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chapter 3: Physics
page 4: Quantum mechanics
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Quantum mechanics
Introduction
Quantum mechanics is a mathematical toolkit for constructing models of the world. Nielsen & Chuang 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.
Quantum mechanics replaces classical mechanics. Classical mechanics sees physical observables like position and energy as values of functions on a real space like ordinary three dimensional space. Quantum mechanics represents physical observables with the eigenvalues of linear operators on complex Hilbert spaces. Hilbert space - Wikipedia, Mathematical formulation of quantum mechanics - Wikipedia, Function (mathematics) - Wikipedia, Kreyszig, Function space - Wikipedia
The history of quantum mechanics
Quantum physics was born in 1900 when Planck found that the interaction between the radiation and matter was quantized, like the natural numbers, rather than a smooth continuous function. Max Planck, Kuhn, Planck's Law - Wikipedia,
It took nearly thirty years to develop Planck's insight into quantum mechanics as we now know it. Dirac, von Neumann Quantum mechanics appears to describe the world exactly to the limits of our computational and observational ability. Feynman
The axioms of quantum mechanics
The heart of quantum mechanics can be compressed into 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
where H is the energy (or Hamiltonian) operator. 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 || q >|2, where |q > is the preexisting state of the system.
A network interpretation of the axioms of quantum mechanics
The energy operator may be represented by a square matrix of the same dimension as the state vector |q >. The elements of the matrix encode the energy (frequency) of interaction between the elements of the state vector.
A network comprises a set of nodes or personas 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 in the Hilbert space of the quantum mechanical description.
If we think of each pair of basis vectors of Hilbert space as representing a channel in a network, the Hamiltonian describes (in a probabilistic way) the flow of traffic on this channel.
Then:
Axiom (1) We let each element 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 nodes.
Axiom (2) describes the creation of an internet between two quantum networks. The number of nodes 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.
Observation and quantization
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 as described by axiom (2) Observation occurs when two quantum systems communicate, so that one becomes correlated with the other. This situation is described by axiom (2).
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. Why is the Observable Universe Quantized, Zurek
Axiom (4) attests to the robustness of the observed quantized states.
Axiom(6) establishes that the statistics of q 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 [sigma]i pi = 1.
Continuity, communication and quantization
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.
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. Heath This leads to the naive (but useful) notion that an infinity of infinitesimals add up to a finite magnitude.
Cantor famously set out to find the cardinal of the continuum and was led to the development of the transfinite numbers. Cantor, 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
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 epsilon (no matter how small) map to elements of its range to be found in similarly small interval, delta
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 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. 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, 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. Feynman, Double-slit experiment - Wikipedia This wavelike structure is reflected in the complex exponential functions used in the mathematical formalism of quantum mechanics.
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 23 October 2008)
Further reading
Books
| Beale, R, and T Jackson, Neural Computing: An Introduction, Adam Hilger 1991 Jacket: '... starts from basics and goes on to cover all the most important approaches to the subject. ... The capabilities, advantages and disadvantages of each model are discussed as are possible applications of each. The relationship of the models developed to the brain and its functions are also explored." http://www.amazon.com/exec/obidos/ASIN/0852742622/tnrp">Amazon back |
| Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality ... This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality. Amazon back |
| 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 ...' Amazon back |
| Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 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.' Amazon back |
| 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 Goedel's proof of the consistency of the continuum hypothesis. ..' (i) Amazon back |
| Davies, Paul C W, and David S Betts, Quantum Mechanics, Chapman and Hall 1994-1995 Jacket: 'Quantum mechanics is the key to modern physics and chemistry, yet it is notoriously difficult to understand. This book is designed to overcome that obstacle. Clear and concise, it provides an easily readable introduction intended for science undergraduates with no previous knowledge of quantum theory, leading them through to the advanced topics usually encountered at the final year level. Although the subject matter is standard, novel techniques have been employed that considerably simplify the technical presentation. The authors use their extensive experience of teaching and popularizing science to explain the many difficult, abstract points of the subject in easily comprehensible language. Helpful examples and thorough sets of exercises are also given to enable students to master the subject.. Amazon back |
