page 7: Quantum mechanics
Gravitation is blind to the distinction between potential and kinetic energy. It sees just energy. Integrated over the whole Universe, kinetic and potential energy might add up to zero, but locally they become differentiated due to various processes.. The study of the interaction of potential and kinetic energy is mechanics. Richard Feynman: The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Mechanics - Wikipedia
Classical mechanics is based on continuous processes in continuous spaces. It describes the phenomena well at large scales, but does not explain how things work in detail. For this we turn to quantum mechanics. Classical mechanics - Wikipedia, Quantum mechanics - Wikipedia
Both classical mechanics and quantum mechanics are normally studied in the context of four dimensional space-time. The union of special relativity, which describes flat space-time, and quantum mechanics is quantum field theory. Quantum mechanics itself can be formulated in the single dimension of time or energy. Time and energy are related by the Planck-Einstein equation E = hν where E is energy, ν is frequency, that is inverse time, and h (Planck's constant) is the constant proportionality between them. Quantum mechanics is a one dimensional field theory. Planck-Einstein relation - Wikipedia, Quantum field theory - Wikipedia, Anthony Zee: Quantum Field Theory in a Nutshell
Quantum mechanics is a mathematical
toolkit for constructing models of the world. It's relationship to the physical world is rather like the relationship of arithmetic to accounting. It does not tell us any details, but when we study the details, we find that they always fit the model. If we have 10 sheep and sell 4, we will have 6 left. In a similar way, quantum mechanics tells us in a general way what we can expect if we
start with this and do that. Nielsen & Chuang: Quantum Computation and Quantum Information
Quantum mechanics represents states |ψ> of the world by vectors in complex Hilbert spaces. A Hilbert space is a function space. The complexity of the state is represented by the dimension of the Hilbert space, which may vary from 1 to countable infinity. These states are invisible to us. Hilbert space - Wikipedia, Mathematical formulation of quantum mechanics - Wikipedia, Function (mathematics) - Wikipedia, Erwin Kreyszig; Introductory Functional Analysis with Applications, Function space -Wikipedia, John von Neumann: Mathematical Foundations of Quantum Mechanics, John von Neumann: Mathematical Foundations of Quantum Mechanics [new online edition]
We observe unknown states using a measurement operator or observable M and what we actually observe are eigenvalues corresponding to eigenfunctions (ie the basis vectors) of M. These eigenvectors are fixed points under the operation of M. The eigenfunctions we observe are predetermined by our choice of M. What the theory predicts is the probability of observing each of the eigenvalues. Observable - Wikipedia, Eigenvalues and eigenvectors - Wikipedia
The heart of quantum mechanics can be expressed in six propositions. Three of these propositions are mathematical and embody the linearity and unitarity of quantum systems:
(1) the quantum state of a system is represented by a vector in its Hilbert space;
(2) a complex system is represented by a vector in the tensor product of the Hilbert spaces of the constituent systems;
(3)
the evolution of isolated quantum systems is unitary governed by the
Schrödinger equation:
i∂|ψ> / ∂t = H|ψ >
where H is the energy (or Hamiltonian) operator.
Wojciech Hubert Zurek: Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical, Schrödinger equation - Wikipedia
The other three show how the mathematical formalism couples to the
observed world:
(4) immediate repetition of a measurement yields
the same outcome;
(5) measurement outcomes are restricted to an orthonormal set
{ | sk > } of eigenstates of the measured
observable;
(6) the probability of finding a given outcome is
pk= |<sk||ψ>|2,
where |ψ> is the preexisting state of the system.
Born rule - Wikipedia
The energy operator may be represented by a square matrix of the
same dimension as Hilbert space of the system of interest. The state vector of this system |ψ > is a superposition of the basis vectors of this space. The elements of the matrix encode the energy (rate) of interaction between the elements of the state vector. Richard Feynman, Lectures on Physics, FLP III:08: The Hamiltonian Matrix
A network comprises a set of nodes or agents connected by a set
of channels through which they can communicate. Our first step toward
identifying the quantum axioms with a network is to equate the number
of nodes in the network with the number of dimensions and basis states in the Hilbert space of the quantum mechanical description.
