Momentum vector

The fundamental method [22,24] represents a multidimensional nuclear wavepacket by a multivariate Gaussian with time-dependent width niaUix, A center position vector, R, momentum vector, p and phase, y,... 

To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

Figure 1.5 Direction of the angular momentum vector p for an electron in an orbit...

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

Each electron in an atom has two possible kinds of angular momenta, one due to its orbital motion and the other to its spin motion. The magnitude of the orbital angular momentum vector for a single electron is given, as in Equation (1.44), by... 

There is appreciable coupling between the resultant orbital and resultant spin momenta. This is referred to as LS coupling and is due to spin-orbit interaction. This interaction is caused by the positive charge Ze on the nucleus and is proportional to Z". The coupling between L and S gives the total angular momentum vector J. 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

The directing jet is supplied at a right angle to the mam stream axis with an initial velocity of I os from the nozzle with an inner diameter (d, -) located at the distance (Iq) from the plane of main stream supply and at the distance Yq from its geometrical axis. The momentum vector component along the Y axis remains constant and equal to the initial momentum (Fig. 7.57) ... 

In the case of a nonisothermal directing jet, the above assumptions are true, except that the momentum vector component along the Y axis changes due to the buoyancy force ... 

The state of any particle at any instant is given by its position vector q and its linear momentum vector p, and we say that the state of a particle can be described by giving its location in phase space. For a system of N atoms, this space has 6iV dimensions three components of p and the three components of q for each atom. If we use the symbol F to denote a particular point in this six-dimensional phase space (just as we would use the vector r to denote a point in three-dimensional coordinate space) then the value of a particular property A (such as the mutual potential energy, the pressure and so on) will be a function of r and is often written as A(F). As the system evolves in time then F will change and so will A(F). 

Focus attention on the top left-hand box. As the position and momentum of each particle in this box change, then so do the position and momentum vectors of all the image particles in all the other boxes. In particular, if the grey particle in the top left-hand box leaves its box during a dynamics run, then it is replaced by the movement of its image particle from a corresponding neighbouring box. This is a well-known technique in statistical mechanics, which acts to preserve the density. 

We now introduce the principle of microscopic reversibility. This states that the transition states for any pathway for an elementary reaction in forward and reverse directions are related as mirror images. The atoms are in the same places but the momentum vectors are, of course, reversed since in general the transition state is proceeding in one direction only. In other words, the forward and reverse mechanisms are identical, according to this principle. 

It depends on both the angle a of the angular momentum vector rotation and other Euler angles F and q, which determine the molecule s axis shift. Besides, the angle F is also the azimuth of the change in angular momentum A/ = J(t + 0) — J(t — 0), which is the result of collision. 

In this case, it is expressed via polar coordinates of the angular momentum vector in a plane, perpendicular to the diatomic molecular axis. The function, which is present in (6.6), can be determined by (6.7) as follows ... 

Here r is the radius vector from the origin to a point R in the crystal, t is the electron-pair-bond function in the region near R, Pfc is the momentum vector corresponding to the three quantum numbers k (the density of states being calculated in the usual way), h is Planck s constant, and G is the normalizing factor. 

In addition to CO(v = 0—2,7) populations, Houston and Kable recorded CO Doppler profiles to measure the translational energy release, and the vector correlation between the recoil velocity vector and the angular momentum vector of CO. Together, these data paint a compelling picture that two pathways to CH4 + CO are operative. The rotationally hot CO population (85% of total CO)... 

Figure 5.5 The motion in a magnetic field B of the orbital angular momentum vector L.

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