15304:
15302:
15299:
15297:
15294:
15292:
15289:
15287:
15284:
15282:
15279:
15277:
15274:
15270:
15267:
15266:
15265:
15262:
15260:
15257:
15255:
15252:
15250:
15247:
15243:
15240:
15239:
15238:
15235:
15233:
15230:
15228:
15225:
15223:
15220:
15219:
15217:
15215:
15211:
15205:
15202:
15200:
15197:
15195:
15192:
15190:
15187:
15185:
15182:
15180:
15177:
15175:
15172:
15170:
15167:
15165:
15164:Quantum chaos
15162:
15160:
15157:
15155:
15152:
15151:
15149:
15147:
15143:
15137:
15134:
15132:
15131:Stern–Gerlach
15129:
15127:
15124:
15122:
15119:
15117:
15114:
15112:
15109:
15107:
15104:
15102:
15099:
15097:
15094:
15092:
15089:
15087:
15084:
15083:
15081:
15077:
15071:
15068:
15066:
15065:Transactional
15063:
15061:
15058:
15056:
15055:Quantum logic
15053:
15051:
15048:
15046:
15043:
15037:
15034:
15033:
15032:
15029:
15028:
15027:
15024:
15022:
15019:
15017:
15014:
15012:
15009:
15007:
15004:
15002:
14999:
14998:
14996:
14994:
14990:
14984:
14981:
14979:
14976:
14974:
14971:
14969:
14966:
14964:
14961:
14959:
14956:
14954:
14951:
14949:
14946:
14945:
14943:
14939:
14933:
14930:
14928:
14925:
14923:
14920:
14918:
14915:
14913:
14910:
14908:
14905:
14903:
14900:
14899:
14897:
14893:
14885:
14882:
14880:
14877:
14876:
14875:
14874:Wave function
14872:
14870:
14867:
14865:
14862:
14860:
14857:
14855:
14852:
14850:
14849:Superposition
14847:
14845:
14844:Quantum state
14842:
14840:
14837:
14835:
14832:
14830:
14827:
14825:
14822:
14820:
14817:
14815:
14812:
14808:
14805:
14803:
14800:
14798:
14797:Excited state
14795:
14793:
14790:
14789:
14788:
14785:
14783:
14780:
14778:
14775:
14773:
14770:
14768:
14765:
14764:
14762:
14758:
14752:
14749:
14747:
14744:
14742:
14739:
14735:
14732:
14731:
14730:
14727:
14725:
14722:
14721:
14719:
14715:
14711:
14704:
14699:
14697:
14692:
14690:
14685:
14684:
14681:
14673:
14672:
14667:
14663:
14658:
14654:
14653:
14648:
14644:
14639:
14638:
14635:
14632:
14631:
14628:
14625:
14623:
14602:
14600:
14596:
14592:
14588:
14584:
14580:
14577:
14574:
14573:
14569:
14566:
14562:
14558:
14555:
14554:
14551:
14548:
14547:
14544:
14540:
14537:
14535:
14532:
14530:
14526:
14525:
14516:
14513:
14512:
14509:
14506:
14505:
14501:
14497:
14494:
14490:
14486:
14482:
14476:
14472:
14468:
14464:
14460:
14454:
14449:
14448:
14442:
14438:
14434:
14428:
14424:
14420:
14416:
14414:
14413:0-19-530573-6
14410:
14406:
14404:
14398:
14395:
14391:
14387:
14383:
14377:
14373:
14372:
14367:
14366:Omnès, Roland
14363:
14361:
14357:
14353:
14350:
14346:
14340:
14336:
14332:
14328:
14325:
14324:0-486-43517-2
14321:
14317:
14313:
14312:George Mackey
14310:
14308:
14307:0-08-203204-1
14304:
14300:
14296:
14292:
14286:
14282:
14278:
14274:
14272:
14267:
14261:
14257:
14253:
14249:
14248:E.M. Lifshitz
14245:
14241:
14239:
14235:
14232:
14228:
14224:
14221:
14218:
14214:
14211:
14206:
14202:
14198:
14192:
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14187:
14182:
14178:
14175:
14171:
14167:
14164:
14160:
14154:
14150:
14149:
14144:
14140:
14136:
14132:
14128:
14124:
14120:
14116:
14112:
14108:
14104:
14100:
14099:Everett, Hugh
14096:
14094:
14093:0-691-08131-X
14090:
14086:
14082:
14079:
14075:
14069:
14065:
14064:
14059:
14054:
14050:
14044:
14040:
14036:
14035:Binney, James
14032:
14028:
14022:
14017:
14016:
14010:
14006:
14002:
13996:
13992:
13991:
13990:Quantum Leaps
13986:
13982:
13981:
13980:
13974:
13973:philosophical
13970:
13966:
13962:
13959:
13956:
13952:
13949:
13946:
13942:
13938:
13934:
13930:
13927:
13924:
13920:
13919:0-691-08388-6
13916:
13912:
13911:
13906:
13903:
13899:
13893:
13889:
13888:
13883:
13882:Forshaw, Jeff
13879:
13875:
13873:
13872:0-486-42878-8
13869:
13865:
13861:
13858:
13857:
13856:
13829:
13822:
13815:
13808:
13796:
13792:
13788:
13784:
13778:
13767:
13762:
13757:
13753:
13749:
13748:
13740:
13733:
13725:
13723:0-691-01243-1
13719:
13714:
13713:
13707:
13706:Pais, Abraham
13701:
13693:
13689:
13684:
13679:
13675:
13671:
13667:
13663:
13662:
13657:
13653:
13647:
13640:
13634:
13627:
13621:
13613:
13611:0-19-851791-2
13607:
13603:
13599:
13595:
13594:
13589:
13583:
13575:
13571:
13567:
13561:
13557:
13556:
13551:
13545:
13530:
13526:
13520:
13512:
13511:
13506:
13502:
13495:
13487:
13483:
13479:
13473:
13468:
13467:
13458:
13456:
13441:
13440:
13439:Physics World
13435:
13431:
13425:
13417:
13411:
13407:
13403:
13399:
13391:
13387:
13383:
13380:(in German).
