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Topic: Annihilation operator

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Creation operator

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	Creation and annihilation operators - Wikipedia, the free encyclopedia
		An annihilation operator is the operator in that lowers the number of particles in a given state by one.
		A creation operator is an operator that increases the number of particles in a given state by one, and it is the Hermitian conjugate of the annihilation operator.
		The mathematics behind the creation and the annihilation operators is identical as the formulae for ladder operators that appear in the quantum harmonic oscillator.
	en.wikipedia.org /wiki/Creation_and_annihilation_operators (1335 words)

	Raising operator - Wikipedia, the free encyclopedia
		In linear algebra (and its application to quantum mechanics), a raising (or lowering) operator is an operator that increases (or decreases) the eigenvalue of another operator.
		In quantum mechanics, the raising operator is sometimes called the "creation" operator, and the loweing operator the "annihilation" operator; collectively they are sometimes known as "ladder operators." Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
		Suppose that two operators X and N have the commutation relation [N,X]=cX for some scalar c.
	en.wikipedia.org /wiki/Raising_operator (115 words)

	The Fermi Vacuum and the Particle-Hole Formalism (Site not responding. Last check: 2007-10-09)
		This nomenclature is based upon the determinant produced when annihilation-creation operator strings act on the Fermi vacuum.
		Therefore, a string of second-quantized operators is normal ordered relative to the Fermi vacuum if all q-annihilation operators lie to the right of all q-creation operators.
		Whereas before, the only nonzero pairwise contraction required the annihilation operator to be to the left of the creation operator (cf.
	zopyros.ccqc.uga.edu /lec_top/cc/html/node14.html (343 words)

	[No title]
		Before giving the definition of the correlation functions in terms of the field operators (1.1), let us discuss how the correlation functions come naturally from the calculation of the mean values of operators.
		(the operator is diagonal in the momentum representation).
		operators are conveniently expressed in terms of the field operators (1.1).
	www.science.unitn.it /~astra/PhD/node7.html (363 words)

	Read about Quantum mechanics at WorldVillage Encyclopedia. Research Quantum mechanics and learn about Quantum mechanics ... (Site not responding. Last check: 2007-10-09)
		operator corresponding to the total energy of the system, generates time evolution.
		During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states.
		John von Neumann formulated the rigorous mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces, as described in his likewise famous 1932 textbook.
	encyclopedia.worldvillage.com /s/b/Quantum_mechanics (3412 words)

	Annihilation Article, Annihilation Information (Site not responding. Last check: 2007-10-09)
		Annihilation occurs when a particle collides with an antiparticle.
		When a proton annihilates an antiproton they produce gamma rays and a swarm of secondaryparticles, like pairs of top-anti-top quarks.
		Annihilation is the most powerful reaction in the universe, and is one of the few ways matter can be converted into energy.Scientist think that annihilation was the reason there is empty space in the universe.
	www.anoca.org /energy/rays/annihilation.html (290 words)

	Brian Hoffman's Quantum Chemistry Notes (Site not responding. Last check: 2007-10-09)
		In the second quantization formalism, the wave function is written as a product of creation and annihilation operators which act upon a vacuum reference state.
		In order to be physically realistic, a creation operator which acts upon a wave function in which the spin orbital is already occupied must yield a zero result.
		Moreover, an annihilation operator which destroys an electron which does not exist must produce a zero result.
	www.cat.cc.md.us /~bhoffm30/instructional/ci/node5.html (509 words)

	Annihilation
		It means that, contrary to a popular belief, the annihilation produces more than just photons.
		When an electron annihilates a positron (anti-electron) the process yields pure energy in a form of gamma rays, see Electron-positron annihilation.
		When a proton annihilates an antiproton they produce gamma rays and a swarm of secondary particles, like pairs of top-anti-top quarks.
	www.brainyencyclopedia.com /encyclopedia/a/an/annihilation.html (183 words)

	18. Selected CI
		Once the default/user-input reference configurations have been determined additional reference functions may be generated by applying multiple sets of creation-annihilation operators, permitting for instance, the ready specification of complete or restricted active spaces.
		Finally, a uniform level of excitation from the current set of configurations into all orbitals may be applied, enabling, for instance, the simple creation of single or single+double excitation spaces from an MCSCF reference.
		If orbitals 3 and 4 were initially doubly occupied, and orbitals 5 and 6 initially unoccupied, then the application of this set of operators four times in succession is sufficient to generate the four electron in four orbital complete active space.
	www.emsl.pnl.gov /docs/nwchem/doc/user/node20.html (1449 words)

	[No title]
		The basic idea was to leave the position operator alone, but replace the derivative in the momentum operator by a q-derivative.
		There are annihilation and creation operators a and a* which push us up and down this ladder of states.
		In these terms, the creation operator a* is just multiplication by x, while the annihilation operator a is differentiation.
	math.ucr.edu /home/baez/twf_ascii/week185 (3446 words)