If we think of each pair of vectors in a Hilbert space as defining a channel in a network, the Hamiltonian describes (in a complex way) the flow of traffic on this channel. The Born rule produces the probability of traffic between the states represented by two vectors.
Then:
Axiom (1) We let each basis vector in a state vector correspond to a
unique node. In computer terms, the state vector represents the state
of the network memory, which is distributed among the basis vectors.
Axiom (2) describes the creation of an internet between two
quantum networks. The number of links in the new network is the
product of the nodes in the constituent networks since each node in
one of the product networks has access to all the nodes in the other.
Axiom (3) describes the evolution of network traffic subject to
the constraint that total traffic in a particular network is constant
and conventionally normalized to 1. If traffic on one channel
increases, it must decrease on another. This situation is a
consequence of linearity of quantum mechanics and the conservation of
energy in a network, since the frequency of communication is measured
by energy.
The three mathematical axioms above describe a system which is not
directly observable and therefore to some extent hypothetical, to be
verified by its observable consequences. The mathematical system
described is not quantized but evolves continuously and unitarily as described by axiom (3) constrained by the Schrödinger equation. Observation occurs when two quantum systems communicate, so that one becomes correlated with the other. In laboratory systems one system is usually designated as the observer and the other as the observed, but in nature quantum systems are continually "observing" one another to produce the classical world. Unitarity is broken. This situation is described by axiom (5).
Axiom (5) introduces the idea that what we see when we observe a quantum system depends on the operator we use to look at it. So we might use a momentum operator to measure momentum, or an energy operator to measure energy. More generally, the results we obtain are restricted to the orthonormal eigenstates
of the measurement operator. This introduces quantization. Such
quantization appears necessary to enable the error free transmission
of information from one quantum system to another. page 8: Why is the Observable
Universe Quantized?, Zurek [ref above]
Axiom (4) attests to the robustness of the observed fixed points. From the network point of view, these fixed points represent the halted states of the Turing machines that are represented by the eigenfunctions of quantum mechanics.
Axiom (6) establishes that the statistics of a quantum observable
are constrained by the same normalization that we find in the
mathematical description of communication sources. Communication
theory requires for a source A that the probabilities pi for the emission of letters ai of the source alphabet be normalized so that Σi pi = 1.
Historically, the first three postulates of quantum mechanics have
been considered uncontroversial. but there has been endless debate
about the interpretation of the mathematical formalism encapsulated
in postulates (4) - (6). The paper by Zurek referred to above has
clarified the situation slightly by showing that if we regard a
quantum observation as an act of communication the mathematical
postulates of quantum mechanics imply the observational postulates.
Modern scientific epistemology accepts Einstein's view that we can trust only knowledge obtained by direct contact with the entity we wish to know. The foundations of physical knowledge are observed
events. Heisenberg sought to free quantum mechanics from classical
misconceptions by insisting that only the phenomena need be
explained; a theory has no standing except insofar as it does this (or at least promises to do it). Werner Heisenberg
The success of continuous formalism does not therefore guarantee
that the Universe itself is continuous. In practical physics all our
computations are implemented logically and digitally, and it is known
that our digital approximations to continuous systems are limited
only by the computing resources available. Even in the current state
of the art they far exceed the precision of any practical experiment.
The study of continuity and its close relations the infinite and
the infinitesimal raise many questions that physicists generally
answer with the observation that the methods of calculus work and
that is good enough for us.
The situation is not so easy for
mathematicians. Euclid defined a point as an entity with position but
no magnitude, and conceived of a line as an entity of measurable
magnitude constructed from points. This leads to the naive
(but useful) notion that an infinity of infinitesimals add up to a
finite magnitude. Heath
Cantor famously
set out to find the cardinal of the continuum and was led to the
development of the transfinite numbers. Model page 3; Immensity
His method, set theory, ultimately led to function spaces (including
Hilbert spaces) and many other wonderful developments in mathematics,
but it failed to deliver the result Cantor wanted. Later Cohen showed
that Cantor's continuum hypothesis is independent of set theory.