13379:
13378:
13373:
13367:
13358:
13354:
13353:Stachel, John
13350:Reprinted in
13346:
13341:
13337:
13333:
13329:
13325:
13324:
13319:
13312:
13304:
13298:
13294:
13290:
13286:
13282:
13281:Stachel, John
13276:
13261:
13260:
13259:Physics World
13255:
13251:
13245:
13237:
13231:
13227:
13223:
13217:
13201:
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13066:
13062:
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13054:
13050:
13046:
13042:
13038:
13037:
13032:
13025:
13017:
13013:
13009:
13007:0-521-64222-1
13003:
12999:
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12993:
12989:
12983:
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12974:
12969:
12962:
12954:
12950:
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12922:
12918:
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12912:
12908:
12902:
12894:
12890:
12885:
12880:
12876:
12872:
12866:
12859:
12855:
12849:
12845:
12838:
12831:
12823:
12819:
12815:
12811:
12807:
12803:
12799:
12795:
12788:
12780:
12776:
12772:
12768:
12764:
12760:
12755:
12750:
12746:
12742:
12735:
12727:
12725:0-691-08131-X
12721:
12718:. p. v.
12717:
12713:
12709:
12708:DeWitt, Bryce
12705:
12704:Cooper, L. N.
12701:
12700:DeWitt, B. S.
12697:
12693:
12692:Everett, Hugh
12687:
12679:
12678:
12673:
12666:
12658:
12654:
12647:
12638:
12633:
12629:
12625:
12624:
12619:
12615:
12609:
12601:
12594:
12586:
12582:
12578:
12574:
12569:
12564:
12560:
12556:
12552:
12548:
12544:
12537:
12529:
12525:
12521:
12517:
12513:
12509:
12502:
12494:
12490:
12486:
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12478:
12474:
12469:
12464:
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12456:
12455:
12450:
12443:
12435:
12431:
12427:
12423:
12419:
12415:
12410:
12405:
12401:
12397:
12390:
12382:
12378:
12374:
12370:
12366:
12362:
12358:
12354:
12349:
12344:
12341:(1): 015005.
12340:
12336:
12332:
12325:
12317:
12313:
12309:
12305:
12301:
12297:
12293:
12289:
12282:
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12129:
12125:
12121:
12114:
12106:
12102:
12098:
12094:
12090:
12086:
12081:
12076:
12073:(6): 062116.
12072:
12068:
12061:
12053:
12051:0-262-56003-8
12047:
12043:
12042:
12037:
12031:
12023:
12017:
12013:
12012:
12004:
11996:
11990:
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11831:
11823:
11817:
11813:
11812:
11804:
11788:
11782:
11768:
11767:
11762:
11758:
11757:Baez, John C.
11752:
11744:
11742:0-7923-2549-4
11738:
11734:
11733:
11728:
11722:
11720:
11718:
11709:
11703:
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11316:
11308:
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11300:
11296:
11292:
11286:
11282:
11278:
11273:
11268:
11264:
11257:
11249:
11243:
11240:. MIT Press.
11239:
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11212:
11208:
11202:
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11168:0-521-49730-2
11164:
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11135:
11131:
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11115:
11107:
11103:
11099:
11093:
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11085:
11084:
11079:
11075:
11069:
11061:
11059:0-471-16433-X
11055:
11051:
11047:
11041:
11039:
11037:
11035:
11026:
11020:
11016:
11012:
11008:
11001:
10993:
10989:
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10961:
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10942:
10934:
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10926:
10922:
10918:
10914:
10910:
10906:
10902:
10898:
10894:
10887:
10879:
10873:
10869:
10865:
10864:
10856:
10848:
10844:
10838:
10834:
10830:
10823:
10816:
10812:
10810:0-345-40946-9
10806:
10802:
10801:
10796:
10790:
10783:
10781:
10768:
10764:
10760:
10759:Baez, John C.
10754:
10739:
10738:
10733:
10729:
10723:
10721:
10712:
10708:
10704:
10700:
10695:
10690:
10686:
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10655:
10649:
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10641:
10636:
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10627:
10623:
10619:
10613:
10611:
10602:
10601:
10596:
10592:
10586:
10572:
10568:
10561:
10553:
10549:
10544:
10539:
10535:
10531:
10527:
10523:
10519:
10512:
10504:
10502:0-13-124405-1
10498:
10494:
10493:
10488:
10482:
10474:
10468:
10464:
10463:
10455:
10447:
10441:
10437:
10436:
10428:
10420:
10414:
10410:
10409:
10401:
10399:
10397:
10388:
10382:
10379:. MIT Press.
10378:
10374:
10368:
10366:
10364:
10362:
10360:
10358:
10356:
10354:
10352:
10350:
10341:
10337:
10333:
10329:
10325:
10321:
10316:
10311:
10308:(7): 071801.
10307:
10303:
10296:
10288:
10284:
10280:
10276:
10272:
10268:
10264:
10260:
10255:
10250:
10247:(2): 023901.
