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Topic: Annihilation and creation operators

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Creation operator

Annihilation operator

	Creation and annihilation operators - Wikipedia, the free encyclopedia
		A creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint 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.
		(f) is the creation operator, and a(f) is the annihilation operator.
	en.wikipedia.org /wiki/Creation_and_annihilation_operators (1331 words)

	Fock space - Wikipedia, the free encyclopedia
		is the operator which symmetrizes or antisymmetrizes the space, depending on whether the Hilbert space describes particles obeying bosonic (ν = +) or fermionic (ν = −) statistics respectively.
		Two operators of paramount importance are the annihilation and creation operators, which upon acting on a Fock state respectively remove and add a particle, in the ascribed quantum state.
		It is often convenient to work with states of the basis of H so that these operators remove and add exactly one particle in the given state.
	en.wikipedia.org /wiki/Fock_space (499 words)

	Rhett Herman-Chapter 2
		The Fourier transform of the operator $\underline{\phi}(x)$ is given by, \begin{equation} \underline{\phi}(x)=\int{d^{3}k\over \sqrt{(2\pi)^{3}2\omega_{k}}} \left[ \underline{a}(k)e^{-ik_{\alpha}x^{\alpha}}+ \underline{b}^{\dagger}(k)e^{ik_{\alpha}x^{\alpha}} \right], \label{phifourier}\end{equation} where $k^{\alpha}$ is the wavenumber of the basis functions $e^{\pm ik_{\alpha}x^{\alpha}}$, $\omega_{\vec{k}}\equiv k_{0}=\sqrt{\vec{k}\cdot\vec{k}+m^{2}}$, and the limits of integration extend from zero to infinity in each wavenumber integral.
		The action of the particle annihilation and creation operators on the vacuum state $0\rangle$ is given by \begin{equation} \underline{a}(\vec{k})0\rangle=0\hspace{1.0cm},\hspace{1.0cm} \underline{a}^{\dagger}(\vec{k})0\rangle=1(\vec{k})\rangle_{part}, \label{partbasisvac}\end{equation} where $1(\vec{k})\rangle_{part}$ is a state with one particle of wavenumber $\vec{k}$.
		The action of the antiparticle annihilation and creation operators on the vacuum state $0\rangle$ is given by \begin{equation} \underline{b}(\vec{k})0\rangle=0\hspace{1.0cm},\hspace{1.0cm} \underline{b}^{\dagger}(\vec{k})0\rangle=1(\vec{k})\rangle_{anti}\.
	www.runet.edu /~rherman/thesis/chap2.html (1123 words)

	Wick's Theorem for the Evaluation of Matrix Elements (Site not responding. Last check: 2007-10-12)
		[2.19], [2.20], and [2.21], an arbitrary string of annihilation and creation operators can be written as a linear combination of normal-ordered strings (most of which contain reduced numbers of operators) multiplied by Kronecker delta functions.
		The final combination where A is an annihilation operator and B is a creation operator is not zero, however, due to the anticommutation relations in Eq.
		The composite string of annihilation and creation operators may then be rewritten using Wick's theorem as an expansion of normal-ordered strings.
	zopyros.ccqc.uga.edu /lec_top/cc/html/node13.html (808 words)

	Normal-Ordered Second-Quantized Operators (Site not responding. Last check: 2007-10-12)
		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.
		Since the left- and right-hand states may be written simply as single annihilation and creation operators acting on the vacuum, the desired matrix element of
		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)

	Normal-Ordered Operators
		As discussed in appendix A many-electron operators are often conveniently expressed in terms of annihilation and creation operators -- that is, in second quantization.
		A normal-ordered product of annihilation and creation operators is one in which all annihilation operators lie to the right of all creation operators; any such product may be expressed in normal-ordered form using the anti-commutation relations given in appendix A.
		Thus, a zero result is obtained only when those annihilation operators corresponding to states unoccupied in the reference (which we will hereafter refer to as ``particle'' states) or creation operators corresponding to states occupied in the reference (which we will hereafter refer to as ``hole'' states) are applied to
	www.ua.es /cuantica/docencia/otros/cc/node14.html (300 words)

	[No title]
		There are annihilation and creation operators a and a* which push us up and down this ladder of states.
		Now let's invent a creation operation A* on structure types that reduces to the usual creation operator a* when we take their generating functions.
		The creation and annihilation operators are linear: A(F+G) = AF + AG A(F+G) = AF + A*G where the equals sign is secretly an isomorphism...
	math.ucr.edu /home/baez/twf_ascii/week185 (3446 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		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.
		The relationship between the true mode and quasi mode annihilation, creation operators is determined and shown to involve a Bogolubov transformation.
		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)

