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Topic: Mathematical formulation of quantum mechanics

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List of mathematical topics in quantum theory

Quantum mechanics

Quantum theory

Quantum field theory

Hamiltonian (quantum mechanics)

Hilbert space

Quantum harmonic oscillator

Functional analysis

Operator algebra

Hamiltonian mechanics

Hamiltonian

Particle in a ring

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Symmetry

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	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations; probability theory was used in statistical mechanics.
		The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures.
		A quantum description consists of a Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=mathematical_formulation_of_quantum_mechanics (3233 words)

	Wikinfo \| Quantum mechanics (Site not responding. Last check: 2007-11-06)
		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.
		Quantum mechanics as such omits the electromagnetic force, the strong nuclear force, and gravity.
		The quantum field theory describing the strong nuclear force is quantum chromodynamics, which describes the interactions of the subnuclear particles: quarks and gluons.
	www.wikinfo.org /wiki.php?title=Quantum_mechanics (1644 words)

	Wavefunction collapse
		In quantum mechanics, the collapse of the wavefunction is a name given to one of two processes by which quantum systems[?] evolve.
		Why the wavefunction collapses is a fundamental question in the interpretation of quantum mechanics, and is addressed directly by both the Copenhagen interpretation (which asserts that it is collapsed by "measurement") and the Everett many-worlds interpretation (which claims that the collapse is merely a result of quantum decoherence).
		See also mathematical formulation of quantum mechanics; the collapse of the wavefunction is postulate (3).
	www.ebroadcast.com.au /lookup/encyclopedia/qu/Quantum_collapse.html (219 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.
		In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system).
	www.crystalinks.com /quantumechanics.html (4216 words)

	Mathematical formulation of quantum mechanics - Definition, explanation
		One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces.
		This formulation of quantum mechanics, called canonical quantization, continues to be used today, and still forms the basis of ab-initio calculations in atomic, molecular and solid-state physics.
		This is related to quantization and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework.
	www.calsky.com /lexikon/en/txt/m/ma/mathematical_formulation_of_quantum_mechanics.php (3071 words)

	quantum theory. The Columbia Encyclopedia, Sixth Edition. 2001-05
		Quantum mechanics, the final mathematical formulation of the quantum theory, was developed during the 1920s.
		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.
		Quantum mechanics was combined with the theory of relativity in the formulation of P. Dirac (1928), which, in addition, predicted the existence of antiparticles.
	www.bartleby.com /65/qu/quantumt.html (795 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.
		And, for Everett, this restriction on the applicability of quantum mechanics was unacceptable.
		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 (6664 words)

	Interpretation and Philosophical Foundation of Quantum Mechanics
		It is suggested that the objective randomness of the individual quantum event is a necessity of a description of the world in view of the significant influence the observer in quantum mechanics has.
		That is because the epistemological problems of quantum mechanics are immune against a variation of the magnitude of the quantum of action over a wide range, yet again, the fact that a quantum of action exists at all surely is significant in the quest for the new paradigm.
		Furthermore, any position that would necessitate a change of the quantum formalism[37] in the sense that it leads to a change of its predictions in may opinion is, at the least, highly improbable in view of the excellent agreement of methods of experiments with theoretical prediction.
	www.quantum.univie.ac.at /zeilinger/philosop.html (6432 words)

	Hans Reichenbach [Internet Encyclopedia of Philosophy]
		Mathematical geometry, a branch of mathematics, is a purely formal system and it does not deal with the truth of axioms, but with the proof of theorems, ie it only search for the consequences of axioms.
		Also the mathematical formulation of the special theory of relativity acknowledges the difference between space and time: the equation that defines the metric is dx^2 + dy^2 + dz^2 - dt^2 = ds^2 and the time coordinate is distinguishable from the space coordinates by the negative sign.
		The mathematical formulation of the special theory of relativity uses a four-dimensional space-time known as the Minkowski space (mathematician Hermann Minkowski, b 1864 d 1909, gave a mathematical formulation of Einstein's special theory of relativity), in which three coordinates are the space coordinates and one coordinate is the time coordinate.
	www.utm.edu /research/iep/r/reichenb.htm (8071 words)

	Mathematical formulation of quantum mechanics - Wikipedia, the free encyclopedia
		Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations; probability theory was used in statistical mechanics.
		The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures.
		A quantum description consists of a Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.
	en.wikipedia.org /wiki/Mathematical_formulation_of_quantum_mechanics (3464 words)

	Quantum Physics Quackery (Skeptical Inquirer January 1997)
		Quantum mechanics is thought, even by many physicists, to be suffused with mysteries and paradoxes.
		In conventional quantum mechanics, the wave properties of particles are formally represented by a mathematical quantity called the wave function, used to compute the probability that the particle will be found at a particular position.
		Quantum mechanics, the centerpiece of modern physics, is misinterpreted as implying that the human mind controls reality and that the universe is one connected whole that cannot be understood by the usual reduction to parts.
	www.csicop.org /si/9701/quantum-quackery.html (2265 words)

	fUSION Anomaly. Quantum Mechanics
		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.
		According to the quantum theory, energy is emitted and absorbed in a small packet, called a quantum (pl. quanta), which in some situations behaves as particles of matter do; particles exhibit certain wavelike properties when in motion and are no longer viewed as localized in a given region but as spread out to some degree.
		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.
	fusionanomaly.net /quantummechanics.html (1928 words)