| 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.' Amazon back |
| 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 mechaincs, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature) Amazon back |
| 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 simplext and most intelligible form, and on the whole, in the sequence and connectionin which they actually originated.' page 3 Amazon back |
| 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 Amazon back |
| Heath, Thomas L, 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.' Amazon back |
| 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.' Amazon back |
| Hille, Einar , Analytic Function Theory, Volume 1 , Chelsea 1973 Foreword: 'This book represents an effort to integrate the theory of analytic functions with modern analysis as a whole, in particular to present it as a branch of functional anlysis, to which it gives concrete illustrations, problems and motivation. Amazon back |
| 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 Amazon back |
| 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.' Amazon back |
| Lo, Hoi-Kwong, and Tim Spiller, Sandra Popescu, Introduction to Quantum Computation and Information, World Scientific 1998 Jacket: 'This book provides a pedagogical introduction to the subjects of quantum information and computation. Topics include non-locality of quantum mechanics, quantum computation, quantum cryptography, quantum error correction, fault tolerant quantum computation, as well as some experimental aspects of quantum computation and quantum cryptography. A knowledge of basic quantum mechanics is assumed.' Amazon back |
| Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators), The Principia : Mathematical 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. Amazon back |
| 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. Amazon back |
| 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.' Amazon back |
| Pais 2, Abraham, Inward Bound: Of Matter and Forces in the Physical World, Clarendon Press, Oxford University Press 1986 Jacket: 'With his deep sense of history Pais has perception regarding the long range development of physics. He couples this with the aim of introducing a new style of presenting the history of science. The result is an epic story of a heroic perion in the human endeavour to discover the secrets of nature. This is a book that will endure.' href="http://www.physics.sunysb.edu/Physics/research_groups/p006_itp.htm">Chen Ning Yang Amazon back |
| 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.' Amazon back |
| 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 vN regards as the definitive form of quantum mechanics. ... Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.' Amazon back |
Papers
| Einstein, Albert, B. Podolsky, and N. Rosen,, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?", Phys. Rev., 47, , 1935, page 777-780. Abstract: 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the oher. Then either (1) the description of reality given by the wave function is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that has previously interacted with it leads to the reasult that if (1) is false then (2) is also false. One thus is led to conclude that the description of reality as given by a wave function is not complete.'. back |
| Heisenberg, Werner, "English: On Quantum Mechanical Reinterpretation of Kinematic and Mechanical Relations", Zeitschrift fur Physik, , 33, 1925, page 879. translated in B L van der Waerden, Sources of Quantum Mechanics, Dover Publications, New York, 1968, pp 261-276. . back |
| Shannon, Claude E, "Communication in the Presence of Noise", Proceedings of the IEEE, 86, 2, February 1998, page 447-457. Reprint of Shannon, Claude E. "Communication in the Presence of Noise." Proceedings of the IEEE, 37 (January 1949) : 10-21. 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of "ideal" systems which transmit this maximum rate are discussed. The equivalent number of binary digits per second of certain information sources is calculated.' . back |
| Zurek, Wojciech Hubert, "Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical", Physical Review A, 76, 052110, 16 November 2007, page . Abstract: 'Measurements transfer information about a system to the apparatus and then, further on, to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide a framework for "wave-packet collapse," designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment--the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them--into becoming a witness.'. back |
Links
| Bolzano-Weierstrass theorem - Wikipedia Bolzano-Weierstrass theorem - Wikipedia, the free encyclopedia 'Formal statement A subset A of Rn is sequentially compact if and only if it is both closed and bounded. [edit] Understanding the theorem A set is sequentially compact if every sequence of points in the set has a convergent subsequence which converges to a point in that set. Intuitively, a sequence of points converges to a point if it becomes arbitrarily close to that point for elements of the set with large enough index. A subsequence is a sequence that omits some members, for instance a2, a5, a13, ... Keep in mind that although we are taking away some members of the sequence, the number of elements in the subsequence is still infinite. The sequence a1, a2, a3, ... is bounded if there exists a real number L such that the norm ||an|| is less than L for every index n. For the real numbers, the norm is just the absolute value function, and the concept can be illustrated graphically in 2 dimensions. If ai is plotted on a 2-dimensional graph, with i on the horizontal axis and the value on the vertical, the sequence then travels to the right as it progresses. It is bounded if we can draw a horizontal strip which encloses all of the points. A set is closed if every convergent sequence of points within the set converges to a point in the set. Note that this is not saying that every sequence within the set converges. While it may seem that every set must be closed by the above definition, this is in fact not the case. Consider the set of all points on the real line less than 1. Then it is easy to define a sequence that approaches 1 arbitrarily closely, but never exceeds or equals 1. This sequence converges to 1. But 1 is not in the set, so this is an example of a set that is not closed. (It is actually open.)' back |
