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Topic: Matrix mechanics

	mechanics. The Columbia Encyclopedia, Sixth Edition. 2001-05
		Mechanics was studied by a number of ancient Greek scientists, most notably Aristotle, whose ideas dominated the subject until the late Middle Ages, and Archimedes, who made several contributions and whose approach was quite modern compared to other ancient scientists.
		In 1905, Albert Einstein showed that Newton’s mechanics was an approximation, valid for cases involving speeds much less than the speed of light; for very great speeds the relativistic mechanics of his theory of relativity was required.
		In the quantum mechanics developed during the 1920s as part of the quantum theory, the motions of very tiny particles, such as the electrons in an atom, were explained using the fact that both matter and energy have a dual nature—sometimes behaving like particles and other times behaving like waves.
	www.bartleby.com /65/me/mechanics.html (694 words)

	Quantum mechanics (Site not responding. Last check: 2007-10-22)
		Quantum mechanics is a theory of mechanics, a branch of physics that deals with the motion of bodies and associated physical quantities such as energy and momentum.
		Quantum mechanics predicts at least three classes of phenomena that classical mechanics and classical electrodynamics cannot account for: (i) the quantization (discretization) of certain physical quantities, (ii) wave-particle duality, and (iii) quantum entanglement.
		An alternative formulation of Quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics.
	quantum-mechanics.iqnaut.net (4031 words)

	Matrix mechanics (Site not responding. Last check: 2007-10-22)
		Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg in 1925.
		Matrix mechanics involve associating the properties of matter with matrices.
		Though Matrix mechanics do not include concepts such as the wave function of Erwin Schrödinger's wave equation, the two approaches were proven to be mathematically equivalent by mathematician David Hilbert.
	matrix-mechanics.setcom.sk (73 words)

	Highbeam Encyclopedia - Search Results for Matrix
		The size n of the square matrix, as determined from the number of entries in any row or column, is called the order of the determinant.
		The cement is the chemically active element, or matrix; the sand and stone are the inert elements, or aggregate.
		Matrix management in hospitals: testing theories of matrix structure and development.
	www.encyclopedia.com /SearchResults.aspx?Q=Matrix (693 words)

	ScienceWeek - In Focus: Pascual Jordan (Site not responding. Last check: 2007-10-22)
		Of the triumvirate Pascual Jordan, Max Born, and Werner Heisenberg that formulated matrix quantum mechanics in 1925, Jordan was the principal architect of the theory.
		Note #1: What is known as "Heisenberg's matrix mechanics" (matrix quantum mechanics) is a particular formulation of quantum mechanics in which the vector aspect of quantum theory is emphasized, whereas the wave aspects of quantum phenomena play a secondary role.
		Given wave mechanics, the invention of matrix mechanics might be viewed as inevitable, since the set of all solutions of a linear differential equation can be regarded as a vector space.
	scienceweek.com /focus001.htm (1380 words)

	Quantum Mechanics -- from Eric Weisstein's World of Physics
		Quantum mechanics is the description of motion and interaction of particles at the small scales where the discrete nature of the physical world becomes important.
		Quantum mechanics represented a fundamental break with classical physics, in which energies and angular momenta were regarded as continuous quantities that could change by arbitrary amounts.
		Quantum mechanics contains many counterintuitive properties, including the uncertainty principle, which states that it is not possible to simultaneously determine a particle's position and momentum.
	scienceworld.wolfram.com /physics/QuantumMechanics.html (723 words)

	Introduction to quantum mechanics - Wikipedia, the free encyclopedia
		Quantum mechanics began with the discovery that energy is delivered in packets whose size is related to the frequencies of all electromagnetic waves (and to the color of visible light since in that case frequency determines color).
		Multiplying matrix A by matrix B is not the same as multiplying matrix B by matrix A. In symbols, AxB is in general not equal to BxA.
		Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of the electron.
	en.wikipedia.org /wiki/Basics_of_quantum_mechanics (9410 words)

	Matrix mechanics - Wikipedia, the free encyclopedia
		It has been shown to be exactly equivalent to the Schroedinger wave formulation of quantum mechanics and is the basis of the bra-ket notation used to summarize quantum mechanical wave functions.
		In atomic physics, through spectroscopy, it was known that observational data related to atomic transitions arise from interactions of the atoms with light quanta.
		An observable in this case could be the position of the particle, represented by the matrix X. Since the particle could be anywhere on the line, the possible outcome of a measurement of its position could be any one of an infinite set of eigenvalues of X, denoted by x.
	en.wikipedia.org /wiki/Matrix_mechanics (1111 words)

	MIT OpenCourseWare \| Health Sciences and Technology \| HST.523J Cell-Matrix Mechanics, Spring 2004 \| Home
		Mechanical forces play a decisive role during development of tissues and organs, during remodeling following injury as well as in normal function.
		A stress field influences cell function primarily through deformation of the extracellular matrix to which cells are attached.
		The unit cell process paradigm combined with topics in connective tissue mechanics form the basis for discussions of several topics from cell biology, physiology, and medicine.
	ocw.mit.edu /OcwWeb/Health-Sciences-and-Technology/HST-523JSpring-2004/CourseHome/index.htm (145 words)

