Two-particle problem

The center of mass of the two-particle system is located by the vector R with cartesian components, X, Y, Z 

If we restrict our interest to systems for which the potential energy 7 is a function only of the relative position vector r, then the classical Hamiltonian function H is given by 

These momenta may be expressed in terms of the time derivatives of R and r by substitution of equation (6.3) 

The momenta pi and p, corresponding to the center of mass position R and the relative position variable r, respectively, may be defined as 

In terms of these momenta, the classical Hamiltonian becomes 

By definition, the center of mass is related to ri and r2 by zwirj + OT2r2 

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Equation (4.1) is a two-particle problem. In Section 1.12 we showed that we can separate a two-particle problem into two one-particle problems, 142... 

Before discussing the statistics and kinetics of many-particle systems, which requires the use of rather refined and complicated formalism, let us start first of all from a two-particle problem. Although it is presented from the methodological point of view, it is realistic. For example, it can describe... 

As we have mentioned in Chapter 2, the accuracy of the kinetic equations derived using the superposition approximation cannot be checked up in the framework of the same theory. It is the analysis of the limiting case of the infinitely diluted system, no —> 0, which nevertheless permits us to compare approximate results obtained in the linearized approximation with the exact solution of the two-particle problem (Chapter 3). 

Now we apply the eigenfunction of the isolated two-particle problem, and we get, in the coordinate representation,... 

Let us consider a system of H atoms the elementary particles of which are electrons (e) and protons (p). We assume that we have solved the two-particle problem... 

The dynamics of the two-particle problem can be separated into center-of-mass motion and relative motion with the reduced mass /i = morn s/(rnp + me), of the two particles. The kinetic energy of the relative motion is a conserved quantity. The outcome of the elastic collision is described by the deflection angle of the trajectory, and this is the main quantity to be determined in the following. The deflection angle, X, gives the deviation from the incident straight line trajectory due to attractive and repulsive forces. Thus, x is the angle between the final and initial directions of the relative velocity vector for the two particles. 

At first sight, it may look like the two particle problem of describing the motion of a molecule 2 relative to molecule 1 is equivalent to the one particle problem of describing the motion of a single molecule 2 with reduced mass /i acted on by the force F21. With respect to a fixed particle 1 the scattering process appears as shown in Fig. 2.2. 

As we noted in Section 18.7, the variances of the mean plume dimensions can be expressed in terms of the motion of single particles released from the source. (At a particular instant the plume outline is defined by the statistics of the trajectories of two particles released simultaneously at the source. We have not considered the two-particle problem here.) In an effort to overcome the practical difficulties associated with using (18.72) to obtain results for Gy and gz, Pasquill (1971) suggested an alternate definition that retained the essential features of Taylor s statistical theory but that is more amenable to parametrization in terms of readily measured Eulerian quantities. As adopted by Draxler (1976), the American Meteorological Society (1977), and Irwin (1979), the Pasquill representation leads to... 

Thus, when the system is composed of two noninteracting particles, we can reduce the two-particle problem to two separate one-particle problems by solving... 

Section 6.3 Reduction of the Two-Particle Problem to Two One-Particle Problems 127... 

The hydrogen atom contains two particles, the proton and the electron. For a system of two particles 1 and 2 with coordinates Zi) and X2,y2, z-, the potential energy of interaction between the particles is usually a function of only the relative coordinates X2 Xi,y2 y, and Z2 Zi of the particles. In this case the two-particle problem can be simplified to two separate one-particle problems, as we now prove. 

The solution to the two-particle problem can then be composed of solutions of one-variable Schrddinger equations. 

The solution of two-particle problem of the interaction of two material points with masses m and obtained under the condition of no external forces available corresponds to the interactions taking place by the gradient, the positive work is performed by the system (similar to attraction process in the gravitation field). 

They also compared their approximation method with the usual perturbation theory. This comparison is repeated in table 1.7. Bacher and Goudsmit immediately concluded from the table that if the approximation used was a pair-approximation or better, and if the exact energies of the two-particle problem were used in that approximation, then all the higher orders of the pair interaction have been included. This conclusion of Bacher and Goudsmit very clearly explains the result (2) above. It also appears from table 1.7 that the triple... 

This shape equation is of fourth order, but it is linear. Unfortunately, in the present context we must solve it for a two-particle problem with finite-sized particles, and therein lies the mb the operator in square brackets is not separable in any simple coordinate system, so we have to deal with the fact that this equation is indeed a partial differential equation. 

It is now clear what the mean field approximation is the two-particle problem is reduced to a sing/e-particle one (denoted as number 1), and the influence of the second particle is averaged over its positions (it = Wx ) = (< >(2) x < >(2)) . 

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