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Topic: Pauli matrices

	Wolfgang Pauli
		Pauli was born in Vienna, Austria on August 25, 1900.
		Pauli moved to the United States in 1940, where he was Professor of Theoretical Physics at Princeton.
		In 1945, Pauli received the Nobel Prize in Physics for his "decisive contribution through his discovery in 1925 of a new law of Nature, the exclusion principle or Pauli principle." He had been nominated for the prize by Einstein.
	www.jewishvirtuallibrary.org /jsource/biography/pauli.html (647 words)

	Pauli matrices - Wikipedia, the free encyclopedia
		The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.
		The determinants and traces of the Pauli matrices are:
		The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states (2 × 2 positive semidefinite matrices with trace 1).
	en.wikipedia.org /wiki/Pauli_matrices (811 words)

	TPauliMatrix
		This package has particular members to facilitate a quantum mechanical calculation in which the Pauli spinor describes the spin-state of a fermion and the QM operators are described by Pauli matrices.
		Pauli matrices are also used to describe mixed states, ensembles that contain mixtures of particles described by more than one Pauli spinor.
		The standard Pauli matrices are generated by invoking the construc- to with an argument of enum type EPauliIndex.
	zeus.phys.uconn.edu /refs/root/TPauliMatrix.html (424 words)

	Search ScienceWorld
		The Dirac matrices are a class of 4x4 matrices which arise in quantum electrodynamics.
		The number of mxn binary matrices is 2^(mn), so the number of square nxn binary matrices is 2^(n^2) which, for n==1, 2,..., gives 2, 16, 512, 65536, 33554432,...
		Two square matrices A and B that are related by B==X^(-1)AX, where X is a square nonsingular matrix are said to be similar.
	scienceworld.wolfram.com /search/index.cgi?num=&q=Matrices (461 words)

	Pauli Matrices (Site not responding. Last check: 2007-10-12)
		Pauli's ability to describe complex relationships clearly was demonstrated after the publication of the Encyclopedia article in 1921 on the theory of relativity, and a second time in his chapter Quantum Theory in the "Handbuch der Physik" of 1926.
		While Pauli was working on this article throughout almost the whole of 1925, the "new quantum mechanics" arose, which was introduced by Werner Heisenberg's fundamental work on matrix mechanics.
		In May 1927 Pauli published the work "Zur Quantenmechanik des magnetischen Elektrons", in which he introduced the so-called "Pauli matrices".
	www.ethbib.ethz.ch /exhibit/pauli/matrizen_e.html (228 words)

	Maths - Alternative Quaternion Notations Pauli Matricies - Martin Baker
		I think Pauli matrices are interesting though, because instead of having to learn the rules for multiplying the operators i, j and k these multiplication rules come automatically provided you know how to multiply 2x2 matrices.
		If you want to calculate the spin of fundamental particles in many dimensions then Pauli matrices may be the only way to do it.
		Pauli matrices were developed for physics (quantum mechanics) so that may be why they are formulated with this i factor difference from quaternions.
	www.euclideanspace.com /maths/algebra/realNormedAlgebra/quaternions/notations/pauli (477 words)

	Voltage Reflection Matrices Based Decompositions
		The Pauli basis mentioned previously (3.14), is such an approach and the vectorization of the [S]-matrix can be interpreted as the description of a scatterer in terms of deterministic scattering mechanisms, expressed by the Pauli matrices.
		Apart from the physical importance these particular mechanisms have in Radar imagery, the Pauli decomposition has the further advantage that the scattering mechanisms are orthogonal and so their separation is possible, even in the case of second order statistics where noise and depolarisation effects can be introduced.
		The elements of the two matrices are in function of the scattering matrix components.
	epsilon.nought.de /tutorials/polsmart/node19.html (1398 words)

	Gamma matrices - Wikipedia, the free encyclopedia
		The gamma matrices, also known as Dirac matrices, were developed by P.A.M. Dirac in order to serve as coefficients of the Dirac equation.
		This defining property is considered to be more fundamental than the numerical values used in the gamma matrices, so other sign conventions for the metric necessitate a change in the definitions of the gamma matrices.
		The gamma matrices we have written so far are appropriate for acting on Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices.
	en.wikipedia.org /wiki/Gamma_matrices (669 words)

	Pauli two-component formalism
		It is conventional to represent the eigenstates of spin angular momentum as column (or row) matrices.
		These matrices, which are called the Pauli matrices, can easily be evaluated using the explicit forms for the spin operators given in Eqs.
		Rotation matrices act on spinors in much the same manner as the corresponding rotation operators act on state kets.
	farside.ph.utexas.edu /teaching/qm/lectures/node45.html (613 words)

