Dear Skyler Saleebyan

Any problem involving more than 2 particles can not be solved analytically. However, in the following paper my my colleagues, a 3-body problem in 1-dimension is solved analytically. Please note that all 3 particles move along a line, and this is not a general 3-particle problem.

Path integral for a three body problem

D C Khandekar and S V Lawande

Journal of Physics A: Volume 5 Number 6, 812, 1972

Abstract: A one dimensional three body problem is treated by the path integral approach of Feynman. When expressed in suitable coordinates, it is possible to extract the symmetry from the path integral of the problem. The resulting propagator is evaluated in a closed analytical form.

best wishes, SVG Menon

Thank you for the response,

By "can not be solved analytically" do you mean that it's been shown that  no analytical solution CAN exist, or we just haven't discovered a mathematical technique to allow us to solve these complex problems (yet)?

What makes correlated electrons so difficult mathematically speaking?

Analytical solutions of PDE are very difficult because of the complexity of math involved. It is a well known fact that all PDEs cannot be solved analytically (There is no proof that they cannot be solved, they are just too complex to solve let alone apply boundary conditions). May be a mathematician will rise who will come up with a general algorithm to solve any PDE. The world would be a much better place with exact solutions.

Difficulty is simply many degrees of freedom, Skyler. Similar question concerning algebraic equations has led to group theory 200 years ago.

Regards,

Eugene.

General analytical solution of 3-body problem can not be obtained in principle as the general solution is unstable (highly sensitive to small changes of the initial conditions). The understanding of this fact lead to the Dynamical Chaos paradigm and development of statistical methods of analysis of such solution (e.g. S. M. Ulam, On some statistical properties of dynamical system, 1961 and 55 years of active investigations)

All analitical solutions of TD Schrodinger equation are possible in so-called exatly soluble problems. This means that you need to involve special potentials like (x^2 - harmonic oscillator) in which you can simulate interaction between two electrons. But for these interaction you need to use also exactly soluble model. After this, semi-analitical scheme could be used for general solution.

This is just my thinking. Some example you can find in this paper (but it is in russian):

Tunneling of interacting particles through the potential barriers: computer simulation by quantum molecular dynamics method  (Yu.E. Lozovik, A.V. Filinov, A.S. Arkhipov)