	Quantum Mechanics
		Quantum mechanics is needed to explain many properties of matter, such as the temperature dependence of the specific heat of solids, as well as when very small quantities of matter or energy are involved, as in the interaction of elementary particles and fields, but the theory of
		The wave mechanics of Erwin Schrödinger (1926) involves the use of a mathematical entity, the wave function, which is related to the probability of finding a particle at a given point in space.
		The matrix mechanics of Werner Heisenberg (1925) makes no mention of wave functions or similar concepts but was shown to be mathematically equivalent to Schrödinger's theory.
	www.neutron.anl.gov /hyper-physics/Quantum-mechanics.html (611 words)

	Heisenberg - Quantum Mechanics, 1925-1927: The Uncertainty Principle
		Most of the young men who created matrix mechanics were ready to move into teaching positions as professors, and the older generation of theoretical physicists was beginning to vacate positions at German universities.
		Heisenberg's family was exerting pressure on the young man to capture one of the vacancies at the same time that his best work, matrix mechanics, seemed to be overshadowed by wave mechanics.
		the equivalence of the matrix and wave versions of quantum mechanics, and Born presented a statistical interpretation of the wave function, Jordan in Göttingen and Paul Dirac in Cambridge, England, created unified equations known as "transformation theory." These formed the basis of what is now regarded as quantum mechanics.
	www.aip.org /history/heisenberg/p08.htm (958 words)

	Quantum Mechanics (Text Only)
		Today quantum mechanics is said to be a theory set in "Hilbert Space." At the International Congress of Mathematicians in Paris (1900) Hilbert presented the now famous 23 problems which he challenged 20th century mathematicians to solve.
		Born's interpretation of the wave equation proved to be of fundamental importance in the new theory of quantum mechanics.
		After hearing Heisenberg lecture on matrix mechanics, von Neumann decided to develop his own version of quantum mechanics--the matrices of Heisenberg were "too imprecise." In his book (1932), The mathematical Foundations of Quantum Mechanics, von Neumann invented operator theory (now called Neumann algebras) to explain certain aspects of quantum mechanics.
	web.fccj.org /~ethall/quantum/quant2.htm (2445 words)

	Quantum Mechanics, 1925-1927: The Quantum Mechanic
		The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.
		The wave mechanics appealed to many physicists because it seemed to do everything that matrix mechanics could do but much more easily and seemingly without giving up the visualization of orbits within the atom.
		This unleashed an intense debate between the followers of the alternative versions of quantum mechanics that formed the background to the later uncertainty relations and the Copenhagen Interpretation.
	www.aip.org /history/heisenberg/p07.htm (527 words)

	anoscmm.html (Site not responding. Last check: 2007-10-22)
		Matrix mechanics is the original form of quantum mechanics that Werner Heisenberg initiated in 1925 and elaborated in association with Max Born and Pascual Jordan.
		Alternative procedures in Maple to generate energies of vibration-rotational states of a diatomic molecule, based on BKW quasi-classical theory and hypervirial perturbation theory in wave mechanics, are available in separate worksheets.
		We illustrate here how matrix mechanics can be effectively used to solve an important problem in quantum mechanics using perturbation theory.
	www.cecm.sfu.ca /~ogilvie/anoscmm1.html (1059 words)

	Matrix Mechanics Tutorial (Site not responding. Last check: 2007-10-22)
		The basic principles of quantum theory can be demonstrated very simply by exploring the properties of electron spin using Heisenberg's formulation of quantum mechanics which is usually referred to as matrix mechanics.
		The matrix formulation provides clear illustrations of the following essential quantum mechanical concepts: eigenvector, operator, eigenvalue, expectation value, the linear superposition, and the commutation relations.
		In quantum mechanics states are vectors and operators are matrices.
	www.users.csbsju.edu /~frioux/matmech/matmech.htm (1377 words)

	Quantum Mechanics: Matrix Mechanics and Wave Mechanics - Numericana
		Matrix Mechanics: Neither measurements nor matrices can be switched at will.
		This bothers [or should bother] everybody who deals with quantum mechanics: The so-called Schrödinger's Cat theme is often used to discuss the problem, in the guise of a system that includes a cat (a "qualified" observer) in the presence of a quantum device which could trigger a lethal device.
		This hindered neither the early development of Schrödinger's theory of "Wave Mechanics", nor the derivation of the nonrelativistic equation at its core.
	home.att.net /~numericana/answer/quantum.htm (4570 words)

	fUSION Anomaly. Quantum Mechanics (Site not responding. Last check: 2007-10-22)
		quantum mechanics or quantum theory, branch of mathematical physics that deals with the emission and absorption of energy by matter and with the motion of material particles.
		Quantum mechanics was combined with the theory of relativity in the formulation of P.A.M. DIRAC (1928), which also predicted the existence of ANTIPARTICLES.
		Matrix mechanics showed that there were an infinite number of matrices that represented the position and momentum of an electron inside an atom.
	fusionanomaly.net /quantummechanics.html (1928 words)