	The Standard Error Models for Qubits
		To be able to think of the model as randomly applied Pauli matrices, it is crucial that the environment states labeling the different Pauli matrices be orthogonal.
		To determine the equivalent random Pauli operator error model, it is necessary to rewrite the total effect of the procedure using an environment labeled sum involving orthogonal environment states and Pauli operators.
		To obtain the standard depolarizing error model with equal probabilities for the Pauli matrices, it is necessary to strengthen the randomization procedure by applying a random member
	www.eskimo.com /~knill/qip/ecprhtml/node14.html (960 words)

	Springer Online Reference Works
		the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can be expanded.
		Quite generally, the Pauli matrices are used not only to describe isotopic space, but also in the formalism of the group of inner symmetries
		The formulas (1) and (2) make it possible to generalize the Pauli matrices covariantly to an arbitrary curved space:
	eom.springer.de /p/p071860.htm (351 words)

	CONTEXTS FOR SIMPLE SPINOR ALGEBRA
		Constraining the general complex matrices of M(2, C) to have vanishing trace reduces the number of free parameters from 8 to 6.
		Since a linear combination of matrices of trace 0 also has trace 0 by virtue of of the linarity of the trace functional, the Râ¶ image subspace is linear in Râ¸ (isomorphic to the Câ´ of M(2, C).
		The "special orthogonal group in three dimensions", SO(3) is one of the classical Lie groups, and it is therefore simultaneously: an analytic manifold, a group and a measure space, with a left invariant Haar measure, invariant under the left action of the group on itself.
	graham.main.nc.us /~bhammel/PHYS/spinor.html (5134 words)

	Non-local densities
		The density matrices in the spin and isospin spaces can be expressed as linear combinations of the unity and Pauli matrices.
		The densities are traces in spin and isospin indices of the following combinations of the density and the Pauli matrices:
		Since the p-h density matrix and the Pauli matrices are both hermitian, all the p-h densities are hermitian too,
	www.fuw.edu.pl /~dobaczew/nppair60w/node4.html (356 words)

	Pauli Spin Matrices in Domain Walls (Site not responding. Last check: 2007-10-12)
		The problem I think that I'm having is that I don't really understand the pauli matrices (my background in quantum is virtually nonexistent).
		All I know about the region is that electrons are injected into the domain wall with spins parallel to the wall (pointed in the y direction, as they travel in the x direction) and that the magnetization within the wall is as defined above.
		I don't understand how this defines axes for the pauli matrices and thus how he arrives at the matrix in the Hamiltonian at all.
	www.lns.cornell.edu /spr/2005-07/msg0070152.html (297 words)

	Unitary Matrices
		is the subgroup for which the determinant is 1 (unimodular matrices).
		Physicists prefer to work with the Pauli spin matrices instead of the quaternions.
		The Pauli matrices are just the Hermitian counterparts to i, j, and k:
	math.ucr.edu /home/baez/lie/node6.html (170 words)

	Pauli Spin Matrices in Domain Walls
		All I know about the region is that electrons are\ninjected into the domain wall with spins parallel to the wall (pointed\nin the y direction, as they travel in the x direction) and that the\nmagnetization within the wall is as defined above.
		I don\'t understand\nhow this defines axes for the pauli matrices and thus how he arrives at\nthe matrix in the Hamiltonian at all.
		Hope this is an\nandgt; appropriate place for the following post.\nandgt;\nandgt; I\'m trying to follow a paper about spin flip across domain walls.\n\nsnip\n\nandgt; I don\'t understand\nandgt; how this defines axes for the pauli matrices and thus how he arrives at\nandgt; the matrix in the Hamiltonian at all.
	www.physicsforums.com /showthread.php?t=81674 (961 words)

	Qubit dynamics: single qubit gates
		This equation can be derived using quite elementary reasoning that is partly based on classical electrodynamics, and partly on seeking a simplest possible evolutionary equation for the quantum state.
		Since Pauli matrices are 1,1-spinors, then so is this newly redefined magnetic dipole moment of a proton.
		With a little of extra care, especially when interference is going to be involved, we should be able to live with it.
	beige.ucs.indiana.edu /M743-talk-2/node6.html (1394 words)

	Pauli matrices and quaternions (Site not responding. Last check: 2007-10-12)
		Pauli matrices do not form a division algebra.
		multiplies the Pauli matrix and makes it identical to a quaternion.
		Call a duck a duck, and an i Pauli matrix a quaternion.
	world.std.com /~sweetser/quaternions/spr/pauli.html (173 words)