10246:
10242:
10235:
10227:
10223:
10219:
10215:
10211:
10207:
10203:
10199:
10192:
10184:
10180:
10176:
10172:
10168:
10164:
10157:
10142:
10136:
10132:
10131:
10123:
10121:
10119:
10117:
10108:
10104:
10100:
10096:
10092:
10088:
10084:
10080:
10076:
10072:
10068:
10062:
10058:
10041:
10035:
10034:
10029:
10028:
10023:
10022:
10015:
10008:
10004:
10000:
9996:
9992:
9988:
9984:
9978:
9971:
9965:
9961:
9950:
9947:
9945:
9942:
9940:
9937:
9935:
9932:
9930:
9927:
9925:
9922:
9920:
9917:
9916:
9909:
9907:
9903:
9899:
9895:
9891:
9887:
9883:
9879:
9875:
9871:
9867:
9863:
9859:
9855:
9854:David Hilbert
9850:
9848:
9844:
9840:
9836:
9831:
9826:
9824:
9820:
9816:
9812:
9808:
9807:specific heat
9804:
9800:
9796:
9792:
9784:
9780:
9775:
9771:
9768:
9764:
9760:
9756:
9752:
9751:realistically
9747:
9743:
9739:
9716:
9713:
9710:
9707:
9700:
9699:
9698:
9696:
9692:
9688:
9684:
9680:
9675:
9671:
9666:
9664:
9663:J. J. Thomson
9660:
9656:
9652:
9648:
9644:
9640:
9636:
9632:
9628:
9624:
9620:
9619:atomic theory
9616:
9612:
9608:
9603:
9601:
9597:
9593:
9590:
9586:
9582:
9578:
9569:
9565:
9560:
9559:Atomic theory
9556:
9546:
9544:
9540:
9536:
9533:
9529:
9525:
9520:
9517:
9513:
9511:
9507:
9502:
9498:
9494:
9490:
9486:
9482:
9478:
9474:
9470:
9466:
9462:
9458:
9454:
9450:
9447:
9446:complementary
9442:
9438:
9434:
9430:
9427:The views of
9425:
9423:
9419:
9415:
9411:
9407:
9403:
9399:
9398:philosophical
9392:
9386:
9382:
9381:superposition
9369:
9359:
9357:
9356:Planck length
9353:
9349:
9348:spin networks
9345:
9341:
9336:
9334:
9330:
9326:
9322:
9318:
9314:
9310:
9306:
9302:
9301:string theory
9297:
9295:
9291:
9287:
9283:
9273:
9271:
9267:
9263:
9259:
9255:
9251:
9247:
9243:
9239:
9235:
9234:Quantum field
9231:
9229:
9225:
9221:
9202:
9195:
9192:
9187:
9183:
9180:
9173:
9167:
9163:
9159:
9150:
9146:
9145:hydrogen atom
9142:
9138:
9133:
9131:
9127:
9123:
9119:
9115:
9111:
9101:
9099:
9094:
9089:
9087:
9086:absolute zero
9082:
9078:
9074:
9070:
9068:
9067:quantum chaos
9064:
9059:
9057:
9053:
9049:
9045:
9040:
9038:
9034:
9030:
9026:
9021:
9017:
9016:Hilbert space
9000:
8995:
8993:
8988:
8986:
8981:
8980:
8978:
8977:
8967:
8961:
8960:
8954:
8953:
8946:
8943:
8941:
8938:
8936:
8933:
8931:
8928:
8926:
8923:
8921:
8918:
8916:
8913:
8911:
8908:
8906:
8903:
8901:
8900:J. J. Thomson
8898:
8896:
8893:
8891:
8888:
8886:
8883:
8881:
8878:
8876:
8873:
8871:
8868:
8866:
8863:
8861:
8858:
8856:
8853:
8851:
8848:
8846:
8843:
8841:
8838:
8836:
8833:
8831:
8828:
8826:
8823:
8821:
8818:
8816:
8813:
8811:
8808:
8806:
8803:
8801:
8798:
8796:
8793:
8791:
8788:
8786:
8783:
8781:
8778:
8776:
8773:
8771:
8768:
8766:
8763:
8761:
8758:
8756:
8753:
8751:
8748:
8746:
8745:van der Waals
8743:
8741:
8738:
8736:
8733:
8731:
8728:
8726:
8723:
8721:
8718:
8716:
8713:
8711:
8708:
8706:
8703:
8701:
8698:
8696:
8693:
8691:
8688:
8686:
8683:
8681:
8678:
8676:
8673:
8671:
8668:
8666:
8663:
8661:
8658:
8656:
8653:
8651:
8648:
8646:
8643:
8641:
8638:
8636:
8633:
8631:
8628:
8626:
8623:
8621:
8618:
8616:
8613:
8612:
8606:
8605:
8598:
8595:
8593:
8590:
8588:
8585:
8583:
8580:
8578:
8575:
8573:
8570:
8568:
8565:
8563:
8560:
8558:
8555:
8553:
8550:
8548:
8545:
8543:
8540:
8538:
8535:
8533:
8530:
8528:
8525:
8523:
8520:
8518:
8515:
8513:
8510:
8508:
8505:
8503:
8500:
8498:
8495:
8493:
8490:
8486:
8483:
8481:
8478:
8476:
8473:
8472:
8471:
8468:
8466:
8463:
8461:
8458:
8456:
8453:
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8446:
8443:
8442:
8436:
8435:
8428:
8425:
8423:
8420:
8418:
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8395:
8393:
8390:
8388:
8385:
8383:
8380:
8378:
8375:
8373:
8370:
8369:
8363:
8362:
8355:
8352:
