Editing Euler's three-body problem (section)

==Quantum mechanical version==
A special case of the quantum mechanical three-body problem is the [[hydrogen molecule ion]], {{chem|H|2|+}}.  Two of the three bodies are nuclei and the third is a fast moving electron.  The two nuclei are 1800 times heavier than the electron and thus modeled as fixed centers.  It is well known that the Schrödinger wave equation is separable in [[prolate spheroidal coordinates]] and can be decoupled into two ordinary differential equations coupled by the energy eigenvalue and a separation constant.<ref>
G.B. Arfken, ''Mathematical Methods for Physicists'', 2nd ed., Academic Press, New York (1970).
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However, solutions required series expansions from basis sets.  Nonetheless, through [[experimental mathematics]], it was found that the energy eigenvalue was mathematically a ''generalization'' of the Lambert W function (see [[Lambert W function]] and references therein for more details). The hydrogen molecular ion in the case of clamped nuclei can be completely worked out within a [[Computer algebra system]].  The fact that its solution is an [[implicit function]] is revealing in itself.  One of the successes of theoretical physics is not simply a matter that it is amenable to a mathematical treatment but that the algebraic equations involved can be symbolically manipulated until an analytical solution, preferably a closed form solution, is isolated.  This type of solution for a special case of the three-body problem shows us the possibilities of what is possible as an analytical solution for the quantum three-body and many-body problem.