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Topic: Hamiltonian (quantum mechanics)

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	Wikinfo \| Quantum mechanics (Site not responding. Last check: 2007-11-03)
		Quantum mechanics or quantum physics is a physical theory formulated in the first half of the twentieth century which successfully describes the behavior of matter at small distance scales.
		The quantum field theory describing electromagnetism is quantum electrodynamics; it is, at least in principle, capable of explaining chemical interactions as well as the interaction of matter and electromagnetic radiation.
		The quantum field theory describing the strong nuclear force is quantum chromodynamics, which describes the interactions of the subnuclear particles: quarks and gluons.
	wikinfo.org /wiki.php?title=Quantum_mechanics (1644 words)

	Quantum Theory - Mechanics - Crystalinks
		Quantum mechanics is a fundamental branch of theoretical physics that replaces Newtonian mechanics and classical electromagnetism at the atomic and subatomic levels.
		Quantum mechanics is a more fundamental theory than Newtonian mechanics and classical electromagnetism, in the sense that it provides accurate and precise descriptions for many phenomena that these "classical" theories simply cannot explain on the atomic and subatomic level.
		For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.
	www.crystalinks.com /quantumechanics.html (4216 words)

	PowerPedia:Quantum mechanics - PESWiki
		Quantum electrodynamics is a quantum theory of electrons, positrons, and the electromagnetic field, and served as a role model for subsequent quantum field theories.
		In the formalism of quantum mechanics, the state of a system at a given time is described by a complex number wave functions (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space.
		The transactional interpretation of quantum mechanics (TIQM) by John Cramer is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves.
	peswiki.com /index.php/PowerPedia:Quantum_mechanics (6984 words)

	Hamiltonian
		In classical mechanics, it is a function which describes the state of a mechanical system in terms of position[?] and momentum variables, which is the basis for a re-formulation of classical mechanics known as Hamiltonian mechanics.
		As explained in the article mathematical formulation of quantum mechanics, the physical state of a system may be characterized as a vector in an abstract Hilbert space, and physically observable quantities as Hermitian operators acting on these vectors.
		The quantum Hamiltonian H is the observable corresponding to the total energy of the system.
	www.ebroadcast.com.au /lookup/encyclopedia/ha/Hamiltonian.html (982 words)

	TAU Quantum Group - Topological Effects in Quantum Mechanics (Site not responding. Last check: 2007-11-03)
		Actually, the passive nonlocality of quantum correlations is distinct from the active nonlocality of the AB effect (action at a distance); Aharonov, Pendleton and Petersen [6] have shown how the AB effect implies nonlocal quantum equations of motion.
		Quantum topological and geometrical phases are ubiquitous in modern physics---in cosmology, particle physics, modern string theory, condensed matter, chemical and molecular physics, laser dynamics, and classical dynamical systems.
		Topological quantum phases are crucial in explaining superconductivity, the quantum Hall effect, the Josephson junction, flux quantization and many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behavior.
	www.tau.ac.il /~quantum/publicat/topo-effects.html (1118 words)

	Operators in Quantum Mechanics (Site not responding. Last check: 2007-11-03)
		Such operators arise because in quantum mechanics you are describing nature with waves (the wavefunction) rather than with discrete particles whose motion and dymamics can be described with the deterministic equations of Newtonian physics.
		Part of the development of quantum mechanics is the establishment of the operators associated with the parameters needed to describe the system.
		It is part of the basic structure of quantum mechanics that functions of position are unchanged in the Schrodinger equation, while momenta take the form of spatial derivatives.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/qmoper.html (107 words)

	Hamiltonian mechanics - Wikipedia, the free encyclopedia
		Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton.
		The Hamiltonian is the Legendre transform of the Lagrangian:
		The Hamiltonian induces a special vector field on the symplectic manifold, known as the symplectic vector field.
	en.wikipedia.org /wiki/Hamiltonian_mechanics (1322 words)

	Quantum Mechanics
		The quantum state of the universe can be approximated to the desired precision by a vector in a finite-dimensional vector space and can be stated exactly either as a sequence of such vectors by using increasingly detailed approximations or as a vector in an infinite dimensional Euclidean space.
		The Hamiltonian H is the sum of the kinetic and potential part.
		In non-relativistic mechanics, the stationary states are non-normalizable: The wavefunction density must be the same everywhere in space and hence cannot be normalized to one.
	web.mit.edu /dmytro/www/QuantumMechanics.htm (1376 words)

	Quantum Mechanics (Text Only)
		Hamiltonians are a method for finding the minimum value of a given equation and are used to calculate the path of least action such as orbits and trajectories.
		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.
	web.fccj.org /~ethall/quantum/quant2.htm (2445 words)