8350:
8347:
8345:
8342:
8340:
8337:
8335:
8332:
8330:
8327:
8325:
8322:
8320:
8317:
8315:
8312:
8310:
8307:
8305:
8302:
8300:
8297:
8296:
8290:
8289:
8285:
8281:
8263:
8260:
8256:
8251:
8247:
8242:
8239:
8235:
8228:
8223:
8220:
8216:
8189:
8181:
8177:
8165:
8162:
8158:
8150:
8147:
8138:
8130:
8126:
8111:
8101:
8100:
8097:
8094:
8093:
8085:
8083:
8079:
8075:
8071:
8067:
8063:
8059:
8055:
8051:
8047:
8043:
8039:
8035:
8030:
8028:
8024:
8020:
8016:
8012:
8008:
8002:
7992:
7955:
7951:
7947:
7944:
7938:
7932:
7929:
7923:
7917:
7907:
7905:
7886:
7881:
7868:
7863:
7859:
7855:
7850:
7835:
7831:
7827:
7824:
7821:
7818:
7813:
7809:
7797:
7791:
7785:
7778:
7764:
7759:
7746:
7741:
7737:
7733:
7728:
7713:
7709:
7705:
7702:
7699:
7696:
7691:
7687:
7675:
7669:
7663:
7656:
7655:
7654:
7637:
7632:
7623:
7619:
7606:
7603:
7593:
7589:
7576:
7573:
7570:
7564:
7557:
7553:
7543:
7539:
7535:
7532:
7527:
7523:
7519:
7516:
7513:
7506:
7505:
7504:
7490:
7470:
7450:
7441:
7402:
7388:
7384:
7378:
7371:
7366:
7360:
7355:
7352:
7330:
7324:
7320:
7298:
7292:
7288:
7266:
7260:
7255:
7248:
7243:
7237:
7229:
7225:
7220:
7217:
7209:
7204:
7190:
7187:
7182:
7172:
7164:
7159:
7149:
7124:
7121:
7115:
7112:
7109:
7086:
7083:
7080:
7058:
7054:
7050:
7047:
7042:
7038:
7034:
7031:
7028:
7006:
7000:
6993:
6987:
6982:
6977:
6973:
6950:
6944:
6937:
6931:
6926:
6921:
6917:
6894:
6884:
6881:
6871:
6869:
6865:
6861:
6852:
6843:
6841:
6822:
6818:
6812:
6809:
6804:
6801:
6797:
6793:
6787:
6782:
6778:
6770:
6769:
6768:
6751:
6747:
6740:
6736:
6732:
6728:
6724:
6715:
6711:
6707:
6701:
6697:
6687:
6683:
6678:
6672:
6664:
6661:
6655:
6649:
6641:
6637:
6629:
6628:
6627:
6626:
6622:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6566:
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:
6182:
6174:
6170:
6166:
6163:
6156:
6152:
6146:
6142:
6135:
6127:
6123:
6119:
6116:
6109:
6105:
6099:
6095:
6089:
6085:
6078:
6073:
6069:
6061:
6060:
6059:
6045:
6022:
6019:
6016:
6013:
6010:
6007:
6004:
6001:
5998:
5995:
5988:
5984:
5981:
5975:
5972:
5965:
5964:
5963:
5949:
5929:
5926:
5906:
5903:
5897:
5894:
5888:
5885:
5865:
5845:
5822:
5816:
5813:
5807:
5804:
5801:
5798:
5795:
5792:
5786:
5780:
5773:
5772:
5771:
5757:
5754:
5751:
5731:
5728:
5725:
5702:
5699:
5693:
5687:
5684:
5681:
5678:
5672:
5666:
5663:
5660:
5657:
5654:
5651:
5645:
5639:
5632:
5631:
5630:
5616:
5613:
5610:
5590:
5570:
5567:
5564:
5544:
5541:
5538:
5518:
5498:
5495:
5492:
5489:
5465:
5459:
5456:
5450:
5447:
5444:
5441:
5435:
5432:
5426:
5423:
5420:
5417:
5411:
5405:
5398:
5397:
5396:
5394:
5372:
5369:
5362:
5358:
5352:
5348:
5341:
5338:
5331:
5328:
5325:
5322:
5318:
5314:
5311:
5306:
5303:
5300:
5296:
5292:
5289:
5283:
5277:
5270:
5269:
5268:
5265:
5251:
5231:
5208:
5205:
5202:
5197:
5192:
5182:
5172:
5169:
5165:
5156:
5153:
5134:
5131:
5127:
5119:
5116:
5113:
5108:
5098:
5087:
5086:
5085:
5068:
5065:
5062:
5059:
5051:
5047:
5043:
5038:
5033:
5029:
5019:
5016:
5010:
5006:
5000:
4993:
4992:
4991:
4977:
4969:
4965:
4958:
4949:
4940:
4936:
4922:
4902:
4879:
4872:
4866:
4862:
4858:
4852:
4848:
4841:
4836:
4833:
4827:
4821:
4818:
4815:
4803:
4793:
4792:
4791:
4769:
4766:
4760:
4756:
4750:
4746:
4738:
4733:
4730:
4725:
4720:
4714:
4711:
4708:
4702:
4695:
4694:
4693:
4691:
4686:
4669:
4666:
4663:
4657:
4634:
4631:
4628:
4616:
4593:
4587:
4584:
4559:
4553:
4550:
4543:
4539:
4535:
4529:
4526:
4523:
4517:
4513:
4505:
4486:
4483:
4470:
4464:
4461:
4454:
4450:
4446:
4440:
4437:
4434:
4428:
4424:
4417:
4414:
4411:
4399:
4383:
4379:
4372:
4369:
4365:
4360:
4354:
4351:
4348:
4342:
4335:
4334:
4333:
4316:
4308:
4304:
4300:
4294:
4290:
4281:
4278:
4272:
4268:
4262:
4259:
4254:
4250:
