The two-body problem

The classical energy of a system of two point masses, and m2, has the form 

Replacing the momenta by the corresponding quantum mechanical operators yields the Hamiltonian operator 

In general, the potential energy is not separable into terms involving only certain sets of the six Cartesian coordinates but depends on the internal coordinates of the system. So to simplify the problem we transform these six Cartesian coordinates into three coordinates for the center of mass X, 7, and Z and three internal coordinates, x, y, z. 

The center-of-mass coordinates (X, Y, Z) are determined by the condition that the sum of the first moments of mass about the center of mass vanish for each axis that is, 

In the second expression, the derivatives have been evaluated from the definitions of X and X. Then, 

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In the quantum mechanical applications of the two-body problem, the classical energy of the system becomes the Hamiltonian operator The conversion... 

Show that the kinetic energy in the two-body problem in terms of momenta is given by Eq. (44). 

It will be identified in Chapter 6 as the azimuthal quantum number, which is characteristic of the two-body problem. 

At this stage the assumption that the wave function ip can be factorized into com and relative-motion (rm) components, by defining E = Ecom + Erm, is commonly made. In terms of ip = body problem is decoupled into two one-body problems ... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

The higher-order two-loop corrections are to be calculated within the so-called external filed approximation (i. e. neglecting by the nuclear motion), while the recoil effects require an essential two-body treatment. There are a few approaches to solve the two-body problem (see e.g. [31]). Most start with the Green function of the two-body system which has to have a pole at the energy of the bound state... 

By referring the motion to the centre of mass, the two-body problem has been reduced to a one-body problem of the vibrational motion of a particle of mass p against a fixed point, under the restraining influence of a spring of length R with a force constant k. 

In the beginning of the twentieth century, a mechanics in small world was constructed, namely quantum mechanics. In the early stage of making quantum mechanics, Bohr found a rule to obtain some part of the energy spectrum of a hydrogen atom (i.e., the two-body problem one electron and one nucleus) [4]. 

In ref. 162 the numerical integration of Hamiltonian systems is investigated. Trigonometrically fitted S5miplectic partinioned Runge-Kutta methods of second, third and fourth orders are obtained. The methods are tested on the numerical integration of the harmonic oscillator, the two body problem and an orbital problem studied by Stiefel and Bettis. 

The calculation of the matrix element of S G2 proceeds along the same lines. Note that, as was the case with the two-body problem, a real symmetric matrix is obtained. 

Although the conclusions of Dirac s theory agree very well, generally speaking, with experiment, yet even in it serious difficulties occur, which up till now have only in part been successfully overcome. There is in the first place the theory of the many-electron problem, which has not so far been successfully brought under the scheme of Dirac s theory this is ultimately connected with the fact that even classically no satisfactory formulation of the two-body problem has yet been found within the ambit of the theory of relativity. 

We consider first a body of negligible mass moving around a body of finite mass m in an elliptic orbit. It can be proved (Murray and Dermott, 1999) that the action-angle variables of the two-body problem in the inertial frame, in the plane, are the Delaunay variables defined by... 

If the system has symmetries (as is the case with the restricted problem), usually the symmetric periodic orbits survive (but not always ). The resonant fixed points that survive correspond to monoparametric families of elliptic periodic orbits, in the rotating frame. These families bifurcate from the circular family, at the corresponding circular resonant orbits. From the above analysis we come to the conclusion that out of the infinite set of resonant elliptic periodic orbits of the two-body problem, with the same semimajor axes and the same eccentricities, but different orientations, as shown in Figure 15, only a finite number survive as periodic orbits in the rotating frame, and in most cases only two, usually, but not always, are symmetric. 

Recall that the Hamiltonian function describing the two-body problem is given (in normalized units of measure) by... 

For each planet, we may transform / , At, r, p into the elements a, e, A, zu employing the same transformations used to define the ordinary osculating heliocentric elements aosc, eosc, AOSc, wOSc of the two-body problem as functions of m, G(M + m), r, mi. However, the equations giving the osculating heliocentric elements depend on m only through fi. In order to use always the same routines, the above equations may be transformed. 

For the sake of future calculations, let us recall some series expansions of the two-body problem. These expansions are helpful in the task of... 

For a many-electron system, even if one is concerned with only the lowest ionisation threshold and even if one neglects the spin-orbit interaction, it is no longer true that a single Rydberg series occurs. As noted in section 2.1, it is a remarkable feature of the two-body problem that, although... 

The effect of the nuclear mass was already mentioned in the introduction. In the Furry picture which is employed in the calculations of QED effects on bound electron states a static external field is assumed which corresponds to an infinitely heavy nucleus. In a non-relativistic approximation its finite mass is encountered by the reduced mass correction similar to the two-body problem in classical mechanics. In a relativistic treatment, however, this approach is oversimplyfied. Recently Artemyev et al. [42, 43] almost solved the whole problem by considering the nucleus as a simple Dirac particle with spin 1 /2, mass M and charge Ze. The interaction of the two Dirac particles electron and nucleus leads to a quasipotential equation in the center-of-mass system,... 

Atomic spectra famish material for testing theories of atomic structure. Since hydrogen is the simplest kind of atom, the interpretation of its spectrum has been of the greatest interest to theorists, particularly since all but the most recent theories can be applied with mathematical rigour to the two-body problem which the hydrogen atom presents. 

We consider now the case of a system which is subject to internal forces only. The above considerations are then applicable to the axis of the resultant angular momentum, where, in place of , the angle denoted above by ifi appears and the quantum condition (8) applies. The polarisation of the light cannot be observed, however, since the atoms or molecules of a gas have all possible orientations. The case mentioned above, where all the particles of the system move in planes perpendicular to the axis, is of frequent occurrence, e.g. in the case of the two-body problem (atom with one electron) and in that of the rigid rotator (dumb-bell model of the molecule) the transition j->j is then impossible. 

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