Cohen, Cantor
Mathematical
analysis in its entirety is based on the notion of continuity by
proximity. We prove classical results like the Bolzano-Weierstrass
theorem by crowding points closer and closer together into ever more
confined spaces. Bolzano-Weierstrass theorem - Wikipedia
We define and prove the continuity of functions by similar processes, showing that the elements of the domain of a continuous function lying in an interval ε (no matter how small) map to elements of its range to be found in similarly small interval,
δ. Continuous function - Wikipedia
The proximity definition of continuity underlies calculus, the
notion of unitary evolution of quantum systems and the general notion
of an 'argument from continuity' ubiquitous in mathematics and
physics. A little reflection reveals, however, that arguments from
proximity have no real force since in general closeness implies
nothing but the absence of change and the majority of mathematical functions are not continuous.
Logical continuity
Instead we must turn to the concept of logical continuity embodied
in the idea of mathematical proof and formalized as the propositional
calculus used by computers. Model page 4: Logical continuity
Turing formalized the notion of proof in a
machine which performs a deterministic sequence of logical operations
moving from some initial state (the premisses) to some final state
(conclusion). He showed that such a machine (a turing machine)
was capable of performing anything which could reasonably called a
computation. Further, a turing machine could have an initial state
that led to no final state. Such initial states establish the
mathematical existence incomputable functions. Davis, Model page 5: Computation
Much of the
literature of quantum mechanics speaks of 'wave-particle' duality.
This duality, however, is a 'broken'. We observe particles. We do not
observe waves, but rather find periodic structures (suggestive of
waves) in repeated observations of certain systems like the
paradigmatic two slit experiment. This wavelike structure is
reflected in the complex exponential functions used in the
mathematical formalism of quantum mechanics. Double-slit experiment - Wikipedia
The theory of computation is also cyclic, recursive or wavelike.
The power of a computer lies in its ability to perform very simple
operations repetitively at very great speed. The fundamental operator
in a practical computer is the clock, which in effect implements the
logical operation not, where tick = not-tock. The clock pulses serve
to order the operations of the computer.
Conclusion
On this site, we wish to use the properties of communication
networks to model the whole world, including the physical world. This
project is possible because the rather abstract and counterintuitive
mathematical machinery of quantum mechanics fits neatly into the
network paradigm: the quantum world is a world of computation and
communication. Since we are naturally thinking (computing) and
communicative beings, this approach helps us to see quantum mechanics
as a description of communication between the independent particles
that constitute the Universe.
(revised 20 February 2021)
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Further readingBooks
Brandt, Siegmund, and Hans Dieter Dahmen, The Picture Book of Quantum Mechanics, Springer-Verlag 1995 Jacket: 'This book is an introduction to the basic concepts and phenomena of quantum mechanics. Computer-generated illustrations are used extensively throughout the text, helping to establish the relation between quantum mechanics on one side and classical physics . . . on the other side. Even more by studying the pictures in parallel with the text, readers develop an intuition for notoriously abstract quantum phenomena . . .'
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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1895, 1897, 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.'
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Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'
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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.'
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Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)
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Einstein, Albert, and Robert W Lawson (translator) Roger Penrose (Introduction), Robert Geroch (Commentary), David C Cassidy (Historical Essay), Relativity: The Special and General Theory, Pi Press 2005 Preface: 'The present book is intended, as far as possible, to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. ... The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3
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Everett III, Hugh, and Bryce S Dewitt, Neill Graham (editors), The Many Worlds Interpretation of Quantum Mechanics, Princeton University Press 1973 Jacket: 'A novel interpretation of quantum mechanics, first proposed in brief form by Hugh Everett in 1957, forms the nucleus around which this book has developed. The volume contains Dr Everett's short paper from 1957, "'Relative State' formulation of quantum mechanics" and a far longer exposition of his interpretation entitled "The Theory of the Universal Wave Function" never before published. In addition other papers by Wheeler, DeWitt, Graham, Cooper and van Vechten provide further discussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.'
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Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. . . . In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands
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Gamow, George, and Roger Penrose (Designer), Mr Tomkins in Paperback, Cambridge University Press 1993 Amazon customer review: 'This is one of the best introductions to the concepts of relativity and quantum theory I have ever read. Not only does it have an excellent nonmathmatical and easy to understand description of these areas of modern physics, but it has an interesting and funny story to move it along. It also includes a more technical description for those who are up to it (even the technical description is nothing too difficult, and also nonmathmatical, it can be skipped)' Edward Wendt III
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.'