	Prof. Dr. Surkay D. Akbarov
		scientific conference of young scientists of the institute of mechanics of the ac.
		Non-classical and mixed problem of deformable body mechanics: proceedings of the seminar of young scientists, Kiev, May 21-23, Institute of mechanics of the ac.
		scientific conference of young scientists of the institute of mechanics of the acad.
	www.yildiz.edu.tr /~akbarov/yayinlar.html (3890 words)

	Thall's History of Quantum Mechanics
		The development of quantum mechanics was perhaps the greatest intellectual achievement of the 20th century.
		The concept of the electron as a point particle moving in a well-defined path around the nucleus is replaced in wave mechanics by clouds that describe the probable locations of electrons in different states.
		Schrödinger's wave mechanics eventually became the method of choice because it is less abstract and easier to understand than Heisenberg's matrix mechanics.
	web.fccj.org /~ethall/quantum/quant.htm (2212 words)

	Matrix Methods in Quantum mechanics (Site not responding. Last check: 2007-10-22)
		Because matrix multiplication is in general not commutative AB - BA is not zero.
		This is known as the completeness relation and is widely used in quantum theory, general inthe reverse process of what we have just done, that is to break up a product of operators into a sum of the two operators separately.
		This is a direct link between Schrödinger's Wave Mechanics and Heisenberg's Matrix Mechanics.
	www.cobalt.chem.ucalgary.ca /ziegler/educmat/chm386/rudiment/quanmath/matrix.htm (434 words)

	Tendon Fiber-Matrix Mechanics in Normal and Healing Tissue (Site not responding. Last check: 2007-10-22)
		The causes of the decreased tendon mechanical properties are largely unknown, but may be related to disruptions in the mechanical integrity of tendon microstructures such as fibers, matrix or fiber-matrix interactions.
		A combination of theoretical modeling and tensile experiments are being used to quantitatively determine the contributions of the fiber, matrix, and fiber-matrix interactions to tendon mechanics.
		In the theoretical model, a general three-dimensional constitutive model of tendon is being developed which includes anisotropy due to the contribution of the collagen fibers within an isotropic matrix.
	www.uphs.upenn.edu /orl/research/bioengineering/fmatrix.htm (291 words)

	matrix mechanics - HighBeam Encyclopedia (Site not responding. Last check: 2007-10-22)
		Find newspaper and magazine articles plus images and maps related to "matrix mechanics" at HighBeam.
		Fracture mechanics for the reconstruction of Noto Cathedral.
		Modifications to FBG sensor spectra due to matrix cracking in a GFRP composite.
	www.encyclopedia.com /html/X/X-matrixme.asp (194 words)

	Why Configuration Interaction? (Site not responding. Last check: 2007-10-22)
		0 In the first paper on quantum mechanics, Heisenberg used matrix mechanics to calculate the frequencies and intensities of spectral lines [2].
		Later, when Schrödinger discovered wave mechanics, it was quickly shown that the Schrödinger and Heisenberg approaches are mathematically equivalent [3, 4].
		Matrix mechanics requires that we choose a vector space for the expansion of the problem.
	zopyros.ccqc.uga.edu /lec_top/ci/node5.html (812 words)

	partwav (Site not responding. Last check: 2007-10-22)
		Born pointed out that Heisenberg's strange multiplication law could be understood in terms of matrix multiplication.
		Together with P. Jordan he transposed Heisenberg's theory into a systematic matrix language.
		In 1926 W. Pauli applied the Heisenberg theory to the hydrogen atom problem and did not only deduce the spectrum of hydrogen but also the additional lines produced by electric and magnetic fields.
	rugth30.phys.rug.nl /quantummechanics/mech.htm (267 words)

	Matrix Theory - Search Results - MSN Encarta (Site not responding. Last check: 2007-10-22)
		Matrix Theory, a branch of pure mathematics, introduced by Arthur Cayley in 1858, associated with the solution of systems of linear equations, which...
		Simultaneously with the development of wave mechanics, Heisenberg evolved a different mathematical analysis known as matrix mechanics.
		In spite of these difficulties, game theorists have been able to identify certain important features of certain kinds of game situation.
	uk.encarta.msn.com /Matrix_Theory.html (102 words)

	ELEMENTARY QUANTUM MECHANICS (Site not responding. Last check: 2007-10-22)
		Quantum mechanics is a difficult subject for students to learn after years of rigorous training in classical physics.
		The matrix formulation presented allows quantum theory to be generalized to new physical systems such as electron spin, which cannot be done by the Schrödinger approach.
		The result is a textbook which promises to provide a future generation of students a clear, usable and authoritative resource to study the fundamentals of quantum mechanics.
	www.worldscibooks.com /physics/5802.html (209 words)

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