Precessing orbit

The vector L is so strongly coupled to the electrostatic field and the consequent frequency of precession about the intemuclear axis is so high that the magnitude of L is not defined in other words L is not a good quantum number. Only the component H of the orbital angular momentum along the intemuclear axis is defined, where the quantum number A can take the values... 

A qualitative explanation of these abnormally large diamagnetic susceptibilities as arising from the Larmor precession of electrons in orbits including many nuclei3 has come to be generally accepted. With the aid of simple assumptions, I have now developed this idea into an approximate quantitative treatment, described below. 

The high sensitivity of atomic Sagnac interferometers to rotation rates will enable HYPER to measure the modulation of the precession due to the Lense-Thirring effect while the satellite orbits around the Earth. In a Sun-synchronous, circular orbit at 700 km altitude, HYPER will detect how the direction of the Earth s drag varies over the course of the near-polar orbit as a function of the latitudinal position 9 ... 

Figure 7.1 Precession of the orbital angular momentum vector L and the spin angular momentum vector S about their vector sum J.

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

The precession observed in a plane of an electron orbit of an atom when that atom hes within a uniform magnetic field. The plane will precess about the direction of the field such that a cone is traced out having an axis normal to the direction of the field. The frequency of this precession is usually symbolized by j l and is equal to eBI (Aimtc) where e is the electron charge, B is the field strength, m is the mass, and c is the velocity of the electron. Larmor precession is also important in NMR studies of enzymes. 

The interaction of an electron, as a charged particle, with an applied magnetic field causes a precession of its orbital motion which acts to cancel the field so that /, and hence x in equation (52), is negative. The effect is small but omnipresent. The spin and angular momenta of an electron... 

Since the s orbitals do not have orbital magnetic moment, g — ge — 2.0023. A is an eneigy, which in SI units is expressed in joule. Sometimes it is convenient to have it expressed in frequency units, i.e. in hertz or in radians per second, and then we have to divide A by h or by h (ft = h/2n). In magnetic resonance there often is a factor 2jt which complicates life. The frequency which we refer to is the angular frequency or the Larmor precession frequency. Such frequency co is... 

Figure 19. It is assumed here that the concept of the ellipsoids of observation apply to all fields moving with the velocity of light, such as electric or gravitational fields. Thus, precession of the perihelion of Mercury (the rotation of its elliptic orbit) can be explained by the asymmetry of the gravitational forces as the planet advances toward (resp. retreats from) the sun.

The frequency of cyclotron motion, that is, how rapidly an ion precesses about the orbit, is m/z dependent. Applying Newton s Second Law... 

Eccentricity of the Earth s orbit, which varies from almost circular to strongly elliptical with a periodicity of about 95,800 years (these changes modulate precession). 

Rubincam D. P. (2004). Black body temperature, orbital elements, the Milankovitch precession index, and the Seversmith psychroterms. Theoretical and Applied Climatology, 79(1-2), 111-131. 

The Jeffery orbits are deterministic, and the particles will precess indefinitely in the flow. The following effects can perturb a particle and deflect it from an orbit and send it into a new one ... 

Figure 9.14. The hourly scan pattern of COBE, WMAP and Planck from left to right. In each panel the anti-solar direction is in the center, and plus signs denote the North and South ecliptic poles. An equiangular azimuthal projection is used. COBE scanned this area in one orbit of about 103 minutes, WMAP scans its pattern every hour, while Planck spends several hours integrating on one scan circle of radius 70° radius shown offset from anti-Sun by a 15° precession angle.

However, a special relativistic, or kinematical, correction, is necessary it is the Thomas precession. The electron orbiting around the nucleus with speed v (where v is a reasonably large fraction of the speed of light c) causes the period of one full rotation around the nucleus to be T in the fast-moving electron rest frame, but a longer time T (time dilatation) in the stationary rest frame of nucleus [see Eq. (2.13.11)] ... 

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