| Carl R Nave The wave nature of the electron 'This experiment demonstrated the wave nature of the electron, confirming the earlier hypothesis of deBroglie. Putting wave-particle duality on a firm experimental footing, it represented a major step forward in the development of quantum mechanics. The Bragg law for diffraction had been applied to x-ray diffraction, but this was the first application to particle waves.' back |
| CERN LHC Homepage 'explain it in 60 seconds The Large Hadron Collider is currently being installed in a 27-kilometer ring buried deep below the countryside on the outskirts of Geneva, Switzerland. When its operation begins in 2007, the LHC will be the world's most powerful particle accelerator. High-energy protons in two counter-rotating beams will be smashed together in a search for signatures of supersymmetry, dark matter and the origins of mass. The beams are made up of bunches containing billions of protons. Traveling at a whisker below the speed of light they will be injected, accelerated, and kept circulating for hours, guided by thousands of powerful superconducting magnets. For most of the ring, the beams travel in two separate vacuum pipes, but at four points they collide in the hearts of the main experiments, known by their acronyms: ALICE, ATLAS, CMS, and LHCb. The experiments' detectors will watch carefully as the energy of colliding protons transforms fleetingly into a plethora of exotic particles. The detectors could see up to 600 million collision events per second, with the experiments scouring the data for signs of extremely rare events such as the creation of the much-sought Higgs boson.' back |
| Davis Associates Wave Mechanics - Prince Louis de Broglie 'In 1923, while still a graduate student at the University of Paris, Louis de Broglie published a brief note in the journal Comptes rendus containing an idea that was to revolutionize our understanding of the physical world at the most fundamental level. He had been troubled by a curious "contradiction" arising from Einstein's special theory of relativity. ' back |
| Davisson and Germer Diffraction of Electrons by a Crystal of Nickel A pdf version of the Physical Review paper. 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 |
| Einstein, Podolsky and Rosen Can the Quantum Mechanical Description of Physical Reality be Considered Complete A PDF of the classic paper back |
| EPR Paradox = Wikipedia EPR Paradox = Wikipedia, the free encyclopedia 'In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. "EPR" stands for Einstein, Podolsky, and Rosen, who introduced the thought experiment in a 1935 paper to argue that quantum mechanics is not a complete physical theory.' 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 |
| Hamiltonian (quantum mechanics) - Wikipedia Hamiltonian (quantum mechanics) - Wikipedia, the free encyclopedia 'In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system. Like any other self-adjoint operator, the spectrum of the Hamiltonian can be decomposed, via its spectral measures, into pure point, absolutely continuous, and singular parts. The pure point spectrum can be associated to eigenvectors, which in turn are the bound states of the system. The absolutely continuous spectrum corresponds to the free states.' back |
| Hilbert space - Wikipedia Hilbert space - Wikipedia, the free encyclopedia 'The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. In more formal terms, a Hilbert space is an inner product space -- an abstract vector space in which distances and angles can be measured -- which is "complete", meaning that if a sequence of vectors approaches a limit, then that limit is guaranteed to be in the space as well.' 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 |
| Louis de Broglie Radiation: Waves and Quanta Note of Louis de Broglie, presented by Jean Perrin. (Translated from Comptes rendus, Vol. 177, 1923, pp. 507-510) 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 '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. In any case the theory requires a correction, and I shall attempt in the following to accomplish this on the basis of the theory of electromagnetic radiation which I developed.' back |
| NIST Fundamental Physical Constants "... working with industry to develop and apply technology, measurements and standards" back |
| Planck's Law - Wikipedia Planck's Law - Wikipedia, the free encyclopedia 'In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature T. As a function of frequency nu, back |
| Quantum entanglement - Wikipedia Quantum entanglement - Wikipedia, the free encyclopedia 'Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems.' back |
| Quantum mechanics - Wikipedia Quantum mechanics - Wikipedia, the free encyclopedia 'Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics. The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level. Quantum theory generalizes all classical theories, including mechanics and electromagnetism, and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits.' back |
| Schrödinger equation - Wikipedia Schrödinger equation - Wikipedia, the free encyclopedia 'In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei.' back |
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