	BBC - h2g2 - Quantum Mechanics
		Quantum mechanics is simply a formulation of mechanics in which the assumption of arbitrary precision has been thrown out.
		There is a quantum phenomenon known as tunnelling, which states that it is possible for a particle to end up on the opposite side of a barrier even if it doesn't have enough energy to pass over the barrier.
		The rest of quantum mechanics is just the maths necessary to describe events in terms of probabilities, instead of certainties.
	www.bbc.co.uk /dna/h2g2/A781823 (2141 words)

	ACM Ubiquity - On the Realizability of Quantum Computers
		The quantum Turing machine and the quantum cellular automata models are equivalent to the circuit model and, therefore, face the same difficulties.
		The quantum circuit model converts the physical problem to a circuit theoretic form but it does not map all the physical constraints required by the laws of quantum mechanics.
		A realistic model of quantum computing must ensure that the questions of preparation of pure states and that of boson/fermion statistics for a quantum state are not ignored.
	www.acm.org /ubiquity/views/v7i11_quantumcomputers.html (2268 words)

	Everett's Relative-State Formulation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)
		Everett's relative-state formulation of quantum mechanics is an attempt to solve the measurement problem by dropping the collapse dynamics from the standard von Neumann-Dirac theory of quantum mechanics.
		Everett's proposal was to drop the collapse postulate from the standard formulation of quantum mechanics then deduce the empirical predictions of the standard theory as the subjective experiences of observers who are themselves treated as physical systems described by his theory.
		On the standard collapse formulation of quantum mechanics, somehow during the measurement interaction the state would collapse to either the first term of this expression (with probability equal to a squared) or to the second term of this expression (with probability equal to b squared).
	plato.stanford.edu /entries/qm-everett (6648 words)

	Marsfind (Site not responding. Last check: 2007-11-03)
		The Hamiltonian of a system is defined as (1) where q is a generalized coordinate, p is a generalized momentum, L is the...
		The Hamiltonian operator is defined as the operator such the energy E of a...
		Hamiltonian mechanics Hamiltonian mechanics is a re-formulation of classical...
	www.marsfind.com /search.html?Keywords=hamiltonian (366 words)

	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.
		If the Hamiltonian is time-independent, {U(t)} form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.
	en.wikipedia.org /wiki/Hamiltonian_(quantum_mechanics) (988 words)

	The Hamiltonian in Quantum Mechanics (Site not responding. Last check: 2007-11-03)
		Associated with each measurable parameter in a physical system is a quantum mechanical operator, and the operator associated with the system energy is called the Hamiltonian.
		Operating on the wavefunction with the Hamiltonian produces the Schrodinger equation.
		The full role of the Hamiltonian is shown in the time dependent Shrodinger equation where both its spatial and time operations manifest themselves.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/hamil.html (143 words)

	Common Applications \| The n-Category Café
		Quantum mathematics could mean, if we get that far, ‘understanding properly the analysis, geometry, topology, algebra of various non-linear function spaces’, and by ‘understanding properly’ I mean understanding it in such a way as to get quite rigorous proofs of all the beautiful things the physicists have been speculating about.
		Noting that statistical mechanics and quantum mechanics can be largely constructed by considering them as ways of manipulating information, Caticha goes on to take on general relativity.
		Interestingly, while quantum mechanics is in a way nothing but statistical mechanics analytically continued to the complex plane, we usually tend to regard not just the Hamiltonian in quantum mechanics as encoding information about nature, but also the rest of the formalism.
	golem.ph.utexas.edu /category/2006/12/common_applications.html (1263 words)

	Amazon.com: Quantum Mechanics (Physics): Books: Albert Messiah (Site not responding. Last check: 2007-11-03)
		Chapter 4 is an overview of the statistical interpretation of quantum mechanics.
		Any textbook on quantum mechanics at this level in the 21st century should include a very detailed introduction to numerical methods so as to prepare the student early on to techniques that will be used more and more in the decades ahead.
		Although the presentation of the material still assumes a knowledge of classical mechanics and magnetism (an approach that has since been abandoned in quantum mechanics texts), the book is remarkably self-contained (the exercises, however, are not).
	www.amazon.com /Quantum-Mechanics-Physics-Albert-Messiah/dp/0486409244 (2298 words)