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Heisenberg, Werner , Physical Principles of the Quantum Theory (translated by Carl Eckart and Frank C Hoyt), Dover 1949 Jacket: 'In this classic, based on lectures delivered at the University of Chicago, Heisenberg presents a complete physical picture of quantum theory. He covers not only his own contributions, but also those of Bohr, Dirac, Bose, de Broglie, Fermi, Einstein, Pauli, Schroedinger, Sommerfeld, Rupp, Wilson, Germer and others in a text written for the physical scientist who is not a specialist in quantum theory or in modern mathematics.'
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Kreyszig, Erwin, Introductory Functional Analysis with Applications, John Wiley and Sons 1989 Amazon: 'Kreyszig's "Introductory Functional Analysis with Applications", provides a great introduction to topics in real and functional analysis. This book is part of the Wiley Classics Library and is extremely well written, with plenty of examples to illustrate important concepts. It can provide you with a solid base in these subjects, before one takes on the likes of Rudin and Royden. I had purchased a copy of this book, when I was taking a graduate course on real analysis and can only strongly recommend it to anyone else.' Krishnan S. Kartik
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Kuhn, Thomas S, Black-Body Theory and the Quantum Discontinuity 1894-1912, University of Chicago Press 1987 Jacket: '[This book] traces the emergence of discontinuous physics during the early years of this century. Breaking with historiographic tradition, Kuhn maintains that, though clearly due to Max Planck, the concept of discontinuous energy change does not originate in his work. Instead it was introduced by physicists trying to understand the success of his brilliant new theory of black-body radiation.'
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Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators), The Principia Mathematica: I Principles of Natural Philosophy, University of California Press 1999 This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. . . . The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students.
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Nielsen, Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.
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Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this . . . major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. . . . Running through the book is a completely non-scientific biography . . . including many letters which appear in English for the first time, as well as other information not published before.'
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van der Waerden, B L, Sources of Quantum Mechanics, Dover Publications 1968 Amazon Book Description: 'Seventeen seminal papers, dating from the years 1917-26, in which the quantum theory as wenow know it was developed and formulated. Among the scientists represented: Einstein,Ehrenfest, Bohr, Born, Van Vleck, Heisenberg, Dirac, Pauli and Jordan. All 17 papers translatedinto English.'
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von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.'
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Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on.
Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena.
Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory.
Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.
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Links
Bolzano-Weierstrass theorem - Wikipedia, Bolzano-Weierstrass theorem - Wikipedia, the free encyclopedia, 'In mathematics, specifically in real analysis, real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states that each bounded sequence in Rn has a convergent subsequence. An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded.' back |
Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then
the measured result will be one of the eigenvalues λ of A, and
the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'. back |
Classical mechanics - Wikipedia, Classical mechanics - Wikipedia, the free encyclopedia, 'Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science and technology.' back |
Continuous function - Wikipedia, Continuous function - Wikipedia, the free encyclopedia, 'IIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called "bicontinuous".' back |
Derek Homeier, The Origin of Light and the Nature of Matter, 'Welcome to the class homepage of Astronomy 1020! In this place I will post lecture notes, solutions for the homework problems and general announcements and updates on the exams and other stuff. ' back |
Determinism - Wikipedia, Determinism - Wikipedia, the free encyclopedia, 'Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen.' back |