4243:
4240:
4236:
4231:
4228:
4221:
4220:
4219:
4212:
4207:
4202:
4201:Free particle
4195:Free particle
4187:
4185:
4181:
4178:
4174:
4170:
4154:
4134:
4114:
4094:
4074:
4054:
4034:
4014:
3994:
3986:
3970:
3950:
3930:
3907:
3901:
3881:
3855:
3851:
3848:
3845:
3842:
3838:
3834:
3828:
3822:
3814:
3798:
3788:
3778:
3776:
3772:
3768:
3764:
3760:
3757:(invented by
3756:
3752:
3748:
3738:
3736:
3730:
3728:
3723:
3710:
3708:
3688:
3682:
3678:
3674:
3669:
3665:
3661:
3656:
3652:
3648:
3643:
3639:
3634:
3626:
3622:
3612:
3611:
3610:
3596:
3574:
3570:
3547:
3543:
3522:
3500:
3496:
3473:
3469:
3446:
3443:
3411:
3406:
3402:
3398:
3393:
3389:
3381:
3380:
3379:
3363:
3359:
3336:
3332:
3308:
3303:
3291:
3286:
3274:
3269:
3266:
3250:
3249:
3248:
3232:
3201:
3176:
3166:
3150:
3146:
3142:
3134:
3115:
3100:
3097:
3075:
3071:
3062:
3042:
3038:
3029:
3010:
3006:
2992:
2989:
2986:
2972:
2965:
2962:
2956:
2951:
2947:
2941:
2937:
2929:
2928:
2927:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2886:
2883:
2880:
2870:
2869:
2868:
2866:
2850:
2830:
2807:
2802:
2799:
2794:
2789:
2785:
2779:
2775:
2767:
2766:
2765:
2748:
2741:
2736:
2733:
2730:
2725:
2721:
2716:
2712:
2708:
2702:
2697:
2693:
2685:
2684:
2683:
2666:
2657:
2652:
2649:
2646:
2641:
2637:
2632:
2628:
2624:
2616:
2611:
2607:
2599:
2598:
2597:
2595:
2579:
2559:
2536:
2530:
2527:
2515:
2509:
2500:
2487:
2486:
2485:
2483:
2461:
2432:
2414:
2412:
2408:
2403:
2401:
2397:
2393:
2392:hydrogen atom
2389:
2385:
2381:
2377:
2372:
2370:
2366:
2364:
2359:
2355:
2351:
2343:
2339:
2335:
2331:
2327:
2323:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2290:
2286:
2269:
2263:
2240:
2234:
2226:
2225:deterministic
2222:
2200:
2196:
2193:
2190:
2187:
2183:
2179:
2173:
2167:
2160:The operator
2144:
2138:
2132:
2123:
2119:
2116:
2113:
2110:
2106:
2102:
2096:
2090:
2083:
2082:
2081:
2078:
2076:
2072:
2053:
2045:
2029:
2021:
2017:
2001:
1978:
1972:
1966:
1963:
1960:
1954:
1948:
1942:
1927:
1920:
1919:
1918:
1910:
1908:
1907:
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:
15746:Astrophysics
15695:
15560:Experimental
15194:Quantum mind
15106:Franck–Hertz
14948:Klein–Gordon
14902:Formulations
14895:Formulations
14824:Interference
14814:Entanglement
14792:Ground state
14787:Energy level
14760:Fundamentals
14724:Introduction
14709:
14669:
14650:
14570:
14507:On Wikibooks
14499:
14470:
14446:
14425:. Springer.
14422:
14400:
14370:
14355:
14334:
14315:
14298:
14280:
14251:
14226:
14216:
14184:
14173:
14170:K. Hentschel
14147:
14106:
14102:
14084:
14081:Bryce DeWitt
14062:
14038:
14014:
13989:
13978:
13969:cosmological
13964:
13954:
13941:trigonometry
13932:
13908:
13886:
13863:
13854:
13835:. Retrieved
13828:the original
13814:
13806:
13799:. Retrieved
13777:
13751:
13745:
13732:
13711:
13700:
13665:
13659:
13646:
13638:
13633:
13625:
13620:
13592:
13582:
13554:
13544:
13532:. Retrieved
13528:
13519:
13508:
13494:
13465:
13443:. Retrieved
13437:
13430:Ball, Philip
13424:
13397:
13381:
13375:
13366:
13356:
13327:
13321:
13311:
13284:
13275:
13263:. Retrieved
13257:
13250:Kragh, Helge
13244:
13225:
13216:
13204:. Retrieved
13190:
13171:
13162:
13142:
13135:
13127:
13113:
13107:
13097:30 September
13095:. Retrieved
13080:
13073:
13040:
13034:
13024:
12996:
12982:
12971:
12961:
12920:
12914:
12901:
12874:
12871:Kent, Adrian
12865:
12857:
12846:. Springer.
12843:
12830:
12797:
12793:
12787:
12744:
12740:
12734:
12711:
12686:
12675:
12665:
12656:
12646:
12627:
12621:
12608:
12599:
12593:
12550:
12546:
12536:
12511:
12507:
12501:
12458:
12452:
12442:
12399:
12395:
12389:
12338:
12334:
12324:
12291:
12287:
12281:
12254:
12248:
12221:
12217:
12207:
12177:(1): 26–57.
