Relative momentum

This dissipative force is proportional to the relative velocity of the two beads and acts so as tc reduce their relative momentum, v is tire difference between the two velocities (Vy = v, — v ) and vP rjj) is a weight function that depends upon the distemce and disappears for interbead distances greater than unity (i.e. r ). 

The interaction of two particles was analyzed classically in Section 6.3.1. Hie resulting expression for the relative momentum can be written in vector notation as... 

X(b, g) Angle of deflection b Impact parameter pgb Angular momentum g Initial relative speed pg Relative momentum 9t(a) Phase shift l Angular momentum quantum number h ft l +1) Angular momentum a — pgffi = 2jr/k Wavenumber of the deBroglie wave ah Relative momentum... 

We have recently introduced the Wigner intracule (2), a two-electron phase-space distribution. The Wigner intracule, W ( , v), is related to the probability of finding two electrons separated by a distance u and moving with relative momentum v. This reduced function provides a means to interpret the complexity of the wavefunction without removing all of the explicit multi-body information contained therein, as is the case in the one-electron density. 

Electron correlation is inherently a multi-electron phenomenon, and we believe that the retention of explicit two-electron information in the Wigner intracule lends itself to its description (i). It has been well established that electron correlation is related to the inter-electronic distance, but it has also been suggested (4) that the relative momentum of two electrons should be considered which led us to suggest that the Hartree-Fock (HF) Wigner intracule contains information which can yield the electron correlation energy. The calculation of this correlation energy, like HF, formally scales as N. ... 

The function tr describes the relative translational motion of the photofragments. This motion can be described in the semi-classical approximation (except in the turning point region) by an oscillating wavefunction for which the number of oscillations increases with an increase of the relative momentum. An increase of the number of oscillations results in a decrease of the FC factor. Hence, the semiclassical behavior of the translational wavefunction makes a transition to a state with large momentum less favorable. Because of conservation of energy, the resulting state is characterized by a small vibrational quantum number. 

Here y is the component of the transition-dipole operator in the direction of the light s electric field vector E, Jj, M, and p are the energy, total angular momentum, its space-fixed projection, and the parity of the initial bound state k, v, j, and irij are the relative momentum, vibrational quantum number, rotational angular momentum, and its space-fixed projection for the scattering state. 

To describe how the required superposition state [Eq. (7.5)] can be constructed in the laboratory requires some introductory remarks. Note first that Eqs. (7.2) to (7.7) and the E, q, m 0) states are understood to be in the center-of-mass coordinate system and describe the relative translational motion as well as the internal state of A and B. In typical A-B scattering, separating out the center-of-mass motion comes about in a straightforward way. That is, let rA and rB denote the laboratory position of A and B and ikA, Mr denote their laboratory momenta. The relative momentum k, relative coordinate r, center-of-mass momentum K and position Rcm are defined as... 

The experimental setup shown in Fig. 12 has a relative momentum resolution of about IMeV/c. The channel is connected to the accelerator vacuum pipe without any partition and is shielded against the accelerator and the Earth magnetic fields. The channel ends with a vacuum chamber placed between the spectrometer magnet poles (B = 0.85 T). 

The components of these forces are of course determined by the magnitude of the instantaneous momenta of the species B and C at the moment of collision. F is proportional to the component of relative momentum of the pair B and C along the line of centers BC. A more complex case, in which we assume rough spheres, can be taken. In this case the components of momentum tangent to the spheres B and C at contact also 2ontribute to changes in rotation and vibration. 

The sytem is observed before and after the collision in time-dependent channel states ,(t)). The channel index i stands not only for the channel quantum numbers n, j, m, v but also for the relative momentum kj. The entrance channel is denoted by i = 0. The Schrodinger equation of motion for the channel i is, in atomic units. 

We must first find the density of final states, which we characterise in terms of the relative momentum /c . The permitted values of k in the normalisation box are given by (4.7). 

The asymptotic phase is given in terms of the relative momentum k( and relative position r, of each pair i by... 

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

In the limit where scattering is centered around the specular angle, the relative momentum transferred is related to the incident angle. 

Fig. 10. Predicted relative momentum transferred to a satellite surface by an incident particle as a function of the incident angle. Each curve corresponds to a different surface-to-particle atom mass ratio.

This is not consistent with the relative diffusion coefficient of noninteracting particles Dab = >a + because the tensor relationship between and D is broken upon reduction of the FPE to one dimension, that is, Dj Dab except for particles with /ka = Wb a = b- Nevertheless, (Jab from Eq. (2.34) is appropriate for relative momentum relaxation, whereas Dab appropriate for configurational relaxation at large separations. The problem is resolved by splitting configuration space into two regions at a separation R such that for I > / the diffusion coefficient is given by Dab- For R < R, the friction can... 

It is to be hoped that we may soon be able to give an account of the nature of the processes by which these changes of properties are effected but that task can only be entered upon when we have obtained exact determinations of the relative momentum of atoms in various compounds, the proportion of which to their masses determines their physical and chemical properties. 

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