	Algebraic Construction of the CCD Equations
		Now we must use the generalized Wick's theorem to re-write the operator strings in this half of the commutator.
		A simple rule that helps us to evaluate the sign quickly is that the term is negative if there is an odd number of operators between the contraction pair, and positive if there is an even number.
		operators, and the single non-contracted term will cancel with a similar term in the left half of the commutator (this is because all operators in this electron-conserving theory have even numbers of operators).
	www.ua.es /cuantica/docencia/otros/cc/node17.html (1009 words)

	H
		Those authors expressed an awareness that the annihilation operator method they used, which is a valuable one and clearly useful here, could be generalized but gave no sense this could include nonlinear equations.
		The annihilation operator technique employed in Reference 4 uses linear constant-coefficient differential operators of first and second order and the integrands can be expressed as a sum of separable polynomial functions.
		Most of the generalizations of the annihilation operator technique in this section can be further generalized and are more advanced than what is needed to investigate the Rayleigh scattering problem.
	www.science.gmu.edu /~rgomez/GuestLecture1.htm (3937 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.
		A Heisenberg picture approach to quantum scattering theory is applied, in which the input and output operators that are related via the scattering operator are linked to quantum optical measurements described via multitime quantum correlation functions.
		Analytic properties are also examined and it is found that the annihilation, creation operators times the square root of the angular frequency are analytic functions of the variables specifying the modes.
	www.physics.uq.edu.au /qonews/011199.txt (2836 words)

	II. TRUNCATED CREATION & ANNIHILATION OPERATORS
		The technical device that is met in the usual realizations by unbounded operators on an infinite dimensional Hilbert space, but not met in the finite dimensional cases is that the domain of definition Z(n) is clearly not dense in Hilb(n).
		The existence of a genuine "phase operator", the lost (in QM) conjugate to the number operator, that would be used to construct a time operator for the harmonic oscillator is recaptured.
		The fact that the various decompositions (2.15), (2.17), (2.18) and (2.21) exist may be of physical significance in distinguishing a transient process passing through a spacetime patch from a captured or cyclic process that is confined to the patch.
	graham.main.nc.us /~bhammel/FCCR/II.html (2141 words)

	Second Quantization
		Here we will only summarize the anticommutation relations between creation and annihilation operators, and then proceed to express the Hamiltonian in second quantized form for spatial orbitals, rather than for spin orbitals.
		Now we use the anticommutation relation between a creation and an annihilation operator, which is given by (5.3).
		This is the Hamiltonian in terms of replacement operators.
	vergil.chemistry.gatech.edu /notes/ci/node22.html (504 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Operators are defined as objects for which OperatorQ[x] is true.
		Operators are ordered based on their priority which must be specified by Priority[x].
		The order of operators that is considered normal depends on the priority of the operator.
	www.mit.edu /~mforbes/projects/NormalOrder/NormalOrder.m (778 words)

	Exponential Operator Algebra (Site not responding. Last check: 2007-10-09)
		As a preliminary task, we shall establish some operator identities that prove useful both in understanding the eigenstates of a and in later work.
		This identity is only true for operators A, B whose commutator c is a number.
		Recall that the annihilation operator a applied to a simple harmonic oscillator energy eigenstate moves down one step of the ladder, and since the energy eigenvalues cannot be negative, a annihilates the lowest state,
	landau1.phys.virginia.edu /classes/751.mf1i.fall02/CoherentStates.htm (1110 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		The second ones, which are eigenstates of the C{sub{lambda}}-extended oscillator annihilation operator, extend to higher{lambda} values the paraboson coherent states, to which they reduce for{lambda}=2.
		They give rise to Bargmann representations of the latter, wherein the generators of the C{sub{lambda}}-extended oscillator algebra are realized as differential-operator-valued matrices (or differential operators).
		The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20421021 (264 words)

	Normal-Ordered Second-Quantized Operators (Site not responding. Last check: 2007-10-09)
		21, §4), a normal-ordered string of second-quantization operators is one in which we find ``all annihilation operators standing to the right of all creation operators.'' Normal ordering of such strings provides a bookkeeping system by which the nonzero matrix elements of second-quantized operators may be more easily identified.
		Note also that all of the operator strings on the right-hand side of the final equality are normal ordered by Merzbacher's definition.
		By rearranging a given string of annihilation and creation operators into a normal-ordered form, matrix elements of such operators between determinantal wavefunctions may be evaluated in a relatively algorithmic manner.
	zopyros.ccqc.uga.edu /lec_top/cc/html/node12.html (352 words)