12174:
12170:
12160:
12127:
12123:
12113:
12070:
12066:
12060:
12040:
12030:
12010:
12003:
11984:
11975:
11956:
11946:
11934:. Retrieved
11928:
11915:
11903:. Retrieved
11893:
11842:
11836:
11830:
11810:
11803:
11791:. Retrieved
11781:
11770:. Retrieved
11764:
11751:
11731:
11727:Peres, Asher
11697:
11691:
11679:. Retrieved
11675:
11665:
11653:. Retrieved
11625:
11621:
11611:
11584:
11575:
11532:
11526:
11520:
11477:
11471:
11465:
11442:
11432:
11413:
11407:
11395:. Retrieved
11393:(8): 683–695
11390:
11384:
11368:
11325:
11321:
11315:
11262:
11256:
11236:
11194:
11157:
11125:
11121:
11114:
11082:
11074:Landau, L.D.
11068:
11049:
11006:
11000:
10955:
10951:
10941:
10903:(1): 71–77.
10900:
10896:
10886:
10862:
10855:
10847:the original
10832:
10822:
10814:
10799:
10789:
10779:
10777:
10770:. Retrieved
10766:
10753:
10741:. Retrieved
10735:
10684:
10678:
10621:
10599:
10591:Bub, Jeffrey
10585:
10574:. Retrieved
10570:
10560:
10525:
10521:
10511:
10491:
10481:
10461:
10454:
10434:
10427:
10407:
10376:
10305:
10301:
10295:
10244:
10240:
10234:
10201:
10197:
10191:
10166:
10162:
10156:
10144:. Retrieved
10129:
10074:
10070:
10061:
10040:
10031:
10025:
10019:
10014:
10009:, pp. 9–13).
10006:
9998:
9990:
9977:
9964:
9851:
9827:
9819:diamagnetism
9788:
9745:
9737:
9735:
9694:
9686:
9678:
9667:
9659:cathode rays
9609:research by
9604:
9592:Thomas Young
9577:Robert Hooke
9573:
9537:
9531:
9527:
9521:
9514:
9489:Nathan Rosen
9451:
9440:
9436:
9426:
9395:
9337:
9298:
9279:
9232:
9134:
9107:
9090:
9071:
9060:
9041:
9037:quantization
9013:
8577:Astrophysics
8557:Neurophysics
8469:
8460:Mathematical
8450:Experimental
8064:such as the
8031:
8004:
7995:Applications
7908:
7901:
7652:
7442:
7205:
7073:for complex
6872:
6857:
6840:bound states
6837:
6766:
6617:
6615:
6354:
6280:
6274:
6271:energy level
6213:flash memory
6197:
6037:
5837:
5717:
5481:
5390:
5266:
5223:
5083:
4967:
4963:
4960:
4937:
4894:
4789:
4687:
4501:
4331:
4216:
4169:Emmy Noether
3984:
3812:
3790:
3744:
3731:
3711:
3704:
3426:
3323:
3172:
3025:
2925:
2822:
2763:
2681:
2551:
2420:
2404:
2373:
2362:
2347:
2337:
2309:
2305:
2301:
2297:
2159:
2079:
2020:total energy
2014:denotes the
1993:
1916:
1904:
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:
15914:Mathematics
15878:Electronics
15649:Statistical
15565:Theoretical
15542:Engineering
15452:EPR paradox
15232:Quantum bus
15101:Double-slit
15079:Experiments
15045:Many-worlds
14983:Schrödinger
14932:Phase space
14922:Schrödinger
14912:Interaction
14869:Uncertainty
14839:Nonlocality
14834:Measurement
14829:Decoherence
14819:Hamiltonian
14419:Shankar, R.
14271:Online copy
14244:L.D. Landau
14009:Bohm, David
13837:24 November
13801:24 November
13600:. pp.
13558:. Pearson.
13384:: 121–128.
13265:12 December
13222:Kuhn, T. S.
12614:Bell, J. S.
11905:16 December
10795:Sagan, Carl
10772:19 December
10146:19 December
9987:transistors
9906:superfluids
9898:macroscopic
9862:measurement
9803:Peter Debye
9611:John Dalton
9499:. In 1964,
9461:determinism
9329:vibrational
9262:Abdus Salam
8840:von Neumann
8705:Schrödinger
8537:Statistical
8455:Theoretical
8407:Information
8280:Schrödinger
7904:phase shift
7021:, that is,
6357:eigenstates
5224:with state
4690:wave packet
4504:plane waves
4211:wave packet
2400:closed form
2394:. Even the
2016: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
15766:Geophysics
15756:Biophysics
15600:Analytical
15553:Approaches
15380:Extensions
15214:Technology
15060:Relational
15011:Copenhagen
14907:Heisenberg
14854:Tunnelling
14717:Background
14633:Philosophy
14543:Hans Bethe
14213:Max Jammer
13878:Cox, Brian
13534:1 February
13445:2024-03-23
13016:1151058062
12992:Wolf, Emil
12461:(2): 125.
11936:10 October
11772:2024-03-23
11735:. Kluwer.
11335:1911.06282
10965:1801.09977
10958:(4): 040.
10743:8 February
10576:2024-06-07
10315:2209.13084
10254:1501.04899
10053:References
9837:developed
9568:Max Planck
9522:Everett's
9429:Niels Bohr
8957:Categories
8750:Heisenberg
8710:de Broglie
8660:Rutherford
8655:Sommerfeld
8609:Scientists
8552:Biophysics
8402:Randomness
8339:Paul Dirac
8309:Niels Bohr
8299:Max Planck
8058:transistor
4173:Lagrangian
3751:Paul Dirac
3716:or system
2865:commutator
2596:, we have
2390:, 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
15866:Chemistry
15854:Astronomy
15716:Molecular
15617:Acoustics
15610:Continuum
15605:Celestial
15595:Newtonian
15582:Classical
15525:Divisions
15086:Bell test
14941:Equations
14767:Born rule
14583:Chemistry
13692:0002-9505
13574:929609283
13206:18 August
13168:Mehra, J.
