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Unfortunately, the expression of the Hamiltonian \eqref{eq:completeHamiltonian} is too complex to deal with in this form. Indeed, the entire system is coupled, it is a many-body problem. In this form, the movement of one electron is coupled to the movement of all other electrons, but also to the displacement of the nuclei. To better appreciate the complexity of the problem, let us take an analogy consisting of solid balls evolving inside of a force field. First consider a single ball (A) and a force field which acts on A. If the way the force field varies - with respect to time or to the space position - is independent of the ball position, then, the problem is uncoupled and the trajectory of the ball constitutes a one-body problem. Within classical mechanics, this system is solved thanks to the Newtonian equations.
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Second, consider the introduction of a second ball (B). If the balls do not interact, the system is still easily solved as ball A and B can be analysed separately. What happens if the balls are linked through a spring? In this case, the problem is said to be coupled; trajectory of A can not be obtained without considering the trajectory of B. If a large number of such balls is introduced, then the problem rapidly becomes unsolvable. The difficulty rises from the coupled character of the problem. One speaks of a ''many-body problem``.
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Second, consider the introduction of a second ball (B). If the balls do not interact, the system is still easily solved as ball A and B can be analysed separately. What happens if the balls are linked through a spring? In this case, the problem is said to be coupled; trajectory of A can not be obtained without considering the trajectory of B. If a large number of such balls is introduced, then the problem rapidly becomes unsolvable. The difficulty rises from the coupled character of the problem. One speaks of a ``many-body problem''.
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Last, imagine that atop of the interactions occurring between the balls, the field in which they evolve depends on their position. The complexity of such a system can be compared to what the Schrödinger equation describes with the Hamiltonian of \eqref{eq:completeHamiltonian}. The problem is composed of to different kinds of particles which interact at all levels. These interactions are described by the different operators constituting the Hamiltonian.
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