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Topic: Ladder operators

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	Quantum harmonic oscillator
		where x is the position operator, and p is the momentum operator (p = - iℏ ∂ /∂x).
		The "ladder operator" method, due to Paul Dirac, allows us to extract the energy eigenvalues without directly solving the differential equation.
		This task may be simplified by using the ladder operators to rewrite the anharmonic term as
	www.ebroadcast.com.au /lookup/encyclopedia/la/Ladder_operator.html (1756 words)

	Institute of Industrial Relations
		Extensive career ladders, in which significant pay increases are possible as an employee progresses through the grades associated with a particular job, can be a principle mechanism by which a fab tries to minimize costs associated with high levels of employee turnover.
		To construct ladders for operators and technicians (Figures 4-1, 4-2), the minimum pay level in the lowest grade was designated the entry wage, and the maximum pay level in the highest grade was designated the top wage in the ladder.
		In the third fab with a combined job ladder, operators were encouraged by the company to get on-site and off-site training (a technical AA degree), and a significant number of operators were promoted to the technician job category.
	socrates.berkeley.edu /~iir/worktech/csm-hr/chap4/index.html (2035 words)

	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.
		In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as
	en.wikipedia.org /wiki/Creation_and_annihilation_operators (1312 words)

	Phonon
		It is not a priori obvious that these excitations generated by the a operators are literally waves of lattice displacement, but one may convince oneself of this by calculating the position-position correlation function[?].
		The energy spectrum of this Hamiltonian is easily obtained by the method of ladder operators, similar to the quantum harmonic oscillator problem.
		This may be seen from the fact that the ladder operators contain sums over the position and momentum operators of every atom.
	www.ebroadcast.com.au /lookup/encyclopedia/ph/Phonon.html (2148 words)

	Ladder operators
		Two such operators are Q = / 0 1 \ and P = / 0 -i \ \ 1 0 / \ i 0 /.
		So this is the main duty of ladder operators: they take us from one state to another (in a nice way).
		Since operators are defined by what they do to states, we can expand out any arbitrary operator into coefficients times the ladder operators.
	www.lns.cornell.edu /spr/2000-03/msg0022704.html (1383 words)

	LECTURE 31--April 8, 2002 (Monday)
		We are talking about the algebra of angular momentum operators, deriving commutation relations, and relations among the orbital angular momentum squared and the ladder operators.
		Definitions of ladder operators and derivation of commutation relations of the ladder operators with L
		Ladder operators were defined and their matrix representations were obtained from the algebra.
	www.colorado.edu /physics/phys3220/phys3220_sp02/lecture31.html (361 words)

	GamePlan Software User Manual
		The purpose of a tennis challenge ladder is to allow all tennis players, from the beginning novice to the serious, experienced player, to meet and play other tennis enthusiasts at their own skill levels.
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	www.greencourtsoftware.com /gameplan/userman/ladderGuide.html (889 words)

	Universal Symmetry Groups
		The operator which determines the residue class of this least common multiple commutes with all the operators of his von Neumann algebra, and consequently their number determines the multiplicity of the von Neumann algebra, and in consequence, of the unitary modular group which is formed from its operators.
		It is found that satisfactory ladder operators exist, which reduce to Dulock's operators in the classical limit, and furthermore, as is typical of the cases which have been studied, the ladder operators reduce to the ones found by Infeld and Hull.
		The mechanism seems to be that the wave functions are expressed in a certain coordinate system, polar for example, where the ladder operators depend upon the (polar) action-angle variables.
	delta.cs.cinvestav.mx /~mcintosh/comun/symm/node14.html (1963 words)

	Ladder operators - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator.
		Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.
		A raising operator for N is an operator X for which c is real and positive and a lowering operator is one for which c is real and negative.
	en.wikipedia.org /wiki/Ladder_operators (237 words)

	[No title]
		Non-linear problems are characterized by (i) multiple conjunctive goals, (ii) a set of operators such that the application of a sequence of operators for achieving one goal makes the achievement of another goal impossible.
		So, the goals and operators in this example are such that the plan (a sequence of operators) for achieving one goal interacts negatively with the other goal(s).
		Preconditions are achieved by application of operators (plans) that result in the needed assertions becoming True.
	www.cc.gatech.edu /classes/cs3361_98_winter/planning.txt (659 words)