	Footnotes (Site not responding. Last check: 2007-10-12)
		Note that this is only true if the 46#46 operators are defined in terms of spin-orbital annihilation and creation operators and if the annihilation and creation spaces are disjoint.
		Note, however, that the reference wavefunction, which is sometimes referred to as the Fermi vacuum, may be written as a string of creation operators acting on the true vacuum.
		Authors sometimes refer to a normal-ordered string of annihilation and creation operators without indicating whether this is defined with respect to the vacuum or the reference wavefunction.
	www.ua.es /cuantica/docencia/otros/cc/footnode.html (239 words)

	creation myth - Hutchinson encyclopedia article about creation myth (Site not responding. Last check: 2007-10-12)
		All cultures have ancient stories of the creation of the Earth or its inhabitants.
		Often these involve the violent death of a primordial being from whose body everything then arises; the giant Ymir in Scandinavian mythology is an example.
		This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
	encyclopedia.farlex.com /creation+myth (116 words)

	Jamila Douari (Site not responding. Last check: 2007-10-12)
		Thus excited annihilation and creation operators are defined and an excited particles algebra is given as unified symmetry of bosonic, fermionic and anyonic algebras, depending on the kind of statistics of the space in consideration.
		If the dimension of space d is greater than or equal to 3, and the statistical parameter \nu=0,1, we refind the bosonic and fermionc algebras respectively.
		Then, the Heisenberg algebra is extended by a polynomial in some disorder operator denoted K_i and deformed in terms of the statistical parameter.
	www.math.ist.utl.pt /~rpicken/om/omxiii/jdouari.html (236 words)

	18. Selected CI (Site not responding. Last check: 2007-10-12)
		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)

	NormalOrdering.m (Site not responding. Last check: 2007-10-12)
		The operators p and q are Hermitian and their commutator [q,p] equals i hbar.
		The commutator of the creation and annihilation operators equals 1.
		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)

	[No title]
		[last update 2000-2-4] feynman diagrams and categorification the mathematical formalism of feynman diagrams can be understood as arising from the possibility of expressing physically important linear operators as algebraic combinations of so-called "annihilation and creation operators".
		the groupoids fibered over the groupoid of finite sets are included among kelly's objects, corresponding to operations on groupoids built out of basic tensor product (actually just the usual cartesian product of groupoids, but treated as just a tensor product not assumed to be cartesian) and 2-colimit operations (with distributivity of tensor product over 2-colimits).
		this means that we can, much more easily than before, create de-categorified things with interesting properties by first creating categorified things with analogous properties and then systematicly de-categorifying.
	math.ucr.edu /~jdolan/feynman (748 words)

	II. TRUNCATED CREATION & ANNIHILATION OPERATORS
		This is the analog of a proper pseudopolar decomposition for the creation annihilation pair.
		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)

	Re: domain of definition for boson annihilation and creation operators
		In article , nobody wrote: >On Mon, 11 Aug 2003 John Baez wrote: >>In article <200308090619.h796JUc15738@localhost.localdomain>, >>nobody wrote: >>>John Baez wrote: >>>>The unbounded operators of interest in quantum physics are >>>>almost always self-adjoint or at least normal.
		Any complex Hilbert space H has a Fock space K(H) on which there are self-adjoint field operators phi(z) for all vectors z in H, such that [phi(z), phi(z')] = -i Im on the domain of definition of the left-hand side.
		The annihilation and creation operators are then defined as a(z) = (phi(z) + i phi(iz))/sqrt(2) a*(z) = (phi(z) - i phi(iz))/sqrt(2) with their domains being the intersection of the domains of phi(z) and phi(iz).
	www.lns.cornell.edu /spr/2003-12/msg0057252.html (335 words)

	Detlef Lehmann's Research Report (Site not responding. Last check: 2007-10-12)
		The interacting part of the hamiltonian, which is quartic in the annihilation and creation operators, comes, because of conservation of momentum, with 3 independent momentum sums.
		The result is a hamiltonian, which is still quartic in the annihilation and creation operators, but which has only 2 independent momentum sums.
		It is obtained from the quartic BCS model by substituting the product of two annihilation or creation operators by a number, which is choosen to be the expectation value of these operators with respect to the quadratic hamiltonian, to be determined selfconsistently.
	www.math.tu-berlin.de /~lehmann/resrep.html (1799 words)