13065:0031-921X
12988:Born, Max
12945:0015-9018
12884:0905.0624
12822:121747282
12585:1355-2198
12563:CiteSeerX
12468:0706.2661
12409:1301.1069
12373:0034-6861
12348:1407.2930
12273:743037461
12191:1063-6145
12144:0031-8248
12105:119273840
12080:1109.6462
11852:1205.5368
11793:18 August
11603:442351498
11567:119261326
11542:1005.3976
11360:208006050
11299:973404322
11272:1106.1445
11215:844974180
11150:120024263
10992:2542-4653
10925:1050-2947
10703:0028-0836
10635:1302.1864
10226:203638258
10107:119896026
10099:1434-6001
9849:in 1927.
9795:heuristic
9777:The 1927
9717:ν
9691:frequency
9501:John Bell
9437:temporary
9352:spin foam
9188:ϵ
9184:π
9160:−
9093:molecules
9077:coherence
8945:Uhlenbeck
8935:Zeilinger
8895:Gell-Mann
8625:Becquerel
8582:Cosmology
8264:ν
8261:μ
8252:κ
8243:ν
8240:μ
8232:Λ
8224:ν
8221:μ
8196:⟩
8178:ψ
8154:ℏ
8145:⟩
8127:ψ
8115:^
7979:Φ
7976:Δ
7878:Φ
7875:Δ
7869:
7841:⟩
7832:ψ
7810:ψ
7806:⟨
7756:Φ
7753:Δ
7747:
7719:⟩
7710:ψ
7688:ψ
7684:⟨
7616:Φ
7613:Δ
7607:
7586:Φ
7583:Δ
7577:
7571:−
7550:Φ
7547:Δ
7524:ψ
7428:Φ
7425:Δ
7395:Φ
7392:Δ
7173:β
7150:α
7119:⟩
7116:ψ
7110:ψ
7107:⟨
7087:β
7081:α
7055:ψ
7051:β
7039:ψ
7035:α
7029:ψ
6974:ψ
6918:ψ
6885:∈
6882:ψ
6794:ω
6791:ℏ
6733:−
6662:−
6598:…
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
6096:π
6086:ℏ
6020:…
5985:π
5950:π
5889:
5866:ψ
5838:in which
5808:
5781:ψ
5688:
5667:
5640:ψ
5591:ψ
5451:
5427:
5406:ψ
5391:or, from
5349:ℏ
5323:−
5278:ψ
5232:ψ
5186:^
5123:ℏ
5117:−
5102:^
5066:ψ
5039:ψ
5007:ℏ
5001:−
4853:−
4837:π
4807:^
4804:ψ
4751:−
4731:π
4703:ψ
4658:ψ
4620:^
4617:ψ
4591:ℏ
4536:ℏ
4530:−
4447:ℏ
4441:−
4403:^
4400:ψ
4392:∞
4387:∞
4384:−
4380:∫
4373:π
4343:ψ
4269:ℏ
4263:−
3985:conserved
3860:ℏ
3843:−
3813:generator
3707:entangled
3679:ϕ
3675:⊗
3666:ϕ
3653:ψ
3649:⊗
3640:ψ
3571:ϕ
3544:ψ
3497:ϕ
3470:ψ
3403:ψ
3399:⊗
3390:ψ
3360:ψ
3333:ψ
3292:⊗
3147:ℏ
3143:−
3113:∂
3109:∂
3104:ℏ
3098:−
3047:ℏ
2957:≥
2948:σ
2938:σ
2902:−
2800:ℏ
2795:≥
2786:σ
2776:σ
2726:−
2694:σ
2642:−
2608:σ
2534:ℏ
2519:^
2504:^
2465:^
2436:^
2336:and take
2334:quantized
2326:frequency
2264:ψ
2235:ψ
2205:ℏ
2188:−
2133:ψ
2128:ℏ
2111:−
2091:ψ
2057:ℏ
2030:ℏ
1967:ψ
1949:ψ
1940:∂
1936:∂
1931: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.
15960:Category
15475:Category
15269:Timeline
15021:Ensemble
15001:Bayesian
14963:Majorana
14879:Collapse
14751:Glossary
14734:Timeline
14605:5
14498:(2003),
14489:34661512
14469:(1996).
14443:(2013).
14421:(1994).
14399:, 2006.
14390:39849482
14368:(1999).
14354:, 1966.
14333:(1998).
14314:(2004).
14279:(2002).
14250:(1977).
14234:Archived
14225:, 2004.
14215:, 1966.
14205:40251748
14183:(2004).
14131:17178479
14060:(1985).
14011:(1989).
13987:(2009).
13963:, 2000.
13931:, 2004.
13907:, 1985.
13884:(2011).
13862:(1987).
13795:Archived
13766:Archived
13708:(1997).
13626:Synthese
13590:(1996).
13486:66-29628
13224:(1978).
13200:Archived
12994:(1999).
12953:17217776
12493:32755624
12434:55537196
12381:34901303
12316:22433857
12199:57559199
12038:(1967).
11983:(2009).
11885:14712017
11877:23005618
11729:(1993).
11655:31 March
11650:19113681
11583:(2006).
11512:13963928
11376:(1987).
11328:: 1–57.
11193:(2010).
11080:(1977).
10797:(1996).
10711:26503054
10593:(2019).
10552:24039543
10489:(1995).
10375:(2022).
10340:36867820
10287:18463042
10279:25582917
10067:Born, M.
