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Topic: Conjugate momentum

	Calphysics Institute: Introduction to Zero-Point Energy
		A parallel uncertainty exists between measurements involving time and energy (and other so-called conjugate variables in quantum mechanics).
		However these waves do carry energy (and momentum), and each wave has a specific direction, frequency and polarization state.
		The fact that the orbital angular momentum is zero in the quantum ground state is mirrored in the SED orbiting-electron interpretation by random changes in the orbital plane (due to the zero-point fluctuations) yielding a time averaged zero net angular momentum.
	www.calphysics.org /zpe.html (3519 words)

	Quantum Gravity Concept Map - Operators
		In such a representation, the real part (ie., the amplitude) of the operator value is typically a function of momentum while the imaginary part (ie., the phase) is then a function of both the coordinate and momentum.
		One mathematical result of this use of conjugate variables is the "uncertainty relation": the range of values of one variable required to describe the object is inversely proportional to the range of values required of its conjugate.
		For sufficiently complicated collections of interacting quanta, the phase relationships (except in extrordinary "macroscopic quantum systems") vary rapidly, with the result that the overall collection possesses only a single degree of freedom, the amplitude (the phase of the collection is essentially random, and averages to zero).
	www.rwc.uc.edu /koehler/qg/ops.html (342 words)

	Why is 4-momentum usually given as a covector?
		One motivation is that, mechanically speaking, momentum is conjugate to position.
		Momentum is naturally dual to velocity since it usually makes sense to perform the matrix multiplication between momentum (as row vector) and velocity (as column vector) and gets a scalar with units of energy.
		Momentum as a scalar conserved quantity is naturally a covector.
	www.physicsforums.com /showthread.php?threadid=135193 (1513 words)

	[No title]
		By internal is meant that quantities L and p are both internal factors of the momentum, and by conjugate is meant that the product of wavelength L and momentum p has the dimensions of and is numerically equal to action.
		In this description, the momentum is an internal complex of the fundamental dimensions of the action p.
		As the parsecs and years pass, the action gradually fades from the momentum to the length which is, of course, the wavelength and which is aligned along the path in the cosmological dimension of distance.
	www.eskimo.com /~mikel137/power.htm (2202 words)

	Hamiltonian mechanics - Wikipedia, the free encyclopedia
		In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum.
		For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.
		Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton:
	en.wikipedia.org /wiki/Hamiltonian_mechanics (1321 words)

	Momentum - Wikipedia, the free encyclopedia
		Momentum is a conserved quantity, meaning that the total momentum of any closed system (one not affected by external forces) cannot be changed.
		The treatment of the momentum of a field is usually accomplished by considering the so-called energy-momentum tensor and the change in time of the Poynting vector integrated over some volume.
		In this case, using the principle of least coupling, the momentum of the particle should be
	en.wikipedia.org /wiki/Momentum (1542 words)

	[No title]
		In space alone, it loses momentum to distance, which is the action conjugate of momentum; and the total action remains constant.
		Momentum and energy cannot be acquired by the photon as it moves through space, so no Blue Shift of a kind equivalent to the Red Shift occurs in nature.
		Momentum is lost from the diffusion of momentum into its conjugate dimension of wavelength, which increases behind the front.
	www.eskimo.com /~mikel137/assumpti.htm (2554 words)

	No Title
		Thus the corresponding Noether charge, angular momentum, is conserved.
		-- precisely equal to l, the magnitude of the angular momentum.
		Kepler derived this law for the elliptical orbits caused by gravity; here we see that it is a very general consequence of gravity being a central force.
	www.emory.edu /PHYSICS/Faculty/Benson/361/notes/32/32.html (421 words)

	Re: Canonical Conjugate of Force
		Then the conjugate of velocity is (q/m^2) sqrt(m^2 + p^2/c^2)^3, since the derivative of v with respect to p is m^2/sqrt(m^2 + p^2/c^2)^3.
		The derivative of F with respect to q is -V''(q), so the conjugate momentum of -V'(q) is -p/V''(q).
		In general, if X is a function of q, the conjugate momentum of X is p/X'(q).
	www.lns.cornell.edu /spr/2000-02/msg0021984.html (210 words)

	[No title]
		Wavelength is conjugate to the momentum of the photon.
		Photon decay does not violate the action path integral, which merely states that the total action along the path is the integral of the momentum along the distance; and equal, also, to the integral of the energy along the time.
		Complex processes involve conservation of momentum, which appears to be about the same for atoms as it is for billiard balls, freight trains, rifle bullets and planets, stars and galaxies.
	www.eskimo.com /~mikel137/general.htm (4077 words)

	Barnes: Integrals of Motion (Site not responding. Last check: 2007-10-15)
		One is the magnitude of the angular momentum vector.
		The integral that disappears is the magnitude of the angular momentum vector.
		For example, from the Lagrangian of a particle moving in a spherical potential written in cartesian coordinates, it is not at all obvious that the z-component of angular momentum is conserved.
	www.physics.rutgers.edu /~barnesy/iom.html (359 words)