	Re: notation for creation/annihilation operators
		> jeffery_winkler@hotmail.com (Jeffery) wrote: > > I always thought "a dagger" is the creation operator, and "a" is the > > annihilation operator, such as what you see here.
		> > > > http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node14.html > > > > But then other times I have seen "a dagger" to mean the annihilation > > operator, and "a" to mean the creation operator, such as here.
		As far as I could see, they were > both using the notation > a for annihilation and a^\dagger for creation.
	www.lns.cornell.edu /spr/2003-01/msg0048134.html (250 words)

	Re: orthogonal polynomials and creation/annihilation operators
		Moreover, the functions wf_0, wf_1, wf_2 tend to be eigenfunctions of nice 2nd-order differential operators.
		Now, we can write the harmonic oscillator Hamiltonian as H = aa* where the creation operator a* maps each Hermite function to a multiple of the next one, and the annihilation operator a maps each one to a multiple of the previous one.
		Even better, the annihilation and creation operators satisfy a cool commutation relation: [a,a*] = 1 In some sense this formula is all you need to know to recover the whole theory of the harmonic oscillator.
	www.lns.cornell.edu /spr/2003-12/msg0057391.html (621 words)

	Re: notation for creation/annihilation operators (Site not responding. Last check: 2007-10-09)
		jeffery_winkler@hotmail.com (Jeffery) wrote: > I always thought "a dagger" is the creation operator, and "a" is the > annihilation operator, such as what you see here.
		> > http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node14.html > > But then other times I have seen "a dagger" to mean the annihilation > operator, and "a" to mean the creation operator, such as here.
		As far as I could see, they were both using the notation a for annihilation and a^\dagger for creation.
	www.lns.cornell.edu /spr/2003-01/msg0048121.html (129 words)

	Normal Ordering Of Quantum Operators -- from Mathematica Information Center
		NormalOrdering.m normal orders a polynomial in creation and annihilation operators using the commutation relation.
		In normal ordering the annihilation operators are to the right of the creation operators.
		This program can normal order an expression of creation (denoted by 'c') and annihilation (denoted by 'a') operators.
	library.wolfram.com /infocenter/MathSource/625 (100 words)

	Quantum Field Theory
		It is the transition probability expressed in an expansion as the result of the iteration procedure on the transition operator, which transforms the system from an initial state (at time negative infinity) to a final state (at time positive infinity) as shown in the formula below.
		where the symbol N is the normal-order operator, which shifts all the creation operators to the left (to avoid infinite vacuum energy), while T is the time-order operator, which re-arranges the fields so that the one associated with later time is on the left (to take care of the integration limits in Eq.(11)).
		It represents the process of the annihilation of a pair of nucleon and anti-nucleon and the creation of a pion.
	www.zamandayolculuk.com /cetinbal/quantumfieldtheory.htm (3167 words)

	APP B31/9, p. 2075 abstract (Site not responding. Last check: 2007-10-09)
		, where a is annihilation operator and \psi is a constant, and (ii) the normally ordered characteristic function, \chi
		is creation operator and k is a positive real constant.
		Examples of generation of phase-coherent and amplitude-coherent fields are given.
	th-www.if.uj.edu.pl /acta/vol31/abs/v31p2075.htm (157 words)

	Harmonic oscillator (Site not responding. Last check: 2007-10-09)
		These have a fundamental importance, as bosons like photons and phonons are harmonic oscillator eigenstates, and the operator approach is very important in describing boson ``fields''.
		an annihilation operator, as it gives a state with one quantum of energy less, and
		a creation operator, as this gives a state with one quantum of energy more.
	www.astro.cf.ac.uk /undergrad/module/PX4114/aqm/node17.html (326 words)

	7.5 Bound Particles
		When evaluating commutators, it is helpful to include a wavefunction as a reminder that the operators operate on functions to the right.
		We will proceed to use operator algebra to find the energy spectrum of the system.
		Starting with the ground state wavefunction, we can construct all of the others using the raising operator.
	webpages.ursinus.edu /lriley/courses/p212/lectures/node38.html (899 words)

	Explicit Remainder Estimates
		The model sequence (4.1) has another undisputable advantage: There is a systematic way of constructing a sequence transformation which is exact for this model sequence.
		This annihilation operator approach, which was introduced in [23, section 3.2,] in connection with the sequence transformations discussed in this section, was recently used by Brezinski and Redivo Zaglia [43,44] and by Brezinski and Matos [45] also in the case of other sequence transformations
		A fairly complete survey of the older literature on this subject can be found in books by Nielsen [48] and Nörlund [49].
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/report/node4.html (1073 words)

	NormalOrdering.m (Site not responding. Last check: 2007-10-09)
		The operators p and q are Hermitian and their commutator [q,p] equals i hbar.
		The expression can be any polynomial in creation (denoted by c) and annihilation (denoted by a) operators.
		Apart from these two operators the expression may also conatin coefficients, plus and minus signs, powers, brackets and double stars, that is the built-in operator for non-commutative multiplication.
	home.datacomm.ch /atair/physics/no (192 words)

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