	Re: Ladder operators
		The 1s,2s,2p_z, etc. states are simultaneous eigenstates of the operators named Hamiltonian (time translation), and J_z (z-axis angle translation) with eigenvalues respectively named Energy, and angular momentum about z-axis.
		Those operators commute with one another, and their eigenvalues serve as labels for the usual basis of states.
		Two such operators are >> >> Q = / 0 1 \ and P = / 0 -i \ >> \ 1 0 / \ i 0 /.
	www.lns.cornell.edu /spr/2000-03/msg0022869.html (1370 words)

	Other Possible Systems and Symmetries
		The principal additional symmetry operations which it includes allow one to view the universe in a mirror expanding at the velocity of light, as well as from a rotated or uniformly moving coordinate frame.
		Angular momentum operators may be written as differential operators acting on a function space, so that there result differential equations which must be satisfied by the components of the putative vector or tensor operator.
		Their solution admits a slight generalization of the Runge vector of the hydrogen atom, or the tensor operator of the harmonic oscillator, but considerations of single valuedness seem to rule out all such operators with the exception of the ones already known.
	delta.cs.cinvestav.mx /~mcintosh/comun/symm/node13.html (1725 words)

	[No title]
		The good thing about ladder operators (or the Cartan canonical form as the incrowd says) is that they offer a simple picture of the structure of a group.
		You can take one of the generators, call it the operator for spin in the z-direction, and rewrite the others so they are "ladder operators".
		The good thing about ladder >operators (or the Cartan canonical form as the in crowd >says) is that they offer a simple picture of the structure >of a group.
	www.math.niu.edu /~rusin/known-math/00_incoming/dynkin (1593 words)

	Angular Momentum Operators
		In fact, the operator creating such a state from the ground state is a translation operator.
		We have written the wave function as a ket here to emphasize the parallels between this operation and some later ones, but it is simpler at this point to just work with the wave function as a function, so we will drop the ket bracket for now.
		We have established that the momentum operator is the generator of spatial translations (the generalization to three dimensions is trivial).
	galileo.phys.virginia.edu /classes/751.mf1i.fall02/AngularMomentum.htm (1755 words)

	Virtual International Players Ladder - Terms of Service (Site not responding. Last check: 2007-11-07)
		The VIP Ladder reserves the right to change the terms, conditions, and notices under which the VIP Ladder services are offered, including but not limited to the charges associated with the use of the VIP Ladder.
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	www.vipladder.com /vptos.shtml (4037 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Its Hamiltonian operator is H=P2/2m where m is the particle mass. (a) Show that H commutes with the momentum operator, P. (b) Verify that the momentum eigenkets, k>, are eigenkets of H and show that the eigenvalues are k2/2m.
		Similarly, the factorization method (ladder operators, or step-up and step-down operators) for angular momentum achieves a first order D.E. for the angular momentum wave functions. (a) Express the ladder operator L+ in the Schroedinger representaion.
		The commutator of the linear momentum operator P and its conjugate position operator Q is [P,Q] = -ih/2p. (a) Verify that the Schroedinger representation, P= (-ih/2p) d/dq and Q = q, obeys this commutation relation.
	www.poshusta.chem.wsu.edu /ch536/hw01.doc (1042 words)

	More on Quantum Harmonic Oscillator
		The operator a is not Hermitian since it and its adjoint a† are not equal.
		In deriving the form of a†, we have used the fact that the operators x and p, which represent observables, are Hermitian.
		In quantum field theory, a and a† are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.
	www.artilifes.com /quantum-harmonic-oscillator.htm (2215 words)

	Office of the Budget Analyst: 1.9 Vehicle Accidents
		On an annual basis, the total number, rate, cost and severity of vehicle accidents appear to be moving downward at the same time that the proportion of accidents attributable to Fire Department personnel is increasing.
		Aerial ladder trucks, on the other hand, represent less than 10 percent of the fleet, but account for 20 percent of the accidents and nearly one-half of the related settlement cost.
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	www.ci.sf.ca.us /site/budanalyst_page.asp?id=4855 (2099 words)

	Conserved Vector Current Hypothesis
		Because the positive and neutral pions are members of an isospin multiplet, it is possible to describe them mathematically with the ladder operators in isospin space.
		) is the absorption (emission) operator for the neutral pion, and
		Using the isospin ladder operator formalism, the vector part of the interaction for nuclear beta decay [9] becomes:
	pibeta.web.psi.ch /docs/publications/penny_diss/node5.html (403 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		* Theorem: The eigenvalues of a Hermitian operator must be Real. Theorem: The product of two Hermitian operators is Hermitian if and only if the two operators commute.
		Chemistry 371/14 Fall, 2004 IPFW Hence, the Hamiltonian is the linear Hermitian operator corresponding to the physical observable known as the Total Energy of the system.
		There is a fundamental incompatibility in the measurement of physical properties that are represented by NONCOMMUTING operators; a measurement of one causes an uncertainty in the other (i.e.
	www.ipfw.edu /chem/371/duchovic/Lecture_6.doc (1392 words)