	Quantum harmonic oscillator (Site not responding. Last check: 2007-11-06)
		The quantum harmonic oscillator is a quantum mechanical analogue of the classical harmonic oscillator.
		It is one of the most problems in quantum mechanics because (i) a exact solution exists and (ii) a wide of physical situations can be reduced to In particular a system near an equilibrium can often be described in terms of or more harmonic oscillators.
		The following discussion of the quantum harmonic relies on the article Mathematical formulation of quantum mechanics.
	www.freeglossary.com /Quantum_harmonic_oscillator (1860 words)

	Copenhagen Interpretation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)
		Hence in the search for a theory of quantum mechanics it became a methodological requirement to Bohr that any further theory of the atom should predict values in domains of large quantum numbers that should be a close approximation to the values of classical physics.
		The quantum mechanical description of the object differs from the classical description of the measuring apparatus, and this requires that the object and the measuring device should be separated in the description, but the line of separation is not the one between macroscopic instruments and microscopic objects.
		In general, Bohr considered the demands of complementarity in quantum mechanics to be logically on a par with the requirements of relativity in the theory of relativity.
	plato.stanford.edu /entries/qm-copenhagen (4422 words)

	Objective Science - Quantum Mechanics and Dissidents By Eric Dennis
		Beyond that, entanglement is the root of most of the uniquely quantum phenomena discovered over the last century--superconductivity, superfluidity, atomic bose condensation, lasers, electron configurations in atoms and in certain exotic semiconductors, and (perhaps someday) quantum computers.
		This result is consistent with the original predictions of quantum mechanics, in which the entanglement of the two particles is what makes these influences possible.
		The failure of TEW, however, must not be taken to support the sophistry connected with the standard interpretation of quantum mechanics, from the idea that entities lose their attributes until we observe them to the supposed victory of indeterminism in physics.
	objectivescience.com /articles/ed1_quantum_dissidents.htm (2062 words)

	fUSION Anomaly. Quantum Mechanics
		According to the quantum theory, energy is emitted and absorbed in a small packet, called a quantum (pl. quanta), which in some situations behaves as particles of matter do; particles exhibit certain wavelike properties when in motion and are no longer viewed as localized in a given region but as spread out to some degree.
		Quantum mechanics, the final mathematical formulation of the quantum theory, was developed during the 1920s.
		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.
	www.fusionanomaly.net /quantummechanics.html (1928 words)

	Open Questions: Quantum Theory
		The irony is that rather than discovering a logical inconsistency in quantum mechanics, Einstein and the others motivated the confirmation of aspects of quantum mechanics they regarded as absurd: entanglement and nonlocality.
		The author's work, which requires some mathematical sophistication, is a modern study of the interpretation of quantum mechanics with a view to resolving the "measurement problem".
		An introduction to the foundations of quantum mechanics with relatively mild mathematical prerequisites.
	www.openquestions.com /oq-ph002.htm (1989 words)

	Everett's Relative-State Formulation of Quantum Mechanics
		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.
		He then intended to deduce the statistical predictions of quantum mechanics (the predictions that depend on rule 4b in the standard theory) as the subjective experiences of observers who are themselves treated as ordinary physical systems within the new theory.
		Everett said that on his formulation of quantum mechanics "the formal theory is objectively continuous and causal, while subjectively discontinuous and probabilistic" (1973, 9).
	www.seop.leeds.ac.uk /archives/fall2002/entries/qm-everett (4563 words)

	More on Wavefunction
		In the mathematical formulation of quantum mechanics, the state of any system is represented by an object called a ket, which is an element of an abstract mathematical structure called a Hilbert space.
		In mathematical terms, such continuous orthonormal bases are referred to as diagonalizations, because mathematically they correspond to representing certain commutative algebras of operators as algebras of multiplication operators.
		Two common diagonalizations used in quantum mechanics are the configuration (position) space representation (which diagonalizes the position operators) and the momentum space representation (which diagonalizes the momentum operators).
	www.artilifes.com /wavefunction.htm (853 words)

	Electron indeterminism Problem
		The wavefunction is an approximation of the path integral formulation, which is used to formulate quantum field theories.
		The simplest way to interpret the path integral formulation is that the particle actually does travel along all the possible paths from A to B. This approach says that the path integral describes all the paths the particle takes from A to B in the multiverse.
		The MWI has many implications for the quantum nature of spacetime which might have to be considered when formulating a quantum theory of gravity.
	www.physicsforums.com /showthread.php?threadid=134901 (2160 words)

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