	Introduction
		Newtonian mechanics (1687) explained the motion of mechanical objects on both celestial and terrestrial scales.
		The old quantum theory, resulting in Bohr's orbital model of the atom could point to certain real successes: Derivation of the Balmer formula, quantum numbers and selection rules for energy states in an atom, explanation of the periodic table and the Pauli exclusion principle.
		The theory of quantum mechanics asserts that with every possibility for an event in nature to take place, there is a quantity called amplitude associated with each alternative.
	rugth30.phys.rug.nl /quantummechanics/intro.htm (896 words)

	Improving students' understanding of quantum mechanics - Physics Today August 2006
		Most physicists, as students, are introduced to quantum mechanics in a modern-physics course, take quantum mechanics as advanced undergraduates, and then take it again in their first year of graduate school.
		And patterns of incorrect notions of quantum mechanics are analogous to those that have been well-documented for introductory physics courses.
		Shared misconceptions in quantum mechanics can be traced in large part to incorrect generalizations of concepts learned earlier, either in classical mechanics or in quantum mechanics.
	www.physicstoday.org /vol-59/iss-8/p43.html (3667 words)

	Thall's History of Quantum Mechanics
		Today quantum mechanics is said to be a theory set in "Hilbert Space." At the InternationalCongress of Mathematicians in Paris (1900) Hilbert presented the now famous 23 problems
		wave equation proved to be of fundamental importance in the new theory of quantum mechanics.
		Dirac laid the foundations for quantum electrodynamics (1927) with his discovery of an equation incorporating both the quantum theory and the theory of special relativity.
	mooni.fccj.org /~ethall/quantum/quant.htm (2212 words)

	LAGRANGIAN AND HAMILTONIAN MECHANICS
		Connections with other areas of physics which the student is likely to be studying at the same time, such as electromagnetism and quantum mechanics, are made where possible.
		There is thus a discussion of electromagnetic field momentum and mechanical "hidden" momentum in the quasi-static interaction of an electric charge and a magnet.
		There is also a brief introduction to path integrals and their connection with Hamilton's principle, and the relation between the Hamilton—Jacobi equation of mechanics, the eikonal equation of optics, and the Schrödinger equation of quantum mechanics.
	www.worldscibooks.com /physics/3111.html (247 words)

	Research
		For example, one needs ab initio quantum mechanical method at the crack tip where chemical bonds are breaking between neighboring atoms thereby bringing marked deformation in the electron density but far from the crack tip where deformation is less, atoms are described using classical potentials.
		However, even at the quantum level each possible choice for the ab initio method entails approximations used to optimize the accuracy and speed of the calculations.
		It is found that the two quantum rods have noticeable differences, although the embedding method for the equilibrium composite rods is faithful to the underlying quantum method in each case.
	www.phys.ufl.edu /~aditi/research.htm (1536 words)

	More on Hamiltonian
		* In classical mechanics, the Hamiltonian is a function describing the state of a mechanical system in terms of position and momentum variables.
		* In quantum mechanics, the Hamiltonian is an operator corresponding to the total energy of a system.
		Both the Hamiltonian operator in physics and Hamiltonian cycles in graph theory are named after Sir William Rowan Hamilton.
	www.artilifes.com /hamiltonian.htm (298 words)

	Dirac Theory
		The theory of Paul Dirac represents an attempt to unify the theories of quantum mechanics and special relativity.
		That is, one seeks a formulation of quantum mechanics which is Lorentz invariant, and hence consistent with special relativity.
		It can be added by hand to the non-relativistic Schödinger theory with satisfactory results, but spin is a natural consequence of treating quantum mechanics in a completely relativistic fashion.
	www.pha.jhu.edu /~rt19/hydro/node6.html (637 words)

	Lagrangian and Hamiltonian Mechanics
		The correspondence between the conservation of energy and the Lagrangian equations of motion suggests that there might be a convenient variational formulation of mechanics in terms of the total energy E = T + V (as opposed to the Lagrangian L = T - V).
		Notice that the partial derivative of L with respect to x' is the momentum of the particle.
		The Lagrangian and Hamiltonian formulations of mechanics are also notable for the fact that they express the laws of mechanics without reference to any particular coordinate system for the configuration space.
	www.mathpages.com /home/kmath523/kmath523.htm (1247 words)

	Some remarks on hamiltonian systems and quantum mechanics (Site not responding. Last check: 2007-11-03)
		These notes contain some remarks on the general structure of a class of physical systems called Hamiltonian, and on quantum mechanical systems in particular.
		The distinguishing features of classical and quantum mechanical systems are pointed out.
		Then in §4 we study the dynamics of classical and quantum mechanics--we endeavor to show that both systems are Hamiltonian, when the latter condition is interpreted from the modern point of view of symplectic manifolds (see [2]).
	www.cds.caltech.edu /~marsden/bib/1977/01-ChMa1977 (232 words)

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