	Virtual International Players Ladder - FAQ (Site not responding. Last check: 2007-11-07)
		The VIP Ladder rules webpage describes how points are transferred from the losing player to the winning player.
		Because of the structure of the VIP Ladder, it is imperative that matches be corrected as soon as possible.
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	www.vipladder.com /vpfaq.shtml (1978 words)

	Angular Momentum
		The operator associated with the coordinate transformation is
		This equation establishes the connection between the operator associated with a coordinate transformation and the angular momentum operator.
		It is, therefore, evident that the Hamiltonian is an invariant operator (i.e.,
	xbeams.chem.yale.edu /~batista/vvv/node16.html (416 words)

	Ruffing
		Ladder operator formalisms typically arise in factorization approaches to Schrödinger operators.
		So far, typical scenarios when ladder operators arise are briefly sketched.
		In recent contributions it has become apparent that methods involving ladder operators can also be used to deal with moment problems in context of special functions in analysis.
	www.camtp.uni-mb.si /chaos/2002/programme/Ruffing (268 words)

	Quantum harmonic oscillator - Wikipedia, the free encyclopedia
		The operator a is not Hermitian since it and its adjoint a
		These observable operators can be expressed as a linear combination of the ladder operators as
		As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant.
	en.wikipedia.org /wiki/Quantum_harmonic_oscillator (1761 words)

	Ladder Operators for a SHO
		You two guys seems to think very mathematically...where my error came from was my interpretation of the worded definititon of the two ladder operators.
		If the physical action of the operator X is to map the physical state represented by b> onto the physical state represented by c> then the only requirement is that each element of the ray ub> is mapped upon an element of the ray v c>
		You'd figure that the operator R that rotates a system over 360 degrees must be mapping every state upon itself.
	www.physicsforums.com /showthread.php?t=74975 (1267 words)

	Electric and Magnetic Fields due to Photon
		If my interpretation is right, the vector field operator, when sandwiched between two kets in Dirac's notation, should yield the expected value of the vector field for the system described by the ket, just as how one would expect ordinary operators in quantum mechanics to behave.
		Unfortunately, the bra and the ket states each contribute a single ladder operator and the field operator contributes one for each term.
		The expectation values of the E and B quantum field operators in any fixed number photon state is zero.
	www.physicsforums.com /showthread.php?t=123776 (2010 words)

	NIST: Methane Symmetry Operations - Triply Degenerate Vibra.
		In order to apply the full continuous pure rotation group to triply degenerate vibrational coordinates, it is convenient to express the latter in spherical polar form
		The vibrational angular momentum operators given in (eq.
		is identical to the form of the operators J
	physics.nist.gov /Pubs/Methane/chap13.html (583 words)

	NIST: Methane Symmetry Operations - Wave functions and Cosines
		It is in the determination of symmetry properties of functions of the Eulerian angles, and in particular in the question of how to apply sense-reversing point-group operations to these functions, that the principal differences arise in group-theoretical discussions of methane.
		The two other most commonly followed treatments, due to Jahn [5-8] and Moret-Bailly [9-11], respectively, will be discussed briefly in Section 12.
		It can be shown, by direct application of the differential operators, that symmetric top rotational basis functions
	physics.nist.gov /Pubs/Methane/chap07.html (450 words)

	Factorization Method for Solving a Partial Differential Equation: Spherical Harmonics
		into a pair of first order operators which are adjoints of each other.
		at the top and the bottom, and to use the ladder operators to generate any element in between.
		The fact that these operators are self-adjoint relative to the inner product, Eq.(5.75), implies that these eigenvectors (a.k.a spherical harmonics) are orthonormal:
	www.math.ohio-state.edu /~gerlach/math/BVtypset/node135.html (1349 words)

	LECTURE 30--April 5, 2002 (Friday) (Site not responding. Last check: 2007-11-07)
		Angular momentum operators in quantum mechanics are defined in terms of linear momentum operators, just as they are defined as vectors in classical mechanics, the cross product of radius vector with momentum vector.
		and the ladder operators, and uses them to show that when you apply the ladder operator to a state l m), then l is unchanged while m is increased or decreased by 1.
		Lastly, one can derive expressions for the ladder operators and for the angular momentum squared, in terms of spherical polar coordinates.
	www.colorado.edu /physics/phys3220/phys3220_sp02/lecture30.html (328 words)

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