Double-slit experiment - Wikipedia, Double-slit experiment - Wikipedia, the free encyclopedia, 'In the double-slit experiment, light is shone at a solid thin plate that has two slits cut into it. A photographic plate is set up to record what comes through those slits. One or the other slit may be open, or both may be open. . . . The most baffling part of this experiment comes when only one photon at a time is fired at the barrier with both slits open. The pattern of interference remains the same as can be seen if many photons are emitted one at a time and recorded on the same sheet of photographic film. The clear implication is that something with a wavelike nature passes simultaneously through both slits and interferes with itself — even though there is only one photon present. (The experiment works with electrons, atoms, and even some molecules too.)' back |
Eigenvalues and eigenvectors - Wikipedia, Eigenvalues and eigenvectors - Wikipedia, the free encyclopedia, 'An eigenvector of a square matrix A is a non-zero vector vthat, when the matrix multiplies yields a constant multiple of v, the latter multiplier being commonly denoted by λ. That is: Av = λv' back |
Erwin Schroedinger, The Present Situation in Quantum Mechanics, 'A TRANSLATION OF SCHRÖDINGER'S "CAT PARADOX PAPER" Translator: John D. Trimmer. This translation was originally published in Proceedings of the American Philosophical Society, 124, 323-38. [And then appeared as Section I.11 of Part I of Quantum Theory and Measurement (J.A. Wheeler and W.H. Zurek, eds., Princeton UP 1983).]' back |
Feynman, Leighton & Sands FLP III:08, Chapter 8: The Hamiltonian Matrix, 'One problem then in describing nature is to find a suitable representation for the base states. But that’s only the beginning. We still want to be able to say what “happens.” If we know the “condition” of the world at one moment, we would like to know the condition at a later moment. So we also have to find the laws that determine how things change with time. We now address ourselves to this second part of the framework of quantum mechanics—how states change with time.' back |
Function (mathematics) - Wikipedia, Function (mathematics) - Wikipedia, the free encyclopedia, 'The mathematical concept of a function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output"). A function associates a single output with every input element drawn from a fixed set, such as the real numbers.' back |
Function space - Wikipedia, Function space - Wikipedia, the free encyclopedia, 'In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both' back |
Hamiltonian (quantum mechanics) - Wikipedia, Hamiltonian (quantum mechanics) - Wikipedia, the free encyclopedia, 'I In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system (this addition is the total energy of the system in most of the cases under analysis). It is usually denoted by H . . .. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.
The Hamiltonian is named after William Rowan Hamilton, who created a revolutionary reformulation of Newtonian mechanics, now called Hamiltonian mechanics, which is also important in quantum physics. ' back |
Hilbert space - Wikipedia, Hilbert space - Wikipedia, the free encyclopedia, ' The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows defining lengths and angles. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used. ' back |
Imperial College, Welcome to the Newton Project Homepage, 'The magnitude of Newton's accomplishments places him in the very first rank of scientists and mathematicians. However, although most early modern scientists have been honoured with comprehensive editions of their collected works, there is no similar tribute to Newton. Throughout the nineteenth and twentieth centuries, this has been seen as a gaping lacuna and even a national disgrace by scientists and statesmen alike. There are excellent editions of his mathematical and scientific papers, as well as of his correspondence, but very few of his non-scientific writings have ever appeared in print. The Newton Project will place these writings in their relevant contexts, which will be made accessible by means of hyperlinks.' back |
Inverter (logic gate) - Wikipedia, Inverter (logic gate) - Wikipedia, the free encyclopedia, 'In digital logic, an inverter or NOT gate is a logic gate which implements logical negation. . . . The digital inverter is considered the base building block for all digital electronics. Memory (1 bit register) is built as a latch by feeding the output of two serial inverters together. Multiplexers, decoders, state machines, and other sophisticated digital devices all rely on the basic inverter.' back |
J J O'Connor and E F Robertson, A History of quantum mechanics, 'It is hard to realise that the electron was only discovered about 100 years ago in 1897. That it was not expected is illustrated by a remark made by J J Thomson, the discoverer of the electron. He said I was told long afterwards by a distinguished physicist who had been present at my lecture that he thought I had been pulling their leg. The neutron was not discovered until 1932 so it is against this background that we trace the beginnings of quantum theory back to 1859.' back |
John von Neumann, The Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford.
Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC
back |
mathacademy.com, Zeno's paradox of the Tortoise and Achilles, 'Zeno of Elea (circa 450 b.c.) is credited with creating several famous paradoxes, but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad.) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles.' back |
Mathematical formulation of quantum mechanics - Wikipedia, Mathematical formulation of quantum mechanics - Wikipedia - the free encyclopedia, 'The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures were drawn from functional analysis, a research area within pure mathematics that developed in parallel with, and was influenced by, the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues of linear operators.' back |
Max Planck, On the Law of Distribution of Energy in the Normal Spectrum, Annalen der Physik, vol. 4, p. 553 ff (1901)
'The recent spectral measurements made by O. Lummer and E. Pringsheim and even more notable those by H. Rubens and F. Kurlbaum which together confirmed an earlier result obtained by H. Beckmann show that the law of energy distribution in the normal spectrum, first derived by W. Wien from molecular-kinetic considerations and later by me from the theory of electromagnetic radiation, is not valid generally.' back |
Mechanics - Wikipedia, Mechanics - Wikipedia, the free encyclopedia, 'Mechanics (Greek μηχανική) is an area of science concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes.' back |
MIT, Spectroscopy and quantum mechanics, 'Quantum mechanics and atomic/molecular structure
During the latter half of the nineteenth century a tremendous amount of atomic spectral data were collected. Characteristic lines were assigned to each element and their wavelengths were measured precisely. Regularities among lines of the simpler spectra were noted, and several attempts were made to represent a series of lines as harmonics of one or more vibrations, all without success. Finally in 1885, J.J. Balmer showed that the wavelengths of the visible spectral lines of atomic hydrogen, now known as the Balmer series, could be represented by the simple mathematical formula ... ' back |
NIST, Fundamental Physical Constants, 'Founded in 1901, NIST is a non-regulatory federal agency within the U.S. Department of Commerce. NIST's mission is to promote U.S. innovation and industrial competitiveness by advancing measurement science, standards, and technology in ways that enhance economic security and improve our quality of life.' back |
NIST, Planck's constant, 'Fundamental physical constants: Planck constant Value 6.626 068 76 x 10-34 J s. Standard uncertainty 0.000 000 52 x 10-34 J s . Relative standard uncertainty 7.8 x 10-8. Concise form 6.626 068 76(52) x 10-34 J s.' back |
Nobel Foundation, Werner Heisneberg, 'Heisenberg's name will always be associated with his theory of quantum mechanics, published in 1925, when he was only 23 years old. For this theory and the applications of it which resulted especially in the discovery of allotropic forms of hydrogen, Heisenberg was awarded the Nobel Prize for Physics for 1932.' back |
Observable - Wikipedia, Observable - Wikipedia, the free encyclopedia, 'In physics, particularly in quantum physics, a system observable is a measurable operator, or gauge, where the property of the system state can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states.' back |
Planck-Einstein relation - Wikipedia, Planck-Einstein relation - Wikipedia, the free encyclopedia, 'The Planck–Einstein relation. . . refers to a formula integral to quantum mechanics, which states that the energy of a photon (E) is proportional to its frequency (ν). E = hν. The constant of proportionality, h, is known as the Planck constant.' back |
Quantum field theory - Wikipedia, Quantum field theory - Wikipedia, the free encyclopedia, 'Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or (especially in a condensed matter context) of many-body systems. . . . In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). back |
Quantum mechanics - Wikipedia, Quantum mechanics - Wikipedia, the free encyclopedia, 'Quantum mechanics (QM; also known as quantum physics or quantum theory), including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons. In such processes, said to be quantized, the action has been observed to be only in integer multiples of the Planck constant. This is utterly inexplicable in classical physics.'' back |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, 'IIn quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . .
In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Swedish Academy, Max Planck Biography, 'Experimental observations on the wavelength distribution of the energy emitted by a black body as a function of temperature were at variance with the predictions of classical physics. Planck was able to deduce the relationship between the ener gy and the frequency of radiation. In a paper published in 1900, he announced his derivation of the relationship: this was based on the revolutionary idea that the energy emitted by a resonator could only take on discrete values or quanta. The energy for a resonator of frequency v is hv where h is a universal constant, now called Planck's constant. back |
V Gavryushin and A Zukauskas, Quantum mechanics, 'Quantum mechanics, the branch of mathematical physics that deals with atomic and subatomic systems and their interaction with radiation in terms of observable quantities. It is an outgrowth of the concept that all forms of energy are released in discrete units or bundles called quanta.' back |
Werner Heisenberg, Quantum-theoretical re-interpretation of kinematic and mechanical relations, 'The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.' back |
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