	Wick's Theorem (Site not responding. Last check: 2007-10-12)
		In short, this version of Wick's theorem says that given a string of annihilation and creation operators, this may be re-written as the normal-ordered product of the string, plus the normal-ordered product after all single-contractions among operator pairs, plus all double contractions, etc., plus all full contractions.
		If the string is already normal-ordered, which is the case if the right-hand operator is a particle index, then this contraction is zero, since the normal-ordering does nothing to the last term on the RHS.
		One implication of Wick's theorem is that the reference expectation value of an operator written as a string of annihilation and creation operators is zero unless all operators in the string may be eliminated via contractions:
	www.uam.es /docencia/quimcursos/Docs/Knowledge/Fundamental_Theory/cc/node15.html (380 words)

	Creation and Annihilation Operators (Site not responding. Last check: 2007-10-12)
		For this purpose we shall define what are called the Creation and Annihilation operators.
		(4) goes onto say that we cannot create two particles of the same colour in the same box.
		These operators, operate on configurations, which we have constructed in Section (2).
	physics.unipune.ernet.in /~nandy/module/node3.html (169 words)

	Creation and annihilation operators for SU(3) in an SO(6,2) model
		Creation and annihilation operators are defined which are Wigner operators (tensor shift operators) for SU(3).
		While the annihilation operators are simply boson operators, the creation operators are cubic polynomials in boson operators.
		Other Wigner operators for SU(3) can be constructed simply as products of the new creation and annihilation operators, or sums of such products.
	stacks.iop.org /0305-4470/17/2581 (302 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)

	Magnetic moments in IBA-I (Site not responding. Last check: 2007-10-12)
		In the IBA-I, the transition operators can be explicitly expressed in terms of the model operators.
		is a generator operator in O(3) and of all the subgroups compositions of U(6), and is therefore diagonal in any of these bases.
		Higher-order terms in the operator are needed to alter this result, e.g.
	www.nscl.msu.edu /~mertzime/Thesis/node43.html (164 words)

	Abstract (Site not responding. Last check: 2007-10-12)
		The annihilation and creation operators are introduced by means of a time-dependent Bogoliubov transformation.
		With the latter non-hermitian interaction hat-Hamiltonian, the general expression of the stochastic semi-free time-evolution generator is derived for a {\it non-stationary} Gaussian white quantum stochastic process.
		The correlation of the random force operators are also derived generally.
	www.icmp.lviv.ua /journal/zbirnyk.04/002/abstract.html (339 words)

	Quantum Physics
		Define the annihilation and creation operators for the simple harmonic oscillator in one dimension, using either notation A and A
		Familiarize yourself with the properties of angular momentum operators and the first few spherical harmonics, by reading the first parts of either Gasiorowicz chapter 11, p188-197 or Liboff chapter 9, p362-378, in conjunction with your lecture notes.
		are the raising and lowering operators for spin, and A > 0.
	venables.asu.edu /quant/prob013.html (664 words)

	Re: orthogonal polynomials and creation/annihilation operators
		> I'm interested in generalizations of the usual annihilation > and creation operators > > a, a* such that [a,a*] = 1 > > to Hamiltonians other than the harmonic oscillator Hamiltonian.
		> > 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/msg0057277.html (586 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		I emphasize that the operator for interaction with an external source must be an effective Bose operator in all cases.
		To accomplish this for parabose, parafermi and quon operators, I introduce parabose, parafermi and quon Grassmann numbers, respectively.
		I also discuss interactions of non-relativistic quons, quantization of quon fields with antiparticles, calculation of vacuum matrix elements of relativistic quon fields, demonstration of the TCP theorem, cluster decomposition, and Wick's theorem for relativistic quon fields, and the failure of local commutativity of observables for relativistic quon fields.
	www.ma.utexas.edu /mp_arc/a/93-7 (113 words)

	domain of definition for boson annihilation and creation operators (Site not responding. Last check: 2007-10-12)
		On Mon, 11 Aug 2003 John Baez wrote: >In article <200308090619.h796JUc15738@localhost.localdomain>, >nobody wrote: > >>John Baez wrote: >> >>>The unbounded operators of interest in quantum physics are >>>almost always self-adjoint or at least normal.
		Re: domain of definition for boson annihilation and creation operators
		Next by thread: Re: domain of definition for boson annihilation and creation operators
	www.lns.cornell.edu /spr/2003-12/msg0057036.html (242 words)

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