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Three body coulomb problem

It is well known that the three body problem does not have a general analytical solution, and it has been the subject of considerable academic efforts since the 17th century[*]. The history as well as recent progress on the oldest three body problem, the Moon-Earth-Sun system have been reviewed by Gutzwiller [87].

The Coulomb three body problem, in particular, has been a difficult one to solve accurately, due in part to the long range nature of the interaction, in comparison to few nucleon systems in which the interaction is short-ranged. It is an active area of research, as indicated by a number of recent developments and refinements of theoretical techniques with the help of ever-increasing computing resources[*]. Efforts are being made to extend the calculations to full four-body muonic problems [92,93].

In general, the three body Hamiltonian can be written as

 \begin{displaymath}
{\cal H = T + V} = \sum ^{N=3}_{i=1} \frac {1}{2m_{i}} \Del...
...+ \sum ^{N=3}_{i<j} V _{ij} (\vert\rho _{i} - \rho _{j}\vert),
\end{displaymath} (15)

where $\rho _{i}$ is the particle position vector [3]. The center of mass motion can be separated out, leaving a six dimensional problem in r 1, r 2 space.

The exact form of the Hamiltonian depends on the choice of the co-ordinates r 1, r 2. In terms of kinematics, there is no unique choice of the co-ordinates, hence choosing suitable ones, which will give accurate results, is one of the challenges that theorists face.


next up previous contents
Next: Adiabatic approaches Up: Muonic molecule and the Previous: The challenge