	Dr. James M. Feagin - Research Activities (Site not responding. Last check: 2007-10-15)
		In the case of double photoionization, electron-pair states with total angular momentum L = 1 are characterized by a matrix diabatic potential, whose only nonvanishing off-diagonal elements appear in the 1/R^2 contribution due to angular momentum couplings.
		The momentum representation of the Wannier state is naturally described by the momentum vectors conjugate to the coordinate vectors R and r, rather than the detector coordinates k1 and k2.
		In the case of photon excitation, the momentum carried off by the electron-pair is balanced by the momentum of the recoiling ion, i.e.
	chaos.fullerton.edu /jmfprojects.html (1180 words)

	Poisson bracket - tScholars.com (Site not responding. Last check: 2007-10-15)
		The corresponding Lie group is the group of symplectomorphisms of the symplectic manifold (also known as canonical transformations).
		Given a differentiable vector field X on the tangent bundle, let $P_X$ be its conjugate momentum.
		The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:
	tscholars.com /encyclopedia/Poisson_bracket (534 words)

	TIQM: 3.3 The Transactional Model and Relativistic Quantum Mechanics
		Landau and Peierls (1931) have argued that these relativistic limits on determinations of position and momentum irretrievably compromise the utility of these dynamical variables for measurement in the sense of non-relativistic quantum mechanics.
		Neither the position nor the momentum of a particle can, even in principle, be determined to arbitrary accuracy in a finite time interval nor can either be considered to have a particular value at a particular time.
		This would seem, in effect, to invalidate Born's statistical interpretation of quantum mechanics (CI2), in that the description of the state vector as a mathematical representation of the probability of finding a definite value of a particular observable as a result of a measurement made at a given instant is untenable.
	mist.npl.washington.edu /npl/int_rep/tiqm/TI_33.html (3611 words)

	Unifying the Representation of Spin and Angular Momentum
		For the first conjugate, the first term will have the correct sign after a 2 pi journey, but the scalar, third and forth terms will point the opposite way.
		Under an exchange, the identity and conjugate commutators form a distinct group from the commutators formed with the first and second conjugates.
		The behavior in a commutator under exchange of the identity automorphism and the anti-automorphic conjugate are identical.
	world.std.com /~sweetser/quaternions/quantum/spin/spin.html (941 words)

	Momentum Operator -- from Eric Weisstein's World of Physics
		As a trial solution for the momentum operator, use the classical result
		Using the time-dependent Schrödinger equation and its complex conjugate
		Conclude that the momentum operator is given by
	scienceworld.wolfram.com /physics/MomentumOperator.html (49 words)

	The Equations
		In the Newtonian limit, the momentum density, pressure, and shear stress are negligible compared with the density.
		The conjugate momentum has the property that it is simply the spatial part of the 4-momentum with lower indices, i.e., for a particle of mass m,
		The last piece needed for inclusion of the massive neutrinos is the means by which we go from the phase space distribution to the density, velocity, and shear stress fluctuations appearing in equations (2).
	www.astro.princeton.edu /~bode/SC95/eqn_mn.html (641 words)

	Quantum Dynamics of Morphing Psy Trance Formations > GaianXaos
		The momentum attribute is associated with the spatial sine waveform family.
		An example of such a conjugate relationship is found between the sine waveform family and the impulse waveform family.
		The result of this relation is that we can know either position or momentum with perfect accuracy; however, since position and momentum are conjugate attributes, we can not define both attributes at the same time with perfect accuracy.
	www.gaianxaos.com /dynamics_of_quantum_psy-trance.htm (14151 words)

	Noether (Site not responding. Last check: 2007-10-15)
		For a particle moving in a central force field theta is an ignorable coordinate so that the conjugate momentum p(theta) is a constant.
		We then derive the correct laws of physics from the assumption that they must be the same for all observers in inertial reference frames.
		Similarly, conservation of momentum comes from symmetries in spatial translation or rotation [eqn.(23)].These should be present for even more complicated systems, e.g.
	www.mbhs.edu /~halperin/Noether.htm (494 words)

	Formulation of Hamilton's equations for the problem
		The canonically conjugate momentum coordinates are related to, but not the same as, the actual momentum (defined as
		The relation between canonical momentum coordinates and momentum coordinates depends on the particular coordinate system chosen.
		Thus the computed momentum components are velocity components.
	www.physics.uq.edu.au /people/jones/ph362/cphys/node16.html (488 words)