9912:See also
9783:Brussels
9661:, which
9607:chemical
9589:polymath
9465:locality
9333:graviton
9240:and the
8940:Goudsmit
8930:Lawrence
8925:Chadwick
8920:Shockley
8880:'t Hooft
8830:Millikan
8815:Poincaré
8805:Lemaître
8675:Einstein
8517:Particle
8439:Branches
8422:Particle
8372:Topology
8366:Concepts
8314:Max Born
8293:Founders
8072:such as
8015:neutrons
6623:are the
4190:Examples
4180:symmetry
3983:will be
3001:⟩
2979:⟨
2737:⟩
2731:⟨
2722:⟩
2709:⟨
2653:⟩
2647:⟨
2638:⟩
2625:⟨
2318:acoustic
2292: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
15938:Science
15926:Physics
15840:Portals
15800:Related
15684:General
15679:Special
15537:Applied
15428:Related
15407:History
15146:Science
14978:Rydberg
14729:History
14668:(ed.).
14649:(ed.).
14620:
14587:Physics
14111:Bibcode
13937:algebra
13670:Bibcode
13507:(ed.).
13386:Bibcode
13332:Bibcode
13045:Bibcode
12925:Bibcode
12889:Bibcode
12802:Bibcode
12779:1921913
12759:Bibcode
12555:Bibcode
12516:Bibcode
12473:Bibcode
12414:Bibcode
12353:Bibcode
12226:Bibcode
12152:9454552
12085:Bibcode
11857:Bibcode
11630:Bibcode
11547:Bibcode
11492:Bibcode
11397:13 June
11340:Bibcode
11307:2515538
11130:Bibcode
11106:2284121
10970:Bibcode
10933:9908895
10905:Bibcode
10640:Bibcode
10543:3768233
10320:Bibcode
10259:Bibcode
10206:Bibcode
10171:Bibcode
10079:Bibcode
10045:theory.
9888:during
9740:is the
9679:quantum
9549:History
9317:strings
9137:charged
9116:or the
9023:is the
8915:Bardeen
8885:Veltman
8855:Feynman
8825:Penrose
8820:Wheeler
8810:Thomson
8795:Hawking
8770:Hilbert
8765:Moseley
8725:Compton
8700:Kramers
8680:Wilczek
8630:Lorentz
8620:Röntgen
8522:Nuclear
8445:Applied
8412:Entropy
8019:photons
8011:protons
6277:energy.
4968:outside
4127:, then
3767:Feynman
3609:, then
2365:orbital
2221: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
15890:Energy
15711:Atomic
15666:Modern
15516:Major
15121:Popper
14589:. See
14487:
14477:
14455:
14429:
14411:
14388:
14378:
14360:online
14341:
14322:
14305:
14287:
14262:
14203:
14193:
14155:
14129:
14091:
14070:
14045:
14023:
13997:
13943:, and
13917:
13894:
13870:
13720:
13690:
13608:
13572:
13562:
13484:
13474:
13412:
13299:
13232:
13178:
13150:
13120:
13088:
13063:
13014:
13004:
12951:
12943:
12850:
12820:
12777:
12722:
12583:
12565:
12491:
12432:
12379:
12371:
12314:
12308:986572
12306:
12271:
12261:
12218:Nature
12197:
12189:
12150:
12142:
12103:
12048:
12018:
11991:
11963:
11883:
11875:
11818:
11739:
11704:
11681:18 May
11648:
11601:
11591:
11565:
11510:
11453:
11420:
11358:
11305:
11297:
11287:
11244:
11213:
11203:
11165:
11148:
11104:
11094:
11056:
11021:
10990:
10931:
10923:
10874:
10839:
10807:
10709:
10701:
10680:Nature
10650:
10550:
10540:
10499:
10469:
10442:
10415:
10383:
10338:
10285:
10277:
10224:
10137:
10105:
10097:
10030:, and
10003:lasers
9876:, and
9821:, and
9736:where
9720:
9412:, and
9325:charge
9254:gluons
9250:quarks
8870:Lenard
8790:Wigner
8780:Jordan
8760:Zeeman
8735:Walton
8635:Planck
8615:Witten
8512:Atomic
8392:Matter
8387:Energy
8052:, the
6866:, the
6616:where
6243:spring
5583:where
4964:inside
2863:. The
2396:helium
2386:, the
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3304:B
3298:H
3287:A
3281:H
3275:=
3270:B
3267:A
3261:H
3233:B
3227:H
3202:A
3196:H
3183:B
3179:A
3151:2
3116:x
3101:i
3076:i
3072:p
3043:/
3039:i
3011:.
3007:|
2996:]
2993:B
2990:,
2987:A
2984:[
2973:|
2966:2
2963:1
2952:B
2942:A
2911:,
2908:A
2905:B
2899:B
2896:A
2893:=
2890:]
2887:B
2884:,
2881:A
2878:[
2851:B
2831:A
2808:.
2803:2
2790:P
2780:X
2749:.
2742:2
2734:P
2717:2
2713:P
2703:=
2698:P
2667:,
2658:2
2650:X
2633:2
2629:X
2617:=
2612:X
2580:P
2560:X
2537:.
2531:i
2528:=
2525:]
2516:P
2510:,
2501:X
2495:[
2462:P
2433:X
2367:(
2363:s
2310:d
2306:p
2302:s
2298:n
2273:)
2270:t
2267:(
2244:)
2241:0
2238:(
2201:/
2197:t
2194:H
2191:i
2184:e
2180:=
2177:)
2174:t
2171:(
2168:U
2145:.
2142:)
2139:0
2136:(
2124:/
2120:t
2117:H
2114:i
2107:e
2103:=
2100:)
2097:t
2094:(
2054:i
2002:H
1979:.
1976:)
1973:t
1970:(
1964:H
1961:=
1958:)
1955:t
1952:(
1943:t
1928: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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