	HamiltonianGen (Site not responding. Last check: 2007-10-12)
		" used for generation of detection matrices, initial density operators, or any other spin function you wish to know from an input string.
		You are now alowed to perform basic functions on either the numbers or the matrices.
		NOTE:: all functions on the matrices are Element-By-Element...for example, the expresion 'exp(Ix)' will NOT return the matrix exponential, but is equivilient to exp(Ix(i,j)) for each element in the matrix.
	waugh.cchem.berkeley.edu /blochlib/single.php?classes=40 (332 words)

	Quaternion and Pauli matrix
		The Hamilton multiplication rules differ from the Pauli matrix rules only by a factor of i.
		It is possible to formulate special relativity with Hamilton quaternions having complex coefficients(called biquaternions) and indeed it was first done that way(Silberstein).
		It turns out that the formulae of general relativity are simpler with the Pauli quaternions.
	www.physicsforums.com /showthread.php?t=87233 (252 words)

	Interlude
		We shall demonstrate the equivalence with the Pauli matrix algebra explicitly in a companion paper[24], but here it suffices to note that the matrices
		Indeed, since we can represent our algebra by these matrices, it should now be obvious that we can indeed add together the various different geometric objects in the algebra - we just add the corresponding matrices.
		The algebra of 3-dimensional space, the Pauli algebra, is central to physics, and deserves further emphasis.
	www.mrao.cam.ac.uk /~clifford/introduction/intro/node8.html (608 words)

	[No title]
		(*****************************************************************************) ( :Title: Spin Matrices ) ( :Author: Jeff Olson ) ( :Summary: Provides definitions for the Pauli spin matrices and the Dirac gamma matrices.
		Spin matrices are Hermitian, unitary, traceless and have a determinant of -1." PauliMatrix::usage = "PauliMatrix[n] gives the 2x2 Pauli spin matrix for n = 1, 2, 3.
		PauliMatrix[0] gives the 2x2 identity matrix." PauliMatrices::usage = "PauliMatrices is a list of the 4 2x2 Pauli matrices." (****************************************************************************) (****************************************************************************) Begin["`Private`"] ClearAll[stindex, sindex] stindex = (0
	www.ph.utexas.edu /~jdolson/math/SpinMatrices.m (216 words)

	Higher-dimensional extensions of Pauli spin matrices
		The principle of basis set representation in terms of coordinate interchange matrices, of which the Pauli spin matrices are an example in two dimensions, are extended to three and four dimensions.
		The four-dimensional basis set of coordinate interchange matrices satisfies the usual conditions of completeness, but the three-dimensional basis set cannot be complete under any circumstances and an 'anticomplete' property is assigned to it.
		The coefficients of the basis set, when used to represent an arbitrary matrix, form a Hadamard transform of the cyclically interchanged arbitrary matrix.
	stacks.iop.org /0305-4470/12/1667 (219 words)

	pauli saukkonen - ResearchIndex document query (Site not responding. Last check: 2007-10-12)
		(10) where ~oe is the usual vector of the three Pauli matrices.
		n y ri for i =e where oe 2 is the second Pauli spin matrix.
		a oe a a =1 2 3 being the usual Pauli matrices, and the (anti)self-dual matrices (oe
	citeseer.ist.psu.edu /cis?q=Pauli+Saukkonen (480 words)

	Re: Pauli Spin Matrices in Domain Walls (Site not responding. Last check: 2007-10-12)
		snip > I don't understand > how this defines axes for the pauli matrices and thus how he arrives at > the matrix in the Hamiltonian at all.
		I've read a couple books where > they say the matrices are defined so as to diagonalize S^2 and Sz, but > I don't have any idea what that means.
		Next by thread: Re: Pauli Spin Matrices in Domain Walls
	www.lns.cornell.edu /spr/2005-07/msg0070159.html (176 words)

	Jones-Matrix Analysis with Pauli Matrices and Trace Operations: Applications to Ellipsometry (Site not responding. Last check: 2007-10-12)
		We present a new mathematical technique for analyzing optical systems that are described by Jones matrices.
		We represent individual Jones matrices as sum of Pauli matrices and the identity matrix, and intensities as traces of coherency matrices.
		This approach not only allows us to treat partial polarizations explicitly but also to take advantage of various identities to reduce operations necessary for system analysis to algebra, thereby simplifying both analytical derivations and numerical calculations.
	flux.aps.org /meetings/YR00/MAR00/abs/S4310006.html (148 words)

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