	Degrees of freedom (physics and chemistry) - Wikipedia, the free encyclopedia
		This is because for a quantum micro-state, defining a precise value of both the position and momentum of a particle violates the Heisenberg uncertainty principle.
		In 3D, there are 6 degrees of freedom associated to the movement of a mechanical particle, 3 for its position, and 3 for its momentum.
		For a roughly dumbbell-shaped hydrogen molecule, described by two mechanical particles linked by a spring, 6 such independent directions (or modes) of movement would be translation (hurtling through space, 3 modes), rotation (twirling, 2 modes), and vibration (the two dumbbell "balls" bouncing together and apart, 1 mode).
	en.wikipedia.org /wiki/Degrees_of_freedom_(physics_and_chemistry) (1397 words)

	No Title
		Classically, a Lagrangian determines all the conjugate momenta as well as the Hamiltonian, which gives Hamilton's equations (which are equivalent to the Euler-Lagrange equations).
		We quantize the system by 1) making all observables operators; 2) imposing canonical commutation relations between operators for the particles' coordinates and their conjugate momenta; and 3) imposing the Heisenberg equations of motion on all operators.
		Here the coordinates and their conjugate momenta -- and hence the commutation relations that must be imposed -- are clear.
	www.emory.edu /PHYSICS/Faculty/Benson/380-96/notes/29/29.html (581 words)

	Quantum Field Theory (Stanford Encyclopedia of Philosophy)
		A generalized notion of momentum (the conjugate or canonical momentum) is defined by setting p = ∂L/∂q̇, where L is the Lagrange function L = T − V (T is the kinetic energy and V the potential) and q̇ ≡ dq/dt.
		The field φ and its conjugate field π are the direct analogues of the canonical coordinate q and the generalized (canonical or conjugate) momentum p in classical mechanics of point particles.
		The first one is that quantities like total charge, total energy or total momentum of a field are unobservable since their measurement would have to take place in the whole universe.
	plato.stanford.edu /entries/quantum-field-theory (16460 words)

	This Week in Computational Astrophysics » Recent Techniques and Applications (Site not responding. Last check: 2007-10-15)
		allowing the momentum of one fluid to be a linear combination of the velocities of all fluids).
		Maximum use is made of mass, energy, and linear and angular momentum conservation to specify the equations of motion.
		Also used extensively are insights gleaned from a convective variational action principle, key being the distinction between each velocity and its canonically conjugate momentum.
	www.cita.utoronto.ca /~ljdursi/thisweekcomp/entry_102.php (396 words)

	Time in Quantum Mechanics
		This is analogous to the case of a single particle where the total momentum P(q,p) coincides with the momentum p.
		The momentum of the particle is a linear function of t and furnishes a time-variable.
		Similarly, since the wavefunctions of an angle variable must obey a periodic boundary condition, the eigenvalues of the corresponding angular momentum operator are discrete.
	www.phys.uu.nl /~wwwgrnsl/publications/time.html (4019 words)

	Quantum Gravity (Stanford Encyclopedia of Philosophy)
		For example, in quantum mechanics, the position of an electron may be specified with arbitrarily high accuracy only at the cost of a loss of specificity in the description of its momentum, hence its velocity.
		Quantization proceeds by treating the configuration and momentum variables as operators on a quantum state space (a Hilbert space) obeying certain commutation relations analogous to the classical Poisson-bracket relations, which effectively encode the quantum fuzziness associated with Heisenberg's uncertainty principle.
		Although advocates of the canonical approach often accuse string theorists of relying too heavily on classical background spacetime, the canonical approach does something which is arguably quite similar, in that one begins with a theory that conceives time-evolution in terms of evolving some data given on a spacelike surface, and then quantizing the theory.
	plato.stanford.edu /entries/quantum-gravity (6049 words)

	Canonical coordinates
		) with the x 's or q 's denoting the coordinates on the underlying manifold and the p 's denoting the conjugate momentum, which are 1-forms in the cotangent bundle at point q in the manifold.
		This article attempts to provide a rigorous definition of the looser, simpler idea presented in the article canonical conjugate variables.
		A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the canonical one form to be written in the form
	www.danceage.com /biography/sdmc_Conjugate_momentum (354 words)

	Markov Chain Monte Carlo - Hamiltonian Method
		is introduced, which represents the parameter's conjugate momentum variable [4].
		The momentum aspects of the extended pdf, exp(-H), are marginalised out because they are independent of the x dependence.
		For each trajectory, the momentum is drawn from the assumed Gaussian momentum distribution (vertical jumps), which is followed by several steps along a trajectory of constant Hamiltonian value (circular paths).
	public.lanl.gov /kmh/publications/samo01a.html (1456 words)

	Zeno and Uncertainty (Site not responding. Last check: 2007-10-15)
		The problem involves not just velocity but momentum, i.e., the persistence of a definite state of motion over a finite period of time.
		Conversely, in order for the entire inertial motion of an object to be definitely realized in the present instant, so that it's momentum is fully determinate, it cannot have any definite spatial position at all.
		Position and momentum are conjugate observables in this sense, and so they obey the Heisenberg uncertainty relation, which says the product of the indeterminacies of position and of momentum cannot be less than a certain irreducible value.
	www.mathpages.com /home/kmath158.htm (470 words)

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